Table of Clebsch–Gordan coefficients
# Table of Clebsch–Gordan coefficients

This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant $j_{1}$, $j_{2}$, $j$ is arbitrary to some degree and has been fixed according to the Condon–Shortley and Wigner sign convention as discussed by Baird and Biedenharn. Tables with the same sign convention may be found in the Particle Data Group's Review of Particle Properties and in online tables.

## Formulation

The Clebsch–Gordan coefficients are the solutions to

    $|j_{1},j_{2};J,M\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1},m_{1};j_{2},m_{2}\rangle \langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle$

Explicitly:

    ${\begin{aligned}&\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle \\[6pt]=&\delta _{M,m_{1}+m_{2}}{\sqrt {\frac {(2J+1)(J+j_{1}-j_{2})!(J-j_{1}+j_{2})!(j_{1}+j_{2}-J)!}{(j_{1}+j_{2}+J+1)!}}}\ \times\\[6pt]&{\sqrt {(J+M)!(J-M)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times\\[6pt]&\sum _{k}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-J-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(J-j_{2}+m_{1}+k)!(J-j_{1}-m_{2}+k)!}}.\end{aligned}}$

The summation is extended over all integer k for which the argument of every factorial is nonnegative.

For brevity, solutions with M < 0 and $j_1$ < $j_2$ are omitted. They may be calculated using the simple relations

    $\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{1},j_{2};-m_{1},-m_{2}\mid j_{1},j_{2};J,-M\rangle .$

and

    $\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{2},j_{1};m_{2},m_{1}\mid j_{2},j_{1};J,M\rangle .$

## Specific values

The Clebsch–Gordan coefficients for j values less than or equal to 5/2 are given below.

### $j_2$ = 0

When $j_2$ = 0, the Clebsch–Gordan coefficients are given by $\delta _{j,j_{1}}\delta _{m,m_{1}}$.

### $j_1$ = 1/2,  $j_2$ = 1/2

m = 1 :

| $jm_1,$ $m_2$ | 1 |
| - | - |
| 1/2, 1/2 | $1$ |

m = −1 :

| $jm_1,$ $m_2$ | 1 |
| - | - |
| −1/2, −1/2 | $1$ |

m = 0 :

| $jm_1,$ $m_2$ | 1 | 0 |
| - | - | - |
| 1/2, −1/2 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| −1/2, 1/2 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

### $j_1$ = 1,  $j_2$ = 1/2

m = 3/2 :

| $jm_1,$ $m_2$ | 3/2 |
| - | - |
| 1, 1/2 | $1$ |

m = 1/2 :

| $jm_1,$ $m_2$ | 3/2 | 1/2 |
| - | - | - |
| 1, −1/2 | ${\sqrt {\frac {1}{3}}}$ | ${\sqrt {\frac {2}{3}}}$ |
| 0, 1/2 | ${\sqrt {\frac {2}{3}}}$ | $-{\sqrt {\frac {1}{3}}}$ |

### $j_1$ = 1,  $j_2$ = 1

m = 2 :

| $jm_1,$ $m_2$ | 2 |
| - | - |
| 1, 1 | $1$ |

m = 1 :

| $jm_1,$ $m_2$ | 2 | 1 |
| - | - | - |
| 1, 0 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| 0, 1 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 2 | 1 | 0 |
| - | - | - | - |
| 1, −1 | ${\sqrt {\frac {1}{6}}}$ | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{3}}}$ |
| 0, 0 | ${\sqrt {\frac {2}{3}}}$ | $0$ | $-{\sqrt {\frac {1}{3}}}$ |
| −1, 1 | ${\sqrt {\frac {1}{6}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{3}}}$ |

### $j_1$ = 3/2,  $j_2$ = 1/2

m = 2 :

| $jm_1,$ $m_2$ | 2 |
| - | - |
| 3/2, 1/2 | $1$ |

m = 1 :

| $jm_1,$ $m_2$ | 2 | 1 |
| - | - | - |
| 3/2, −1/2 | ${\frac {1}{2}}$ | ${\sqrt {\frac {3}{4}}}$ |
| 1/2, 1/2 | ${\sqrt {\frac {3}{4}}}$ | $-{\frac {1}{2}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 2 | 1 |
| - | - | - |
| 1/2, −1/2 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| −1/2, 1/2 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

### $j_1$ = 3/2,  $j_2$ = 1

m = 5/2 :

| $jm_1,$ $m_2$ | 5/2 |
| - | - |
| 3/2, 1 | $1$ |

m = 3/2 :

| $jm_1,$ $m_2$ | 5/2 | 3/2 |
| - | - | - |
| 3/2, 0 | ${\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {3}{5}}}$ |
| 1/2, 1 | ${\sqrt {\frac {3}{5}}}$ | $-{\sqrt {\frac {2}{5}}}$ |

m = 1/2 :

| $jm_1,$ $m_2$ | 5/2 | 3/2 | 1/2 |
| - | - | - | - |
| 3/2, −1 | ${\sqrt {\frac {1}{10}}}$ | ${\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| 1/2, 0 | ${\sqrt {\frac {3}{5}}}$ | ${\sqrt {\frac {1}{15}}}$ | $-{\sqrt {\frac {1}{3}}}$ |
| −1/2, 1 | ${\sqrt {\frac {3}{10}}}$ | $-{\sqrt {\frac {8}{15}}}$ | ${\sqrt {\frac {1}{6}}}$ |

### $j_1$ = 3/2,  $j_2$ = 3/2

m = 3 :

| $jm_1,$ $m_2$ | 3 |
| - | - |
| 3/2, 3/2 | $1$ |

m = 2 :

| $jm_1,$ $m_2$ | 3 | 2 |
| - | - | - |
| 3/2, 1/2 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| 1/2, 3/2 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

m = 1 :

| $jm_1,$ $m_2$ | 3 | 2 | 1 |
| - | - | - | - |
| 3/2, −1/2 | ${\sqrt {\frac {1}{5}}}$ | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {3}{10}}}$ |
| 1/2, 1/2 | ${\sqrt {\frac {3}{5}}}$ | $0$ | $-{\sqrt {\frac {2}{5}}}$ |
| −1/2, 3/2 | ${\sqrt {\frac {1}{5}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {3}{10}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 3 | 2 | 1 | 0 |
| - | - | - | - | - |
| 3/2, −3/2 | ${\sqrt {\frac {1}{20}}}$ | ${\frac {1}{2}}$ | ${\sqrt {\frac {9}{20}}}$ | ${\frac {1}{2}}$ |
| 1/2, −1/2 | ${\sqrt {\frac {9}{20}}}$ | ${\frac {1}{2}}$ | $-{\sqrt {\frac {1}{20}}}$ | $-{\frac {1}{2}}$ |
| −1/2, 1/2 | ${\sqrt {\frac {9}{20}}}$ | $-{\frac {1}{2}}$ | $-{\sqrt {\frac {1}{20}}}$ | ${\frac {1}{2}}$ |
| −3/2, 3/2 | ${\sqrt {\frac {1}{20}}}$ | $-{\frac {1}{2}}$ | ${\sqrt {\frac {9}{20}}}$ | $-{\frac {1}{2}}$ |

### $j_1$ = 2,  $j_2$ = 1/2

m = 5/2 :

| $jm_1,$ $m_2$ | 5/2 |
| - | - |
| 2, 1/2 | $1$ |

m = 3/2 :

| $jm_1,$ $m_2$ | 5/2 | 3/2 |
| - | - | - |
| 2, −1/2 | ${\sqrt {\frac {1}{5}}}$ | ${\sqrt {\frac {4}{5}}}$ |
| 1, 1/2 | ${\sqrt {\frac {4}{5}}}$ | $-{\sqrt {\frac {1}{5}}}$ |

m = 1/2 :

| $jm_1,$ $m_2$ | 5/2 | 3/2 |
| - | - | - |
| 1, −1/2 | ${\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {3}{5}}}$ |
| 0, 1/2 | ${\sqrt {\frac {3}{5}}}$ | $-{\sqrt {\frac {2}{5}}}$ |

### $j_1$ = 2,  $j_2$ = 1

m = 3 :

| $jm_1,$ $m_2$ | 3 |
| - | - |
| 2, 1 | $1$ |

m = 2 :

| $jm_1,$ $m_2$ | 3 | 2 |
| - | - | - |
| 2, 0 | ${\sqrt {\frac {1}{3}}}$ | ${\sqrt {\frac {2}{3}}}$ |
| 1, 1 | ${\sqrt {\frac {2}{3}}}$ | $-{\sqrt {\frac {1}{3}}}$ |

m = 1 :

| $jm_1,$ $m_2$ | 3 | 2 | 1 |
| - | - | - | - |
| 2, −1 | ${\sqrt {\frac {1}{15}}}$ | ${\sqrt {\frac {1}{3}}}$ | ${\sqrt {\frac {3}{5}}}$ |
| 1, 0 | ${\sqrt {\frac {8}{15}}}$ | ${\sqrt {\frac {1}{6}}}$ | $-{\sqrt {\frac {3}{10}}}$ |
| 0, 1 | ${\sqrt {\frac {2}{5}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{10}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 3 | 2 | 1 |
| - | - | - | - |
| 1, −1 | ${\sqrt {\frac {1}{5}}}$ | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {3}{10}}}$ |
| 0, 0 | ${\sqrt {\frac {3}{5}}}$ | $0$ | $-{\sqrt {\frac {2}{5}}}$ |
| −1, 1 | ${\sqrt {\frac {1}{5}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {3}{10}}}$ |

### $j_1$ = 2,  $j_2$ = 3/2

m = 7/2 :

| $jm_1,$ $m_2$ | 7/2 |
| - | - |
| 2, 3/2 | $1$ |

m = 5/2 :

| $jm_1,$ $m_2$ | 7/2 | 5/2 |
| - | - | - |
| 2, 1/2 | ${\sqrt {\frac {3}{7}}}$ | ${\sqrt {\frac {4}{7}}}$ |
| 1, 3/2 | ${\sqrt {\frac {4}{7}}}$ | $-{\sqrt {\frac {3}{7}}}$ |

m = 3/2 :

| $jm_1,$ $m_2$ | 7/2 | 5/2 | 3/2 |
| - | - | - | - |
| 2, −1/2 | ${\sqrt {\frac {1}{7}}}$ | ${\sqrt {\frac {16}{35}}}$ | ${\sqrt {\frac {2}{5}}}$ |
| 1, 1/2 | ${\sqrt {\frac {4}{7}}}$ | ${\sqrt {\frac {1}{35}}}$ | $-{\sqrt {\frac {2}{5}}}$ |
| 0, 3/2 | ${\sqrt {\frac {2}{7}}}$ | $-{\sqrt {\frac {18}{35}}}$ | ${\sqrt {\frac {1}{5}}}$ |

m = 1/2 :

| $jm_1,$ $m_2$ | 7/2 | 5/2 | 3/2 | 1/2 |
| - | - | - | - | - |
| 2, −3/2 | ${\sqrt {\frac {1}{35}}}$ | ${\sqrt {\frac {6}{35}}}$ | ${\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {2}{5}}}$ |
| 1, −1/2 | ${\sqrt {\frac {12}{35}}}$ | ${\sqrt {\frac {5}{14}}}$ | $0$ | $-{\sqrt {\frac {3}{10}}}$ |
| 0, 1/2 | ${\sqrt {\frac {18}{35}}}$ | $-{\sqrt {\frac {3}{35}}}$ | $-{\sqrt {\frac {1}{5}}}$ | ${\sqrt {\frac {1}{5}}}$ |
| −1, 3/2 | ${\sqrt {\frac {4}{35}}}$ | $-{\sqrt {\frac {27}{70}}}$ | ${\sqrt {\frac {2}{5}}}$ | $-{\sqrt {\frac {1}{10}}}$ |

### $j_1$ = 2,  $j_2$ = 2

m = 4 :

| $jm_1,$ $m_2$ | 4 |
| - | - |
| 2, 2 | $1$ |

m = 3 :

| $jm_1,$ $m_2$ | 4 | 3 |
| - | - | - |
| 2, 1 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| 1, 2 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

m = 2 :

| $jm_1,$ $m_2$ | 4 | 3 | 2 |
| - | - | - | - |
| 2, 0 | ${\sqrt {\frac {3}{14}}}$ | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {2}{7}}}$ |
| 1, 1 | ${\sqrt {\frac {4}{7}}}$ | $0$ | $-{\sqrt {\frac {3}{7}}}$ |
| 0, 2 | ${\sqrt {\frac {3}{14}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {2}{7}}}$ |

m = 1 :

| $jm_1,$ $m_2$ | 4 | 3 | 2 | 1 |
| - | - | - | - | - |
| 2, −1 | ${\sqrt {\frac {1}{14}}}$ | ${\sqrt {\frac {3}{10}}}$ | ${\sqrt {\frac {3}{7}}}$ | ${\sqrt {\frac {1}{5}}}$ |
| 1, 0 | ${\sqrt {\frac {3}{7}}}$ | ${\sqrt {\frac {1}{5}}}$ | $-{\sqrt {\frac {1}{14}}}$ | $-{\sqrt {\frac {3}{10}}}$ |
| 0, 1 | ${\sqrt {\frac {3}{7}}}$ | $-{\sqrt {\frac {1}{5}}}$ | $-{\sqrt {\frac {1}{14}}}$ | ${\sqrt {\frac {3}{10}}}$ |
| −1, 2 | ${\sqrt {\frac {1}{14}}}$ | $-{\sqrt {\frac {3}{10}}}$ | ${\sqrt {\frac {3}{7}}}$ | $-{\sqrt {\frac {1}{5}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 4 | 3 | 2 | 1 | 0 |
| - | - | - | - | - | - |
| 2, −2 | ${\sqrt {\frac {1}{70}}}$ | ${\sqrt {\frac {1}{10}}}$ | ${\sqrt {\frac {2}{7}}}$ | ${\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {1}{5}}}$ |
| 1, −1 | ${\sqrt {\frac {8}{35}}}$ | ${\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {1}{14}}}$ | $-{\sqrt {\frac {1}{10}}}$ | $-{\sqrt {\frac {1}{5}}}$ |
| 0, 0 | ${\sqrt {\frac {18}{35}}}$ | $0$ | $-{\sqrt {\frac {2}{7}}}$ | $0$ | ${\sqrt {\frac {1}{5}}}$ |
| −1, 1 | ${\sqrt {\frac {8}{35}}}$ | $-{\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {1}{14}}}$ | ${\sqrt {\frac {1}{10}}}$ | $-{\sqrt {\frac {1}{5}}}$ |
| −2, 2 | ${\sqrt {\frac {1}{70}}}$ | $-{\sqrt {\frac {1}{10}}}$ | ${\sqrt {\frac {2}{7}}}$ | $-{\sqrt {\frac {2}{5}}}$ | ${\sqrt {\frac {1}{5}}}$ |

### $j_1$ = 5/2,  $j_2$ = 1/2

m = 3 :

| $jm_1,$ $m_2$ | 3 |
| - | - |
| 5/2, 1/2 | $1$ |

m = 2 :

| $jm_1,$ $m_2$ | 3 | 2 |
| - | - | - |
| 5/2, −1/2 | ${\sqrt {\frac {1}{6}}}$ | ${\sqrt {\frac {5}{6}}}$ |
| 3/2, 1/2 | ${\sqrt {\frac {5}{6}}}$ | $-{\sqrt {\frac {1}{6}}}$ |

m = 1 :

| $jm_1,$ $m_2$ | 3 | 2 |
| - | - | - |
| 3/2, −1/2 | ${\sqrt {\frac {1}{3}}}$ | ${\sqrt {\frac {2}{3}}}$ |
| 1/2, 1/2 | ${\sqrt {\frac {2}{3}}}$ | $-{\sqrt {\frac {1}{3}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 3 | 2 |
| - | - | - |
| 1/2, −1/2 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| −1/2, 1/2 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

### $j_1$ = 5/2,  $j_2$ = 1

m = 7/2 :

| $jm_1,$ $m_2$ | 7/2 |
| - | - |
| 5/2, 1 | $1$ |

m = 5/2 :

| $jm_1,$ $m_2$ | 7/2 | 5/2 |
| - | - | - |
| 5/2, 0 | ${\sqrt {\frac {2}{7}}}$ | ${\sqrt {\frac {5}{7}}}$ |
| 3/2, 1 | ${\sqrt {\frac {5}{7}}}$ | $-{\sqrt {\frac {2}{7}}}$ |

m = 3/2 :

| $jm_1,$ $m_2$ | 7/2 | 5/2 | 3/2 |
| - | - | - | - |
| 5/2, −1 | ${\sqrt {\frac {1}{21}}}$ | ${\sqrt {\frac {2}{7}}}$ | ${\sqrt {\frac {2}{3}}}$ |
| 3/2, 0 | ${\sqrt {\frac {10}{21}}}$ | ${\sqrt {\frac {9}{35}}}$ | $-{\sqrt {\frac {4}{15}}}$ |
| 1/2, 1 | ${\sqrt {\frac {10}{21}}}$ | $-{\sqrt {\frac {16}{35}}}$ | ${\sqrt {\frac {1}{15}}}$ |

m = 1/2 :

| $jm_1,$ $m_2$ | 7/2 | 5/2 | 3/2 |
| - | - | - | - |
| 3/2, −1 | ${\sqrt {\frac {1}{7}}}$ | ${\sqrt {\frac {16}{35}}}$ | ${\sqrt {\frac {2}{5}}}$ |
| 1/2, 0 | ${\sqrt {\frac {4}{7}}}$ | ${\sqrt {\frac {1}{35}}}$ | $-{\sqrt {\frac {2}{5}}}$ |
| −1/2, 1 | ${\sqrt {\frac {2}{7}}}$ | $-{\sqrt {\frac {18}{35}}}$ | ${\sqrt {\frac {1}{5}}}$ |

### $j_1$ = 5/2,  $j_2$ = 3/2

m = 4 :

| $jm_1,$ $m_2$ | 4 |
| - | - |
| 5/2, 3/2 | $1$ |

m = 3 :

| $jm_1,$ $m_2$ | 4 | 3 |
| - | - | - |
| 5/2, 1/2 | ${\sqrt {\frac {3}{8}}}$ | ${\sqrt {\frac {5}{8}}}$ |
| 3/2, 3/2 | ${\sqrt {\frac {5}{8}}}$ | $-{\sqrt {\frac {3}{8}}}$ |

m = 2 :

| $jm_1,$ $m_2$ | 4 | 3 | 2 |
| - | - | - | - |
| 5/2, −1/2 | ${\sqrt {\frac {3}{28}}}$ | ${\sqrt {\frac {5}{12}}}$ | ${\sqrt {\frac {10}{21}}}$ |
| 3/2, 1/2 | ${\sqrt {\frac {15}{28}}}$ | ${\sqrt {\frac {1}{12}}}$ | $-{\sqrt {\frac {8}{21}}}$ |
| 1/2, 3/2 | ${\sqrt {\frac {5}{14}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{7}}}$ |

m = 1 :

| $jm_1,$ $m_2$ | 4 | 3 | 2 | 1 |
| - | - | - | - | - |
| 5/2, −3/2 | ${\sqrt {\frac {1}{56}}}$ | ${\sqrt {\frac {1}{8}}}$ | ${\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| 3/2, −1/2 | ${\sqrt {\frac {15}{56}}}$ | ${\sqrt {\frac {49}{120}}}$ | ${\sqrt {\frac {1}{42}}}$ | $-{\sqrt {\frac {3}{10}}}$ |
| 1/2, 1/2 | ${\sqrt {\frac {15}{28}}}$ | $-{\sqrt {\frac {1}{60}}}$ | $-{\sqrt {\frac {25}{84}}}$ | ${\sqrt {\frac {3}{20}}}$ |
| −1/2, 3/2 | ${\sqrt {\frac {5}{28}}}$ | $-{\sqrt {\frac {9}{20}}}$ | ${\sqrt {\frac {9}{28}}}$ | $-{\sqrt {\frac {1}{20}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 4 | 3 | 2 | 1 |
| - | - | - | - | - |
| 3/2, −3/2 | ${\sqrt {\frac {1}{14}}}$ | ${\sqrt {\frac {3}{10}}}$ | ${\sqrt {\frac {3}{7}}}$ | ${\sqrt {\frac {1}{5}}}$ |
| 1/2, −1/2 | ${\sqrt {\frac {3}{7}}}$ | ${\sqrt {\frac {1}{5}}}$ | $-{\sqrt {\frac {1}{14}}}$ | $-{\sqrt {\frac {3}{10}}}$ |
| −1/2, 1/2 | ${\sqrt {\frac {3}{7}}}$ | $-{\sqrt {\frac {1}{5}}}$ | $-{\sqrt {\frac {1}{14}}}$ | ${\sqrt {\frac {3}{10}}}$ |
| −3/2, 3/2 | ${\sqrt {\frac {1}{14}}}$ | $-{\sqrt {\frac {3}{10}}}$ | ${\sqrt {\frac {3}{7}}}$ | $-{\sqrt {\frac {1}{5}}}$ |

### $j_1$ = 5/2,  $j_2$ = 2

m = 9/2 :

| $jm_1,$ $m_2$ | 9/2 |
| - | - |
| 5/2, 2 | $1$ |

m = 7/2 :

| $jm_1,$ $m_2$ | 9/2 | 7/2 |
| - | - | - |
| 5/2, 1 | ${\frac {2}{3}}$ | ${\sqrt {\frac {5}{9}}}$ |
| 3/2, 2 | ${\sqrt {\frac {5}{9}}}$ | $-{\frac {2}{3}}$ |

m = 5/2 :

| $jm_1,$ $m_2$ | 9/2 | 7/2 | 5/2 |
| - | - | - | - |
| 5/2, 0 | ${\sqrt {\frac {1}{6}}}$ | ${\sqrt {\frac {10}{21}}}$ | ${\sqrt {\frac {5}{14}}}$ |
| 3/2, 1 | ${\sqrt {\frac {5}{9}}}$ | ${\sqrt {\frac {1}{63}}}$ | $-{\sqrt {\frac {3}{7}}}$ |
| 1/2, 2 | ${\sqrt {\frac {5}{18}}}$ | $-{\sqrt {\frac {32}{63}}}$ | ${\sqrt {\frac {3}{14}}}$ |

m = 3/2 :

| $jm_1,$ $m_2$ | 9/2 | 7/2 | 5/2 | 3/2 |
| - | - | - | - | - |
| 5/2, −1 | ${\sqrt {\frac {1}{21}}}$ | ${\sqrt {\frac {5}{21}}}$ | ${\sqrt {\frac {3}{7}}}$ | ${\sqrt {\frac {2}{7}}}$ |
| 3/2, 0 | ${\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {2}{7}}}$ | $-{\sqrt {\frac {1}{70}}}$ | $-{\sqrt {\frac {12}{35}}}$ |
| 1/2, 1 | ${\sqrt {\frac {10}{21}}}$ | $-{\sqrt {\frac {2}{21}}}$ | $-{\sqrt {\frac {6}{35}}}$ | ${\sqrt {\frac {9}{35}}}$ |
| −1/2, 2 | ${\sqrt {\frac {5}{42}}}$ | $-{\sqrt {\frac {8}{21}}}$ | ${\sqrt {\frac {27}{70}}}$ | $-{\sqrt {\frac {4}{35}}}$ |

m = 1/2 :

| $jm_1,$ $m_2$ | 9/2 | 7/2 | 5/2 | 3/2 | 1/2 |
| - | - | - | - | - | - |
| 5/2, −2 | ${\sqrt {\frac {1}{126}}}$ | ${\sqrt {\frac {4}{63}}}$ | ${\sqrt {\frac {3}{14}}}$ | ${\sqrt {\frac {8}{21}}}$ | ${\sqrt {\frac {1}{3}}}$ |
| 3/2, −1 | ${\sqrt {\frac {10}{63}}}$ | ${\sqrt {\frac {121}{315}}}$ | ${\sqrt {\frac {6}{35}}}$ | $-{\sqrt {\frac {2}{105}}}$ | $-{\sqrt {\frac {4}{15}}}$ |
| 1/2, 0 | ${\sqrt {\frac {10}{21}}}$ | ${\sqrt {\frac {4}{105}}}$ | $-{\sqrt {\frac {8}{35}}}$ | $-{\sqrt {\frac {2}{35}}}$ | ${\sqrt {\frac {1}{5}}}$ |
| −1/2, 1 | ${\sqrt {\frac {20}{63}}}$ | $-{\sqrt {\frac {14}{45}}}$ | $0$ | ${\sqrt {\frac {5}{21}}}$ | $-{\sqrt {\frac {2}{15}}}$ |
| −3/2, 2 | ${\sqrt {\frac {5}{126}}}$ | $-{\sqrt {\frac {64}{315}}}$ | ${\sqrt {\frac {27}{70}}}$ | $-{\sqrt {\frac {32}{105}}}$ | ${\sqrt {\frac {1}{15}}}$ |

### $j_1$ = 5/2,  $j_2$ = 5/2

m = 5 :

| $jm_1,$ $m_2$ | 5 |
| - | - |
| 5/2, 5/2 | $1$ |

m = 4 :

| $jm_1,$ $m_2$ | 5 | 4 |
| - | - | - |
| 5/2, 3/2 | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {1}{2}}}$ |
| 3/2, 5/2 | ${\sqrt {\frac {1}{2}}}$ | $-{\sqrt {\frac {1}{2}}}$ |

m = 3 :

| $jm_1,$ $m_2$ | 5 | 4 | 3 |
| - | - | - | - |
| 5/2, 1/2 | ${\sqrt {\frac {2}{9}}}$ | ${\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {5}{18}}}$ |
| 3/2, 3/2 | ${\sqrt {\frac {5}{9}}}$ | $0$ | $-{\frac {2}{3}}$ |
| 1/2, 5/2 | ${\sqrt {\frac {2}{9}}}$ | $-{\sqrt {\frac {1}{2}}}$ | ${\sqrt {\frac {5}{18}}}$ |

m = 2 :

| $jm_1,$ $m_2$ | 5 | 4 | 3 | 2 |
| - | - | - | - | - |
| 5/2, −1/2 | ${\sqrt {\frac {1}{12}}}$ | ${\sqrt {\frac {9}{28}}}$ | ${\sqrt {\frac {5}{12}}}$ | ${\sqrt {\frac {5}{28}}}$ |
| 3/2, 1/2 | ${\sqrt {\frac {5}{12}}}$ | ${\sqrt {\frac {5}{28}}}$ | $-{\sqrt {\frac {1}{12}}}$ | $-{\sqrt {\frac {9}{28}}}$ |
| 1/2, 3/2 | ${\sqrt {\frac {5}{12}}}$ | $-{\sqrt {\frac {5}{28}}}$ | $-{\sqrt {\frac {1}{12}}}$ | ${\sqrt {\frac {9}{28}}}$ |
| −1/2, 5/2 | ${\sqrt {\frac {1}{12}}}$ | $-{\sqrt {\frac {9}{28}}}$ | ${\sqrt {\frac {5}{12}}}$ | $-{\sqrt {\frac {5}{28}}}$ |

m = 1 :

| $jm_1,$ $m_2$ | 5 | 4 | 3 | 2 | 1 |
| - | - | - | - | - | - |
| 5/2, −3/2 | ${\sqrt {\frac {1}{42}}}$ | ${\sqrt {\frac {1}{7}}}$ | ${\sqrt {\frac {1}{3}}}$ | ${\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {1}{7}}}$ |
| 3/2, −1/2 | ${\sqrt {\frac {5}{21}}}$ | ${\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {1}{30}}}$ | $-{\sqrt {\frac {1}{7}}}$ | $-{\sqrt {\frac {8}{35}}}$ |
| 1/2, 1/2 | ${\sqrt {\frac {10}{21}}}$ | $0$ | $-{\sqrt {\frac {4}{15}}}$ | $0$ | ${\sqrt {\frac {9}{35}}}$ |
| −1/2, 3/2 | ${\sqrt {\frac {5}{21}}}$ | $-{\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {1}{30}}}$ | ${\sqrt {\frac {1}{7}}}$ | $-{\sqrt {\frac {8}{35}}}$ |
| −3/2, 5/2 | ${\sqrt {\frac {1}{42}}}$ | $-{\sqrt {\frac {1}{7}}}$ | ${\sqrt {\frac {1}{3}}}$ | $-{\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {1}{7}}}$ |

m = 0 :

| $jm_1,$ $m_2$ | 5 | 4 | 3 | 2 | 1 | 0 |
| - | - | - | - | - | - | - |
| 5/2, −5/2 | ${\sqrt {\frac {1}{252}}}$ | ${\sqrt {\frac {1}{28}}}$ | ${\sqrt {\frac {5}{36}}}$ | ${\sqrt {\frac {25}{84}}}$ | ${\sqrt {\frac {5}{14}}}$ | ${\sqrt {\frac {1}{6}}}$ |
| 3/2, −3/2 | ${\sqrt {\frac {25}{252}}}$ | ${\sqrt {\frac {9}{28}}}$ | ${\sqrt {\frac {49}{180}}}$ | ${\sqrt {\frac {1}{84}}}$ | $-{\sqrt {\frac {9}{70}}}$ | $-{\sqrt {\frac {1}{6}}}$ |
| 1/2, −1/2 | ${\sqrt {\frac {25}{63}}}$ | ${\sqrt {\frac {1}{7}}}$ | $-{\sqrt {\frac {4}{45}}}$ | $-{\sqrt {\frac {4}{21}}}$ | ${\sqrt {\frac {1}{70}}}$ | ${\sqrt {\frac {1}{6}}}$ |
| −1/2, 1/2 | ${\sqrt {\frac {25}{63}}}$ | $-{\sqrt {\frac {1}{7}}}$ | $-{\sqrt {\frac {4}{45}}}$ | ${\sqrt {\frac {4}{21}}}$ | ${\sqrt {\frac {1}{70}}}$ | $-{\sqrt {\frac {1}{6}}}$ |
| −3/2, 3/2 | ${\sqrt {\frac {25}{252}}}$ | $-{\sqrt {\frac {9}{28}}}$ | ${\sqrt {\frac {49}{180}}}$ | $-{\sqrt {\frac {1}{84}}}$ | $-{\sqrt {\frac {9}{70}}}$ | ${\sqrt {\frac {1}{6}}}$ |
| −5/2, 5/2 | ${\sqrt {\frac {1}{252}}}$ | $-{\sqrt {\frac {1}{28}}}$ | ${\sqrt {\frac {5}{36}}}$ | $-{\sqrt {\frac {25}{84}}}$ | ${\sqrt {\frac {5}{14}}}$ | $-{\sqrt {\frac {1}{6}}}$ |

## SU(N) Clebsch–Gordan coefficients

Algorithms to produce Clebsch–Gordan coefficients for higher values of $j_{1}$ and $j_{2}$, or for the su(N) algebra instead of su(2), are known. A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.

