# Properties

 Label 1176.2.q.i 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: $$x^{2} - x + 1$$ 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.

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

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

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 4$$ $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ $$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 8 T + 41 T^{2} - 184 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4}$$
$59$ $$1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 4 T + 83 T^{2} )^{2}$$
$89$ $$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 2 T + 97 T^{2} )^{2}$$