# Properties

 Label 1200.2.h.f Level $1200$ Weight $2$ Character orbit 1200.h Analytic conductor $9.582$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + 5 q^{13} + ( -5 + 10 \zeta_{6} ) q^{19} + 3 q^{21} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -5 + 10 \zeta_{6} ) q^{31} + 10 q^{37} + ( -5 + 10 \zeta_{6} ) q^{39} + ( -7 + 14 \zeta_{6} ) q^{43} + 4 q^{49} -15 q^{57} -13 q^{61} + ( -3 + 6 \zeta_{6} ) q^{63} + ( 9 - 18 \zeta_{6} ) q^{67} -10 q^{73} + ( -10 + 20 \zeta_{6} ) q^{79} + 9 q^{81} + ( 5 - 10 \zeta_{6} ) q^{91} -15 q^{93} + 5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{9} + O(q^{10})$$ $$2q - 6q^{9} + 10q^{13} + 6q^{21} + 20q^{37} + 8q^{49} - 30q^{57} - 26q^{61} - 20q^{73} + 18q^{81} - 30q^{93} + 10q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0 0 1.73205i 0 −3.00000 0
1151.2 0 1.73205i 0 0 0 1.73205i 0 −3.00000 0
 $$n$$: 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 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.h.f yes 2
3.b odd 2 1 CM 1200.2.h.f yes 2
4.b odd 2 1 inner 1200.2.h.f yes 2
5.b even 2 1 1200.2.h.d 2
5.c odd 4 2 1200.2.o.h 4
12.b even 2 1 inner 1200.2.h.f yes 2
15.d odd 2 1 1200.2.h.d 2
15.e even 4 2 1200.2.o.h 4
20.d odd 2 1 1200.2.h.d 2
20.e even 4 2 1200.2.o.h 4
60.h even 2 1 1200.2.h.d 2
60.l odd 4 2 1200.2.o.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.h.d 2 5.b even 2 1
1200.2.h.d 2 15.d odd 2 1
1200.2.h.d 2 20.d odd 2 1
1200.2.h.d 2 60.h even 2 1
1200.2.h.f yes 2 1.a even 1 1 trivial
1200.2.h.f yes 2 3.b odd 2 1 CM
1200.2.h.f yes 2 4.b odd 2 1 inner
1200.2.h.f yes 2 12.b even 2 1 inner
1200.2.o.h 4 5.c odd 4 2
1200.2.o.h 4 15.e even 4 2
1200.2.o.h 4 20.e even 4 2
1200.2.o.h 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 3$$ $$T_{11}$$ $$T_{13} - 5$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$3 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$75 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$75 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$147 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$243 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$300 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -5 + T )^{2}$$
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