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Properties

Label 92.2.h.a.67.3
Level $92$
Weight $2$
Character 92.67
Analytic conductor $0.735$
Analytic rank $0$
Dimension $100$
Inner twists $4$

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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] = [92,2,Mod(7,92)] 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("92.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(92, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 19])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 92.h (of order \(22\), degree \(10\), 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: \(0.734623698596\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 67.3
Character \(\chi\) \(=\) 92.67
Dual form 92.2.h.a.11.3

$q$-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+(-1.04738 + 0.950264i) q^{2} +(1.65123 - 0.237411i) q^{3} +(0.193997 - 1.99057i) q^{4} +(0.934364 - 3.18215i) q^{5} +(-1.50386 + 1.81776i) q^{6} +(-0.148303 + 0.171151i) q^{7} +(1.68838 + 2.26922i) q^{8} +(-0.208284 + 0.0611576i) q^{9} +(2.04525 + 4.22080i) q^{10} +(-3.48603 + 2.24034i) q^{11} +(-0.152250 - 3.33294i) q^{12} +(3.81497 + 4.40271i) q^{13} +(-0.00730929 - 0.320186i) q^{14} +(0.787372 - 5.47629i) q^{15} +(-3.92473 - 0.772328i) q^{16} +(-0.449525 - 0.205291i) q^{17} +(0.160036 - 0.261980i) q^{18} +(0.608905 + 1.33332i) q^{19} +(-6.15303 - 2.47724i) q^{20} +(-0.204249 + 0.317818i) q^{21} +(1.52228 - 5.65913i) q^{22} +(-4.22490 - 2.26941i) q^{23} +(3.32664 + 3.34617i) q^{24} +(-5.04678 - 3.24337i) q^{25} +(-8.17944 - 0.986067i) q^{26} +(-4.88178 + 2.22943i) q^{27} +(0.311917 + 0.328410i) q^{28} +(-0.299010 + 0.654741i) q^{29} +(4.37925 + 6.48395i) q^{30} +(9.23576 + 1.32790i) q^{31} +(4.84459 - 2.92061i) q^{32} +(-5.22436 + 4.52693i) q^{33} +(0.665903 - 0.212150i) q^{34} +(0.406059 + 0.631840i) q^{35} +(0.0813321 + 0.426468i) q^{36} +(-1.33496 - 4.54647i) q^{37} +(-1.90476 - 0.817864i) q^{38} +(7.34464 + 6.36416i) q^{39} +(8.79858 - 3.25239i) q^{40} +(-2.79241 - 0.819926i) q^{41} +(-0.0880851 - 0.526966i) q^{42} +(-0.508752 - 3.53845i) q^{43} +(3.78327 + 7.37380i) q^{44} +0.719934i q^{45} +(6.58160 - 1.63784i) q^{46} -6.02195i q^{47} +(-6.66399 - 0.343516i) q^{48} +(0.988905 + 6.87799i) q^{49} +(8.36794 - 1.39875i) q^{50} +(-0.791007 - 0.232261i) q^{51} +(9.50398 - 6.73984i) q^{52} +(4.56115 + 3.95226i) q^{53} +(2.99451 - 6.97403i) q^{54} +(3.87187 + 13.1864i) q^{55} +(-0.638771 - 0.0475654i) q^{56} +(1.32199 + 2.05705i) q^{57} +(-0.309000 - 0.969899i) q^{58} +(-4.34794 + 3.76751i) q^{59} +(-10.7482 - 2.62970i) q^{60} +(-9.72972 - 1.39892i) q^{61} +(-10.9352 + 7.38560i) q^{62} +(0.0204219 - 0.0447178i) q^{63} +(-2.29876 + 7.66262i) q^{64} +(17.5746 - 8.02607i) q^{65} +(1.17009 - 9.70592i) q^{66} +(-10.2383 - 6.57976i) q^{67} +(-0.495852 + 0.854984i) q^{68} +(-7.51506 - 2.74429i) q^{69} +(-1.02571 - 0.275911i) q^{70} +(2.54626 - 3.96206i) q^{71} +(-0.490442 - 0.369385i) q^{72} +(-2.85291 - 6.24699i) q^{73} +(5.71855 + 3.49330i) q^{74} +(-9.10341 - 4.15739i) q^{75} +(2.77218 - 0.953409i) q^{76} +(0.133554 - 0.928886i) q^{77} +(-13.7402 + 0.313665i) q^{78} +(-1.34769 - 1.55532i) q^{79} +(-6.12479 + 11.7675i) q^{80} +(-6.98379 + 4.48821i) q^{81} +(3.70385 - 1.79476i) q^{82} +(6.97591 - 2.04831i) q^{83} +(0.593015 + 0.468228i) q^{84} +(-1.07329 + 1.23864i) q^{85} +(3.89531 + 3.22264i) q^{86} +(-0.338292 + 1.15212i) q^{87} +(-10.9696 - 4.12805i) q^{88} +(14.1914 - 2.04041i) q^{89} +(-0.684127 - 0.754042i) q^{90} -1.31930 q^{91} +(-5.33704 + 7.96969i) q^{92} +15.5656 q^{93} +(5.72245 + 6.30725i) q^{94} +(4.81175 - 0.691826i) q^{95} +(7.30614 - 5.97276i) q^{96} +(-0.265360 + 0.903735i) q^{97} +(-7.57166 - 6.26412i) q^{98} +(0.589070 - 0.679823i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q - 7 q^{2} - 11 q^{4} - 22 q^{5} - 12 q^{6} - 10 q^{8} - 12 q^{9} - 11 q^{10} - 18 q^{12} - 18 q^{13} - 11 q^{14} + 5 q^{16} - 22 q^{17} - 24 q^{18} - 11 q^{20} - 22 q^{21} - 30 q^{24} - 16 q^{25}+ \cdots - 71 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(47\)
\(\chi(n)\) \(e\left(\frac{13}{22}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.04738 + 0.950264i −0.740607 + 0.671938i
\(3\) 1.65123 − 0.237411i 0.953338 − 0.137069i 0.351943 − 0.936021i \(-0.385521\pi\)
0.601395 + 0.798952i \(0.294612\pi\)
\(4\) 0.193997 − 1.99057i 0.0969984 − 0.995285i
\(5\) 0.934364 − 3.18215i 0.417860 − 1.42310i −0.434749 − 0.900552i \(-0.643163\pi\)
0.852609 − 0.522550i \(-0.175019\pi\)
\(6\) −1.50386 + 1.81776i −0.613947 + 0.742099i
\(7\) −0.148303 + 0.171151i −0.0560533 + 0.0646889i −0.783084 − 0.621916i \(-0.786354\pi\)
0.727030 + 0.686605i \(0.240900\pi\)
\(8\) 1.68838 + 2.26922i 0.596932 + 0.802292i
\(9\) −0.208284 + 0.0611576i −0.0694279 + 0.0203859i
\(10\) 2.04525 + 4.22080i 0.646766 + 1.33474i
\(11\) −3.48603 + 2.24034i −1.05108 + 0.675487i −0.947702 − 0.319156i \(-0.896601\pi\)
−0.103376 + 0.994642i \(0.532964\pi\)
\(12\) −0.152250 − 3.33294i −0.0439507 − 0.962138i
\(13\) 3.81497 + 4.40271i 1.05808 + 1.22109i 0.974452 + 0.224594i \(0.0721055\pi\)
0.0836287 + 0.996497i \(0.473349\pi\)
\(14\) −0.00730929 − 0.320186i −0.00195349 − 0.0855734i
\(15\) 0.787372 − 5.47629i 0.203299 − 1.41397i
\(16\) −3.92473 − 0.772328i −0.981183 − 0.193082i
\(17\) −0.449525 − 0.205291i −0.109026 − 0.0497904i 0.360155 − 0.932892i \(-0.382724\pi\)
−0.469181 + 0.883102i \(0.655451\pi\)
\(18\) 0.160036 − 0.261980i 0.0377208 − 0.0617492i
\(19\) 0.608905 + 1.33332i 0.139692 + 0.305884i 0.966529 − 0.256559i \(-0.0825889\pi\)
−0.826836 + 0.562443i \(0.809862\pi\)
\(20\) −6.15303 − 2.47724i −1.37586 − 0.553928i
\(21\) −0.204249 + 0.317818i −0.0445708 + 0.0693536i
\(22\) 1.52228 − 5.65913i 0.324551 − 1.20653i
\(23\) −4.22490 − 2.26941i −0.880952 − 0.473206i
\(24\) 3.32664 + 3.34617i 0.679047 + 0.683034i
\(25\) −5.04678 − 3.24337i −1.00936 − 0.648674i
\(26\) −8.17944 − 0.986067i −1.60412 − 0.193384i
\(27\) −4.88178 + 2.22943i −0.939498 + 0.429054i
\(28\) 0.311917 + 0.328410i 0.0589468 + 0.0620637i
\(29\) −0.299010 + 0.654741i −0.0555248 + 0.121582i −0.935361 − 0.353695i \(-0.884925\pi\)
0.879836 + 0.475277i \(0.157652\pi\)
\(30\) 4.37925 + 6.48395i 0.799537 + 1.18380i
\(31\) 9.23576 + 1.32790i 1.65879 + 0.238498i 0.907069 − 0.420981i \(-0.138314\pi\)
0.751723 + 0.659479i \(0.229223\pi\)
\(32\) 4.84459 − 2.92061i 0.856410 − 0.516296i
\(33\) −5.22436 + 4.52693i −0.909444 + 0.788038i
\(34\) 0.665903 − 0.212150i 0.114201 − 0.0363834i
\(35\) 0.406059 + 0.631840i 0.0686364 + 0.106800i
\(36\) 0.0813321 + 0.426468i 0.0135554 + 0.0710779i
\(37\) −1.33496 − 4.54647i −0.219467 − 0.747435i −0.993454 − 0.114233i \(-0.963559\pi\)
0.773987 − 0.633201i \(-0.218259\pi\)
\(38\) −1.90476 − 0.817864i −0.308992 − 0.132675i
\(39\) 7.34464 + 6.36416i 1.17608 + 1.01908i
\(40\) 8.79858 − 3.25239i 1.39118 − 0.514249i
\(41\) −2.79241 − 0.819926i −0.436101 − 0.128051i 0.0563103 − 0.998413i \(-0.482066\pi\)
−0.492412 + 0.870362i \(0.663885\pi\)
\(42\) −0.0880851 − 0.526966i −0.0135918 − 0.0813126i
\(43\) −0.508752 − 3.53845i −0.0775839 − 0.539608i −0.991132 − 0.132878i \(-0.957578\pi\)
0.913548 − 0.406730i \(-0.133331\pi\)
\(44\) 3.78327 + 7.37380i 0.570349 + 1.11164i
\(45\) 0.719934i 0.107321i
\(46\) 6.58160 − 1.63784i 0.970404 − 0.241486i
\(47\) − 6.02195i − 0.878392i −0.898391 − 0.439196i \(-0.855263\pi\)
0.898391 − 0.439196i \(-0.144737\pi\)
\(48\) −6.66399 − 0.343516i −0.961864 − 0.0495823i
\(49\) 0.988905 + 6.87799i 0.141272 + 0.982569i
\(50\) 8.36794 − 1.39875i 1.18341 − 0.197812i
\(51\) −0.791007 − 0.232261i −0.110763 − 0.0325230i
\(52\) 9.50398 − 6.73984i 1.31796 − 0.934648i
\(53\) 4.56115 + 3.95226i 0.626522 + 0.542885i 0.909216 − 0.416326i \(-0.136682\pi\)
−0.282693 + 0.959210i \(0.591228\pi\)
\(54\) 2.99451 − 6.97403i 0.407501 − 0.949045i
\(55\) 3.87187 + 13.1864i 0.522083 + 1.77805i
\(56\) −0.638771 − 0.0475654i −0.0853594 − 0.00635620i
\(57\) 1.32199 + 2.05705i 0.175101 + 0.272463i
\(58\) −0.309000 − 0.969899i −0.0405737 − 0.127354i
\(59\) −4.34794 + 3.76751i −0.566053 + 0.490488i −0.890232 − 0.455508i \(-0.849458\pi\)
0.324178 + 0.945996i \(0.394912\pi\)
\(60\) −10.7482 − 2.62970i −1.38759 − 0.339493i
\(61\) −9.72972 − 1.39892i −1.24576 − 0.179114i −0.512278 − 0.858820i \(-0.671198\pi\)
−0.733485 + 0.679706i \(0.762107\pi\)
\(62\) −10.9352 + 7.38560i −1.38877 + 0.937972i
\(63\) 0.0204219 − 0.0447178i 0.00257292 − 0.00563391i
\(64\) −2.29876 + 7.66262i −0.287345 + 0.957827i
\(65\) 17.5746 − 8.02607i 2.17987 − 0.995511i
\(66\) 1.17009 − 9.70592i 0.144028 − 1.19472i
\(67\) −10.2383 − 6.57976i −1.25081 − 0.803846i −0.263810 − 0.964575i \(-0.584979\pi\)
−0.986999 + 0.160729i \(0.948615\pi\)
\(68\) −0.495852 + 0.854984i −0.0601309 + 0.103682i
\(69\) −7.51506 − 2.74429i −0.904707 − 0.330373i
\(70\) −1.02571 − 0.275911i −0.122596 − 0.0329777i
\(71\) 2.54626 − 3.96206i 0.302186 − 0.470210i −0.656640 − 0.754204i \(-0.728023\pi\)
0.958826 + 0.283993i \(0.0916594\pi\)
\(72\) −0.490442 − 0.369385i −0.0577992 − 0.0435325i
\(73\) −2.85291 − 6.24699i −0.333907 − 0.731155i 0.665983 − 0.745967i \(-0.268012\pi\)
−0.999890 + 0.0148119i \(0.995285\pi\)
\(74\) 5.71855 + 3.49330i 0.664768 + 0.406088i
\(75\) −9.10341 − 4.15739i −1.05117 − 0.480054i
\(76\) 2.77218 − 0.953409i 0.317991 − 0.109363i
\(77\) 0.133554 − 0.928886i 0.0152198 − 0.105856i
\(78\) −13.7402 + 0.313665i −1.55578 + 0.0355156i
\(79\) −1.34769 − 1.55532i −0.151627 − 0.174987i 0.674854 − 0.737951i \(-0.264206\pi\)
−0.826482 + 0.562964i \(0.809661\pi\)
\(80\) −6.12479 + 11.7675i −0.684772 + 1.31564i
\(81\) −6.98379 + 4.48821i −0.775977 + 0.498690i
\(82\) 3.70385 − 1.79476i 0.409022 − 0.198198i
\(83\) 6.97591 − 2.04831i 0.765706 − 0.224832i 0.124520 − 0.992217i \(-0.460261\pi\)
0.641186 + 0.767386i \(0.278443\pi\)
\(84\) 0.593015 + 0.468228i 0.0647032 + 0.0510878i
\(85\) −1.07329 + 1.23864i −0.116414 + 0.134349i
\(86\) 3.89531 + 3.22264i 0.420042 + 0.347506i
\(87\) −0.338292 + 1.15212i −0.0362687 + 0.123520i
\(88\) −10.9696 − 4.12805i −1.16936 − 0.440052i
\(89\) 14.1914 − 2.04041i 1.50428 − 0.216283i 0.659626 − 0.751594i \(-0.270715\pi\)
0.844655 + 0.535311i \(0.179806\pi\)
\(90\) −0.684127 − 0.754042i −0.0721133 − 0.0794830i
\(91\) −1.31930 −0.138300
\(92\) −5.33704 + 7.96969i −0.556425 + 0.830898i
\(93\) 15.5656 1.61408
\(94\) 5.72245 + 6.30725i 0.590225 + 0.650544i
\(95\) 4.81175 − 0.691826i 0.493675 − 0.0709798i
\(96\) 7.30614 − 5.97276i 0.745680 − 0.609592i
\(97\) −0.265360 + 0.903735i −0.0269433 + 0.0917604i −0.971867 − 0.235530i \(-0.924317\pi\)
0.944924 + 0.327290i \(0.106136\pi\)
\(98\) −7.57166 − 6.26412i −0.764853 − 0.632772i
\(99\) 0.589070 − 0.679823i 0.0592038 − 0.0683248i
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 92.2.h.a.67.3 yes 100
3.2 odd 2 828.2.u.a.343.8 100
4.3 odd 2 inner 92.2.h.a.67.7 yes 100
12.11 even 2 828.2.u.a.343.4 100
23.11 odd 22 inner 92.2.h.a.11.7 yes 100
69.11 even 22 828.2.u.a.379.4 100
92.11 even 22 inner 92.2.h.a.11.3 ✓ 100
276.11 odd 22 828.2.u.a.379.8 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
92.2.h.a.11.3 ✓ 100 92.11 even 22 inner
92.2.h.a.11.7 yes 100 23.11 odd 22 inner
92.2.h.a.67.3 yes 100 1.1 even 1 trivial
92.2.h.a.67.7 yes 100 4.3 odd 2 inner
828.2.u.a.343.4 100 12.11 even 2
828.2.u.a.343.8 100 3.2 odd 2
828.2.u.a.379.4 100 69.11 even 22
828.2.u.a.379.8 100 276.11 odd 22