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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(2,95)] 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("95.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.r (of order \(36\), degree \(12\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 2.12
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.12

$q$-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+(-0.737954 - 1.05391i) q^{2} +(0.528497 + 6.04075i) q^{3} +(2.17002 - 5.96207i) q^{4} +(-10.9602 + 2.20776i) q^{5} +(5.97638 - 5.01478i) q^{6} +(-3.75768 + 14.0239i) q^{7} +(-17.8268 + 4.77668i) q^{8} +(-9.62148 + 1.69653i) q^{9} +(10.4149 + 9.92180i) q^{10} +(-1.83756 + 3.18276i) q^{11} +(37.1622 + 9.95758i) q^{12} +(-86.2862 - 7.54907i) q^{13} +(17.5528 - 6.38871i) q^{14} +(-19.1290 - 65.0409i) q^{15} +(-20.6931 - 17.3635i) q^{16} +(-90.3792 + 63.2842i) q^{17} +(8.88820 + 8.88820i) q^{18} +(76.2640 + 32.2924i) q^{19} +(-10.6209 + 70.1363i) q^{20} +(-86.7004 - 15.2876i) q^{21} +(4.71037 - 0.412104i) q^{22} +(-37.7162 + 17.5873i) q^{23} +(-38.2761 - 105.163i) q^{24} +(115.252 - 48.3950i) q^{25} +(55.7193 + 96.5086i) q^{26} +(27.0415 + 100.920i) q^{27} +(75.4570 + 52.8355i) q^{28} +(-5.37301 - 30.4719i) q^{29} +(-54.4308 + 68.1574i) q^{30} +(-32.4028 + 18.7078i) q^{31} +(-15.8972 + 181.706i) q^{32} +(-20.1974 - 9.41819i) q^{33} +(133.391 + 48.5505i) q^{34} +(10.2235 - 162.000i) q^{35} +(-10.7640 + 61.0455i) q^{36} +(230.298 - 230.298i) q^{37} +(-22.2462 - 104.206i) q^{38} -525.223i q^{39} +(184.839 - 91.7107i) q^{40} +(-145.724 + 173.667i) q^{41} +(47.8692 + 102.656i) q^{42} +(80.6277 - 172.907i) q^{43} +(14.9883 + 17.8623i) q^{44} +(101.708 - 39.8362i) q^{45} +(46.3683 + 26.7707i) q^{46} +(266.747 - 380.955i) q^{47} +(93.9525 - 134.178i) q^{48} +(114.498 + 66.1057i) q^{49} +(-136.054 - 85.7512i) q^{50} +(-430.049 - 512.512i) q^{51} +(-232.251 + 498.063i) q^{52} +(138.972 + 298.026i) q^{53} +(86.4051 - 102.974i) q^{54} +(13.1133 - 38.9405i) q^{55} -267.950i q^{56} +(-154.765 + 477.758i) q^{57} +(-28.1495 + 28.1495i) q^{58} +(-96.8821 + 549.446i) q^{59} +(-429.289 - 27.0916i) q^{60} +(-323.806 - 117.856i) q^{61} +(43.6280 + 20.3441i) q^{62} +(12.3626 - 141.305i) q^{63} +(16.0815 - 9.28467i) q^{64} +(962.380 - 107.761i) q^{65} +(4.97883 + 28.2363i) q^{66} +(658.325 + 460.964i) q^{67} +(181.180 + 676.175i) q^{68} +(-126.174 - 218.539i) q^{69} +(-178.278 + 108.774i) q^{70} +(79.8965 + 219.514i) q^{71} +(163.416 - 76.2024i) q^{72} +(-579.301 + 50.6822i) q^{73} +(-412.663 - 72.7635i) q^{74} +(353.252 + 670.629i) q^{75} +(358.024 - 384.616i) q^{76} +(-37.7295 - 37.7295i) q^{77} +(-553.536 + 387.590i) q^{78} +(-270.712 - 227.155i) q^{79} +(265.135 + 144.622i) q^{80} +(-843.220 + 306.907i) q^{81} +(290.567 + 25.4213i) q^{82} +(-847.946 - 227.206i) q^{83} +(-279.287 + 483.740i) q^{84} +(850.856 - 893.143i) q^{85} +(-241.727 + 42.6230i) q^{86} +(181.233 - 48.5613i) q^{87} +(17.5549 - 65.5158i) q^{88} +(-778.784 + 653.477i) q^{89} +(-117.039 - 77.7933i) q^{90} +(430.103 - 1181.70i) q^{91} +(23.0123 + 263.031i) q^{92} +(-130.134 - 185.850i) q^{93} -598.338 q^{94} +(-907.162 - 185.558i) q^{95} -1106.04 q^{96} +(-637.494 - 910.435i) q^{97} +(-14.8253 - 169.454i) q^{98} +(12.2805 - 33.7403i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 336 q - 12 q^{2} - 12 q^{3} - 12 q^{5} + 12 q^{6} + 6 q^{7} - 18 q^{8} - 12 q^{10} - 12 q^{11} - 18 q^{12} - 12 q^{13} - 264 q^{15} - 996 q^{16} + 132 q^{17} + 1116 q^{20} - 600 q^{21} + 468 q^{22} + 564 q^{23}+ \cdots + 1290 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.737954 − 1.05391i −0.260906 − 0.372613i 0.667231 − 0.744851i \(-0.267479\pi\)
−0.928137 + 0.372238i \(0.878590\pi\)
\(3\) 0.528497 + 6.04075i 0.101709 + 1.16254i 0.859884 + 0.510490i \(0.170536\pi\)
−0.758174 + 0.652052i \(0.773908\pi\)
\(4\) 2.17002 − 5.96207i 0.271252 − 0.745259i
\(5\) −10.9602 + 2.20776i −0.980309 + 0.197468i
\(6\) 5.97638 − 5.01478i 0.406641 − 0.341212i
\(7\) −3.75768 + 14.0239i −0.202896 + 0.757217i 0.787185 + 0.616717i \(0.211538\pi\)
−0.990081 + 0.140500i \(0.955129\pi\)
\(8\) −17.8268 + 4.77668i −0.787841 + 0.211101i
\(9\) −9.62148 + 1.69653i −0.356351 + 0.0628343i
\(10\) 10.4149 + 9.92180i 0.329348 + 0.313755i
\(11\) −1.83756 + 3.18276i −0.0503679 + 0.0872397i −0.890110 − 0.455746i \(-0.849373\pi\)
0.839742 + 0.542985i \(0.182706\pi\)
\(12\) 37.1622 + 9.95758i 0.893983 + 0.239542i
\(13\) −86.2862 − 7.54907i −1.84088 − 0.161057i −0.887106 − 0.461567i \(-0.847288\pi\)
−0.953779 + 0.300510i \(0.902843\pi\)
\(14\) 17.5528 − 6.38871i 0.335085 − 0.121961i
\(15\) −19.1290 − 65.0409i −0.329272 − 1.11957i
\(16\) −20.6931 − 17.3635i −0.323329 − 0.271305i
\(17\) −90.3792 + 63.2842i −1.28942 + 0.902863i −0.998546 − 0.0538980i \(-0.982835\pi\)
−0.290875 + 0.956761i \(0.593947\pi\)
\(18\) 8.88820 + 8.88820i 0.116387 + 0.116387i
\(19\) 76.2640 + 32.2924i 0.920851 + 0.389915i
\(20\) −10.6209 + 70.1363i −0.118746 + 0.784148i
\(21\) −86.7004 − 15.2876i −0.900933 − 0.158859i
\(22\) 4.71037 − 0.412104i 0.0456479 − 0.00399367i
\(23\) −37.7162 + 17.5873i −0.341929 + 0.159444i −0.585995 − 0.810314i \(-0.699296\pi\)
0.244066 + 0.969759i \(0.421519\pi\)
\(24\) −38.2761 − 105.163i −0.325545 − 0.894427i
\(25\) 115.252 − 48.3950i 0.922012 − 0.387160i
\(26\) 55.7193 + 96.5086i 0.420286 + 0.727957i
\(27\) 27.0415 + 100.920i 0.192746 + 0.719336i
\(28\) 75.4570 + 52.8355i 0.509287 + 0.356606i
\(29\) −5.37301 − 30.4719i −0.0344049 − 0.195120i 0.962761 − 0.270354i \(-0.0871408\pi\)
−0.997166 + 0.0752339i \(0.976030\pi\)
\(30\) −54.4308 + 68.1574i −0.331255 + 0.414793i
\(31\) −32.4028 + 18.7078i −0.187733 + 0.108388i −0.590921 − 0.806730i \(-0.701235\pi\)
0.403188 + 0.915117i \(0.367902\pi\)
\(32\) −15.8972 + 181.706i −0.0878203 + 1.00379i
\(33\) −20.1974 − 9.41819i −0.106543 − 0.0496817i
\(34\) 133.391 + 48.5505i 0.672836 + 0.244892i
\(35\) 10.2235 − 162.000i 0.0493740 − 0.782372i
\(36\) −10.7640 + 61.0455i −0.0498332 + 0.282618i
\(37\) 230.298 − 230.298i 1.02326 − 1.02326i 0.0235419 − 0.999723i \(-0.492506\pi\)
0.999723 − 0.0235419i \(-0.00749431\pi\)
\(38\) −22.2462 − 104.206i −0.0949685 − 0.444852i
\(39\) − 525.223i − 2.15649i
\(40\) 184.839 − 91.7107i 0.730642 − 0.362518i
\(41\) −145.724 + 173.667i −0.555080 + 0.661519i −0.968498 − 0.249023i \(-0.919891\pi\)
0.413417 + 0.910542i \(0.364335\pi\)
\(42\) 47.8692 + 102.656i 0.175866 + 0.377146i
\(43\) 80.6277 − 172.907i 0.285944 − 0.613209i −0.709898 − 0.704305i \(-0.751259\pi\)
0.995842 + 0.0910951i \(0.0290367\pi\)
\(44\) 14.9883 + 17.8623i 0.0513538 + 0.0612011i
\(45\) 101.708 − 39.8362i 0.336927 − 0.131965i
\(46\) 46.3683 + 26.7707i 0.148622 + 0.0858071i
\(47\) 266.747 − 380.955i 0.827853 − 1.18230i −0.153130 − 0.988206i \(-0.548935\pi\)
0.980984 − 0.194091i \(-0.0621757\pi\)
\(48\) 93.9525 − 134.178i 0.282518 − 0.403478i
\(49\) 114.498 + 66.1057i 0.333815 + 0.192728i
\(50\) −136.054 − 85.7512i −0.384820 − 0.242541i
\(51\) −430.049 − 512.512i −1.18076 − 1.40718i
\(52\) −232.251 + 498.063i −0.619372 + 1.32825i
\(53\) 138.972 + 298.026i 0.360174 + 0.772396i 0.999993 + 0.00386962i \(0.00123174\pi\)
−0.639818 + 0.768526i \(0.720990\pi\)
\(54\) 86.4051 − 102.974i 0.217745 − 0.259499i
\(55\) 13.1133 − 38.9405i 0.0321490 − 0.0954680i
\(56\) − 267.950i − 0.639398i
\(57\) −154.765 + 477.758i −0.359633 + 1.11019i
\(58\) −28.1495 + 28.1495i −0.0637278 + 0.0637278i
\(59\) −96.8821 + 549.446i −0.213779 + 1.21240i 0.669233 + 0.743053i \(0.266623\pi\)
−0.883012 + 0.469350i \(0.844488\pi\)
\(60\) −429.289 − 27.0916i −0.923682 − 0.0582918i
\(61\) −323.806 − 117.856i −0.679658 − 0.247375i −0.0209573 − 0.999780i \(-0.506671\pi\)
−0.658701 + 0.752405i \(0.728894\pi\)
\(62\) 43.6280 + 20.3441i 0.0893672 + 0.0416726i
\(63\) 12.3626 − 141.305i 0.0247229 − 0.282584i
\(64\) 16.0815 − 9.28467i 0.0314092 − 0.0181341i
\(65\) 962.380 − 107.761i 1.83644 − 0.205631i
\(66\) 4.97883 + 28.2363i 0.00928563 + 0.0526614i
\(67\) 658.325 + 460.964i 1.20041 + 0.840534i 0.990544 − 0.137197i \(-0.0438095\pi\)
0.209863 + 0.977731i \(0.432698\pi\)
\(68\) 181.180 + 676.175i 0.323108 + 1.20586i
\(69\) −126.174 − 218.539i −0.220138 − 0.381290i
\(70\) −178.278 + 108.774i −0.304404 + 0.185728i
\(71\) 79.8965 + 219.514i 0.133549 + 0.366923i 0.988384 − 0.151977i \(-0.0485640\pi\)
−0.854835 + 0.518900i \(0.826342\pi\)
\(72\) 163.416 − 76.2024i 0.267484 − 0.124730i
\(73\) −579.301 + 50.6822i −0.928795 + 0.0812590i −0.541503 − 0.840699i \(-0.682145\pi\)
−0.387291 + 0.921958i \(0.626589\pi\)
\(74\) −412.663 − 72.7635i −0.648257 − 0.114305i
\(75\) 353.252 + 670.629i 0.543867 + 1.03250i
\(76\) 358.024 − 384.616i 0.540370 − 0.580507i
\(77\) −37.7295 − 37.7295i −0.0558400 − 0.0558400i
\(78\) −553.536 + 387.590i −0.803534 + 0.562640i
\(79\) −270.712 − 227.155i −0.385538 − 0.323505i 0.429334 − 0.903146i \(-0.358748\pi\)
−0.814872 + 0.579641i \(0.803193\pi\)
\(80\) 265.135 + 144.622i 0.370537 + 0.202116i
\(81\) −843.220 + 306.907i −1.15668 + 0.420997i
\(82\) 290.567 + 25.4213i 0.391314 + 0.0342355i
\(83\) −847.946 − 227.206i −1.12138 − 0.300472i −0.349935 − 0.936774i \(-0.613796\pi\)
−0.771440 + 0.636302i \(0.780463\pi\)
\(84\) −279.287 + 483.740i −0.362771 + 0.628337i
\(85\) 850.856 − 893.143i 1.08575 − 1.13971i
\(86\) −241.727 + 42.6230i −0.303094 + 0.0534437i
\(87\) 181.233 − 48.5613i 0.223336 − 0.0598427i
\(88\) 17.5549 − 65.5158i 0.0212654 − 0.0793637i
\(89\) −778.784 + 653.477i −0.927539 + 0.778297i −0.975374 − 0.220558i \(-0.929212\pi\)
0.0478352 + 0.998855i \(0.484768\pi\)
\(90\) −117.039 − 77.7933i −0.137078 − 0.0911126i
\(91\) 430.103 − 1181.70i 0.495462 − 1.36127i
\(92\) 23.0123 + 263.031i 0.0260782 + 0.298075i
\(93\) −130.134 − 185.850i −0.145099 − 0.207223i
\(94\) −598.338 −0.656531
\(95\) −907.162 − 185.558i −0.979715 − 0.200398i
\(96\) −1106.04 −1.17588
\(97\) −637.494 − 910.435i −0.667295 − 0.952997i −0.999951 − 0.00994813i \(-0.996833\pi\)
0.332655 − 0.943049i \(-0.392056\pi\)
\(98\) −14.8253 − 169.454i −0.0152814 − 0.174667i
\(99\) 12.2805 − 33.7403i 0.0124670 − 0.0342528i
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 95.4.r.a.2.12 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.12 yes 336
19.10 odd 18 inner 95.4.r.a.67.12 yes 336
95.48 even 36 inner 95.4.r.a.48.12 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.12 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.12 yes 336 95.48 even 36 inner
95.4.r.a.67.12 yes 336 19.10 odd 18 inner
95.4.r.a.78.12 yes 336 5.3 odd 4 inner