# Properties

 Label 1176.2.q.a Level $1176$ Weight $2$ Character orbit 1176.q Analytic conductor $9.390$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

## Character values

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

 $$n$$ $$295$$ $$589$$ $$785$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −0.500000 + 0.866025i 0 −1.00000 1.73205i 0 0 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 −1.00000 + 1.73205i 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.a 2
3.b odd 2 1 3528.2.s.y 2
4.b odd 2 1 2352.2.q.r 2
7.b odd 2 1 1176.2.q.i 2
7.c even 3 1 1176.2.a.i 1
7.c even 3 1 inner 1176.2.q.a 2
7.d odd 6 1 24.2.a.a 1
7.d odd 6 1 1176.2.q.i 2
21.c even 2 1 3528.2.s.j 2
21.g even 6 1 72.2.a.a 1
21.g even 6 1 3528.2.s.j 2
21.h odd 6 1 3528.2.a.d 1
21.h odd 6 1 3528.2.s.y 2
28.d even 2 1 2352.2.q.l 2
28.f even 6 1 48.2.a.a 1
28.f even 6 1 2352.2.q.l 2
28.g odd 6 1 2352.2.a.i 1
28.g odd 6 1 2352.2.q.r 2
35.i odd 6 1 600.2.a.h 1
35.k even 12 2 600.2.f.e 2
56.j odd 6 1 192.2.a.d 1
56.k odd 6 1 9408.2.a.cc 1
56.m even 6 1 192.2.a.b 1
56.p even 6 1 9408.2.a.h 1
63.i even 6 1 648.2.i.b 2
63.k odd 6 1 648.2.i.g 2
63.s even 6 1 648.2.i.b 2
63.t odd 6 1 648.2.i.g 2
77.i even 6 1 2904.2.a.c 1
84.j odd 6 1 144.2.a.b 1
84.n even 6 1 7056.2.a.q 1
91.s odd 6 1 4056.2.a.i 1
91.bb even 12 2 4056.2.c.e 2
105.p even 6 1 1800.2.a.m 1
105.w odd 12 2 1800.2.f.c 2
112.v even 12 2 768.2.d.d 2
112.x odd 12 2 768.2.d.e 2
119.h odd 6 1 6936.2.a.p 1
133.o even 6 1 8664.2.a.j 1
140.s even 6 1 1200.2.a.d 1
140.x odd 12 2 1200.2.f.b 2
168.ba even 6 1 576.2.a.d 1
168.be odd 6 1 576.2.a.b 1
231.k odd 6 1 8712.2.a.u 1
252.n even 6 1 1296.2.i.m 2
252.r odd 6 1 1296.2.i.e 2
252.bj even 6 1 1296.2.i.m 2
252.bn odd 6 1 1296.2.i.e 2
280.ba even 6 1 4800.2.a.cc 1
280.bk odd 6 1 4800.2.a.q 1
280.bp odd 12 2 4800.2.f.bg 2
280.bv even 12 2 4800.2.f.d 2
308.m odd 6 1 5808.2.a.s 1
336.bo even 12 2 2304.2.d.i 2
336.br odd 12 2 2304.2.d.k 2
364.x even 6 1 8112.2.a.be 1
420.be odd 6 1 3600.2.a.v 1
420.br even 12 2 3600.2.f.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 7.d odd 6 1
48.2.a.a 1 28.f even 6 1
72.2.a.a 1 21.g even 6 1
144.2.a.b 1 84.j odd 6 1
192.2.a.b 1 56.m even 6 1
192.2.a.d 1 56.j odd 6 1
576.2.a.b 1 168.be odd 6 1
576.2.a.d 1 168.ba even 6 1
600.2.a.h 1 35.i odd 6 1
600.2.f.e 2 35.k even 12 2
648.2.i.b 2 63.i even 6 1
648.2.i.b 2 63.s even 6 1
648.2.i.g 2 63.k odd 6 1
648.2.i.g 2 63.t odd 6 1
768.2.d.d 2 112.v even 12 2
768.2.d.e 2 112.x odd 12 2
1176.2.a.i 1 7.c even 3 1
1176.2.q.a 2 1.a even 1 1 trivial
1176.2.q.a 2 7.c even 3 1 inner
1176.2.q.i 2 7.b odd 2 1
1176.2.q.i 2 7.d odd 6 1
1200.2.a.d 1 140.s even 6 1
1200.2.f.b 2 140.x odd 12 2
1296.2.i.e 2 252.r odd 6 1
1296.2.i.e 2 252.bn odd 6 1
1296.2.i.m 2 252.n even 6 1
1296.2.i.m 2 252.bj even 6 1
1800.2.a.m 1 105.p even 6 1
1800.2.f.c 2 105.w odd 12 2
2304.2.d.i 2 336.bo even 12 2
2304.2.d.k 2 336.br odd 12 2
2352.2.a.i 1 28.g odd 6 1
2352.2.q.l 2 28.d even 2 1
2352.2.q.l 2 28.f even 6 1
2352.2.q.r 2 4.b odd 2 1
2352.2.q.r 2 28.g odd 6 1
2904.2.a.c 1 77.i even 6 1
3528.2.a.d 1 21.h odd 6 1
3528.2.s.j 2 21.c even 2 1
3528.2.s.j 2 21.g even 6 1
3528.2.s.y 2 3.b odd 2 1
3528.2.s.y 2 21.h odd 6 1
3600.2.a.v 1 420.be odd 6 1
3600.2.f.r 2 420.br even 12 2
4056.2.a.i 1 91.s odd 6 1
4056.2.c.e 2 91.bb even 12 2
4800.2.a.q 1 280.bk odd 6 1
4800.2.a.cc 1 280.ba even 6 1
4800.2.f.d 2 280.bv even 12 2
4800.2.f.bg 2 280.bp odd 12 2
5808.2.a.s 1 308.m odd 6 1
6936.2.a.p 1 119.h odd 6 1
7056.2.a.q 1 84.n even 6 1
8112.2.a.be 1 364.x even 6 1
8664.2.a.j 1 133.o even 6 1
8712.2.a.u 1 231.k odd 6 1
9408.2.a.h 1 56.p even 6 1
9408.2.a.cc 1 56.k odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} - 8T + 64$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$T^{2} + 6T + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 2T + 4$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T + 2)^{2}$$