Properties

Label 117.4.a.e
Level $117$
Weight $4$
Character orbit 117.a
Self dual yes
Analytic conductor $6.903$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{4} - 4 \beta q^{5} - 22 q^{7} - 9 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{4} - 4 \beta q^{5} - 22 q^{7} - 9 \beta q^{8} - 28 q^{10} - 2 \beta q^{11} + 13 q^{13} - 22 \beta q^{14} - 55 q^{16} + 44 \beta q^{17} - 126 q^{19} + 4 \beta q^{20} - 14 q^{22} + 12 \beta q^{23} - 13 q^{25} + 13 \beta q^{26} + 22 q^{28} + 20 \beta q^{29} - 182 q^{31} + 17 \beta q^{32} + 308 q^{34} + 88 \beta q^{35} - 86 q^{37} - 126 \beta q^{38} + 252 q^{40} - 168 \beta q^{41} + 96 q^{43} + 2 \beta q^{44} + 84 q^{46} + 138 \beta q^{47} + 141 q^{49} - 13 \beta q^{50} - 13 q^{52} - 72 \beta q^{53} + 56 q^{55} + 198 \beta q^{56} + 140 q^{58} - 222 \beta q^{59} + 574 q^{61} - 182 \beta q^{62} + 559 q^{64} - 52 \beta q^{65} - 530 q^{67} - 44 \beta q^{68} + 616 q^{70} + 306 \beta q^{71} - 154 q^{73} - 86 \beta q^{74} + 126 q^{76} + 44 \beta q^{77} - 460 q^{79} + 220 \beta q^{80} - 1176 q^{82} - 122 \beta q^{83} - 1232 q^{85} + 96 \beta q^{86} + 126 q^{88} + 544 \beta q^{89} - 286 q^{91} - 12 \beta q^{92} + 966 q^{94} + 504 \beta q^{95} + 70 q^{97} + 141 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 44 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 44 q^{7} - 56 q^{10} + 26 q^{13} - 110 q^{16} - 252 q^{19} - 28 q^{22} - 26 q^{25} + 44 q^{28} - 364 q^{31} + 616 q^{34} - 172 q^{37} + 504 q^{40} + 192 q^{43} + 168 q^{46} + 282 q^{49} - 26 q^{52} + 112 q^{55} + 280 q^{58} + 1148 q^{61} + 1118 q^{64} - 1060 q^{67} + 1232 q^{70} - 308 q^{73} + 252 q^{76} - 920 q^{79} - 2352 q^{82} - 2464 q^{85} + 252 q^{88} - 572 q^{91} + 1932 q^{94} + 140 q^{97}+O(q^{100}) \) 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
−2.64575
2.64575
−2.64575 0 −1.00000 10.5830 0 −22.0000 23.8118 0 −28.0000
1.2 2.64575 0 −1.00000 −10.5830 0 −22.0000 −23.8118 0 −28.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.a.e 2
3.b odd 2 1 inner 117.4.a.e 2
4.b odd 2 1 1872.4.a.ba 2
12.b even 2 1 1872.4.a.ba 2
13.b even 2 1 1521.4.a.p 2
39.d odd 2 1 1521.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.a.e 2 1.a even 1 1 trivial
117.4.a.e 2 3.b odd 2 1 inner
1521.4.a.p 2 13.b even 2 1
1521.4.a.p 2 39.d odd 2 1
1872.4.a.ba 2 4.b odd 2 1
1872.4.a.ba 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 7 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 112 \) Copy content Toggle raw display
$7$ \( (T + 22)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 28 \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 13552 \) Copy content Toggle raw display
$19$ \( (T + 126)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 1008 \) Copy content Toggle raw display
$29$ \( T^{2} - 2800 \) Copy content Toggle raw display
$31$ \( (T + 182)^{2} \) Copy content Toggle raw display
$37$ \( (T + 86)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 197568 \) Copy content Toggle raw display
$43$ \( (T - 96)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 133308 \) Copy content Toggle raw display
$53$ \( T^{2} - 36288 \) Copy content Toggle raw display
$59$ \( T^{2} - 344988 \) Copy content Toggle raw display
$61$ \( (T - 574)^{2} \) Copy content Toggle raw display
$67$ \( (T + 530)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 655452 \) Copy content Toggle raw display
$73$ \( (T + 154)^{2} \) Copy content Toggle raw display
$79$ \( (T + 460)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 104188 \) Copy content Toggle raw display
$89$ \( T^{2} - 2071552 \) Copy content Toggle raw display
$97$ \( (T - 70)^{2} \) Copy content Toggle raw display
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