# Properties

 Label 1176.2.q.e 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 168) 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 $$q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} + ( - 6 \zeta_{6} + 6) q^{11} + 3 q^{13} - 2 q^{15} + ( - 4 \zeta_{6} + 4) q^{17} - 5 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + q^{27} - 4 q^{29} + ( - 7 \zeta_{6} + 7) q^{31} + 6 \zeta_{6} q^{33} + 9 \zeta_{6} q^{37} + (3 \zeta_{6} - 3) q^{39} + 2 q^{41} - q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 2 \zeta_{6} q^{47} + 4 \zeta_{6} q^{51} + (8 \zeta_{6} - 8) q^{53} + 12 q^{55} + 5 q^{57} + 10 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( - 15 \zeta_{6} + 15) q^{67} - 4 q^{69} - 6 q^{71} + (11 \zeta_{6} - 11) q^{73} + \zeta_{6} q^{75} - \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 6 q^{83} + 8 q^{85} + ( - 4 \zeta_{6} + 4) q^{87} - 8 \zeta_{6} q^{89} + 7 \zeta_{6} q^{93} + ( - 10 \zeta_{6} + 10) q^{95} + 14 q^{97} - 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + 2*z * q^5 - z * q^9 + (-6*z + 6) * q^11 + 3 * q^13 - 2 * q^15 + (-4*z + 4) * q^17 - 5*z * q^19 + 4*z * q^23 + (-z + 1) * q^25 + q^27 - 4 * q^29 + (-7*z + 7) * q^31 + 6*z * q^33 + 9*z * q^37 + (3*z - 3) * q^39 + 2 * q^41 - q^43 + (-2*z + 2) * q^45 + 2*z * q^47 + 4*z * q^51 + (8*z - 8) * q^53 + 12 * q^55 + 5 * q^57 + 10*z * q^61 + 6*z * q^65 + (-15*z + 15) * q^67 - 4 * q^69 - 6 * q^71 + (11*z - 11) * q^73 + z * q^75 - z * q^79 + (z - 1) * q^81 - 6 * q^83 + 8 * q^85 + (-4*z + 4) * q^87 - 8*z * q^89 + 7*z * q^93 + (-10*z + 10) * q^95 + 14 * q^97 - 6 * 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 $$2 q - q^{3} + 2 q^{5} - q^{9} + 6 q^{11} + 6 q^{13} - 4 q^{15} + 4 q^{17} - 5 q^{19} + 4 q^{23} + q^{25} + 2 q^{27} - 8 q^{29} + 7 q^{31} + 6 q^{33} + 9 q^{37} - 3 q^{39} + 4 q^{41} - 2 q^{43} + 2 q^{45} + 2 q^{47} + 4 q^{51} - 8 q^{53} + 24 q^{55} + 10 q^{57} + 10 q^{61} + 6 q^{65} + 15 q^{67} - 8 q^{69} - 12 q^{71} - 11 q^{73} + q^{75} - q^{79} - q^{81} - 12 q^{83} + 16 q^{85} + 4 q^{87} - 8 q^{89} + 7 q^{93} + 10 q^{95} + 28 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 + 2 * q^5 - q^9 + 6 * q^11 + 6 * q^13 - 4 * q^15 + 4 * q^17 - 5 * q^19 + 4 * q^23 + q^25 + 2 * q^27 - 8 * q^29 + 7 * q^31 + 6 * q^33 + 9 * q^37 - 3 * q^39 + 4 * q^41 - 2 * q^43 + 2 * q^45 + 2 * q^47 + 4 * q^51 - 8 * q^53 + 24 * q^55 + 10 * q^57 + 10 * q^61 + 6 * q^65 + 15 * q^67 - 8 * q^69 - 12 * q^71 - 11 * q^73 + q^75 - q^79 - q^81 - 12 * q^83 + 16 * q^85 + 4 * q^87 - 8 * q^89 + 7 * q^93 + 10 * q^95 + 28 * q^97 - 12 * 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.e 2
3.b odd 2 1 3528.2.s.d 2
4.b odd 2 1 2352.2.q.v 2
7.b odd 2 1 168.2.q.b 2
7.c even 3 1 1176.2.a.e 1
7.c even 3 1 inner 1176.2.q.e 2
7.d odd 6 1 168.2.q.b 2
7.d odd 6 1 1176.2.a.d 1
21.c even 2 1 504.2.s.g 2
21.g even 6 1 504.2.s.g 2
21.g even 6 1 3528.2.a.f 1
21.h odd 6 1 3528.2.a.y 1
21.h odd 6 1 3528.2.s.d 2
28.d even 2 1 336.2.q.a 2
28.f even 6 1 336.2.q.a 2
28.f even 6 1 2352.2.a.x 1
28.g odd 6 1 2352.2.a.e 1
28.g odd 6 1 2352.2.q.v 2
56.e even 2 1 1344.2.q.t 2
56.h odd 2 1 1344.2.q.i 2
56.j odd 6 1 1344.2.q.i 2
56.j odd 6 1 9408.2.a.cd 1
56.k odd 6 1 9408.2.a.cs 1
56.m even 6 1 1344.2.q.t 2
56.m even 6 1 9408.2.a.f 1
56.p even 6 1 9408.2.a.bk 1
84.h odd 2 1 1008.2.s.m 2
84.j odd 6 1 1008.2.s.m 2
84.j odd 6 1 7056.2.a.i 1
84.n even 6 1 7056.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 7.b odd 2 1
168.2.q.b 2 7.d odd 6 1
336.2.q.a 2 28.d even 2 1
336.2.q.a 2 28.f even 6 1
504.2.s.g 2 21.c even 2 1
504.2.s.g 2 21.g even 6 1
1008.2.s.m 2 84.h odd 2 1
1008.2.s.m 2 84.j odd 6 1
1176.2.a.d 1 7.d odd 6 1
1176.2.a.e 1 7.c even 3 1
1176.2.q.e 2 1.a even 1 1 trivial
1176.2.q.e 2 7.c even 3 1 inner
1344.2.q.i 2 56.h odd 2 1
1344.2.q.i 2 56.j odd 6 1
1344.2.q.t 2 56.e even 2 1
1344.2.q.t 2 56.m even 6 1
2352.2.a.e 1 28.g odd 6 1
2352.2.a.x 1 28.f even 6 1
2352.2.q.v 2 4.b odd 2 1
2352.2.q.v 2 28.g odd 6 1
3528.2.a.f 1 21.g even 6 1
3528.2.a.y 1 21.h odd 6 1
3528.2.s.d 2 3.b odd 2 1
3528.2.s.d 2 21.h odd 6 1
7056.2.a.i 1 84.j odd 6 1
7056.2.a.bn 1 84.n even 6 1
9408.2.a.f 1 56.m even 6 1
9408.2.a.bk 1 56.p even 6 1
9408.2.a.cd 1 56.j odd 6 1
9408.2.a.cs 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 $$T_{11}^{2} - 6T_{11} + 36$$ T11^2 - 6*T11 + 36 $$T_{13} - 3$$ T13 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} - 7T + 49$$
$37$ $$T^{2} - 9T + 81$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} + 8T + 64$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} - 15T + 225$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 8T + 64$$
$97$ $$(T - 14)^{2}$$