Properties

Label 117.4.a.c
Level $117$
Weight $4$
Character orbit 117.a
Self dual yes
Analytic conductor $6.903$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
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 \(\beta = \sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 7) q^{4} + ( - 2 \beta - 12) q^{5} + 2 \beta q^{7} + (\beta - 27) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + ( - 2 \beta + 7) q^{4} + ( - 2 \beta - 12) q^{5} + 2 \beta q^{7} + (\beta - 27) q^{8} + ( - 10 \beta - 16) q^{10} + ( - 12 \beta + 22) q^{11} - 13 q^{13} + ( - 2 \beta + 28) q^{14} + ( - 12 \beta - 15) q^{16} + (4 \beta - 82) q^{17} + ( - 2 \beta + 24) q^{19} + (10 \beta - 28) q^{20} + (34 \beta - 190) q^{22} + (48 \beta - 4) q^{23} + (48 \beta + 75) q^{25} + ( - 13 \beta + 13) q^{26} + (14 \beta - 56) q^{28} + ( - 24 \beta - 202) q^{29} + (26 \beta + 20) q^{31} + ( - 11 \beta + 63) q^{32} + ( - 86 \beta + 138) q^{34} + ( - 24 \beta - 56) q^{35} + ( - 28 \beta - 50) q^{37} + (26 \beta - 52) q^{38} + (42 \beta + 296) q^{40} + (94 \beta - 100) q^{41} + ( - 52 \beta - 308) q^{43} + ( - 128 \beta + 490) q^{44} + ( - 52 \beta + 676) q^{46} + (32 \beta + 162) q^{47} - 287 q^{49} + (27 \beta + 597) q^{50} + (26 \beta - 91) q^{52} + ( - 120 \beta + 82) q^{53} + (100 \beta + 72) q^{55} + ( - 54 \beta + 28) q^{56} + ( - 178 \beta - 134) q^{58} + (40 \beta - 70) q^{59} + ( - 136 \beta + 314) q^{61} + ( - 6 \beta + 344) q^{62} + (170 \beta - 97) q^{64} + (26 \beta + 156) q^{65} + (170 \beta - 236) q^{67} + (192 \beta - 686) q^{68} + ( - 32 \beta - 280) q^{70} + ( - 84 \beta - 214) q^{71} + ( - 76 \beta - 450) q^{73} + ( - 22 \beta - 342) q^{74} + ( - 62 \beta + 224) q^{76} + (44 \beta - 336) q^{77} + (88 \beta - 216) q^{79} + (174 \beta + 516) q^{80} + ( - 194 \beta + 1416) q^{82} + (64 \beta + 694) q^{83} + (116 \beta + 872) q^{85} + ( - 256 \beta - 420) q^{86} + (346 \beta - 762) q^{88} + ( - 190 \beta - 480) q^{89} - 26 \beta q^{91} + (344 \beta - 1372) q^{92} + (130 \beta + 286) q^{94} + ( - 24 \beta - 232) q^{95} + (220 \beta - 266) q^{97} + ( - 287 \beta + 287) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100}) \) 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
−3.74166
3.74166
−4.74166 0 14.4833 −4.51669 0 −7.48331 −30.7417 0 21.4166
1.2 2.74166 0 −0.483315 −19.4833 0 7.48331 −23.2583 0 −53.4166
\(n\): 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.c 2
3.b odd 2 1 39.4.a.b 2
4.b odd 2 1 1872.4.a.t 2
12.b even 2 1 624.4.a.r 2
13.b even 2 1 1521.4.a.s 2
15.d odd 2 1 975.4.a.j 2
21.c even 2 1 1911.4.a.h 2
24.f even 2 1 2496.4.a.s 2
24.h odd 2 1 2496.4.a.bc 2
39.d odd 2 1 507.4.a.f 2
39.f even 4 2 507.4.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 3.b odd 2 1
117.4.a.c 2 1.a even 1 1 trivial
507.4.a.f 2 39.d odd 2 1
507.4.b.f 4 39.f even 4 2
624.4.a.r 2 12.b even 2 1
975.4.a.j 2 15.d odd 2 1
1521.4.a.s 2 13.b even 2 1
1872.4.a.t 2 4.b odd 2 1
1911.4.a.h 2 21.c even 2 1
2496.4.a.s 2 24.f even 2 1
2496.4.a.bc 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 13 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 13 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 24T + 88 \) Copy content Toggle raw display
$7$ \( T^{2} - 56 \) Copy content Toggle raw display
$11$ \( T^{2} - 44T - 1532 \) Copy content Toggle raw display
$13$ \( (T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 164T + 6500 \) Copy content Toggle raw display
$19$ \( T^{2} - 48T + 520 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T - 32240 \) Copy content Toggle raw display
$29$ \( T^{2} + 404T + 32740 \) Copy content Toggle raw display
$31$ \( T^{2} - 40T - 9064 \) Copy content Toggle raw display
$37$ \( T^{2} + 100T - 8476 \) Copy content Toggle raw display
$41$ \( T^{2} + 200T - 113704 \) Copy content Toggle raw display
$43$ \( T^{2} + 616T + 57008 \) Copy content Toggle raw display
$47$ \( T^{2} - 324T + 11908 \) Copy content Toggle raw display
$53$ \( T^{2} - 164T - 194876 \) Copy content Toggle raw display
$59$ \( T^{2} + 140T - 17500 \) Copy content Toggle raw display
$61$ \( T^{2} - 628T - 160348 \) Copy content Toggle raw display
$67$ \( T^{2} + 472T - 348904 \) Copy content Toggle raw display
$71$ \( T^{2} + 428T - 52988 \) Copy content Toggle raw display
$73$ \( T^{2} + 900T + 121636 \) Copy content Toggle raw display
$79$ \( T^{2} + 432T - 61760 \) Copy content Toggle raw display
$83$ \( T^{2} - 1388 T + 424292 \) Copy content Toggle raw display
$89$ \( T^{2} + 960T - 275000 \) Copy content Toggle raw display
$97$ \( T^{2} + 532T - 606844 \) Copy content Toggle raw display
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