↑

Properties

Label 117.2.h.a.22.6
Level $117$
Weight $2$
Character 117.22
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [117,2,Mod(16,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 22.6
Character \(\chi\) \(=\) 117.22
Dual form 117.2.h.a.16.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.163365 q^{2} +(0.725052 + 1.57299i) q^{3} -1.97331 q^{4} +(-1.55806 + 2.69863i) q^{5} +(-0.118448 - 0.256972i) q^{6} +(0.0682144 - 0.118151i) q^{7} +0.649100 q^{8} +(-1.94860 + 2.28100i) q^{9} +(0.254532 - 0.440862i) q^{10} +4.17771 q^{11} +(-1.43075 - 3.10400i) q^{12} +(0.300890 - 3.59297i) q^{13} +(-0.0111438 + 0.0193017i) q^{14} +(-5.37459 - 0.494158i) q^{15} +3.84058 q^{16} +(2.67892 + 4.64002i) q^{17} +(0.318333 - 0.372636i) q^{18} +(-0.154748 - 0.268032i) q^{19} +(3.07453 - 5.32524i) q^{20} +(0.235309 + 0.0216351i) q^{21} -0.682491 q^{22} +(0.961735 + 1.66577i) q^{23} +(0.470632 + 1.02103i) q^{24} +(-2.35507 - 4.07910i) q^{25} +(-0.0491549 + 0.586967i) q^{26} +(-5.00083 - 1.41128i) q^{27} +(-0.134608 + 0.233148i) q^{28} -2.75870 q^{29} +(0.878021 + 0.0807282i) q^{30} +(2.28432 - 3.95657i) q^{31} -1.92562 q^{32} +(3.02905 + 6.57149i) q^{33} +(-0.437642 - 0.758018i) q^{34} +(0.212564 + 0.368171i) q^{35} +(3.84519 - 4.50112i) q^{36} +(4.48902 - 7.77522i) q^{37} +(0.0252804 + 0.0437870i) q^{38} +(5.86988 - 2.13180i) q^{39} +(-1.01133 + 1.75168i) q^{40} +(-3.56877 - 6.18129i) q^{41} +(-0.0384413 - 0.00353442i) q^{42} +(-5.32209 + 9.21812i) q^{43} -8.24392 q^{44} +(-3.11955 - 8.81247i) q^{45} +(-0.157114 - 0.272129i) q^{46} +(-0.663436 - 1.14910i) q^{47} +(2.78462 + 6.04120i) q^{48} +(3.49069 + 6.04606i) q^{49} +(0.384737 + 0.666383i) q^{50} +(-5.35636 + 7.57817i) q^{51} +(-0.593749 + 7.09006i) q^{52} +7.10873 q^{53} +(0.816961 + 0.230555i) q^{54} +(-6.50910 + 11.2741i) q^{55} +(0.0442780 - 0.0766917i) q^{56} +(0.309411 - 0.437754i) q^{57} +0.450675 q^{58} -4.80940 q^{59} +(10.6057 + 0.975129i) q^{60} +(3.61647 - 6.26392i) q^{61} +(-0.373179 + 0.646365i) q^{62} +(0.136579 + 0.385825i) q^{63} -7.36659 q^{64} +(9.22731 + 6.41004i) q^{65} +(-0.494842 - 1.07355i) q^{66} +(6.53369 + 11.3167i) q^{67} +(-5.28634 - 9.15621i) q^{68} +(-1.92294 + 2.72057i) q^{69} +(-0.0347255 - 0.0601463i) q^{70} +(2.24787 + 3.89343i) q^{71} +(-1.26484 + 1.48060i) q^{72} +1.18478 q^{73} +(-0.733350 + 1.27020i) q^{74} +(4.70884 - 6.66207i) q^{75} +(0.305366 + 0.528910i) q^{76} +(0.284980 - 0.493599i) q^{77} +(-0.958933 + 0.348261i) q^{78} +(-1.41978 - 2.45913i) q^{79} +(-5.98384 + 10.3643i) q^{80} +(-1.40592 - 8.88951i) q^{81} +(0.583013 + 1.00981i) q^{82} +(-5.19008 - 8.98948i) q^{83} +(-0.464338 - 0.0426928i) q^{84} -16.6956 q^{85} +(0.869443 - 1.50592i) q^{86} +(-2.00020 - 4.33941i) q^{87} +2.71175 q^{88} +(6.17396 - 10.6936i) q^{89} +(0.509626 + 1.43965i) q^{90} +(-0.403988 - 0.280643i) q^{91} +(-1.89780 - 3.28709i) q^{92} +(7.87989 + 0.724505i) q^{93} +(0.108382 + 0.187723i) q^{94} +0.964425 q^{95} +(-1.39617 - 3.02898i) q^{96} +(8.05756 - 13.9561i) q^{97} +(-0.570257 - 0.987715i) q^{98} +(-8.14068 + 9.52935i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} - q^{3} + 18 q^{4} - 2 q^{5} - 12 q^{6} + 3 q^{7} - 18 q^{8} - 3 q^{9} + 6 q^{11} - 3 q^{12} + 2 q^{14} + 11 q^{15} + 6 q^{16} + 6 q^{17} - 8 q^{18} - 3 q^{19} - 11 q^{20} - 25 q^{21} - 18 q^{22}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.163365 −0.115517 −0.0577583 − 0.998331i \(-0.518395\pi\)
−0.0577583 + 0.998331i \(0.518395\pi\)
\(3\) 0.725052 + 1.57299i 0.418609 + 0.908167i
\(4\) −1.97331 −0.986656
\(5\) −1.55806 + 2.69863i −0.696783 + 1.20686i 0.272792 + 0.962073i \(0.412053\pi\)
−0.969576 + 0.244791i \(0.921281\pi\)
\(6\) −0.118448 − 0.256972i −0.0483563 − 0.104908i
\(7\) 0.0682144 − 0.118151i 0.0257826 − 0.0446568i −0.852846 − 0.522162i \(-0.825126\pi\)
0.878629 + 0.477505i \(0.158459\pi\)
\(8\) 0.649100 0.229492
\(9\) −1.94860 + 2.28100i −0.649533 + 0.760333i
\(10\) 0.254532 − 0.440862i 0.0804900 − 0.139413i
\(11\) 4.17771 1.25963 0.629813 − 0.776747i \(-0.283132\pi\)
0.629813 + 0.776747i \(0.283132\pi\)
\(12\) −1.43075 − 3.10400i −0.413023 − 0.896048i
\(13\) 0.300890 − 3.59297i 0.0834518 − 0.996512i
\(14\) −0.0111438 + 0.0193017i −0.00297832 + 0.00515860i
\(15\) −5.37459 − 0.494158i −1.38771 − 0.127591i
\(16\) 3.84058 0.960146
\(17\) 2.67892 + 4.64002i 0.649733 + 1.12537i 0.983186 + 0.182604i \(0.0584528\pi\)
−0.333453 + 0.942767i \(0.608214\pi\)
\(18\) 0.318333 − 0.372636i 0.0750318 − 0.0878311i
\(19\) −0.154748 − 0.268032i −0.0355017 − 0.0614907i 0.847729 − 0.530430i \(-0.177970\pi\)
−0.883230 + 0.468939i \(0.844636\pi\)
\(20\) 3.07453 − 5.32524i 0.687486 − 1.19076i
\(21\) 0.235309 + 0.0216351i 0.0513486 + 0.00472117i
\(22\) −0.682491 −0.145508
\(23\) 0.961735 + 1.66577i 0.200536 + 0.347338i 0.948701 − 0.316174i \(-0.102398\pi\)
−0.748166 + 0.663512i \(0.769065\pi\)
\(24\) 0.470632 + 1.02103i 0.0960673 + 0.208417i
\(25\) −2.35507 − 4.07910i −0.471014 − 0.815821i
\(26\) −0.0491549 + 0.586967i −0.00964007 + 0.115114i
\(27\) −5.00083 − 1.41128i −0.962410 − 0.271602i
\(28\) −0.134608 + 0.233148i −0.0254386 + 0.0440609i
\(29\) −2.75870 −0.512278 −0.256139 − 0.966640i \(-0.582450\pi\)
−0.256139 + 0.966640i \(0.582450\pi\)
\(30\) 0.878021 + 0.0807282i 0.160304 + 0.0147389i
\(31\) 2.28432 − 3.95657i 0.410277 − 0.710620i −0.584643 − 0.811291i \(-0.698765\pi\)
0.994920 + 0.100670i \(0.0320988\pi\)
\(32\) −1.92562 −0.340404
\(33\) 3.02905 + 6.57149i 0.527291 + 1.14395i
\(34\) −0.437642 − 0.758018i −0.0750550 − 0.129999i
\(35\) 0.212564 + 0.368171i 0.0359298 + 0.0622322i
\(36\) 3.84519 − 4.50112i 0.640866 − 0.750187i
\(37\) 4.48902 − 7.77522i 0.737991 − 1.27824i −0.215408 − 0.976524i \(-0.569108\pi\)
0.953399 − 0.301713i \(-0.0975586\pi\)
\(38\) 0.0252804 + 0.0437870i 0.00410103 + 0.00710319i
\(39\) 5.86988 − 2.13180i 0.939932 − 0.341361i
\(40\) −1.01133 + 1.75168i −0.159906 + 0.276965i
\(41\) −3.56877 − 6.18129i −0.557349 − 0.965356i −0.997717 − 0.0675387i \(-0.978485\pi\)
0.440368 − 0.897817i \(-0.354848\pi\)
\(42\) −0.0384413 − 0.00353442i −0.00593162 − 0.000545373i
\(43\) −5.32209 + 9.21812i −0.811610 + 1.40575i 0.100126 + 0.994975i \(0.468075\pi\)
−0.911736 + 0.410776i \(0.865258\pi\)
\(44\) −8.24392 −1.24282
\(45\) −3.11955 − 8.81247i −0.465035 − 1.31369i
\(46\) −0.157114 − 0.272129i −0.0231652 − 0.0401233i
\(47\) −0.663436 − 1.14910i −0.0967720 − 0.167614i 0.813575 − 0.581460i \(-0.197518\pi\)
−0.910347 + 0.413846i \(0.864185\pi\)
\(48\) 2.78462 + 6.04120i 0.401926 + 0.871972i
\(49\) 3.49069 + 6.04606i 0.498671 + 0.863723i
\(50\) 0.384737 + 0.666383i 0.0544100 + 0.0942408i
\(51\) −5.35636 + 7.57817i −0.750040 + 1.06116i
\(52\) −0.593749 + 7.09006i −0.0823382 + 0.983214i
\(53\) 7.10873 0.976459 0.488230 − 0.872715i \(-0.337643\pi\)
0.488230 + 0.872715i \(0.337643\pi\)
\(54\) 0.816961 + 0.230555i 0.111174 + 0.0313745i
\(55\) −6.50910 + 11.2741i −0.877687 + 1.52020i
\(56\) 0.0442780 − 0.0766917i 0.00591689 − 0.0102484i
\(57\) 0.309411 − 0.437754i 0.0409825 − 0.0579820i
\(58\) 0.450675 0.0591765
\(59\) −4.80940 −0.626131 −0.313066 − 0.949732i \(-0.601356\pi\)
−0.313066 + 0.949732i \(0.601356\pi\)
\(60\) 10.6057 + 0.975129i 1.36920 + 0.125889i
\(61\) 3.61647 − 6.26392i 0.463042 − 0.802012i −0.536069 − 0.844174i \(-0.680091\pi\)
0.999111 + 0.0421621i \(0.0134246\pi\)
\(62\) −0.373179 + 0.646365i −0.0473938 + 0.0820884i
\(63\) 0.136579 + 0.385825i 0.0172074 + 0.0486094i
\(64\) −7.36659 −0.920823
\(65\) 9.22731 + 6.41004i 1.14451 + 0.795068i
\(66\) −0.494842 − 1.07355i −0.0609108 − 0.132145i
\(67\) 6.53369 + 11.3167i 0.798218 + 1.38255i 0.920776 + 0.390092i \(0.127557\pi\)
−0.122558 + 0.992461i \(0.539110\pi\)
\(68\) −5.28634 − 9.15621i −0.641063 − 1.11035i
\(69\) −1.92294 + 2.72057i −0.231495 + 0.327518i
\(70\) −0.0347255 − 0.0601463i −0.00415049 − 0.00718885i
\(71\) 2.24787 + 3.89343i 0.266773 + 0.462065i 0.968027 − 0.250847i \(-0.0807092\pi\)
−0.701253 + 0.712912i \(0.747376\pi\)
\(72\) −1.26484 + 1.48060i −0.149062 + 0.174490i
\(73\) 1.18478 0.138668 0.0693340 − 0.997594i \(-0.477913\pi\)
0.0693340 + 0.997594i \(0.477913\pi\)
\(74\) −0.733350 + 1.27020i −0.0852502 + 0.147658i
\(75\) 4.70884 − 6.66207i 0.543730 − 0.769270i
\(76\) 0.305366 + 0.528910i 0.0350279 + 0.0606702i
\(77\) 0.284980 − 0.493599i 0.0324764 − 0.0562509i
\(78\) −0.958933 + 0.348261i −0.108578 + 0.0394328i
\(79\) −1.41978 − 2.45913i −0.159737 − 0.276673i 0.775037 − 0.631916i \(-0.217731\pi\)
−0.934774 + 0.355243i \(0.884398\pi\)
\(80\) −5.98384 + 10.3643i −0.669014 + 1.15877i
\(81\) −1.40592 − 8.88951i −0.156214 − 0.987723i
\(82\) 0.583013 + 1.00981i 0.0643830 + 0.111515i
\(83\) −5.19008 − 8.98948i −0.569685 − 0.986724i −0.996597 − 0.0824302i \(-0.973732\pi\)
0.426912 − 0.904293i \(-0.359601\pi\)
\(84\) −0.464338 − 0.0426928i −0.0506634 − 0.00465817i
\(85\) −16.6956 −1.81089
\(86\) 0.869443 − 1.50592i 0.0937545 − 0.162387i
\(87\) −2.00020 − 4.33941i −0.214444 − 0.465233i
\(88\) 2.71175 0.289074
\(89\) 6.17396 − 10.6936i 0.654439 − 1.13352i −0.327595 − 0.944818i \(-0.606238\pi\)
0.982034 − 0.188703i \(-0.0604284\pi\)
\(90\) 0.509626 + 1.43965i 0.0537193 + 0.151752i
\(91\) −0.403988 − 0.280643i −0.0423494 − 0.0294194i
\(92\) −1.89780 − 3.28709i −0.197860 − 0.342703i
\(93\) 7.87989 + 0.724505i 0.817107 + 0.0751276i
\(94\) 0.108382 + 0.187723i 0.0111788 + 0.0193622i
\(95\) 0.964425 0.0989479
\(96\) −1.39617 − 3.02898i −0.142496 − 0.309144i
\(97\) 8.05756 − 13.9561i 0.818121 − 1.41703i −0.0889443 − 0.996037i \(-0.528349\pi\)
0.907065 − 0.420990i \(-0.138317\pi\)
\(98\) −0.570257 − 0.987715i −0.0576047 − 0.0997743i
\(99\) −8.14068 + 9.52935i −0.818169 + 0.957736i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 117.2.h.a.22.6 yes 24
3.2 odd 2 351.2.h.a.334.7 24
9.2 odd 6 351.2.f.a.100.6 24
9.7 even 3 117.2.f.a.61.7 ✓ 24
13.3 even 3 117.2.f.a.94.7 yes 24
39.29 odd 6 351.2.f.a.172.6 24
117.16 even 3 inner 117.2.h.a.16.6 yes 24
117.29 odd 6 351.2.h.a.289.7 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.7 ✓ 24 9.7 even 3
117.2.f.a.94.7 yes 24 13.3 even 3
117.2.h.a.16.6 yes 24 117.16 even 3 inner
117.2.h.a.22.6 yes 24 1.1 even 1 trivial
351.2.f.a.100.6 24 9.2 odd 6
351.2.f.a.172.6 24 39.29 odd 6
351.2.h.a.289.7 24 117.29 odd 6
351.2.h.a.334.7 24 3.2 odd 2