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Properties

Label 117.2.h.a.22.4
Level $117$
Weight $2$
Character 117.22
Analytic conductor $0.934$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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 22.4
Character \(\chi\) \(=\) 117.22
Dual form 117.2.h.a.16.4

$q$-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-1.20426 q^{2} +(1.45393 + 0.941317i) q^{3} -0.549753 q^{4} +(1.89177 - 3.27665i) q^{5} +(-1.75092 - 1.13359i) q^{6} +(-0.150228 + 0.260203i) q^{7} +3.07057 q^{8} +(1.22785 + 2.73722i) q^{9} +(-2.27819 + 3.94594i) q^{10} +1.28462 q^{11} +(-0.799304 - 0.517492i) q^{12} +(0.423704 + 3.58057i) q^{13} +(0.180914 - 0.313352i) q^{14} +(5.83487 - 2.98327i) q^{15} -2.59827 q^{16} +(-2.63848 - 4.56998i) q^{17} +(-1.47865 - 3.29634i) q^{18} +(-0.829906 - 1.43744i) q^{19} +(-1.04001 + 1.80135i) q^{20} +(-0.463355 + 0.236905i) q^{21} -1.54702 q^{22} +(1.67399 + 2.89944i) q^{23} +(4.46441 + 2.89038i) q^{24} +(-4.65760 - 8.06721i) q^{25} +(-0.510251 - 4.31194i) q^{26} +(-0.791390 + 5.13553i) q^{27} +(0.0825883 - 0.143047i) q^{28} -9.63240 q^{29} +(-7.02672 + 3.59264i) q^{30} +(-3.29700 + 5.71056i) q^{31} -3.01215 q^{32} +(1.86775 + 1.20923i) q^{33} +(3.17742 + 5.50346i) q^{34} +(0.568394 + 0.984488i) q^{35} +(-0.675011 - 1.50480i) q^{36} +(1.97702 - 3.42430i) q^{37} +(0.999424 + 1.73105i) q^{38} +(-2.75441 + 5.60475i) q^{39} +(5.80882 - 10.0612i) q^{40} +(-1.43368 - 2.48321i) q^{41} +(0.558000 - 0.285296i) q^{42} +(0.0104220 - 0.0180514i) q^{43} -0.706223 q^{44} +(11.2917 + 1.15499i) q^{45} +(-2.01592 - 3.49168i) q^{46} +(0.954216 + 1.65275i) q^{47} +(-3.77771 - 2.44579i) q^{48} +(3.45486 + 5.98400i) q^{49} +(5.60898 + 9.71503i) q^{50} +(0.465627 - 9.12810i) q^{51} +(-0.232933 - 1.96843i) q^{52} -7.15248 q^{53} +(0.953041 - 6.18453i) q^{54} +(2.43021 - 4.20924i) q^{55} +(-0.461286 + 0.798970i) q^{56} +(0.146458 - 2.87114i) q^{57} +11.5999 q^{58} +8.73826 q^{59} +(-3.20774 + 1.64006i) q^{60} +(-2.56839 + 4.44858i) q^{61} +(3.97045 - 6.87702i) q^{62} +(-0.896689 - 0.0917194i) q^{63} +8.82395 q^{64} +(12.5338 + 5.38529i) q^{65} +(-2.24926 - 1.45623i) q^{66} +(-4.15570 - 7.19788i) q^{67} +(1.45051 + 2.51236i) q^{68} +(-0.295418 + 5.79134i) q^{69} +(-0.684496 - 1.18558i) q^{70} +(4.64070 + 8.03792i) q^{71} +(3.77018 + 8.40484i) q^{72} -4.69504 q^{73} +(-2.38085 + 4.12376i) q^{74} +(0.821952 - 16.1135i) q^{75} +(0.456243 + 0.790236i) q^{76} +(-0.192986 + 0.334261i) q^{77} +(3.31703 - 6.74959i) q^{78} +(6.65302 + 11.5234i) q^{79} +(-4.91533 + 8.51360i) q^{80} +(-5.98479 + 6.72177i) q^{81} +(1.72653 + 2.99044i) q^{82} +(-6.45522 - 11.1808i) q^{83} +(0.254730 - 0.130239i) q^{84} -19.9656 q^{85} +(-0.0125508 + 0.0217386i) q^{86} +(-14.0049 - 9.06714i) q^{87} +3.94451 q^{88} +(3.19907 - 5.54096i) q^{89} +(-13.5982 - 1.39091i) q^{90} +(-0.995325 - 0.427653i) q^{91} +(-0.920281 - 1.59397i) q^{92} +(-10.1691 + 5.19926i) q^{93} +(-1.14913 - 1.99035i) q^{94} -6.27997 q^{95} +(-4.37946 - 2.83539i) q^{96} +(1.88195 - 3.25964i) q^{97} +(-4.16056 - 7.20630i) q^{98} +(1.57731 + 3.51629i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.20426 −0.851542 −0.425771 − 0.904831i \(-0.639997\pi\)
−0.425771 + 0.904831i \(0.639997\pi\)
\(3\) 1.45393 + 0.941317i 0.839429 + 0.543470i
\(4\) −0.549753 −0.274876
\(5\) 1.89177 − 3.27665i 0.846026 − 1.46536i −0.0387007 − 0.999251i \(-0.512322\pi\)
0.884727 − 0.466110i \(-0.154345\pi\)
\(6\) −1.75092 − 1.13359i −0.714809 − 0.462787i
\(7\) −0.150228 + 0.260203i −0.0567809 + 0.0983473i −0.893019 − 0.450020i \(-0.851417\pi\)
0.836238 + 0.548367i \(0.184750\pi\)
\(8\) 3.07057 1.08561
\(9\) 1.22785 + 2.73722i 0.409282 + 0.912408i
\(10\) −2.27819 + 3.94594i −0.720427 + 1.24782i
\(11\) 1.28462 0.387327 0.193664 − 0.981068i \(-0.437963\pi\)
0.193664 + 0.981068i \(0.437963\pi\)
\(12\) −0.799304 − 0.517492i −0.230739 − 0.149387i
\(13\) 0.423704 + 3.58057i 0.117514 + 0.993071i
\(14\) 0.180914 − 0.313352i 0.0483513 − 0.0837469i
\(15\) 5.83487 − 2.98327i 1.50656 − 0.770276i
\(16\) −2.59827 −0.649567
\(17\) −2.63848 − 4.56998i −0.639926 − 1.10838i −0.985449 − 0.169974i \(-0.945632\pi\)
0.345523 − 0.938410i \(-0.387702\pi\)
\(18\) −1.47865 − 3.29634i −0.348520 − 0.776954i
\(19\) −0.829906 − 1.43744i −0.190393 − 0.329771i 0.754987 − 0.655739i \(-0.227643\pi\)
−0.945381 + 0.325968i \(0.894310\pi\)
\(20\) −1.04001 + 1.80135i −0.232553 + 0.402793i
\(21\) −0.463355 + 0.236905i −0.101112 + 0.0516969i
\(22\) −1.54702 −0.329825
\(23\) 1.67399 + 2.89944i 0.349051 + 0.604574i 0.986081 − 0.166265i \(-0.0531706\pi\)
−0.637030 + 0.770839i \(0.719837\pi\)
\(24\) 4.46441 + 2.89038i 0.911293 + 0.589996i
\(25\) −4.65760 − 8.06721i −0.931521 − 1.61344i
\(26\) −0.510251 − 4.31194i −0.100068 − 0.845642i
\(27\) −0.791390 + 5.13553i −0.152303 + 0.988334i
\(28\) 0.0825883 − 0.143047i 0.0156077 − 0.0270334i
\(29\) −9.63240 −1.78869 −0.894346 − 0.447376i \(-0.852359\pi\)
−0.894346 + 0.447376i \(0.852359\pi\)
\(30\) −7.02672 + 3.59264i −1.28290 + 0.655923i
\(31\) −3.29700 + 5.71056i −0.592158 + 1.02565i 0.401783 + 0.915735i \(0.368391\pi\)
−0.993941 + 0.109913i \(0.964943\pi\)
\(32\) −3.01215 −0.532478
\(33\) 1.86775 + 1.20923i 0.325134 + 0.210500i
\(34\) 3.17742 + 5.50346i 0.544924 + 0.943836i
\(35\) 0.568394 + 0.984488i 0.0960762 + 0.166409i
\(36\) −0.675011 − 1.50480i −0.112502 − 0.250799i
\(37\) 1.97702 − 3.42430i 0.325020 − 0.562952i −0.656496 − 0.754329i \(-0.727962\pi\)
0.981517 + 0.191378i \(0.0612955\pi\)
\(38\) 0.999424 + 1.73105i 0.162128 + 0.280814i
\(39\) −2.75441 + 5.60475i −0.441059 + 0.897478i
\(40\) 5.80882 − 10.0612i 0.918455 − 1.59081i
\(41\) −1.43368 − 2.48321i −0.223904 − 0.387813i 0.732086 − 0.681212i \(-0.238547\pi\)
−0.955990 + 0.293399i \(0.905213\pi\)
\(42\) 0.558000 − 0.285296i 0.0861013 − 0.0440221i
\(43\) 0.0104220 − 0.0180514i 0.00158933 − 0.00275281i −0.865230 − 0.501376i \(-0.832827\pi\)
0.866819 + 0.498623i \(0.166161\pi\)
\(44\) −0.706223 −0.106467
\(45\) 11.2917 + 1.15499i 1.68327 + 0.172176i
\(46\) −2.01592 − 3.49168i −0.297232 − 0.514821i
\(47\) 0.954216 + 1.65275i 0.139187 + 0.241079i 0.927189 − 0.374594i \(-0.122218\pi\)
−0.788002 + 0.615672i \(0.788884\pi\)
\(48\) −3.77771 − 2.44579i −0.545265 − 0.353020i
\(49\) 3.45486 + 5.98400i 0.493552 + 0.854857i
\(50\) 5.60898 + 9.71503i 0.793229 + 1.37391i
\(51\) 0.465627 − 9.12810i 0.0652008 − 1.27819i
\(52\) −0.232933 − 1.96843i −0.0323019 − 0.272972i
\(53\) −7.15248 −0.982469 −0.491235 − 0.871027i \(-0.663454\pi\)
−0.491235 + 0.871027i \(0.663454\pi\)
\(54\) 0.953041 − 6.18453i 0.129693 − 0.841608i
\(55\) 2.43021 − 4.20924i 0.327689 − 0.567574i
\(56\) −0.461286 + 0.798970i −0.0616419 + 0.106767i
\(57\) 0.146458 − 2.87114i 0.0193988 − 0.380292i
\(58\) 11.5999 1.52315
\(59\) 8.73826 1.13762 0.568812 − 0.822468i \(-0.307403\pi\)
0.568812 + 0.822468i \(0.307403\pi\)
\(60\) −3.20774 + 1.64006i −0.414117 + 0.211731i
\(61\) −2.56839 + 4.44858i −0.328849 + 0.569583i −0.982284 − 0.187400i \(-0.939994\pi\)
0.653435 + 0.756983i \(0.273327\pi\)
\(62\) 3.97045 − 6.87702i 0.504247 − 0.873382i
\(63\) −0.896689 − 0.0917194i −0.112972 − 0.0115556i
\(64\) 8.82395 1.10299
\(65\) 12.5338 + 5.38529i 1.55463 + 0.667963i
\(66\) −2.24926 − 1.45623i −0.276865 − 0.179250i
\(67\) −4.15570 − 7.19788i −0.507700 − 0.879361i −0.999960 − 0.00891361i \(-0.997163\pi\)
0.492261 − 0.870448i \(-0.336171\pi\)
\(68\) 1.45051 + 2.51236i 0.175901 + 0.304669i
\(69\) −0.295418 + 5.79134i −0.0355641 + 0.697196i
\(70\) −0.684496 − 1.18558i −0.0818129 − 0.141704i
\(71\) 4.64070 + 8.03792i 0.550749 + 0.953926i 0.998221 + 0.0596277i \(0.0189913\pi\)
−0.447471 + 0.894298i \(0.647675\pi\)
\(72\) 3.77018 + 8.40484i 0.444321 + 0.990520i
\(73\) −4.69504 −0.549512 −0.274756 − 0.961514i \(-0.588597\pi\)
−0.274756 + 0.961514i \(0.588597\pi\)
\(74\) −2.38085 + 4.12376i −0.276769 + 0.479377i
\(75\) 0.821952 − 16.1135i 0.0949108 − 1.86062i
\(76\) 0.456243 + 0.790236i 0.0523346 + 0.0906463i
\(77\) −0.192986 + 0.334261i −0.0219928 + 0.0380926i
\(78\) 3.31703 − 6.74959i 0.375580 − 0.764240i
\(79\) 6.65302 + 11.5234i 0.748523 + 1.29648i 0.948530 + 0.316686i \(0.102570\pi\)
−0.200007 + 0.979794i \(0.564097\pi\)
\(80\) −4.91533 + 8.51360i −0.549550 + 0.951849i
\(81\) −5.98479 + 6.72177i −0.664977 + 0.746864i
\(82\) 1.72653 + 2.99044i 0.190663 + 0.330239i
\(83\) −6.45522 − 11.1808i −0.708552 − 1.22725i −0.965394 − 0.260795i \(-0.916015\pi\)
0.256842 − 0.966453i \(-0.417318\pi\)
\(84\) 0.254730 − 0.130239i 0.0277934 − 0.0142103i
\(85\) −19.9656 −2.16558
\(86\) −0.0125508 + 0.0217386i −0.00135339 + 0.00234413i
\(87\) −14.0049 − 9.06714i −1.50148 − 0.972100i
\(88\) 3.94451 0.420486
\(89\) 3.19907 − 5.54096i 0.339101 − 0.587340i −0.645163 − 0.764045i \(-0.723210\pi\)
0.984264 + 0.176705i \(0.0565438\pi\)
\(90\) −13.5982 − 1.39091i −1.43337 − 0.146615i
\(91\) −0.995325 − 0.427653i −0.104338 − 0.0448302i
\(92\) −0.920281 − 1.59397i −0.0959459 − 0.166183i
\(93\) −10.1691 + 5.19926i −1.05448 + 0.539138i
\(94\) −1.14913 − 1.99035i −0.118523 − 0.205289i
\(95\) −6.27997 −0.644311
\(96\) −4.37946 − 2.83539i −0.446977 − 0.289385i
\(97\) 1.88195 − 3.25964i 0.191083 − 0.330966i −0.754526 − 0.656270i \(-0.772133\pi\)
0.945610 + 0.325304i \(0.105467\pi\)
\(98\) −4.16056 − 7.20630i −0.420280 − 0.727946i
\(99\) 1.57731 + 3.51629i 0.158526 + 0.353400i
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.22.4 yes 24
3.2 odd 2 351.2.h.a.334.9 24
9.2 odd 6 351.2.f.a.100.4 24
9.7 even 3 117.2.f.a.61.9 ✓ 24
13.3 even 3 117.2.f.a.94.9 yes 24
39.29 odd 6 351.2.f.a.172.4 24
117.16 even 3 inner 117.2.h.a.16.4 yes 24
117.29 odd 6 351.2.h.a.289.9 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.9 ✓ 24 9.7 even 3
117.2.f.a.94.9 yes 24 13.3 even 3
117.2.h.a.16.4 yes 24 117.16 even 3 inner
117.2.h.a.22.4 yes 24 1.1 even 1 trivial
351.2.f.a.100.4 24 9.2 odd 6
351.2.f.a.172.4 24 39.29 odd 6
351.2.h.a.289.9 24 117.29 odd 6
351.2.h.a.334.9 24 3.2 odd 2