Properties

Label 117.4.a.b
Level $117$
Weight $4$
Character orbit 117.a
Self dual yes
Analytic conductor $6.903$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5 q^{2} + 17 q^{4} + 7 q^{5} - 13 q^{7} + 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{2} + 17 q^{4} + 7 q^{5} - 13 q^{7} + 45 q^{8} + 35 q^{10} + 26 q^{11} + 13 q^{13} - 65 q^{14} + 89 q^{16} - 77 q^{17} - 126 q^{19} + 119 q^{20} + 130 q^{22} + 96 q^{23} - 76 q^{25} + 65 q^{26} - 221 q^{28} + 82 q^{29} + 196 q^{31} + 85 q^{32} - 385 q^{34} - 91 q^{35} - 131 q^{37} - 630 q^{38} + 315 q^{40} - 336 q^{41} - 201 q^{43} + 442 q^{44} + 480 q^{46} + 105 q^{47} - 174 q^{49} - 380 q^{50} + 221 q^{52} + 432 q^{53} + 182 q^{55} - 585 q^{56} + 410 q^{58} + 294 q^{59} - 56 q^{61} + 980 q^{62} - 287 q^{64} + 91 q^{65} + 478 q^{67} - 1309 q^{68} - 455 q^{70} - 9 q^{71} + 98 q^{73} - 655 q^{74} - 2142 q^{76} - 338 q^{77} + 1304 q^{79} + 623 q^{80} - 1680 q^{82} + 308 q^{83} - 539 q^{85} - 1005 q^{86} + 1170 q^{88} + 1190 q^{89} - 169 q^{91} + 1632 q^{92} + 525 q^{94} - 882 q^{95} + 70 q^{97} - 870 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
0
5.00000 0 17.0000 7.00000 0 −13.0000 45.0000 0 35.0000
\(n\): 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 117.4.a.b 1
3.b odd 2 1 13.4.a.a 1
4.b odd 2 1 1872.4.a.k 1
12.b even 2 1 208.4.a.g 1
13.b even 2 1 1521.4.a.a 1
15.d odd 2 1 325.4.a.d 1
15.e even 4 2 325.4.b.b 2
21.c even 2 1 637.4.a.a 1
24.f even 2 1 832.4.a.a 1
24.h odd 2 1 832.4.a.r 1
33.d even 2 1 1573.4.a.a 1
39.d odd 2 1 169.4.a.e 1
39.f even 4 2 169.4.b.a 2
39.h odd 6 2 169.4.c.a 2
39.i odd 6 2 169.4.c.e 2
39.k even 12 4 169.4.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 3.b odd 2 1
117.4.a.b 1 1.a even 1 1 trivial
169.4.a.e 1 39.d odd 2 1
169.4.b.a 2 39.f even 4 2
169.4.c.a 2 39.h odd 6 2
169.4.c.e 2 39.i odd 6 2
169.4.e.e 4 39.k even 12 4
208.4.a.g 1 12.b even 2 1
325.4.a.d 1 15.d odd 2 1
325.4.b.b 2 15.e even 4 2
637.4.a.a 1 21.c even 2 1
832.4.a.a 1 24.f even 2 1
832.4.a.r 1 24.h odd 2 1
1521.4.a.a 1 13.b even 2 1
1573.4.a.a 1 33.d even 2 1
1872.4.a.k 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 5 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 7 \) Copy content Toggle raw display
$7$ \( T + 13 \) Copy content Toggle raw display
$11$ \( T - 26 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T + 77 \) Copy content Toggle raw display
$19$ \( T + 126 \) Copy content Toggle raw display
$23$ \( T - 96 \) Copy content Toggle raw display
$29$ \( T - 82 \) Copy content Toggle raw display
$31$ \( T - 196 \) Copy content Toggle raw display
$37$ \( T + 131 \) Copy content Toggle raw display
$41$ \( T + 336 \) Copy content Toggle raw display
$43$ \( T + 201 \) Copy content Toggle raw display
$47$ \( T - 105 \) Copy content Toggle raw display
$53$ \( T - 432 \) Copy content Toggle raw display
$59$ \( T - 294 \) Copy content Toggle raw display
$61$ \( T + 56 \) Copy content Toggle raw display
$67$ \( T - 478 \) Copy content Toggle raw display
$71$ \( T + 9 \) Copy content Toggle raw display
$73$ \( T - 98 \) Copy content Toggle raw display
$79$ \( T - 1304 \) Copy content Toggle raw display
$83$ \( T - 308 \) Copy content Toggle raw display
$89$ \( T - 1190 \) Copy content Toggle raw display
$97$ \( T - 70 \) Copy content Toggle raw display
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