Properties

Label 117.4.g.f
Level $117$
Weight $4$
Character orbit 117.g
Analytic conductor $6.903$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + 21016320 x^{2} + 24920064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{3} - 5 \beta_{2} - 5) q^{4} + (\beta_{12} + \beta_{9}) q^{5} + ( - \beta_{7} - \beta_{4} + 3 \beta_{2} + 3) q^{7} + (\beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} - 5 \beta_{5} - 5 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{3} - 5 \beta_{2} - 5) q^{4} + (\beta_{12} + \beta_{9}) q^{5} + ( - \beta_{7} - \beta_{4} + 3 \beta_{2} + 3) q^{7} + (\beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} - 5 \beta_{5} - 5 \beta_1) q^{8} + (\beta_{10} - 3 \beta_{6} + \beta_{4} + 4 \beta_{2}) q^{10} + (\beta_{14} - \beta_{9} + \beta_1) q^{11} + (\beta_{10} - \beta_{8} - \beta_{4} - 2 \beta_{3} - 4 \beta_{2}) q^{13} + ( - \beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{12} + \beta_{11} + 2 \beta_{9} + 7 \beta_{5} + \cdots + 7 \beta_1) q^{14}+ \cdots + (8 \beta_{15} + 25 \beta_{13} + 57 \beta_{12} - 33 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + 21016320 x^{2} + 24920064 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1400516976091 \nu^{14} - 70185147718972 \nu^{12} + \cdots - 25\!\cdots\!88 ) / 19\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1058387435 \nu^{14} - 48809305790 \nu^{12} - 1728612264289 \nu^{10} - 25222482239398 \nu^{8} - 300483724727391 \nu^{6} + \cdots + 86\!\cdots\!80 ) / 77\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3479246660543 \nu^{14} + 32435361605764 \nu^{12} + \cdots + 20\!\cdots\!40 ) / 19\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1400516976091 \nu^{15} - 70185147718972 \nu^{13} + \cdots - 25\!\cdots\!88 \nu ) / 19\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1197306588571 \nu^{14} - 60813761007292 \nu^{12} + \cdots - 23\!\cdots\!40 ) / 14\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45779722905 \nu^{14} + 2174587212793 \nu^{12} + 74769775086507 \nu^{10} + \cdots - 12\!\cdots\!00 ) / 31\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28699944525 \nu^{14} + 1329441616294 \nu^{12} + 46874211134535 \nu^{10} + 683949768407370 \nu^{8} + \cdots - 77\!\cdots\!76 ) / 15\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30816719395 \nu^{15} + 1427060227874 \nu^{13} + 50331435663113 \nu^{11} + 734394732886166 \nu^{9} + \cdots - 11\!\cdots\!80 \nu ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 841688410411 \nu^{14} - 44413834261852 \nu^{12} + \cdots - 18\!\cdots\!56 ) / 24\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47750918355 \nu^{15} + 2208009120514 \nu^{13} + 77989231891737 \nu^{11} + \cdots - 35\!\cdots\!04 \nu ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33607973109585 \nu^{15} + \cdots + 70\!\cdots\!40 \nu ) / 51\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8076330538717 \nu^{15} - 411639246983524 \nu^{13} + \cdots - 15\!\cdots\!08 \nu ) / 39\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 292503248765 \nu^{15} + 13760478427434 \nu^{13} + 477731203563991 \nu^{11} + \cdots - 58\!\cdots\!92 \nu ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 147991404382739 \nu^{15} + \cdots + 37\!\cdots\!52 \nu ) / 51\!\cdots\!88 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{3} - 13\beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} - 21\beta_{5} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} - 33\beta_{6} + 273\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -35\beta_{13} - 51\beta_{12} + 525\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -86\beta_{8} + 16\beta_{7} - 993\beta_{3} + 6889 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16\beta_{14} + 1063\beta_{11} - 1799\beta_{9} + 14253\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2862 \beta_{10} + 2862 \beta_{8} - 832 \beta_{7} + 29281 \beta_{6} - 832 \beta_{4} + 29281 \beta_{3} - 188201 \beta_{2} - 188201 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 832 \beta_{15} - 832 \beta_{14} + 31311 \beta_{13} + 56703 \beta_{12} - 31311 \beta_{11} + 56703 \beta_{9} - 401949 \beta_{5} - 401949 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 88014\beta_{10} - 857185\beta_{6} + 30384\beta_{4} + 5325241\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 30384\beta_{15} - 914815\beta_{13} - 1710079\beta_{12} + 11533101\beta_{5} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -2624894\beta_{8} + 977568\beta_{7} - 25018209\beta_{3} + 153050601 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 977568\beta_{14} + 26665535\beta_{11} - 50597391\beta_{9} + 333536637\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 77262926 \beta_{10} + 77262926 \beta_{8} - 29797264 \beta_{7} + 729228897 \beta_{6} - 29797264 \beta_{4} + 729228897 \beta_{3} - 4429748569 \beta_{2} + \cdots - 4429748569 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 29797264 \beta_{15} - 29797264 \beta_{14} + 776694559 \beta_{13} + 1484189759 \beta_{12} - 776694559 \beta_{11} + 1484189759 \beta_{9} - 9680813133 \beta_{5} + \cdots - 9680813133 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−2.69794 4.67298i
−1.84606 3.19747i
−1.37814 2.38701i
−0.643348 1.11431i
0.643348 + 1.11431i
1.37814 + 2.38701i
1.84606 + 3.19747i
2.69794 + 4.67298i
−2.69794 + 4.67298i
−1.84606 + 3.19747i
−1.37814 + 2.38701i
−0.643348 + 1.11431i
0.643348 1.11431i
1.37814 2.38701i
1.84606 3.19747i
2.69794 4.67298i
−2.69794 4.67298i 0 −10.5578 + 18.2867i 13.0421 0 3.21247 5.56416i 70.7705 0 −35.1870 60.9457i
55.2 −1.84606 3.19747i 0 −2.81586 + 4.87721i −18.7574 0 12.0691 20.9044i −8.74396 0 34.6273 + 59.9763i
55.3 −1.37814 2.38701i 0 0.201468 0.348954i 0.313209 0 −14.2830 + 24.7388i −23.1608 0 −0.431646 0.747632i
55.4 −0.643348 1.11431i 0 3.17221 5.49442i 12.4484 0 4.50137 7.79659i −18.4569 0 −8.00866 13.8714i
55.5 0.643348 + 1.11431i 0 3.17221 5.49442i −12.4484 0 4.50137 7.79659i 18.4569 0 −8.00866 13.8714i
55.6 1.37814 + 2.38701i 0 0.201468 0.348954i −0.313209 0 −14.2830 + 24.7388i 23.1608 0 −0.431646 0.747632i
55.7 1.84606 + 3.19747i 0 −2.81586 + 4.87721i 18.7574 0 12.0691 20.9044i 8.74396 0 34.6273 + 59.9763i
55.8 2.69794 + 4.67298i 0 −10.5578 + 18.2867i −13.0421 0 3.21247 5.56416i −70.7705 0 −35.1870 60.9457i
100.1 −2.69794 + 4.67298i 0 −10.5578 18.2867i 13.0421 0 3.21247 + 5.56416i 70.7705 0 −35.1870 + 60.9457i
100.2 −1.84606 + 3.19747i 0 −2.81586 4.87721i −18.7574 0 12.0691 + 20.9044i −8.74396 0 34.6273 59.9763i
100.3 −1.37814 + 2.38701i 0 0.201468 + 0.348954i 0.313209 0 −14.2830 24.7388i −23.1608 0 −0.431646 + 0.747632i
100.4 −0.643348 + 1.11431i 0 3.17221 + 5.49442i 12.4484 0 4.50137 + 7.79659i −18.4569 0 −8.00866 + 13.8714i
100.5 0.643348 1.11431i 0 3.17221 + 5.49442i −12.4484 0 4.50137 + 7.79659i 18.4569 0 −8.00866 + 13.8714i
100.6 1.37814 2.38701i 0 0.201468 + 0.348954i −0.313209 0 −14.2830 24.7388i 23.1608 0 −0.431646 + 0.747632i
100.7 1.84606 3.19747i 0 −2.81586 4.87721i 18.7574 0 12.0691 + 20.9044i 8.74396 0 34.6273 59.9763i
100.8 2.69794 4.67298i 0 −10.5578 18.2867i −13.0421 0 3.21247 + 5.56416i −70.7705 0 −35.1870 + 60.9457i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.f 16
3.b odd 2 1 inner 117.4.g.f 16
13.c even 3 1 inner 117.4.g.f 16
13.c even 3 1 1521.4.a.bc 8
13.e even 6 1 1521.4.a.bd 8
39.h odd 6 1 1521.4.a.bd 8
39.i odd 6 1 inner 117.4.g.f 16
39.i odd 6 1 1521.4.a.bc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.g.f 16 1.a even 1 1 trivial
117.4.g.f 16 3.b odd 2 1 inner
117.4.g.f 16 13.c even 3 1 inner
117.4.g.f 16 39.i odd 6 1 inner
1521.4.a.bc 8 13.c even 3 1
1521.4.a.bc 8 39.i odd 6 1
1521.4.a.bd 8 13.e even 6 1
1521.4.a.bd 8 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 52 T_{2}^{14} + 1899 T_{2}^{12} + 33440 T_{2}^{10} + 424113 T_{2}^{8} + 2869882 T_{2}^{6} + 13705540 T_{2}^{4} + 21016320 T_{2}^{2} + 24920064 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 52 T^{14} + 1899 T^{12} + \cdots + 24920064 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 677 T^{6} + 140795 T^{4} + \cdots + 909792)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 11 T^{7} + 821 T^{6} + \cdots + 1590733456)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 9720 T^{14} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T^{8} - 18 T^{7} + \cdots + 23298085122481)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 29425 T^{14} + \cdots + 90\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{8} + 122 T^{7} + \cdots + 210844339430400)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 46352 T^{14} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + 139365 T^{14} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{4} - 121 T^{3} - 54236 T^{2} + \cdots - 349762400)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 509 T^{7} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 225537 T^{14} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{8} + 37 T^{7} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 157456 T^{6} + \cdots + 36\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 733517 T^{6} + \cdots + 44\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 445056 T^{14} + \cdots + 33\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{8} + 574 T^{7} + \cdots + 46\!\cdots\!21)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 1099 T^{7} + \cdots + 72\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 1791632 T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + 1088 T^{3} + \cdots + 20997802657)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 931 T^{3} + 176852 T^{2} + \cdots - 9861650240)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 870232 T^{6} + \cdots + 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 3056192 T^{14} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{8} - 2185 T^{7} + \cdots + 84\!\cdots\!36)^{2} \) Copy content Toggle raw display
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