Properties

Label 117.4.g.a
Level $117$
Weight $4$
Character orbit 117.g
Analytic conductor $6.903$
Analytic rank $0$
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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 7 q^{5} + 10 \zeta_{6} q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 7 q^{5} + 10 \zeta_{6} q^{7} - 15 q^{8} + ( - 7 \zeta_{6} + 7) q^{10} + (22 \zeta_{6} - 22) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} - 10 q^{14} + (41 \zeta_{6} - 41) q^{16} + 37 \zeta_{6} q^{17} - 30 \zeta_{6} q^{19} - 49 \zeta_{6} q^{20} - 22 \zeta_{6} q^{22} + (162 \zeta_{6} - 162) q^{23} - 76 q^{25} + ( - 39 \zeta_{6} + 52) q^{26} + (70 \zeta_{6} - 70) q^{28} + (113 \zeta_{6} - 113) q^{29} + 196 q^{31} - 161 \zeta_{6} q^{32} - 37 q^{34} - 70 \zeta_{6} q^{35} + (13 \zeta_{6} - 13) q^{37} + 30 q^{38} + 105 q^{40} + ( - 285 \zeta_{6} + 285) q^{41} + 246 \zeta_{6} q^{43} - 154 q^{44} - 162 \zeta_{6} q^{46} + 462 q^{47} + ( - 243 \zeta_{6} + 243) q^{49} + ( - 76 \zeta_{6} + 76) q^{50} + ( - 364 \zeta_{6} + 91) q^{52} + 537 q^{53} + ( - 154 \zeta_{6} + 154) q^{55} - 150 \zeta_{6} q^{56} - 113 \zeta_{6} q^{58} + 576 \zeta_{6} q^{59} + 635 \zeta_{6} q^{61} + (196 \zeta_{6} - 196) q^{62} - 167 q^{64} + (91 \zeta_{6} + 273) q^{65} + (202 \zeta_{6} - 202) q^{67} + (259 \zeta_{6} - 259) q^{68} + 70 q^{70} - 1086 \zeta_{6} q^{71} - 805 q^{73} - 13 \zeta_{6} q^{74} + ( - 210 \zeta_{6} + 210) q^{76} - 220 q^{77} + 884 q^{79} + ( - 287 \zeta_{6} + 287) q^{80} + 285 \zeta_{6} q^{82} - 518 q^{83} - 259 \zeta_{6} q^{85} - 246 q^{86} + ( - 330 \zeta_{6} + 330) q^{88} + ( - 194 \zeta_{6} + 194) q^{89} + ( - 520 \zeta_{6} + 130) q^{91} - 1134 q^{92} + (462 \zeta_{6} - 462) q^{94} + 210 \zeta_{6} q^{95} + 1202 \zeta_{6} q^{97} + 243 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 7 q^{4} - 14 q^{5} + 10 q^{7} - 30 q^{8} + 7 q^{10} - 22 q^{11} - 91 q^{13} - 20 q^{14} - 41 q^{16} + 37 q^{17} - 30 q^{19} - 49 q^{20} - 22 q^{22} - 162 q^{23} - 152 q^{25} + 65 q^{26} - 70 q^{28} - 113 q^{29} + 392 q^{31} - 161 q^{32} - 74 q^{34} - 70 q^{35} - 13 q^{37} + 60 q^{38} + 210 q^{40} + 285 q^{41} + 246 q^{43} - 308 q^{44} - 162 q^{46} + 924 q^{47} + 243 q^{49} + 76 q^{50} - 182 q^{52} + 1074 q^{53} + 154 q^{55} - 150 q^{56} - 113 q^{58} + 576 q^{59} + 635 q^{61} - 196 q^{62} - 334 q^{64} + 637 q^{65} - 202 q^{67} - 259 q^{68} + 140 q^{70} - 1086 q^{71} - 1610 q^{73} - 13 q^{74} + 210 q^{76} - 440 q^{77} + 1768 q^{79} + 287 q^{80} + 285 q^{82} - 1036 q^{83} - 259 q^{85} - 492 q^{86} + 330 q^{88} + 194 q^{89} - 260 q^{91} - 2268 q^{92} - 462 q^{94} + 210 q^{95} + 1202 q^{97} + 243 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 3.50000 6.06218i −7.00000 0 5.00000 8.66025i −15.0000 0 3.50000 + 6.06218i
100.1 −0.500000 + 0.866025i 0 3.50000 + 6.06218i −7.00000 0 5.00000 + 8.66025i −15.0000 0 3.50000 6.06218i
\(n\): e.g. 2-40 or 990-1000
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.a 2
3.b odd 2 1 39.4.e.b 2
12.b even 2 1 624.4.q.c 2
13.c even 3 1 inner 117.4.g.a 2
13.c even 3 1 1521.4.a.h 1
13.e even 6 1 1521.4.a.e 1
39.h odd 6 1 507.4.a.d 1
39.i odd 6 1 39.4.e.b 2
39.i odd 6 1 507.4.a.b 1
39.k even 12 2 507.4.b.d 2
156.p even 6 1 624.4.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 3.b odd 2 1
39.4.e.b 2 39.i odd 6 1
117.4.g.a 2 1.a even 1 1 trivial
117.4.g.a 2 13.c even 3 1 inner
507.4.a.b 1 39.i odd 6 1
507.4.a.d 1 39.h odd 6 1
507.4.b.d 2 39.k even 12 2
624.4.q.c 2 12.b even 2 1
624.4.q.c 2 156.p even 6 1
1521.4.a.e 1 13.e even 6 1
1521.4.a.h 1 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$11$ \( T^{2} + 22T + 484 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 37T + 1369 \) Copy content Toggle raw display
$19$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} + 162T + 26244 \) Copy content Toggle raw display
$29$ \( T^{2} + 113T + 12769 \) Copy content Toggle raw display
$31$ \( (T - 196)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$41$ \( T^{2} - 285T + 81225 \) Copy content Toggle raw display
$43$ \( T^{2} - 246T + 60516 \) Copy content Toggle raw display
$47$ \( (T - 462)^{2} \) Copy content Toggle raw display
$53$ \( (T - 537)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 576T + 331776 \) Copy content Toggle raw display
$61$ \( T^{2} - 635T + 403225 \) Copy content Toggle raw display
$67$ \( T^{2} + 202T + 40804 \) Copy content Toggle raw display
$71$ \( T^{2} + 1086 T + 1179396 \) Copy content Toggle raw display
$73$ \( (T + 805)^{2} \) Copy content Toggle raw display
$79$ \( (T - 884)^{2} \) Copy content Toggle raw display
$83$ \( (T + 518)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 194T + 37636 \) Copy content Toggle raw display
$97$ \( T^{2} - 1202 T + 1444804 \) Copy content Toggle raw display
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