Properties

Label 117.4.g.c
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 4) q^{2} - 8 \zeta_{6} q^{4} - 17 q^{5} - 20 \zeta_{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \zeta_{6} + 4) q^{2} - 8 \zeta_{6} q^{4} - 17 q^{5} - 20 \zeta_{6} q^{7} + (68 \zeta_{6} - 68) q^{10} + (32 \zeta_{6} - 32) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} - 80 q^{14} + ( - 64 \zeta_{6} + 64) q^{16} - 13 \zeta_{6} q^{17} - 30 \zeta_{6} q^{19} + 136 \zeta_{6} q^{20} + 128 \zeta_{6} q^{22} + ( - 78 \zeta_{6} + 78) q^{23} + 164 q^{25} + (156 \zeta_{6} - 208) q^{26} + (160 \zeta_{6} - 160) q^{28} + ( - 197 \zeta_{6} + 197) q^{29} - 74 q^{31} - 256 \zeta_{6} q^{32} - 52 q^{34} + 340 \zeta_{6} q^{35} + ( - 227 \zeta_{6} + 227) q^{37} - 120 q^{38} + (165 \zeta_{6} - 165) q^{41} + 156 \zeta_{6} q^{43} + 256 q^{44} - 312 \zeta_{6} q^{46} + 162 q^{47} + (57 \zeta_{6} - 57) q^{49} + ( - 656 \zeta_{6} + 656) q^{50} + (416 \zeta_{6} - 104) q^{52} - 93 q^{53} + ( - 544 \zeta_{6} + 544) q^{55} - 788 \zeta_{6} q^{58} - 864 \zeta_{6} q^{59} - 145 \zeta_{6} q^{61} + (296 \zeta_{6} - 296) q^{62} - 512 q^{64} + (221 \zeta_{6} + 663) q^{65} + (862 \zeta_{6} - 862) q^{67} + (104 \zeta_{6} - 104) q^{68} + 1360 q^{70} + 654 \zeta_{6} q^{71} + 215 q^{73} - 908 \zeta_{6} q^{74} + (240 \zeta_{6} - 240) q^{76} + 640 q^{77} - 76 q^{79} + (1088 \zeta_{6} - 1088) q^{80} + 660 \zeta_{6} q^{82} - 628 q^{83} + 221 \zeta_{6} q^{85} + 624 q^{86} + (266 \zeta_{6} - 266) q^{89} + (1040 \zeta_{6} - 260) q^{91} - 624 q^{92} + ( - 648 \zeta_{6} + 648) q^{94} + 510 \zeta_{6} q^{95} - 238 \zeta_{6} q^{97} + 228 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 8 q^{4} - 34 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 8 q^{4} - 34 q^{5} - 20 q^{7} - 68 q^{10} - 32 q^{11} - 91 q^{13} - 160 q^{14} + 64 q^{16} - 13 q^{17} - 30 q^{19} + 136 q^{20} + 128 q^{22} + 78 q^{23} + 328 q^{25} - 260 q^{26} - 160 q^{28} + 197 q^{29} - 148 q^{31} - 256 q^{32} - 104 q^{34} + 340 q^{35} + 227 q^{37} - 240 q^{38} - 165 q^{41} + 156 q^{43} + 512 q^{44} - 312 q^{46} + 324 q^{47} - 57 q^{49} + 656 q^{50} + 208 q^{52} - 186 q^{53} + 544 q^{55} - 788 q^{58} - 864 q^{59} - 145 q^{61} - 296 q^{62} - 1024 q^{64} + 1547 q^{65} - 862 q^{67} - 104 q^{68} + 2720 q^{70} + 654 q^{71} + 430 q^{73} - 908 q^{74} - 240 q^{76} + 1280 q^{77} - 152 q^{79} - 1088 q^{80} + 660 q^{82} - 1256 q^{83} + 221 q^{85} + 1248 q^{86} - 266 q^{89} + 520 q^{91} - 1248 q^{92} + 648 q^{94} + 510 q^{95} - 238 q^{97} + 228 q^{98}+O(q^{100}) \) 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
2.00000 + 3.46410i 0 −4.00000 + 6.92820i −17.0000 0 −10.0000 + 17.3205i 0 0 −34.0000 58.8897i
100.1 2.00000 3.46410i 0 −4.00000 6.92820i −17.0000 0 −10.0000 17.3205i 0 0 −34.0000 + 58.8897i
\(n\): 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.c 2
3.b odd 2 1 13.4.c.a 2
12.b even 2 1 208.4.i.b 2
13.c even 3 1 inner 117.4.g.c 2
13.c even 3 1 1521.4.a.b 1
13.e even 6 1 1521.4.a.k 1
39.d odd 2 1 169.4.c.d 2
39.f even 4 2 169.4.e.c 4
39.h odd 6 1 169.4.a.a 1
39.h odd 6 1 169.4.c.d 2
39.i odd 6 1 13.4.c.a 2
39.i odd 6 1 169.4.a.d 1
39.k even 12 2 169.4.b.c 2
39.k even 12 2 169.4.e.c 4
156.p even 6 1 208.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 3.b odd 2 1
13.4.c.a 2 39.i odd 6 1
117.4.g.c 2 1.a even 1 1 trivial
117.4.g.c 2 13.c even 3 1 inner
169.4.a.a 1 39.h odd 6 1
169.4.a.d 1 39.i odd 6 1
169.4.b.c 2 39.k even 12 2
169.4.c.d 2 39.d odd 2 1
169.4.c.d 2 39.h odd 6 1
169.4.e.c 4 39.f even 4 2
169.4.e.c 4 39.k even 12 2
208.4.i.b 2 12.b even 2 1
208.4.i.b 2 156.p even 6 1
1521.4.a.b 1 13.c even 3 1
1521.4.a.k 1 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 17)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$11$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$19$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} - 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} - 197T + 38809 \) Copy content Toggle raw display
$31$ \( (T + 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 227T + 51529 \) Copy content Toggle raw display
$41$ \( T^{2} + 165T + 27225 \) Copy content Toggle raw display
$43$ \( T^{2} - 156T + 24336 \) Copy content Toggle raw display
$47$ \( (T - 162)^{2} \) Copy content Toggle raw display
$53$ \( (T + 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 864T + 746496 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} + 862T + 743044 \) Copy content Toggle raw display
$71$ \( T^{2} - 654T + 427716 \) Copy content Toggle raw display
$73$ \( (T - 215)^{2} \) Copy content Toggle raw display
$79$ \( (T + 76)^{2} \) Copy content Toggle raw display
$83$ \( (T + 628)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 266T + 70756 \) Copy content Toggle raw display
$97$ \( T^{2} + 238T + 56644 \) Copy content Toggle raw display
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