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Properties

Label 117.2.h.a.22.8
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 22.8
Character \(\chi\) \(=\) 117.22
Dual form 117.2.h.a.16.8

$q$-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+0.697564 q^{2} +(-1.39192 - 1.03081i) q^{3} -1.51340 q^{4} +(1.44568 - 2.50399i) q^{5} +(-0.970952 - 0.719053i) q^{6} +(1.58641 - 2.74774i) q^{7} -2.45082 q^{8} +(0.874877 + 2.86960i) q^{9} +(1.00846 - 1.74670i) q^{10} +2.31294 q^{11} +(2.10654 + 1.56003i) q^{12} +(-3.15329 + 1.74835i) q^{13} +(1.10662 - 1.91673i) q^{14} +(-4.59341 + 1.99514i) q^{15} +1.31720 q^{16} +(2.69365 + 4.66554i) q^{17} +(0.610282 + 2.00173i) q^{18} +(-2.58760 - 4.48186i) q^{19} +(-2.18790 + 3.78956i) q^{20} +(-5.04055 + 2.18936i) q^{21} +1.61342 q^{22} +(3.27079 + 5.66518i) q^{23} +(3.41135 + 2.52632i) q^{24} +(-1.67999 - 2.90983i) q^{25} +(-2.19962 + 1.21959i) q^{26} +(1.74024 - 4.89608i) q^{27} +(-2.40088 + 4.15845i) q^{28} +4.02068 q^{29} +(-3.20419 + 1.39174i) q^{30} +(4.23854 - 7.34137i) q^{31} +5.82048 q^{32} +(-3.21942 - 2.38419i) q^{33} +(1.87899 + 3.25451i) q^{34} +(-4.58689 - 7.94473i) q^{35} +(-1.32404 - 4.34286i) q^{36} +(-2.42323 + 4.19715i) q^{37} +(-1.80502 - 3.12638i) q^{38} +(6.19134 + 0.816870i) q^{39} +(-3.54311 + 6.13685i) q^{40} +(-1.25716 - 2.17746i) q^{41} +(-3.51610 + 1.52721i) q^{42} +(-2.99320 + 5.18437i) q^{43} -3.50041 q^{44} +(8.45025 + 1.95784i) q^{45} +(2.28159 + 3.95182i) q^{46} +(0.521283 + 0.902888i) q^{47} +(-1.83344 - 1.35778i) q^{48} +(-1.53340 - 2.65593i) q^{49} +(-1.17190 - 2.02979i) q^{50} +(1.05992 - 9.27068i) q^{51} +(4.77221 - 2.64597i) q^{52} -1.29495 q^{53} +(1.21393 - 3.41532i) q^{54} +(3.34377 - 5.79158i) q^{55} +(-3.88801 + 6.73424i) q^{56} +(-1.01820 + 8.90570i) q^{57} +2.80468 q^{58} -4.70451 q^{59} +(6.95168 - 3.01945i) q^{60} +(-3.71841 + 6.44047i) q^{61} +(2.95665 - 5.12107i) q^{62} +(9.27284 + 2.14842i) q^{63} +1.42575 q^{64} +(-0.180794 + 10.4234i) q^{65} +(-2.24575 - 1.66312i) q^{66} +(-4.18368 - 7.24635i) q^{67} +(-4.07658 - 7.06085i) q^{68} +(1.28702 - 11.2570i) q^{69} +(-3.19965 - 5.54196i) q^{70} +(-0.680710 - 1.17903i) q^{71} +(-2.14417 - 7.03288i) q^{72} +1.41722 q^{73} +(-1.69036 + 2.92778i) q^{74} +(-0.661061 + 5.78200i) q^{75} +(3.91609 + 6.78287i) q^{76} +(3.66927 - 6.35536i) q^{77} +(4.31886 + 0.569819i) q^{78} +(-0.0365793 - 0.0633573i) q^{79} +(1.90426 - 3.29827i) q^{80} +(-7.46918 + 5.02109i) q^{81} +(-0.876947 - 1.51892i) q^{82} +(-1.08808 - 1.88462i) q^{83} +(7.62839 - 3.31338i) q^{84} +15.5766 q^{85} +(-2.08795 + 3.61643i) q^{86} +(-5.59646 - 4.14454i) q^{87} -5.66860 q^{88} +(-0.0891486 + 0.154410i) q^{89} +(5.89459 + 1.36572i) q^{90} +(-0.198393 + 11.4381i) q^{91} +(-4.95003 - 8.57371i) q^{92} +(-13.4672 + 5.84947i) q^{93} +(0.363628 + 0.629822i) q^{94} -14.9634 q^{95} +(-8.10164 - 5.99979i) q^{96} +(-0.0654501 + 0.113363i) q^{97} +(-1.06965 - 1.85268i) q^{98} +(2.02353 + 6.63720i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.697564 0.493252 0.246626 − 0.969111i \(-0.420678\pi\)
0.246626 + 0.969111i \(0.420678\pi\)
\(3\) −1.39192 − 1.03081i −0.803625 − 0.595136i
\(4\) −1.51340 −0.756702
\(5\) 1.44568 − 2.50399i 0.646529 − 1.11982i −0.337417 − 0.941355i \(-0.609553\pi\)
0.983946 − 0.178465i \(-0.0571133\pi\)
\(6\) −0.970952 − 0.719053i −0.396390 − 0.293552i
\(7\) 1.58641 − 2.74774i 0.599607 − 1.03855i −0.393272 − 0.919422i \(-0.628657\pi\)
0.992879 − 0.119128i \(-0.0380098\pi\)
\(8\) −2.45082 −0.866497
\(9\) 0.874877 + 2.86960i 0.291626 + 0.956533i
\(10\) 1.00846 − 1.74670i 0.318902 − 0.552354i
\(11\) 2.31294 0.697377 0.348688 − 0.937239i \(-0.386627\pi\)
0.348688 + 0.937239i \(0.386627\pi\)
\(12\) 2.10654 + 1.56003i 0.608105 + 0.450341i
\(13\) −3.15329 + 1.74835i −0.874566 + 0.484906i
\(14\) 1.10662 − 1.91673i 0.295757 − 0.512267i
\(15\) −4.59341 + 1.99514i −1.18601 + 0.515143i
\(16\) 1.31720 0.329301
\(17\) 2.69365 + 4.66554i 0.653306 + 1.13156i 0.982316 + 0.187232i \(0.0599517\pi\)
−0.329010 + 0.944326i \(0.606715\pi\)
\(18\) 0.610282 + 2.00173i 0.143845 + 0.471812i
\(19\) −2.58760 − 4.48186i −0.593636 − 1.02821i −0.993738 − 0.111738i \(-0.964358\pi\)
0.400101 − 0.916471i \(-0.368975\pi\)
\(20\) −2.18790 + 3.78956i −0.489230 + 0.847371i
\(21\) −5.04055 + 2.18936i −1.09994 + 0.477757i
\(22\) 1.61342 0.343982
\(23\) 3.27079 + 5.66518i 0.682007 + 1.18127i 0.974367 + 0.224963i \(0.0722261\pi\)
−0.292360 + 0.956308i \(0.594441\pi\)
\(24\) 3.41135 + 2.52632i 0.696338 + 0.515684i
\(25\) −1.67999 − 2.90983i −0.335999 − 0.581967i
\(26\) −2.19962 + 1.21959i −0.431382 + 0.239181i
\(27\) 1.74024 − 4.89608i 0.334910 − 0.942250i
\(28\) −2.40088 + 4.15845i −0.453724 + 0.785873i
\(29\) 4.02068 0.746621 0.373311 − 0.927706i \(-0.378223\pi\)
0.373311 + 0.927706i \(0.378223\pi\)
\(30\) −3.20419 + 1.39174i −0.585003 + 0.254095i
\(31\) 4.23854 − 7.34137i 0.761264 − 1.31855i −0.180935 − 0.983495i \(-0.557912\pi\)
0.942199 − 0.335053i \(-0.108754\pi\)
\(32\) 5.82048 1.02893
\(33\) −3.21942 − 2.38419i −0.560429 − 0.415034i
\(34\) 1.87899 + 3.25451i 0.322244 + 0.558144i
\(35\) −4.58689 − 7.94473i −0.775326 − 1.34290i
\(36\) −1.32404 − 4.34286i −0.220674 − 0.723811i
\(37\) −2.42323 + 4.19715i −0.398376 + 0.690008i −0.993526 − 0.113607i \(-0.963759\pi\)
0.595150 + 0.803615i \(0.297093\pi\)
\(38\) −1.80502 − 3.12638i −0.292812 − 0.507166i
\(39\) 6.19134 + 0.816870i 0.991408 + 0.130804i
\(40\) −3.54311 + 6.13685i −0.560215 + 0.970321i
\(41\) −1.25716 − 2.17746i −0.196335 − 0.340062i 0.751002 − 0.660299i \(-0.229571\pi\)
−0.947337 + 0.320237i \(0.896237\pi\)
\(42\) −3.51610 + 1.52721i −0.542547 + 0.235654i
\(43\) −2.99320 + 5.18437i −0.456458 + 0.790609i −0.998771 − 0.0495679i \(-0.984216\pi\)
0.542312 + 0.840177i \(0.317549\pi\)
\(44\) −3.50041 −0.527707
\(45\) 8.45025 + 1.95784i 1.25969 + 0.291857i
\(46\) 2.28159 + 3.95182i 0.336401 + 0.582664i
\(47\) 0.521283 + 0.902888i 0.0760369 + 0.131700i 0.901537 − 0.432703i \(-0.142440\pi\)
−0.825500 + 0.564402i \(0.809107\pi\)
\(48\) −1.83344 − 1.35778i −0.264635 − 0.195979i
\(49\) −1.53340 − 2.65593i −0.219057 − 0.379418i
\(50\) −1.17190 − 2.02979i −0.165732 − 0.287056i
\(51\) 1.05992 − 9.27068i 0.148419 − 1.29815i
\(52\) 4.77221 − 2.64597i 0.661787 − 0.366929i
\(53\) −1.29495 −0.177875 −0.0889376 − 0.996037i \(-0.528347\pi\)
−0.0889376 + 0.996037i \(0.528347\pi\)
\(54\) 1.21393 − 3.41532i 0.165195 − 0.464767i
\(55\) 3.34377 − 5.79158i 0.450874 − 0.780937i
\(56\) −3.88801 + 6.73424i −0.519558 + 0.899900i
\(57\) −1.01820 + 8.90570i −0.134863 + 1.17959i
\(58\) 2.80468 0.368272
\(59\) −4.70451 −0.612475 −0.306238 − 0.951955i \(-0.599070\pi\)
−0.306238 + 0.951955i \(0.599070\pi\)
\(60\) 6.95168 − 3.01945i 0.897459 − 0.389810i
\(61\) −3.71841 + 6.44047i −0.476094 + 0.824618i −0.999625 − 0.0273883i \(-0.991281\pi\)
0.523531 + 0.852006i \(0.324614\pi\)
\(62\) 2.95665 − 5.12107i 0.375495 − 0.650377i
\(63\) 9.27284 + 2.14842i 1.16827 + 0.270676i
\(64\) 1.42575 0.178218
\(65\) −0.180794 + 10.4234i −0.0224247 + 1.29286i
\(66\) −2.24575 − 1.66312i −0.276433 − 0.204716i
\(67\) −4.18368 − 7.24635i −0.511118 − 0.885283i −0.999917 − 0.0128861i \(-0.995898\pi\)
0.488799 − 0.872397i \(-0.337435\pi\)
\(68\) −4.07658 − 7.06085i −0.494358 − 0.856253i
\(69\) 1.28702 − 11.2570i 0.154939 − 1.35519i
\(70\) −3.19965 − 5.54196i −0.382431 − 0.662390i
\(71\) −0.680710 − 1.17903i −0.0807855 − 0.139925i 0.822802 − 0.568328i \(-0.192410\pi\)
−0.903588 + 0.428403i \(0.859076\pi\)
\(72\) −2.14417 − 7.03288i −0.252693 − 0.828833i
\(73\) 1.41722 0.165873 0.0829365 − 0.996555i \(-0.473570\pi\)
0.0829365 + 0.996555i \(0.473570\pi\)
\(74\) −1.69036 + 2.92778i −0.196500 + 0.340348i
\(75\) −0.661061 + 5.78200i −0.0763327 + 0.667648i
\(76\) 3.91609 + 6.78287i 0.449206 + 0.778048i
\(77\) 3.66927 − 6.35536i 0.418152 − 0.724261i
\(78\) 4.31886 + 0.569819i 0.489014 + 0.0645193i
\(79\) −0.0365793 − 0.0633573i −0.00411550 − 0.00712825i 0.863960 − 0.503560i \(-0.167977\pi\)
−0.868076 + 0.496432i \(0.834643\pi\)
\(80\) 1.90426 − 3.29827i 0.212903 − 0.368758i
\(81\) −7.46918 + 5.02109i −0.829909 + 0.557899i
\(82\) −0.876947 − 1.51892i −0.0968426 − 0.167736i
\(83\) −1.08808 − 1.88462i −0.119433 − 0.206863i 0.800110 − 0.599853i \(-0.204774\pi\)
−0.919543 + 0.392989i \(0.871441\pi\)
\(84\) 7.62839 − 3.31338i 0.832326 − 0.361520i
\(85\) 15.5766 1.68952
\(86\) −2.08795 + 3.61643i −0.225149 + 0.389969i
\(87\) −5.59646 − 4.14454i −0.600003 − 0.444341i
\(88\) −5.66860 −0.604275
\(89\) −0.0891486 + 0.154410i −0.00944973 + 0.0163674i −0.870712 − 0.491794i \(-0.836341\pi\)
0.861262 + 0.508161i \(0.169675\pi\)
\(90\) 5.89459 + 1.36572i 0.621344 + 0.143959i
\(91\) −0.198393 + 11.4381i −0.0207973 + 1.19903i
\(92\) −4.95003 − 8.57371i −0.516077 − 0.893871i
\(93\) −13.4672 + 5.84947i −1.39649 + 0.606562i
\(94\) 0.363628 + 0.629822i 0.0375053 + 0.0649612i
\(95\) −14.9634 −1.53521
\(96\) −8.10164 − 5.99979i −0.826870 − 0.612351i
\(97\) −0.0654501 + 0.113363i −0.00664545 + 0.0115103i −0.869329 − 0.494234i \(-0.835449\pi\)
0.862684 + 0.505744i \(0.168782\pi\)
\(98\) −1.06965 − 1.85268i −0.108050 − 0.187149i
\(99\) 2.02353 + 6.63720i 0.203373 + 0.667063i
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.22.8 yes 24
3.2 odd 2 351.2.h.a.334.5 24
9.2 odd 6 351.2.f.a.100.8 24
9.7 even 3 117.2.f.a.61.5 ✓ 24
13.3 even 3 117.2.f.a.94.5 yes 24
39.29 odd 6 351.2.f.a.172.8 24
117.16 even 3 inner 117.2.h.a.16.8 yes 24
117.29 odd 6 351.2.h.a.289.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.5 ✓ 24 9.7 even 3
117.2.f.a.94.5 yes 24 13.3 even 3
117.2.h.a.16.8 yes 24 117.16 even 3 inner
117.2.h.a.22.8 yes 24 1.1 even 1 trivial
351.2.f.a.100.8 24 9.2 odd 6
351.2.f.a.172.8 24 39.29 odd 6
351.2.h.a.289.5 24 117.29 odd 6
351.2.h.a.334.5 24 3.2 odd 2