Properties

Label 117.4.g.e
Level $117$
Weight $4$
Character orbit 117.g
Analytic conductor $6.903$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} - \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{5} + \beta_{3} + 2) q^{5} + (\beta_{7} + \beta_{5} - 4 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{7} + 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 16) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} - \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{5} + \beta_{3} + 2) q^{5} + (\beta_{7} + \beta_{5} - 4 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{7} + 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 16) q^{8} + (2 \beta_{6} + \beta_{4} + 11 \beta_{2} + 9 \beta_1 + 11) q^{10} + ( - \beta_{6} + \beta_{4} + 8 \beta_{2} + 4 \beta_1 + 8) q^{11} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - 25 \beta_{2} + 2 \beta_1 - 20) q^{13} + ( - 6 \beta_{5} - 7 \beta_{3} - 7) q^{14} + ( - 5 \beta_{6} - 2 \beta_{4} - 21 \beta_{2} - 19 \beta_1 - 21) q^{16} + (\beta_{7} - 19 \beta_{5} - 15 \beta_{2} - 19 \beta_1 + 19) q^{17} + (\beta_{7} - 7 \beta_{6} + 6 \beta_{5} + 7 \beta_{3} + 28 \beta_{2} + 6 \beta_1 - 6) q^{19} + (2 \beta_{7} + 9 \beta_{6} + 23 \beta_{5} - 9 \beta_{3} + 105 \beta_{2} + 23 \beta_1 - 23) q^{20} + ( - \beta_{7} + 9 \beta_{6} + 2 \beta_{5} - 9 \beta_{3} + 54 \beta_{2} + 2 \beta_1 - 2) q^{22} + (\beta_{6} + 3 \beta_{4} + 28 \beta_{2} - 4 \beta_1 + 28) q^{23} + ( - 2 \beta_{7} - 19 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 5) q^{25} + ( - \beta_{7} - 7 \beta_{6} - 27 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 47 \beta_{2} + \cdots + 46) q^{26}+ \cdots + (29 \beta_{7} + 84 \beta_{6} - 49 \beta_{5} - 84 \beta_{3} + 501 \beta_{2} + \cdots + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8} + 62 q^{10} + 40 q^{11} - 60 q^{13} - 80 q^{14} - 122 q^{16} + 98 q^{17} - 124 q^{19} - 466 q^{20} - 220 q^{22} + 104 q^{23} - 116 q^{25} - 14 q^{26} + 144 q^{28} + 194 q^{29} + 52 q^{31} + 654 q^{32} + 2124 q^{34} + 88 q^{35} - 102 q^{37} - 664 q^{38} - 1996 q^{40} - 1054 q^{41} - 450 q^{43} + 88 q^{44} + 172 q^{46} + 192 q^{47} - 1070 q^{49} + 996 q^{50} + 2280 q^{52} - 524 q^{53} - 204 q^{55} + 2164 q^{56} - 722 q^{58} + 308 q^{59} + 928 q^{61} + 2780 q^{62} + 2052 q^{64} - 2346 q^{65} + 1134 q^{67} + 1786 q^{68} - 4648 q^{70} + 1064 q^{71} + 1904 q^{73} + 1158 q^{74} + 1708 q^{76} - 5016 q^{77} - 1492 q^{79} - 2922 q^{80} - 1734 q^{82} + 808 q^{83} + 1394 q^{85} - 6336 q^{86} - 3060 q^{88} + 1620 q^{89} + 3278 q^{91} - 664 q^{92} + 772 q^{94} + 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5984 \nu^{7} + 26943 \nu^{6} - 182186 \nu^{5} + 346833 \nu^{4} - 3020182 \nu^{3} + 11402021 \nu^{2} - 10968108 \nu - 5315076 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6325 \nu^{7} + 66824 \nu^{6} - 151547 \nu^{5} - 228206 \nu^{4} + 1218353 \nu^{3} - 1751772 \nu^{2} - 2486484 \nu - 364394082 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4762 \nu^{7} + 120183 \nu^{6} - 812666 \nu^{5} + 3650909 \nu^{4} - 13471942 \nu^{3} + 50860301 \nu^{2} - 245245534 \nu + 145078050 ) / 12613042 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14975 \nu^{7} - 8650 \nu^{6} + 358801 \nu^{5} + 540298 \nu^{4} + 9678629 \nu^{3} + 4147476 \nu^{2} + 5886972 \nu + 74245782 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 56492 \nu^{7} - 274785 \nu^{6} + 1858070 \nu^{5} - 5277333 \nu^{4} + 30802090 \nu^{3} - 116286395 \nu^{2} + 96372822 \nu - 331704750 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 341411 \nu^{7} + 675847 \nu^{6} - 10275913 \nu^{5} - 849943 \nu^{4} - 203391203 \nu^{3} + 61980363 \nu^{2} - 864335982 \nu - 641863404 ) / 37839126 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + 13\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 21\beta_{5} - \beta_{4} - 2\beta_{3} - 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -29\beta_{6} - 2\beta_{4} - 269\beta_{2} - 43\beta _1 - 269 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -29\beta_{7} - 84\beta_{6} - 509\beta_{5} + 84\beta_{3} - 501\beta_{2} - 509\beta _1 + 509 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -84\beta_{7} - 1511\beta_{5} + 84\beta_{4} + 767\beta_{3} + 7960 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2782\beta_{6} + 767\beta_{4} + 18109\beta_{2} + 13077\beta _1 + 18109 \) Copy content Toggle raw display

Character values

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

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

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
−2.11303 3.65987i
−0.733051 1.26968i
1.18088 + 2.04535i
2.66520 + 4.61626i
−2.11303 + 3.65987i
−0.733051 + 1.26968i
1.18088 2.04535i
2.66520 4.61626i
−2.11303 3.65987i 0 −4.92977 + 8.53861i 5.85953 0 12.0627 20.8932i 7.85849 0 −12.3814 21.4451i
55.2 −0.733051 1.26968i 0 2.92527 5.06672i −9.85055 0 −14.9698 + 25.9285i −20.3063 0 7.22095 + 12.5071i
55.3 1.18088 + 2.04535i 0 1.21104 2.09758i −6.42208 0 14.7469 25.5424i 24.6145 0 −7.58371 13.1354i
55.4 2.66520 + 4.61626i 0 −10.2065 + 17.6783i 16.4131 0 −4.83984 + 8.38285i −66.1667 0 43.7441 + 75.7670i
100.1 −2.11303 + 3.65987i 0 −4.92977 8.53861i 5.85953 0 12.0627 + 20.8932i 7.85849 0 −12.3814 + 21.4451i
100.2 −0.733051 + 1.26968i 0 2.92527 + 5.06672i −9.85055 0 −14.9698 25.9285i −20.3063 0 7.22095 12.5071i
100.3 1.18088 2.04535i 0 1.21104 + 2.09758i −6.42208 0 14.7469 + 25.5424i 24.6145 0 −7.58371 + 13.1354i
100.4 2.66520 4.61626i 0 −10.2065 17.6783i 16.4131 0 −4.83984 8.38285i −66.1667 0 43.7441 75.7670i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.e 8
3.b odd 2 1 39.4.e.c 8
12.b even 2 1 624.4.q.i 8
13.c even 3 1 inner 117.4.g.e 8
13.c even 3 1 1521.4.a.v 4
13.e even 6 1 1521.4.a.bb 4
39.h odd 6 1 507.4.a.i 4
39.i odd 6 1 39.4.e.c 8
39.i odd 6 1 507.4.a.m 4
39.k even 12 2 507.4.b.h 8
156.p even 6 1 624.4.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 3.b odd 2 1
39.4.e.c 8 39.i odd 6 1
117.4.g.e 8 1.a even 1 1 trivial
117.4.g.e 8 13.c even 3 1 inner
507.4.a.i 4 39.h odd 6 1
507.4.a.m 4 39.i odd 6 1
507.4.b.h 8 39.k even 12 2
624.4.q.i 8 12.b even 2 1
624.4.q.i 8 156.p even 6 1
1521.4.a.v 4 13.c even 3 1
1521.4.a.bb 4 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + 29T_{2}^{6} + 2T_{2}^{5} + 595T_{2}^{4} - 288T_{2}^{3} + 2526T_{2}^{2} + 1872T_{2} + 6084 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + 29 T^{6} + \cdots + 6084 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 6 T^{3} - 203 T^{2} + 156 T + 6084)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 14 T^{7} + \cdots + 42523388944 \) Copy content Toggle raw display
$11$ \( T^{8} - 40 T^{7} + \cdots + 751198464 \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{7} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 509493017090304 \) Copy content Toggle raw display
$19$ \( T^{8} + 124 T^{7} + \cdots + 2959721107456 \) Copy content Toggle raw display
$23$ \( T^{8} - 104 T^{7} + \cdots + 6612632822016 \) Copy content Toggle raw display
$29$ \( T^{8} - 194 T^{7} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{4} - 26 T^{3} - 80975 T^{2} + \cdots + 328187792)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 738573457717264 \) Copy content Toggle raw display
$41$ \( T^{8} + 1054 T^{7} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{8} + 450 T^{7} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{4} - 96 T^{3} - 434600 T^{2} + \cdots + 42871452048)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 262 T^{3} - 111719 T^{2} + \cdots + 744728256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 308 T^{7} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} - 928 T^{7} + \cdots + 27\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{8} - 1134 T^{7} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} - 1064 T^{7} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} - 952 T^{3} + \cdots - 120133390247)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 746 T^{3} + 184337 T^{2} + \cdots + 680937616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 404 T^{3} + \cdots + 58964273856)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 1620 T^{7} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + 2166 T^{7} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
show more
show less