Properties

Label 117.4.q.c
Level $117$
Weight $4$
Character orbit 117.q
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.q (of order \(6\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 13 \zeta_{6} + 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} + 2) q^{2} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 13 \zeta_{6} + 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8} + ( - 48 \zeta_{6} + 48) q^{10} + (13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} + 78 q^{14} + ( - 80 \zeta_{6} + 80) q^{16} + 27 \zeta_{6} q^{17} + (51 \zeta_{6} - 102) q^{19} + ( - 32 \zeta_{6} + 64) q^{20} + 78 \zeta_{6} q^{22} + ( - 57 \zeta_{6} + 57) q^{23} - 67 q^{25} + (130 \zeta_{6} - 182) q^{26} + (52 \zeta_{6} + 52) q^{28} + (69 \zeta_{6} - 69) q^{29} + ( - 84 \zeta_{6} + 42) q^{31} + ( - 96 \zeta_{6} + 192) q^{32} + (108 \zeta_{6} - 54) q^{34} - 312 \zeta_{6} q^{35} + ( - 23 \zeta_{6} - 23) q^{37} - 306 q^{38} - 192 q^{40} + (227 \zeta_{6} + 227) q^{41} + 85 \zeta_{6} q^{43} + (104 \zeta_{6} - 52) q^{44} + ( - 114 \zeta_{6} + 228) q^{46} + (396 \zeta_{6} - 198) q^{47} + ( - 164 \zeta_{6} + 164) q^{49} + ( - 134 \zeta_{6} - 134) q^{50} + (52 \zeta_{6} - 208) q^{52} - 426 q^{53} + ( - 312 \zeta_{6} + 312) q^{55} - 312 \zeta_{6} q^{56} + (138 \zeta_{6} - 276) q^{58} + ( - 11 \zeta_{6} + 22) q^{59} + 17 \zeta_{6} q^{61} + ( - 252 \zeta_{6} + 252) q^{62} - 64 q^{64} + (208 \zeta_{6} + 520) q^{65} + (95 \zeta_{6} + 95) q^{67} + (108 \zeta_{6} - 108) q^{68} + ( - 1248 \zeta_{6} + 624) q^{70} + (337 \zeta_{6} - 674) q^{71} + (1160 \zeta_{6} - 580) q^{73} - 138 \zeta_{6} q^{74} + ( - 204 \zeta_{6} - 204) q^{76} + 507 q^{77} - 1244 q^{79} + ( - 640 \zeta_{6} - 640) q^{80} + 1362 \zeta_{6} q^{82} + (492 \zeta_{6} - 246) q^{83} + ( - 216 \zeta_{6} + 432) q^{85} + (340 \zeta_{6} - 170) q^{86} + ( - 312 \zeta_{6} + 312) q^{88} + ( - 177 \zeta_{6} - 177) q^{89} + (1183 \zeta_{6} - 338) q^{91} + 228 q^{92} + (1188 \zeta_{6} - 1188) q^{94} + 1224 \zeta_{6} q^{95} + ( - 713 \zeta_{6} + 1426) q^{97} + ( - 328 \zeta_{6} + 656) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 4 q^{4} + 39 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 4 q^{4} + 39 q^{7} + 48 q^{10} + 39 q^{11} - 26 q^{13} + 156 q^{14} + 80 q^{16} + 27 q^{17} - 153 q^{19} + 96 q^{20} + 78 q^{22} + 57 q^{23} - 134 q^{25} - 234 q^{26} + 156 q^{28} - 69 q^{29} + 288 q^{32} - 312 q^{35} - 69 q^{37} - 612 q^{38} - 384 q^{40} + 681 q^{41} + 85 q^{43} + 342 q^{46} + 164 q^{49} - 402 q^{50} - 364 q^{52} - 852 q^{53} + 312 q^{55} - 312 q^{56} - 414 q^{58} + 33 q^{59} + 17 q^{61} + 252 q^{62} - 128 q^{64} + 1248 q^{65} + 285 q^{67} - 108 q^{68} - 1011 q^{71} - 138 q^{74} - 612 q^{76} + 1014 q^{77} - 2488 q^{79} - 1920 q^{80} + 1362 q^{82} + 648 q^{85} + 312 q^{88} - 531 q^{89} + 507 q^{91} + 456 q^{92} - 1188 q^{94} + 1224 q^{95} + 2139 q^{97} + 984 q^{98}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.500000 0.866025i
0.500000 + 0.866025i
3.00000 1.73205i 0 2.00000 3.46410i 13.8564i 0 19.5000 + 11.2583i 13.8564i 0 24.0000 + 41.5692i
82.1 3.00000 + 1.73205i 0 2.00000 + 3.46410i 13.8564i 0 19.5000 11.2583i 13.8564i 0 24.0000 41.5692i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.q.c 2
3.b odd 2 1 13.4.e.a 2
12.b even 2 1 208.4.w.a 2
13.e even 6 1 inner 117.4.q.c 2
13.f odd 12 2 1521.4.a.q 2
39.d odd 2 1 169.4.e.b 2
39.f even 4 2 169.4.c.i 4
39.h odd 6 1 13.4.e.a 2
39.h odd 6 1 169.4.b.b 2
39.i odd 6 1 169.4.b.b 2
39.i odd 6 1 169.4.e.b 2
39.k even 12 2 169.4.a.h 2
39.k even 12 2 169.4.c.i 4
156.r even 6 1 208.4.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 3.b odd 2 1
13.4.e.a 2 39.h odd 6 1
117.4.q.c 2 1.a even 1 1 trivial
117.4.q.c 2 13.e even 6 1 inner
169.4.a.h 2 39.k even 12 2
169.4.b.b 2 39.h odd 6 1
169.4.b.b 2 39.i odd 6 1
169.4.c.i 4 39.f even 4 2
169.4.c.i 4 39.k even 12 2
169.4.e.b 2 39.d odd 2 1
169.4.e.b 2 39.i odd 6 1
208.4.w.a 2 12.b even 2 1
208.4.w.a 2 156.r even 6 1
1521.4.a.q 2 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6T_{2} + 12 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 192 \) Copy content Toggle raw display
$7$ \( T^{2} - 39T + 507 \) Copy content Toggle raw display
$11$ \( T^{2} - 39T + 507 \) Copy content Toggle raw display
$13$ \( T^{2} + 26T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 27T + 729 \) Copy content Toggle raw display
$19$ \( T^{2} + 153T + 7803 \) Copy content Toggle raw display
$23$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$29$ \( T^{2} + 69T + 4761 \) Copy content Toggle raw display
$31$ \( T^{2} + 5292 \) Copy content Toggle raw display
$37$ \( T^{2} + 69T + 1587 \) Copy content Toggle raw display
$41$ \( T^{2} - 681T + 154587 \) Copy content Toggle raw display
$43$ \( T^{2} - 85T + 7225 \) Copy content Toggle raw display
$47$ \( T^{2} + 117612 \) Copy content Toggle raw display
$53$ \( (T + 426)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 33T + 363 \) Copy content Toggle raw display
$61$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$67$ \( T^{2} - 285T + 27075 \) Copy content Toggle raw display
$71$ \( T^{2} + 1011 T + 340707 \) Copy content Toggle raw display
$73$ \( T^{2} + 1009200 \) Copy content Toggle raw display
$79$ \( (T + 1244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 181548 \) Copy content Toggle raw display
$89$ \( T^{2} + 531T + 93987 \) Copy content Toggle raw display
$97$ \( T^{2} - 2139 T + 1525107 \) Copy content Toggle raw display
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