# Properties

 Label 1134.2.t.c Level $1134$ Weight $2$ Character orbit 1134.t Analytic conductor $9.055$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 2 - \zeta_{12}^{2} ) q^{10} + ( -2 \zeta_{12} - \zeta_{12}^{3} ) q^{14} -\zeta_{12}^{2} q^{16} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( 4 + 4 \zeta_{12}^{2} ) q^{19} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} + 6 \zeta_{12}^{3} q^{23} -2 q^{25} + ( 2 + \zeta_{12}^{2} ) q^{28} -9 \zeta_{12} q^{29} + ( -2 - 2 \zeta_{12}^{2} ) q^{31} + \zeta_{12} q^{32} + ( 2 + 2 \zeta_{12}^{2} ) q^{34} + ( -\zeta_{12} - 4 \zeta_{12}^{3} ) q^{35} + ( -4 + 4 \zeta_{12}^{2} ) q^{37} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{38} + ( 1 - 2 \zeta_{12}^{2} ) q^{40} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( 8 - 8 \zeta_{12}^{2} ) q^{43} -6 \zeta_{12}^{2} q^{46} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{50} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{53} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{56} + 9 q^{58} + ( -7 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{59} + ( -4 + 2 \zeta_{12}^{2} ) q^{61} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{62} - q^{64} + ( -14 + 14 \zeta_{12}^{2} ) q^{67} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{68} + ( 1 + 4 \zeta_{12}^{2} ) q^{70} -6 \zeta_{12}^{3} q^{71} + ( -14 + 7 \zeta_{12}^{2} ) q^{73} -4 \zeta_{12}^{3} q^{74} + ( 8 - 4 \zeta_{12}^{2} ) q^{76} -11 \zeta_{12}^{2} q^{79} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{80} + ( -2 - 2 \zeta_{12}^{2} ) q^{82} + ( -10 \zeta_{12} + 20 \zeta_{12}^{3} ) q^{83} + 6 \zeta_{12}^{2} q^{85} + 8 \zeta_{12}^{3} q^{86} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + 6 \zeta_{12} q^{92} + ( 2 + 2 \zeta_{12}^{2} ) q^{94} -12 \zeta_{12} q^{95} + ( 4 + 4 \zeta_{12}^{2} ) q^{97} + ( 5 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 2q^{7} + O(q^{10})$$ $$4q + 2q^{4} + 2q^{7} + 6q^{10} - 2q^{16} + 24q^{19} - 8q^{25} + 10q^{28} - 12q^{31} + 12q^{34} - 8q^{37} + 16q^{43} - 12q^{46} - 26q^{49} + 36q^{58} - 12q^{61} - 4q^{64} - 28q^{67} + 12q^{70} - 42q^{73} + 24q^{76} - 22q^{79} - 12q^{82} + 12q^{85} + 12q^{94} + 24q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1 - \zeta_{12}^{2}$$ $$\zeta_{12}^{2}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
593.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0.500000 0.866025i −1.73205 0 0.500000 + 2.59808i 1.00000i 0 1.50000 0.866025i
593.2 0.866025 0.500000i 0 0.500000 0.866025i 1.73205 0 0.500000 + 2.59808i 1.00000i 0 1.50000 0.866025i
1025.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.73205 0 0.500000 2.59808i 1.00000i 0 1.50000 + 0.866025i
1025.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.73205 0 0.500000 2.59808i 1.00000i 0 1.50000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.k odd 6 1 inner
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.t.c 4
3.b odd 2 1 inner 1134.2.t.c 4
7.d odd 6 1 1134.2.l.b 4
9.c even 3 1 378.2.k.c 4
9.c even 3 1 1134.2.l.b 4
9.d odd 6 1 378.2.k.c 4
9.d odd 6 1 1134.2.l.b 4
21.g even 6 1 1134.2.l.b 4
63.g even 3 1 2646.2.d.a 4
63.i even 6 1 378.2.k.c 4
63.k odd 6 1 inner 1134.2.t.c 4
63.k odd 6 1 2646.2.d.a 4
63.n odd 6 1 2646.2.d.a 4
63.s even 6 1 inner 1134.2.t.c 4
63.s even 6 1 2646.2.d.a 4
63.t odd 6 1 378.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.c 4 9.c even 3 1
378.2.k.c 4 9.d odd 6 1
378.2.k.c 4 63.i even 6 1
378.2.k.c 4 63.t odd 6 1
1134.2.l.b 4 7.d odd 6 1
1134.2.l.b 4 9.c even 3 1
1134.2.l.b 4 9.d odd 6 1
1134.2.l.b 4 21.g even 6 1
1134.2.t.c 4 1.a even 1 1 trivial
1134.2.t.c 4 3.b odd 2 1 inner
1134.2.t.c 4 63.k odd 6 1 inner
1134.2.t.c 4 63.s even 6 1 inner
2646.2.d.a 4 63.g even 3 1
2646.2.d.a 4 63.k odd 6 1
2646.2.d.a 4 63.n odd 6 1
2646.2.d.a 4 63.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1134, [\chi])$$:

 $$T_{5}^{2} - 3$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -3 + T^{2} )^{2}$$
$7$ $$( 7 - T + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$144 + 12 T^{2} + T^{4}$$
$19$ $$( 48 - 12 T + T^{2} )^{2}$$
$23$ $$( 36 + T^{2} )^{2}$$
$29$ $$6561 - 81 T^{2} + T^{4}$$
$31$ $$( 12 + 6 T + T^{2} )^{2}$$
$37$ $$( 16 + 4 T + T^{2} )^{2}$$
$41$ $$144 + 12 T^{2} + T^{4}$$
$43$ $$( 64 - 8 T + T^{2} )^{2}$$
$47$ $$144 + 12 T^{2} + T^{4}$$
$53$ $$81 - 9 T^{2} + T^{4}$$
$59$ $$21609 + 147 T^{2} + T^{4}$$
$61$ $$( 12 + 6 T + T^{2} )^{2}$$
$67$ $$( 196 + 14 T + T^{2} )^{2}$$
$71$ $$( 36 + T^{2} )^{2}$$
$73$ $$( 147 + 21 T + T^{2} )^{2}$$
$79$ $$( 121 + 11 T + T^{2} )^{2}$$
$83$ $$90000 + 300 T^{2} + T^{4}$$
$89$ $$11664 + 108 T^{2} + T^{4}$$
$97$ $$( 48 - 12 T + T^{2} )^{2}$$