Properties

Label 117.4.q.a
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: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{2} -5 \zeta_{6} q^{4} + ( -1 + 2 \zeta_{6} ) q^{5} + ( -16 + 8 \zeta_{6} ) q^{7} + ( -13 + 26 \zeta_{6} ) q^{8} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{2} -5 \zeta_{6} q^{4} + ( -1 + 2 \zeta_{6} ) q^{5} + ( -16 + 8 \zeta_{6} ) q^{7} + ( -13 + 26 \zeta_{6} ) q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} + ( -8 - 8 \zeta_{6} ) q^{11} + ( 39 + 13 \zeta_{6} ) q^{13} + 24 q^{14} + ( -1 + \zeta_{6} ) q^{16} + 117 \zeta_{6} q^{17} + ( -132 + 66 \zeta_{6} ) q^{19} + ( 10 - 5 \zeta_{6} ) q^{20} + 24 \zeta_{6} q^{22} + ( -78 + 78 \zeta_{6} ) q^{23} + 122 q^{25} + ( -26 - 65 \zeta_{6} ) q^{26} + ( 40 + 40 \zeta_{6} ) q^{28} + ( -141 + 141 \zeta_{6} ) q^{29} + ( -90 + 180 \zeta_{6} ) q^{31} + ( 210 - 105 \zeta_{6} ) q^{32} + ( 117 - 234 \zeta_{6} ) q^{34} -24 \zeta_{6} q^{35} + ( -83 - 83 \zeta_{6} ) q^{37} + 198 q^{38} -39 q^{40} + ( -157 - 157 \zeta_{6} ) q^{41} -104 \zeta_{6} q^{43} + ( -40 + 80 \zeta_{6} ) q^{44} + ( 156 - 78 \zeta_{6} ) q^{46} + ( 174 - 348 \zeta_{6} ) q^{47} + ( -151 + 151 \zeta_{6} ) q^{49} + ( -122 - 122 \zeta_{6} ) q^{50} + ( 65 - 260 \zeta_{6} ) q^{52} -93 q^{53} + ( 24 - 24 \zeta_{6} ) q^{55} -312 \zeta_{6} q^{56} + ( 282 - 141 \zeta_{6} ) q^{58} + ( 328 - 164 \zeta_{6} ) q^{59} -145 \zeta_{6} q^{61} + ( 270 - 270 \zeta_{6} ) q^{62} -307 q^{64} + ( -65 + 91 \zeta_{6} ) q^{65} + ( -454 - 454 \zeta_{6} ) q^{67} + ( 585 - 585 \zeta_{6} ) q^{68} + ( -24 + 48 \zeta_{6} ) q^{70} + ( -1220 + 610 \zeta_{6} ) q^{71} + ( -265 + 530 \zeta_{6} ) q^{73} + 249 \zeta_{6} q^{74} + ( 330 + 330 \zeta_{6} ) q^{76} + 192 q^{77} + 1276 q^{79} + ( -1 - \zeta_{6} ) q^{80} + 471 \zeta_{6} q^{82} + ( -456 + 912 \zeta_{6} ) q^{83} + ( -234 + 117 \zeta_{6} ) q^{85} + ( -104 + 208 \zeta_{6} ) q^{86} + ( 312 - 312 \zeta_{6} ) q^{88} + ( 564 + 564 \zeta_{6} ) q^{89} + ( -728 + 208 \zeta_{6} ) q^{91} + 390 q^{92} + ( -522 + 522 \zeta_{6} ) q^{94} -198 \zeta_{6} q^{95} + ( 232 - 116 \zeta_{6} ) q^{97} + ( 302 - 151 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} - 5q^{4} - 24q^{7} + O(q^{10}) \) \( 2q - 3q^{2} - 5q^{4} - 24q^{7} + 3q^{10} - 24q^{11} + 91q^{13} + 48q^{14} - q^{16} + 117q^{17} - 198q^{19} + 15q^{20} + 24q^{22} - 78q^{23} + 244q^{25} - 117q^{26} + 120q^{28} - 141q^{29} + 315q^{32} - 24q^{35} - 249q^{37} + 396q^{38} - 78q^{40} - 471q^{41} - 104q^{43} + 234q^{46} - 151q^{49} - 366q^{50} - 130q^{52} - 186q^{53} + 24q^{55} - 312q^{56} + 423q^{58} + 492q^{59} - 145q^{61} + 270q^{62} - 614q^{64} - 39q^{65} - 1362q^{67} + 585q^{68} - 1830q^{71} + 249q^{74} + 990q^{76} + 384q^{77} + 2552q^{79} - 3q^{80} + 471q^{82} - 351q^{85} + 312q^{88} + 1692q^{89} - 1248q^{91} + 780q^{92} - 522q^{94} - 198q^{95} + 348q^{97} + 453q^{98} + O(q^{100}) \)

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
−1.50000 + 0.866025i 0 −2.50000 + 4.33013i 1.73205i 0 −12.0000 6.92820i 22.5167i 0 1.50000 + 2.59808i
82.1 −1.50000 0.866025i 0 −2.50000 4.33013i 1.73205i 0 −12.0000 + 6.92820i 22.5167i 0 1.50000 2.59808i
\(n\): 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.a 2
3.b odd 2 1 13.4.e.b 2
12.b even 2 1 208.4.w.b 2
13.e even 6 1 inner 117.4.q.a 2
13.f odd 12 2 1521.4.a.o 2
39.d odd 2 1 169.4.e.a 2
39.f even 4 2 169.4.c.h 4
39.h odd 6 1 13.4.e.b 2
39.h odd 6 1 169.4.b.d 2
39.i odd 6 1 169.4.b.d 2
39.i odd 6 1 169.4.e.a 2
39.k even 12 2 169.4.a.i 2
39.k even 12 2 169.4.c.h 4
156.r even 6 1 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 3.b odd 2 1
13.4.e.b 2 39.h odd 6 1
117.4.q.a 2 1.a even 1 1 trivial
117.4.q.a 2 13.e even 6 1 inner
169.4.a.i 2 39.k even 12 2
169.4.b.d 2 39.h odd 6 1
169.4.b.d 2 39.i odd 6 1
169.4.c.h 4 39.f even 4 2
169.4.c.h 4 39.k even 12 2
169.4.e.a 2 39.d odd 2 1
169.4.e.a 2 39.i odd 6 1
208.4.w.b 2 12.b even 2 1
208.4.w.b 2 156.r even 6 1
1521.4.a.o 2 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 11 T^{2} + 24 T^{3} + 64 T^{4} \)
$3$ 1
$5$ \( 1 - 247 T^{2} + 15625 T^{4} \)
$7$ \( 1 + 24 T + 535 T^{2} + 8232 T^{3} + 117649 T^{4} \)
$11$ \( 1 + 24 T + 1523 T^{2} + 31944 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 91 T + 2197 T^{2} \)
$17$ \( 1 - 117 T + 8776 T^{2} - 574821 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 198 T + 19927 T^{2} + 1358082 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 78 T - 6083 T^{2} + 949026 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 141 T - 4508 T^{2} + 3438849 T^{3} + 594823321 T^{4} \)
$31$ \( ( 1 - 308 T + 29791 T^{2} )( 1 + 308 T + 29791 T^{2} ) \)
$37$ \( 1 + 249 T + 71320 T^{2} + 12612597 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 471 T + 142868 T^{2} + 32461791 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 104 T - 68691 T^{2} + 8268728 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 116818 T^{2} + 10779215329 T^{4} \)
$53$ \( ( 1 + 93 T + 148877 T^{2} )^{2} \)
$59$ \( 1 - 492 T + 286067 T^{2} - 101046468 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 145 T - 205956 T^{2} + 32912245 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 1362 T + 919111 T^{2} + 409639206 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 + 1830 T + 1474211 T^{2} + 654977130 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 567359 T^{2} + 151334226289 T^{4} \)
$79$ \( ( 1 - 1276 T + 493039 T^{2} )^{2} \)
$83$ \( 1 - 519766 T^{2} + 326940373369 T^{4} \)
$89$ \( 1 - 1692 T + 1659257 T^{2} - 1192807548 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 348 T + 953041 T^{2} - 317610204 T^{3} + 832972004929 T^{4} \)
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