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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$11$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$13$ \( T^{2} - 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 117T + 13689 \) Copy content Toggle raw display
$19$ \( T^{2} + 198T + 13068 \) Copy content Toggle raw display
$23$ \( T^{2} + 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} + 141T + 19881 \) Copy content Toggle raw display
$31$ \( T^{2} + 24300 \) Copy content Toggle raw display
$37$ \( T^{2} + 249T + 20667 \) Copy content Toggle raw display
$41$ \( T^{2} + 471T + 73947 \) Copy content Toggle raw display
$43$ \( T^{2} + 104T + 10816 \) Copy content Toggle raw display
$47$ \( T^{2} + 90828 \) Copy content Toggle raw display
$53$ \( (T + 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 492T + 80688 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} + 1362 T + 618348 \) Copy content Toggle raw display
$71$ \( T^{2} + 1830 T + 1116300 \) Copy content Toggle raw display
$73$ \( T^{2} + 210675 \) Copy content Toggle raw display
$79$ \( (T - 1276)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 623808 \) Copy content Toggle raw display
$89$ \( T^{2} - 1692 T + 954288 \) Copy content Toggle raw display
$97$ \( T^{2} - 348T + 40368 \) Copy content Toggle raw display
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