Properties

Label 1176.2.q.o
Level $1176$
Weight $2$
Character orbit 1176.q
Analytic conductor $9.390$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + \beta_{2} q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{5} + \beta_{2} q^{9} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{11} + \beta_{3} q^{13} + ( 2 - \beta_{3} ) q^{15} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{23} + ( -1 - 4 \beta_{1} - \beta_{2} ) q^{25} - q^{27} + 6 \beta_{3} q^{29} + ( 8 + 2 \beta_{1} + 8 \beta_{2} ) q^{31} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{33} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{37} -\beta_{1} q^{39} + ( 2 - \beta_{3} ) q^{41} -8 q^{43} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{45} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{51} + ( 2 - 8 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -8 + 6 \beta_{3} ) q^{55} + ( 4 + 2 \beta_{3} ) q^{57} + ( 8 + 2 \beta_{1} + 8 \beta_{2} ) q^{59} + ( 7 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 8 + 8 \beta_{2} ) q^{67} + ( -2 - 2 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{3} ) q^{71} + ( -4 + 5 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{75} + ( -4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{79} + ( -1 - \beta_{2} ) q^{81} + ( -4 - 8 \beta_{3} ) q^{83} + ( -10 + 8 \beta_{3} ) q^{85} -6 \beta_{1} q^{87} + ( -9 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} ) q^{89} + ( 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{93} + ( -4 - 4 \beta_{2} ) q^{95} + ( 12 - 3 \beta_{3} ) q^{97} + ( 2 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{5} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{5} - 2q^{9} - 4q^{11} + 8q^{15} - 4q^{17} + 8q^{19} - 4q^{23} - 2q^{25} - 4q^{27} + 16q^{31} + 4q^{33} + 8q^{37} + 8q^{41} - 32q^{43} + 4q^{45} + 8q^{47} + 4q^{51} + 4q^{53} - 32q^{55} + 16q^{57} + 16q^{59} + 8q^{61} - 4q^{65} + 16q^{67} - 8q^{69} + 8q^{71} - 8q^{73} + 2q^{75} + 16q^{79} - 2q^{81} - 16q^{83} - 40q^{85} + 4q^{89} - 16q^{93} - 8q^{95} + 48q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 0.500000 0.866025i 0 0.292893 + 0.507306i 0 0 0 −0.500000 0.866025i 0
361.2 0 0.500000 0.866025i 0 1.70711 + 2.95680i 0 0 0 −0.500000 0.866025i 0
961.1 0 0.500000 + 0.866025i 0 0.292893 0.507306i 0 0 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 1.70711 2.95680i 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.o 4
3.b odd 2 1 3528.2.s.bd 4
4.b odd 2 1 2352.2.q.bc 4
7.b odd 2 1 1176.2.q.k 4
7.c even 3 1 1176.2.a.j 2
7.c even 3 1 inner 1176.2.q.o 4
7.d odd 6 1 1176.2.a.o yes 2
7.d odd 6 1 1176.2.q.k 4
21.c even 2 1 3528.2.s.bm 4
21.g even 6 1 3528.2.a.bb 2
21.g even 6 1 3528.2.s.bm 4
21.h odd 6 1 3528.2.a.bl 2
21.h odd 6 1 3528.2.s.bd 4
28.d even 2 1 2352.2.q.be 4
28.f even 6 1 2352.2.a.bb 2
28.f even 6 1 2352.2.q.be 4
28.g odd 6 1 2352.2.a.bd 2
28.g odd 6 1 2352.2.q.bc 4
56.j odd 6 1 9408.2.a.dg 2
56.k odd 6 1 9408.2.a.ds 2
56.m even 6 1 9408.2.a.du 2
56.p even 6 1 9408.2.a.ee 2
84.j odd 6 1 7056.2.a.cg 2
84.n even 6 1 7056.2.a.cx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 7.c even 3 1
1176.2.a.o yes 2 7.d odd 6 1
1176.2.q.k 4 7.b odd 2 1
1176.2.q.k 4 7.d odd 6 1
1176.2.q.o 4 1.a even 1 1 trivial
1176.2.q.o 4 7.c even 3 1 inner
2352.2.a.bb 2 28.f even 6 1
2352.2.a.bd 2 28.g odd 6 1
2352.2.q.bc 4 4.b odd 2 1
2352.2.q.bc 4 28.g odd 6 1
2352.2.q.be 4 28.d even 2 1
2352.2.q.be 4 28.f even 6 1
3528.2.a.bb 2 21.g even 6 1
3528.2.a.bl 2 21.h odd 6 1
3528.2.s.bd 4 3.b odd 2 1
3528.2.s.bd 4 21.h odd 6 1
3528.2.s.bm 4 21.c even 2 1
3528.2.s.bm 4 21.g even 6 1
7056.2.a.cg 2 84.j odd 6 1
7056.2.a.cx 2 84.n even 6 1
9408.2.a.dg 2 56.j odd 6 1
9408.2.a.ds 2 56.k odd 6 1
9408.2.a.du 2 56.m even 6 1
9408.2.a.ee 2 56.p even 6 1

Hecke kernels

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

\( T_{5}^{4} - 4 T_{5}^{3} + 14 T_{5}^{2} - 8 T_{5} + 4 \)
\( T_{11}^{4} + 4 T_{11}^{3} + 20 T_{11}^{2} - 16 T_{11} + 16 \)
\( T_{13}^{2} - 2 \)