Properties

Label 1176.2.a.o
Level $1176$
Weight $2$
Character orbit 1176.a
Self dual yes
Analytic conductor $9.390$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( 2 + \beta ) q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 2 + \beta ) q^{5} + q^{9} + ( 2 + 2 \beta ) q^{11} + \beta q^{13} + ( 2 + \beta ) q^{15} + ( -2 - 3 \beta ) q^{17} + ( 4 - 2 \beta ) q^{19} + ( 2 - 2 \beta ) q^{23} + ( 1 + 4 \beta ) q^{25} + q^{27} -6 \beta q^{29} + ( 8 + 2 \beta ) q^{31} + ( 2 + 2 \beta ) q^{33} + ( -4 - 4 \beta ) q^{37} + \beta q^{39} + ( -2 - \beta ) q^{41} -8 q^{43} + ( 2 + \beta ) q^{45} + ( 4 - 2 \beta ) q^{47} + ( -2 - 3 \beta ) q^{51} + ( -2 + 8 \beta ) q^{53} + ( 8 + 6 \beta ) q^{55} + ( 4 - 2 \beta ) q^{57} + ( 8 + 2 \beta ) q^{59} + ( 4 - 7 \beta ) q^{61} + ( 2 + 2 \beta ) q^{65} -8 q^{67} + ( 2 - 2 \beta ) q^{69} + ( 2 + 2 \beta ) q^{71} + ( -4 + 5 \beta ) q^{73} + ( 1 + 4 \beta ) q^{75} + ( -8 - 4 \beta ) q^{79} + q^{81} + ( 4 - 8 \beta ) q^{83} + ( -10 - 8 \beta ) q^{85} -6 \beta q^{87} + ( 2 + 9 \beta ) q^{89} + ( 8 + 2 \beta ) q^{93} + 4 q^{95} + ( -12 - 3 \beta ) q^{97} + ( 2 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 4q^{5} + 2q^{9} + 4q^{11} + 4q^{15} - 4q^{17} + 8q^{19} + 4q^{23} + 2q^{25} + 2q^{27} + 16q^{31} + 4q^{33} - 8q^{37} - 4q^{41} - 16q^{43} + 4q^{45} + 8q^{47} - 4q^{51} - 4q^{53} + 16q^{55} + 8q^{57} + 16q^{59} + 8q^{61} + 4q^{65} - 16q^{67} + 4q^{69} + 4q^{71} - 8q^{73} + 2q^{75} - 16q^{79} + 2q^{81} + 8q^{83} - 20q^{85} + 4q^{89} + 16q^{93} + 8q^{95} - 24q^{97} + 4q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 1.00000 0 0.585786 0 0 0 1.00000 0
1.2 0 1.00000 0 3.41421 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.2.a.o yes 2
3.b odd 2 1 3528.2.a.bb 2
4.b odd 2 1 2352.2.a.bb 2
7.b odd 2 1 1176.2.a.j 2
7.c even 3 2 1176.2.q.k 4
7.d odd 6 2 1176.2.q.o 4
8.b even 2 1 9408.2.a.dg 2
8.d odd 2 1 9408.2.a.du 2
12.b even 2 1 7056.2.a.cg 2
21.c even 2 1 3528.2.a.bl 2
21.g even 6 2 3528.2.s.bd 4
21.h odd 6 2 3528.2.s.bm 4
28.d even 2 1 2352.2.a.bd 2
28.f even 6 2 2352.2.q.bc 4
28.g odd 6 2 2352.2.q.be 4
56.e even 2 1 9408.2.a.ds 2
56.h odd 2 1 9408.2.a.ee 2
84.h odd 2 1 7056.2.a.cx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 7.b odd 2 1
1176.2.a.o yes 2 1.a even 1 1 trivial
1176.2.q.k 4 7.c even 3 2
1176.2.q.o 4 7.d odd 6 2
2352.2.a.bb 2 4.b odd 2 1
2352.2.a.bd 2 28.d even 2 1
2352.2.q.bc 4 28.f even 6 2
2352.2.q.be 4 28.g odd 6 2
3528.2.a.bb 2 3.b odd 2 1
3528.2.a.bl 2 21.c even 2 1
3528.2.s.bd 4 21.g even 6 2
3528.2.s.bm 4 21.h odd 6 2
7056.2.a.cg 2 12.b even 2 1
7056.2.a.cx 2 84.h odd 2 1
9408.2.a.dg 2 8.b even 2 1
9408.2.a.ds 2 56.e even 2 1
9408.2.a.du 2 8.d odd 2 1
9408.2.a.ee 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1176))\):

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