# Properties

 Label 1176.2.q.l Level $1176$ Weight $2$ Character orbit 1176.q Analytic conductor $9.390$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1176.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.39040727770$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{3} ) q^{5} + ( -1 - \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -1 - \beta_{1} + \beta_{3} ) q^{5} + ( -1 - \beta_{1} ) q^{9} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{13} + ( 1 - \beta_{2} ) q^{15} + 4 \beta_{1} q^{17} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{19} + ( -4 - 4 \beta_{1} ) q^{23} + ( 10 \beta_{1} + \beta_{2} - \beta_{3} ) q^{25} + q^{27} + ( 1 + \beta_{2} ) q^{29} + \beta_{1} q^{31} + ( 1 + \beta_{1} - \beta_{3} ) q^{33} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{37} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{39} + ( -4 + 2 \beta_{2} ) q^{41} + ( -4 + \beta_{2} ) q^{43} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{45} + ( -6 - 6 \beta_{1} ) q^{47} + ( -4 - 4 \beta_{1} ) q^{51} + ( 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{53} + ( -15 + \beta_{2} ) q^{55} + ( 2 + \beta_{2} ) q^{57} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{59} + ( 10 + 10 \beta_{1} ) q^{61} + ( -12 - 12 \beta_{1} - 2 \beta_{3} ) q^{65} + ( 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} + 4 q^{69} + 2 q^{71} + ( -\beta_{2} + \beta_{3} ) q^{73} + ( -10 - 10 \beta_{1} + \beta_{3} ) q^{75} + ( -3 - 3 \beta_{1} - 2 \beta_{3} ) q^{79} + \beta_{1} q^{81} + ( -3 - \beta_{2} ) q^{83} + ( 4 - 4 \beta_{2} ) q^{85} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{87} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( -1 - \beta_{1} ) q^{93} + ( -12 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{95} + ( -13 + \beta_{2} ) q^{97} + ( -1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} - q^{5} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} - q^{5} - 2q^{9} + q^{11} - 10q^{13} + 2q^{15} - 8q^{17} - 5q^{19} - 8q^{23} - 19q^{25} + 4q^{27} + 6q^{29} - 2q^{31} + q^{33} - 3q^{37} + 5q^{39} - 12q^{41} - 14q^{43} - q^{45} - 12q^{47} - 8q^{51} - 11q^{53} - 58q^{55} + 10q^{57} + 5q^{59} + 20q^{61} - 26q^{65} - 7q^{67} + 16q^{69} + 8q^{71} - q^{73} - 19q^{75} - 8q^{79} - 2q^{81} - 14q^{83} + 8q^{85} - 3q^{87} - 6q^{89} - 2q^{93} + 26q^{95} - 50q^{97} - 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 25$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 9 \nu + 5$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{3} + 2 \nu^{2} + 8 \nu - 25$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} - 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 14 \beta_{1} + 13$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 19$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$$.

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 −0.500000 + 0.866025i 0 −2.13746 3.70219i 0 0 0 −0.500000 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 1.63746 + 2.83616i 0 0 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 −2.13746 + 3.70219i 0 0 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 0.866025i 0 1.63746 2.83616i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.q.l 4
3.b odd 2 1 3528.2.s.bk 4
4.b odd 2 1 2352.2.q.bf 4
7.b odd 2 1 168.2.q.c 4
7.c even 3 1 1176.2.a.n 2
7.c even 3 1 inner 1176.2.q.l 4
7.d odd 6 1 168.2.q.c 4
7.d odd 6 1 1176.2.a.k 2
21.c even 2 1 504.2.s.i 4
21.g even 6 1 504.2.s.i 4
21.g even 6 1 3528.2.a.bk 2
21.h odd 6 1 3528.2.a.bd 2
21.h odd 6 1 3528.2.s.bk 4
28.d even 2 1 336.2.q.g 4
28.f even 6 1 336.2.q.g 4
28.f even 6 1 2352.2.a.bf 2
28.g odd 6 1 2352.2.a.ba 2
28.g odd 6 1 2352.2.q.bf 4
56.e even 2 1 1344.2.q.x 4
56.h odd 2 1 1344.2.q.w 4
56.j odd 6 1 1344.2.q.w 4
56.j odd 6 1 9408.2.a.ec 2
56.k odd 6 1 9408.2.a.dw 2
56.m even 6 1 1344.2.q.x 4
56.m even 6 1 9408.2.a.dp 2
56.p even 6 1 9408.2.a.dj 2
84.h odd 2 1 1008.2.s.r 4
84.j odd 6 1 1008.2.s.r 4
84.j odd 6 1 7056.2.a.cu 2
84.n even 6 1 7056.2.a.ch 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 7.b odd 2 1
168.2.q.c 4 7.d odd 6 1
336.2.q.g 4 28.d even 2 1
336.2.q.g 4 28.f even 6 1
504.2.s.i 4 21.c even 2 1
504.2.s.i 4 21.g even 6 1
1008.2.s.r 4 84.h odd 2 1
1008.2.s.r 4 84.j odd 6 1
1176.2.a.k 2 7.d odd 6 1
1176.2.a.n 2 7.c even 3 1
1176.2.q.l 4 1.a even 1 1 trivial
1176.2.q.l 4 7.c even 3 1 inner
1344.2.q.w 4 56.h odd 2 1
1344.2.q.w 4 56.j odd 6 1
1344.2.q.x 4 56.e even 2 1
1344.2.q.x 4 56.m even 6 1
2352.2.a.ba 2 28.g odd 6 1
2352.2.a.bf 2 28.f even 6 1
2352.2.q.bf 4 4.b odd 2 1
2352.2.q.bf 4 28.g odd 6 1
3528.2.a.bd 2 21.h odd 6 1
3528.2.a.bk 2 21.g even 6 1
3528.2.s.bk 4 3.b odd 2 1
3528.2.s.bk 4 21.h odd 6 1
7056.2.a.ch 2 84.n even 6 1
7056.2.a.cu 2 84.j odd 6 1
9408.2.a.dj 2 56.p even 6 1
9408.2.a.dp 2 56.m even 6 1
9408.2.a.dw 2 56.k odd 6 1
9408.2.a.ec 2 56.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1176, [\chi])$$:

 $$T_{5}^{4} + T_{5}^{3} + 15 T_{5}^{2} - 14 T_{5} + 196$$ $$T_{11}^{4} - T_{11}^{3} + 15 T_{11}^{2} + 14 T_{11} + 196$$ $$T_{13}^{2} + 5 T_{13} - 8$$