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 $2$

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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
0.500000 + 0.866025i
0.500000 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:

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 \(S_{2}^{\mathrm{new}}(1176, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{11}^{2} + 4 T_{11} + 16 \)
\( T_{13} - 2 \)

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