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Properties

Label 117.2.h.a.22.5
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 22.5
Character \(\chi\) \(=\) 117.22
Dual form 117.2.h.a.16.5

$q$-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-0.867378 q^{2} +(0.973656 - 1.43248i) q^{3} -1.24766 q^{4} +(-0.0324057 + 0.0561283i) q^{5} +(-0.844528 + 1.24250i) q^{6} +(1.96209 - 3.39845i) q^{7} +2.81694 q^{8} +(-1.10399 - 2.78948i) q^{9} +(0.0281080 - 0.0486844i) q^{10} -5.29315 q^{11} +(-1.21479 + 1.78724i) q^{12} +(3.21261 - 1.63680i) q^{13} +(-1.70188 + 2.94774i) q^{14} +(0.0488506 + 0.101070i) q^{15} +0.0519555 q^{16} +(2.28144 + 3.95157i) q^{17} +(0.957575 + 2.41954i) q^{18} +(0.281137 + 0.486944i) q^{19} +(0.0404311 - 0.0700288i) q^{20} +(-2.95780 - 6.11957i) q^{21} +4.59116 q^{22} +(1.42839 + 2.47404i) q^{23} +(2.74274 - 4.03521i) q^{24} +(2.49790 + 4.32649i) q^{25} +(-2.78655 + 1.41973i) q^{26} +(-5.07078 - 1.13456i) q^{27} +(-2.44802 + 4.24009i) q^{28} +6.00595 q^{29} +(-0.0423719 - 0.0876660i) q^{30} +(-4.23254 + 7.33098i) q^{31} -5.67895 q^{32} +(-5.15371 + 7.58232i) q^{33} +(-1.97887 - 3.42750i) q^{34} +(0.127166 + 0.220258i) q^{35} +(1.37740 + 3.48031i) q^{36} +(-0.506751 + 0.877718i) q^{37} +(-0.243852 - 0.422364i) q^{38} +(0.783295 - 6.19568i) q^{39} +(-0.0912850 + 0.158110i) q^{40} +(-0.674907 - 1.16897i) q^{41} +(2.56553 + 5.30798i) q^{42} +(3.45051 - 5.97645i) q^{43} +6.60403 q^{44} +(0.192344 + 0.0284301i) q^{45} +(-1.23895 - 2.14593i) q^{46} +(-2.22815 - 3.85927i) q^{47} +(0.0505868 - 0.0744251i) q^{48} +(-4.19963 - 7.27396i) q^{49} +(-2.16662 - 3.75270i) q^{50} +(7.88187 + 0.579356i) q^{51} +(-4.00823 + 2.04217i) q^{52} +1.68875 q^{53} +(4.39828 + 0.984090i) q^{54} +(0.171528 - 0.297096i) q^{55} +(5.52711 - 9.57324i) q^{56} +(0.971267 + 0.0713929i) q^{57} -5.20943 q^{58} +9.14878 q^{59} +(-0.0609487 - 0.126101i) q^{60} +(-3.17857 + 5.50545i) q^{61} +(3.67121 - 6.35873i) q^{62} +(-11.6460 - 1.72138i) q^{63} +4.82189 q^{64} +(-0.0122360 + 0.233360i) q^{65} +(4.47021 - 6.57674i) q^{66} +(-1.76507 - 3.05718i) q^{67} +(-2.84645 - 4.93019i) q^{68} +(4.93477 + 0.362730i) q^{69} +(-0.110301 - 0.191047i) q^{70} +(-5.02865 - 8.70987i) q^{71} +(-3.10987 - 7.85782i) q^{72} +3.39546 q^{73} +(0.439545 - 0.761314i) q^{74} +(8.62970 + 0.634325i) q^{75} +(-0.350762 - 0.607538i) q^{76} +(-10.3857 + 17.9885i) q^{77} +(-0.679413 + 5.37400i) q^{78} +(5.67573 + 9.83065i) q^{79} +(-0.00168365 + 0.00291617i) q^{80} +(-6.56242 + 6.15911i) q^{81} +(0.585399 + 1.01394i) q^{82} +(-1.87243 - 3.24315i) q^{83} +(3.69031 + 7.63512i) q^{84} -0.295726 q^{85} +(-2.99289 + 5.18384i) q^{86} +(5.84773 - 8.60340i) q^{87} -14.9105 q^{88} +(-2.00609 + 3.47464i) q^{89} +(-0.166835 - 0.0246596i) q^{90} +(0.740862 - 14.1295i) q^{91} +(-1.78214 - 3.08675i) q^{92} +(6.38043 + 13.2009i) q^{93} +(1.93265 + 3.34745i) q^{94} -0.0364418 q^{95} +(-5.52935 + 8.13498i) q^{96} +(-1.67726 + 2.90510i) q^{97} +(3.64266 + 6.30928i) q^{98} +(5.84358 + 14.7652i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.867378 −0.613329 −0.306664 − 0.951818i \(-0.599213\pi\)
−0.306664 + 0.951818i \(0.599213\pi\)
\(3\) 0.973656 − 1.43248i 0.562141 − 0.827042i
\(4\) −1.24766 −0.623828
\(5\) −0.0324057 + 0.0561283i −0.0144923 + 0.0251013i −0.873181 − 0.487397i \(-0.837947\pi\)
0.858688 + 0.512498i \(0.171280\pi\)
\(6\) −0.844528 + 1.24250i −0.344777 + 0.507249i
\(7\) 1.96209 − 3.39845i 0.741602 − 1.28449i −0.210164 − 0.977666i \(-0.567400\pi\)
0.951766 − 0.306826i \(-0.0992669\pi\)
\(8\) 2.81694 0.995940
\(9\) −1.10399 − 2.78948i −0.367996 − 0.929827i
\(10\) 0.0281080 − 0.0486844i 0.00888852 − 0.0153954i
\(11\) −5.29315 −1.59595 −0.797973 − 0.602694i \(-0.794094\pi\)
−0.797973 + 0.602694i \(0.794094\pi\)
\(12\) −1.21479 + 1.78724i −0.350679 + 0.515932i
\(13\) 3.21261 − 1.63680i 0.891018 − 0.453967i
\(14\) −1.70188 + 2.94774i −0.454846 + 0.787816i
\(15\) 0.0488506 + 0.101070i 0.0126132 + 0.0260962i
\(16\) 0.0519555 0.0129889
\(17\) 2.28144 + 3.95157i 0.553330 + 0.958395i 0.998031 + 0.0627166i \(0.0199764\pi\)
−0.444702 + 0.895679i \(0.646690\pi\)
\(18\) 0.957575 + 2.41954i 0.225703 + 0.570290i
\(19\) 0.281137 + 0.486944i 0.0644973 + 0.111713i 0.896471 − 0.443103i \(-0.146122\pi\)
−0.831974 + 0.554815i \(0.812789\pi\)
\(20\) 0.0404311 − 0.0700288i 0.00904068 − 0.0156589i
\(21\) −2.95780 − 6.11957i −0.645444 − 1.33540i
\(22\) 4.59116 0.978839
\(23\) 1.42839 + 2.47404i 0.297840 + 0.515873i 0.975641 − 0.219371i \(-0.0704006\pi\)
−0.677802 + 0.735245i \(0.737067\pi\)
\(24\) 2.74274 − 4.03521i 0.559858 − 0.823684i
\(25\) 2.49790 + 4.32649i 0.499580 + 0.865298i
\(26\) −2.78655 + 1.41973i −0.546487 + 0.278431i
\(27\) −5.07078 − 1.13456i −0.975872 − 0.218346i
\(28\) −2.44802 + 4.24009i −0.462632 + 0.801302i
\(29\) 6.00595 1.11528 0.557639 − 0.830084i \(-0.311708\pi\)
0.557639 + 0.830084i \(0.311708\pi\)
\(30\) −0.0423719 − 0.0876660i −0.00773602 − 0.0160055i
\(31\) −4.23254 + 7.33098i −0.760187 + 1.31668i 0.182567 + 0.983193i \(0.441559\pi\)
−0.942754 + 0.333489i \(0.891774\pi\)
\(32\) −5.67895 −1.00391
\(33\) −5.15371 + 7.58232i −0.897146 + 1.31991i
\(34\) −1.97887 − 3.42750i −0.339373 − 0.587812i
\(35\) 0.127166 + 0.220258i 0.0214950 + 0.0372304i
\(36\) 1.37740 + 3.48031i 0.229566 + 0.580052i
\(37\) −0.506751 + 0.877718i −0.0833094 + 0.144296i −0.904670 − 0.426114i \(-0.859882\pi\)
0.821360 + 0.570410i \(0.193216\pi\)
\(38\) −0.243852 − 0.422364i −0.0395580 − 0.0685165i
\(39\) 0.783295 − 6.19568i 0.125428 − 0.992103i
\(40\) −0.0912850 + 0.158110i −0.0144334 + 0.0249994i
\(41\) −0.674907 − 1.16897i −0.105403 − 0.182563i 0.808500 − 0.588496i \(-0.200280\pi\)
−0.913903 + 0.405933i \(0.866947\pi\)
\(42\) 2.56553 + 5.30798i 0.395869 + 0.819040i
\(43\) 3.45051 − 5.97645i 0.526197 − 0.911400i −0.473337 − 0.880881i \(-0.656951\pi\)
0.999534 − 0.0305189i \(-0.00971599\pi\)
\(44\) 6.60403 0.995595
\(45\) 0.192344 + 0.0284301i 0.0286730 + 0.00423811i
\(46\) −1.23895 − 2.14593i −0.182674 − 0.316400i
\(47\) −2.22815 − 3.85927i −0.325009 − 0.562933i 0.656505 − 0.754322i \(-0.272034\pi\)
−0.981514 + 0.191389i \(0.938701\pi\)
\(48\) 0.0505868 − 0.0744251i 0.00730157 − 0.0107423i
\(49\) −4.19963 − 7.27396i −0.599946 − 1.03914i
\(50\) −2.16662 − 3.75270i −0.306407 − 0.530712i
\(51\) 7.88187 + 0.579356i 1.10368 + 0.0811260i
\(52\) −4.00823 + 2.04217i −0.555842 + 0.283197i
\(53\) 1.68875 0.231968 0.115984 − 0.993251i \(-0.462998\pi\)
0.115984 + 0.993251i \(0.462998\pi\)
\(54\) 4.39828 + 0.984090i 0.598530 + 0.133918i
\(55\) 0.171528 − 0.297096i 0.0231289 − 0.0400604i
\(56\) 5.52711 − 9.57324i 0.738591 − 1.27928i
\(57\) 0.971267 + 0.0713929i 0.128647 + 0.00945622i
\(58\) −5.20943 −0.684032
\(59\) 9.14878 1.19107 0.595535 − 0.803330i \(-0.296940\pi\)
0.595535 + 0.803330i \(0.296940\pi\)
\(60\) −0.0609487 − 0.126101i −0.00786844 − 0.0162795i
\(61\) −3.17857 + 5.50545i −0.406974 + 0.704900i −0.994549 − 0.104270i \(-0.966750\pi\)
0.587575 + 0.809170i \(0.300083\pi\)
\(62\) 3.67121 − 6.35873i 0.466245 − 0.807559i
\(63\) −11.6460 − 1.72138i −1.46726 − 0.216874i
\(64\) 4.82189 0.602736
\(65\) −0.0122360 + 0.233360i −0.00151769 + 0.0289448i
\(66\) 4.47021 − 6.57674i 0.550245 − 0.809541i
\(67\) −1.76507 − 3.05718i −0.215637 − 0.373494i 0.737832 − 0.674984i \(-0.235849\pi\)
−0.953469 + 0.301490i \(0.902516\pi\)
\(68\) −2.84645 − 4.93019i −0.345183 − 0.597874i
\(69\) 4.93477 + 0.362730i 0.594076 + 0.0436675i
\(70\) −0.110301 − 0.191047i −0.0131835 − 0.0228345i
\(71\) −5.02865 − 8.70987i −0.596790 − 1.03367i −0.993292 − 0.115637i \(-0.963109\pi\)
0.396501 − 0.918034i \(-0.370224\pi\)
\(72\) −3.10987 − 7.85782i −0.366502 − 0.926053i
\(73\) 3.39546 0.397409 0.198704 − 0.980059i \(-0.436327\pi\)
0.198704 + 0.980059i \(0.436327\pi\)
\(74\) 0.439545 − 0.761314i 0.0510960 − 0.0885009i
\(75\) 8.62970 + 0.634325i 0.996472 + 0.0732455i
\(76\) −0.350762 − 0.607538i −0.0402352 − 0.0696894i
\(77\) −10.3857 + 17.9885i −1.18356 + 2.04998i
\(78\) −0.679413 + 5.37400i −0.0769284 + 0.608485i
\(79\) 5.67573 + 9.83065i 0.638570 + 1.10604i 0.985747 + 0.168235i \(0.0538069\pi\)
−0.347177 + 0.937800i \(0.612860\pi\)
\(80\) −0.00168365 + 0.00291617i −0.000188238 + 0.000326038i
\(81\) −6.56242 + 6.15911i −0.729158 + 0.684346i
\(82\) 0.585399 + 1.01394i 0.0646465 + 0.111971i
\(83\) −1.87243 − 3.24315i −0.205526 − 0.355982i 0.744774 − 0.667317i \(-0.232557\pi\)
−0.950300 + 0.311335i \(0.899224\pi\)
\(84\) 3.69031 + 7.63512i 0.402646 + 0.833060i
\(85\) −0.295726 −0.0320760
\(86\) −2.99289 + 5.18384i −0.322732 + 0.558988i
\(87\) 5.84773 − 8.60340i 0.626943 − 0.922381i
\(88\) −14.9105 −1.58947
\(89\) −2.00609 + 3.47464i −0.212645 + 0.368312i −0.952541 − 0.304409i \(-0.901541\pi\)
0.739897 + 0.672721i \(0.234874\pi\)
\(90\) −0.166835 − 0.0246596i −0.0175860 − 0.00259935i
\(91\) 0.740862 − 14.1295i 0.0776634 − 1.48117i
\(92\) −1.78214 − 3.08675i −0.185801 − 0.321816i
\(93\) 6.38043 + 13.2009i 0.661619 + 1.36887i
\(94\) 1.93265 + 3.34745i 0.199338 + 0.345263i
\(95\) −0.0364418 −0.00373885
\(96\) −5.52935 + 8.13498i −0.564337 + 0.830273i
\(97\) −1.67726 + 2.90510i −0.170300 + 0.294968i −0.938525 − 0.345212i \(-0.887807\pi\)
0.768225 + 0.640180i \(0.221140\pi\)
\(98\) 3.64266 + 6.30928i 0.367964 + 0.637333i
\(99\) 5.84358 + 14.7652i 0.587302 + 1.48395i
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.22.5 yes 24
3.2 odd 2 351.2.h.a.334.8 24
9.2 odd 6 351.2.f.a.100.5 24
9.7 even 3 117.2.f.a.61.8 ✓ 24
13.3 even 3 117.2.f.a.94.8 yes 24
39.29 odd 6 351.2.f.a.172.5 24
117.16 even 3 inner 117.2.h.a.16.5 yes 24
117.29 odd 6 351.2.h.a.289.8 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.8 ✓ 24 9.7 even 3
117.2.f.a.94.8 yes 24 13.3 even 3
117.2.h.a.16.5 yes 24 117.16 even 3 inner
117.2.h.a.22.5 yes 24 1.1 even 1 trivial
351.2.f.a.100.5 24 9.2 odd 6
351.2.f.a.172.5 24 39.29 odd 6
351.2.h.a.289.8 24 117.29 odd 6
351.2.h.a.334.8 24 3.2 odd 2