Properties

Label 117.2.a.c
Level $117$
Weight $2$
Character orbit 117.a
Self dual yes
Analytic conductor $0.934$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} -2 \beta q^{5} -2 \beta q^{7} + ( 3 + \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 2 \beta ) q^{4} -2 \beta q^{5} -2 \beta q^{7} + ( 3 + \beta ) q^{8} + ( -4 - 2 \beta ) q^{10} + 2 q^{11} - q^{13} + ( -4 - 2 \beta ) q^{14} + 3 q^{16} + ( -2 + 4 \beta ) q^{17} + 2 \beta q^{19} + ( -8 - 2 \beta ) q^{20} + ( 2 + 2 \beta ) q^{22} + 4 q^{23} + 3 q^{25} + ( -1 - \beta ) q^{26} + ( -8 - 2 \beta ) q^{28} -2 q^{29} + ( -4 - 2 \beta ) q^{31} + ( -3 + \beta ) q^{32} + ( 6 + 2 \beta ) q^{34} + 8 q^{35} + ( -2 + 4 \beta ) q^{37} + ( 4 + 2 \beta ) q^{38} + ( -4 - 6 \beta ) q^{40} + ( -8 - 2 \beta ) q^{41} + ( 4 + 4 \beta ) q^{43} + ( 2 + 4 \beta ) q^{44} + ( 4 + 4 \beta ) q^{46} + ( 6 - 4 \beta ) q^{47} + q^{49} + ( 3 + 3 \beta ) q^{50} + ( -1 - 2 \beta ) q^{52} + 2 q^{53} -4 \beta q^{55} + ( -4 - 6 \beta ) q^{56} + ( -2 - 2 \beta ) q^{58} + ( -2 + 4 \beta ) q^{59} + ( 2 - 8 \beta ) q^{61} + ( -8 - 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + 2 \beta q^{65} + ( 4 - 2 \beta ) q^{67} + 14 q^{68} + ( 8 + 8 \beta ) q^{70} -2 q^{71} + ( 6 + 4 \beta ) q^{73} + ( 6 + 2 \beta ) q^{74} + ( 8 + 2 \beta ) q^{76} -4 \beta q^{77} + 8 \beta q^{79} -6 \beta q^{80} + ( -12 - 10 \beta ) q^{82} + ( 2 + 4 \beta ) q^{83} + ( -16 + 4 \beta ) q^{85} + ( 12 + 8 \beta ) q^{86} + ( 6 + 2 \beta ) q^{88} + ( -12 + 2 \beta ) q^{89} + 2 \beta q^{91} + ( 4 + 8 \beta ) q^{92} + ( -2 + 2 \beta ) q^{94} -8 q^{95} + ( -2 - 4 \beta ) q^{97} + ( 1 + \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 6q^{8} - 8q^{10} + 4q^{11} - 2q^{13} - 8q^{14} + 6q^{16} - 4q^{17} - 16q^{20} + 4q^{22} + 8q^{23} + 6q^{25} - 2q^{26} - 16q^{28} - 4q^{29} - 8q^{31} - 6q^{32} + 12q^{34} + 16q^{35} - 4q^{37} + 8q^{38} - 8q^{40} - 16q^{41} + 8q^{43} + 4q^{44} + 8q^{46} + 12q^{47} + 2q^{49} + 6q^{50} - 2q^{52} + 4q^{53} - 8q^{56} - 4q^{58} - 4q^{59} + 4q^{61} - 16q^{62} - 14q^{64} + 8q^{67} + 28q^{68} + 16q^{70} - 4q^{71} + 12q^{73} + 12q^{74} + 16q^{76} - 24q^{82} + 4q^{83} - 32q^{85} + 24q^{86} + 12q^{88} - 24q^{89} + 8q^{92} - 4q^{94} - 16q^{95} - 4q^{97} + 2q^{98} + O(q^{100}) \)

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
−0.414214 0 −1.82843 2.82843 0 2.82843 1.58579 0 −1.17157
1.2 2.41421 0 3.82843 −2.82843 0 −2.82843 4.41421 0 −6.82843
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)

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 117.2.a.c 2
3.b odd 2 1 39.2.a.b 2
4.b odd 2 1 1872.2.a.w 2
5.b even 2 1 2925.2.a.v 2
5.c odd 4 2 2925.2.c.u 4
7.b odd 2 1 5733.2.a.u 2
8.b even 2 1 7488.2.a.cl 2
8.d odd 2 1 7488.2.a.co 2
9.c even 3 2 1053.2.e.e 4
9.d odd 6 2 1053.2.e.m 4
12.b even 2 1 624.2.a.k 2
13.b even 2 1 1521.2.a.f 2
13.d odd 4 2 1521.2.b.j 4
15.d odd 2 1 975.2.a.l 2
15.e even 4 2 975.2.c.h 4
21.c even 2 1 1911.2.a.h 2
24.f even 2 1 2496.2.a.bi 2
24.h odd 2 1 2496.2.a.bf 2
33.d even 2 1 4719.2.a.p 2
39.d odd 2 1 507.2.a.h 2
39.f even 4 2 507.2.b.e 4
39.h odd 6 2 507.2.e.d 4
39.i odd 6 2 507.2.e.h 4
39.k even 12 4 507.2.j.f 8
156.h even 2 1 8112.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 3.b odd 2 1
117.2.a.c 2 1.a even 1 1 trivial
507.2.a.h 2 39.d odd 2 1
507.2.b.e 4 39.f even 4 2
507.2.e.d 4 39.h odd 6 2
507.2.e.h 4 39.i odd 6 2
507.2.j.f 8 39.k even 12 4
624.2.a.k 2 12.b even 2 1
975.2.a.l 2 15.d odd 2 1
975.2.c.h 4 15.e even 4 2
1053.2.e.e 4 9.c even 3 2
1053.2.e.m 4 9.d odd 6 2
1521.2.a.f 2 13.b even 2 1
1521.2.b.j 4 13.d odd 4 2
1872.2.a.w 2 4.b odd 2 1
1911.2.a.h 2 21.c even 2 1
2496.2.a.bf 2 24.h odd 2 1
2496.2.a.bi 2 24.f even 2 1
2925.2.a.v 2 5.b even 2 1
2925.2.c.u 4 5.c odd 4 2
4719.2.a.p 2 33.d even 2 1
5733.2.a.u 2 7.b odd 2 1
7488.2.a.cl 2 8.b even 2 1
7488.2.a.co 2 8.d odd 2 1
8112.2.a.bm 2 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(117))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -8 + T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -28 + 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( 8 + 8 T + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( 56 + 16 T + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( 4 - 12 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( -28 + 4 T + T^{2} \)
$61$ \( -124 - 4 T + T^{2} \)
$67$ \( 8 - 8 T + T^{2} \)
$71$ \( ( 2 + T )^{2} \)
$73$ \( 4 - 12 T + T^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -28 - 4 T + T^{2} \)
$89$ \( 136 + 24 T + T^{2} \)
$97$ \( -28 + 4 T + T^{2} \)
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