# Properties

 Label 1176.2.q.f 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 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -2 \zeta_{6} q^{5} -\zeta_{6} q^{9} + 6 q^{13} -2 q^{15} + ( 2 - 2 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} - q^{27} -10 q^{29} + ( 8 - 8 \zeta_{6} ) q^{31} -6 \zeta_{6} q^{37} + ( 6 - 6 \zeta_{6} ) q^{39} -2 q^{41} -4 q^{43} + ( -2 + 2 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} -2 \zeta_{6} q^{51} + ( 10 - 10 \zeta_{6} ) q^{53} -4 q^{57} + ( -12 + 12 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} -12 \zeta_{6} q^{65} + ( -12 + 12 \zeta_{6} ) q^{67} + 4 q^{69} -12 q^{71} + ( 14 - 14 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + 8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} -4 q^{85} + ( -10 + 10 \zeta_{6} ) q^{87} + 2 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( -8 + 8 \zeta_{6} ) q^{95} + 10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 2q^{5} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{5} - q^{9} + 12q^{13} - 4q^{15} + 2q^{17} - 4q^{19} + 4q^{23} + q^{25} - 2q^{27} - 20q^{29} + 8q^{31} - 6q^{37} + 6q^{39} - 4q^{41} - 8q^{43} - 2q^{45} - 8q^{47} - 2q^{51} + 10q^{53} - 8q^{57} - 12q^{59} + 2q^{61} - 12q^{65} - 12q^{67} + 8q^{69} - 24q^{71} + 14q^{73} - q^{75} + 8q^{79} - q^{81} + 24q^{83} - 8q^{85} - 10q^{87} + 2q^{89} - 8q^{93} - 8q^{95} + 20q^{97} + O(q^{100})$$

## 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.f 2
3.b odd 2 1 3528.2.s.w 2
4.b odd 2 1 2352.2.q.d 2
7.b odd 2 1 1176.2.q.d 2
7.c even 3 1 168.2.a.a 1
7.c even 3 1 inner 1176.2.q.f 2
7.d odd 6 1 1176.2.a.f 1
7.d odd 6 1 1176.2.q.d 2
21.c even 2 1 3528.2.s.g 2
21.g even 6 1 3528.2.a.v 1
21.g even 6 1 3528.2.s.g 2
21.h odd 6 1 504.2.a.e 1
21.h odd 6 1 3528.2.s.w 2
28.d even 2 1 2352.2.q.w 2
28.f even 6 1 2352.2.a.c 1
28.f even 6 1 2352.2.q.w 2
28.g odd 6 1 336.2.a.e 1
28.g odd 6 1 2352.2.q.d 2
35.j even 6 1 4200.2.a.t 1
35.l odd 12 2 4200.2.t.j 2
56.j odd 6 1 9408.2.a.be 1
56.k odd 6 1 1344.2.a.b 1
56.m even 6 1 9408.2.a.da 1
56.p even 6 1 1344.2.a.m 1
84.j odd 6 1 7056.2.a.bq 1
84.n even 6 1 1008.2.a.b 1
112.u odd 12 2 5376.2.c.bb 2
112.w even 12 2 5376.2.c.d 2
140.p odd 6 1 8400.2.a.y 1
168.s odd 6 1 4032.2.a.bh 1
168.v even 6 1 4032.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 7.c even 3 1
336.2.a.e 1 28.g odd 6 1
504.2.a.e 1 21.h odd 6 1
1008.2.a.b 1 84.n even 6 1
1176.2.a.f 1 7.d odd 6 1
1176.2.q.d 2 7.b odd 2 1
1176.2.q.d 2 7.d odd 6 1
1176.2.q.f 2 1.a even 1 1 trivial
1176.2.q.f 2 7.c even 3 1 inner
1344.2.a.b 1 56.k odd 6 1
1344.2.a.m 1 56.p even 6 1
2352.2.a.c 1 28.f even 6 1
2352.2.q.d 2 4.b odd 2 1
2352.2.q.d 2 28.g odd 6 1
2352.2.q.w 2 28.d even 2 1
2352.2.q.w 2 28.f even 6 1
3528.2.a.v 1 21.g even 6 1
3528.2.s.g 2 21.c even 2 1
3528.2.s.g 2 21.g even 6 1
3528.2.s.w 2 3.b odd 2 1
3528.2.s.w 2 21.h odd 6 1
4032.2.a.bc 1 168.v even 6 1
4032.2.a.bh 1 168.s odd 6 1
4200.2.a.t 1 35.j even 6 1
4200.2.t.j 2 35.l odd 12 2
5376.2.c.d 2 112.w even 12 2
5376.2.c.bb 2 112.u odd 12 2
7056.2.a.bq 1 84.j odd 6 1
8400.2.a.y 1 140.p odd 6 1
9408.2.a.be 1 56.j odd 6 1
9408.2.a.da 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 $$S_{2}^{\mathrm{new}}(1176, [\chi])$$:

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}$$ $$T_{13} - 6$$

## 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
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}$$
$17$ $$1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4}$$
$19$ $$1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$1 + 6 T - T^{2} + 222 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4}$$
$59$ $$1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 12 T + 83 T^{2} )^{2}$$
$89$ $$1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 10 T + 97 T^{2} )^{2}$$