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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 2.20
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.20

$q$-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+(1.07308 + 1.53251i) q^{2} +(-0.00470693 - 0.0538005i) q^{3} +(1.53906 - 4.22853i) q^{4} +(-11.1517 - 0.799155i) q^{5} +(0.0773991 - 0.0649456i) q^{6} +(5.96461 - 22.2602i) q^{7} +(22.5887 - 6.05261i) q^{8} +(26.5869 - 4.68799i) q^{9} +(-10.7420 - 17.9478i) q^{10} +(6.95091 - 12.0393i) q^{11} +(-0.234741 - 0.0628987i) q^{12} +(-0.984331 - 0.0861178i) q^{13} +(40.5146 - 14.7461i) q^{14} +(0.00949559 + 0.603731i) q^{15} +(5.93808 + 4.98264i) q^{16} +(-29.9819 + 20.9936i) q^{17} +(35.7143 + 35.7143i) q^{18} +(-79.7732 - 22.2539i) q^{19} +(-20.5424 + 45.9255i) q^{20} +(-1.22569 - 0.216122i) q^{21} +(25.9093 - 2.26677i) q^{22} +(188.209 - 87.7633i) q^{23} +(-0.431957 - 1.18679i) q^{24} +(123.723 + 17.8239i) q^{25} +(-0.924287 - 1.60091i) q^{26} +(-0.754760 - 2.81680i) q^{27} +(-84.9482 - 59.4814i) q^{28} +(24.5488 + 139.223i) q^{29} +(-0.915036 + 0.662402i) q^{30} +(-249.098 + 143.817i) q^{31} +(15.0415 - 171.925i) q^{32} +(-0.680439 - 0.317294i) q^{33} +(-64.3459 - 23.4200i) q^{34} +(-84.3052 + 243.474i) q^{35} +(21.0955 - 119.639i) q^{36} +(-97.1259 + 97.1259i) q^{37} +(-51.4985 - 146.134i) q^{38} +0.0533629i q^{39} +(-256.740 + 49.4453i) q^{40} +(-0.359759 + 0.428744i) q^{41} +(-0.984048 - 2.11030i) q^{42} +(16.1811 - 34.7005i) q^{43} +(-40.2108 - 47.9213i) q^{44} +(-300.237 + 31.0322i) q^{45} +(336.461 + 194.256i) q^{46} +(120.076 - 171.486i) q^{47} +(0.240118 - 0.342924i) q^{48} +(-162.895 - 94.0476i) q^{49} +(105.449 + 208.733i) q^{50} +(1.27059 + 1.51423i) q^{51} +(-1.87909 + 4.02973i) q^{52} +(195.037 + 418.257i) q^{53} +(3.50687 - 4.17933i) q^{54} +(-87.1360 + 128.705i) q^{55} -538.931i q^{56} +(-0.821783 + 4.39659i) q^{57} +(-187.019 + 187.019i) q^{58} +(-102.303 + 580.192i) q^{59} +(2.56751 + 0.889025i) q^{60} +(601.536 + 218.941i) q^{61} +(-487.703 - 227.419i) q^{62} +(54.2249 - 619.794i) q^{63} +(333.323 - 192.444i) q^{64} +(10.9082 + 1.74700i) q^{65} +(-0.243907 - 1.38326i) q^{66} +(-503.491 - 352.548i) q^{67} +(42.6280 + 159.090i) q^{68} +(-5.60760 - 9.71264i) q^{69} +(-463.593 + 132.067i) q^{70} +(172.689 + 474.458i) q^{71} +(572.189 - 266.816i) q^{72} +(370.409 - 32.4066i) q^{73} +(-253.070 - 44.6231i) q^{74} +(0.376582 - 6.74024i) q^{75} +(-216.877 + 303.073i) q^{76} +(-226.539 - 226.539i) q^{77} +(-0.0817793 + 0.0572625i) q^{78} +(275.324 + 231.024i) q^{79} +(-62.2380 - 60.3105i) q^{80} +(684.814 - 249.252i) q^{81} +(-1.04311 - 0.0912600i) q^{82} +(1075.90 + 288.287i) q^{83} +(-2.80028 + 4.85023i) q^{84} +(351.128 - 210.155i) q^{85} +(70.5426 - 12.4386i) q^{86} +(7.37473 - 1.97605i) q^{87} +(84.1423 - 314.023i) q^{88} +(-123.514 + 103.641i) q^{89} +(-369.735 - 426.818i) q^{90} +(-7.78816 + 21.3978i) q^{91} +(-81.4449 - 930.920i) q^{92} +(8.90990 + 12.7247i) q^{93} +391.655 q^{94} +(871.826 + 311.921i) q^{95} -9.32046 q^{96} +(-862.977 - 1232.46i) q^{97} +(-30.6700 - 350.559i) q^{98} +(128.363 - 352.675i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.07308 + 1.53251i 0.379390 + 0.541825i 0.962752 − 0.270387i \(-0.0871517\pi\)
−0.583361 + 0.812213i \(0.698263\pi\)
\(3\) −0.00470693 − 0.0538005i −0.000905850 − 0.0103539i 0.995728 − 0.0923315i \(-0.0294320\pi\)
−0.996634 + 0.0819776i \(0.973876\pi\)
\(4\) 1.53906 − 4.22853i 0.192382 − 0.528566i
\(5\) −11.1517 − 0.799155i −0.997442 − 0.0714786i
\(6\) 0.0773991 − 0.0649456i 0.00526634 − 0.00441899i
\(7\) 5.96461 − 22.2602i 0.322059 − 1.20194i −0.595177 − 0.803595i \(-0.702918\pi\)
0.917236 − 0.398345i \(-0.130415\pi\)
\(8\) 22.5887 − 6.05261i 0.998287 − 0.267490i
\(9\) 26.5869 − 4.68799i 0.984701 − 0.173629i
\(10\) −10.7420 − 17.9478i −0.339691 − 0.567558i
\(11\) 6.95091 − 12.0393i 0.190525 − 0.329999i −0.754899 − 0.655841i \(-0.772314\pi\)
0.945424 + 0.325841i \(0.105648\pi\)
\(12\) −0.234741 − 0.0628987i −0.00564699 − 0.00151311i
\(13\) −0.984331 − 0.0861178i −0.0210003 − 0.00183729i 0.0766507 − 0.997058i \(-0.475577\pi\)
−0.0976511 + 0.995221i \(0.531133\pi\)
\(14\) 40.5146 − 14.7461i 0.773428 − 0.281505i
\(15\) 0.00949559 + 0.603731i 0.000163450 + 0.0103922i
\(16\) 5.93808 + 4.98264i 0.0927824 + 0.0778537i
\(17\) −29.9819 + 20.9936i −0.427746 + 0.299511i −0.767555 − 0.640983i \(-0.778527\pi\)
0.339808 + 0.940495i \(0.389638\pi\)
\(18\) 35.7143 + 35.7143i 0.467663 + 0.467663i
\(19\) −79.7732 − 22.2539i −0.963223 − 0.268705i
\(20\) −20.5424 + 45.9255i −0.229671 + 0.513463i
\(21\) −1.22569 − 0.216122i −0.0127365 − 0.00224579i
\(22\) 25.9093 − 2.26677i 0.251085 − 0.0219671i
\(23\) 188.209 − 87.7633i 1.70627 − 0.795648i 0.710510 − 0.703687i \(-0.248464\pi\)
0.995763 − 0.0919614i \(-0.0293136\pi\)
\(24\) −0.431957 − 1.18679i −0.00367387 − 0.0100939i
\(25\) 123.723 + 17.8239i 0.989782 + 0.142591i
\(26\) −0.924287 − 1.60091i −0.00697183 − 0.0120756i
\(27\) −0.754760 − 2.81680i −0.00537976 − 0.0200775i
\(28\) −84.9482 − 59.4814i −0.573346 − 0.401461i
\(29\) 24.5488 + 139.223i 0.157193 + 0.891487i 0.956753 + 0.290901i \(0.0939549\pi\)
−0.799560 + 0.600586i \(0.794934\pi\)
\(30\) −0.915036 + 0.662402i −0.00556873 + 0.00403125i
\(31\) −249.098 + 143.817i −1.44320 + 0.833234i −0.998062 − 0.0622264i \(-0.980180\pi\)
−0.445141 + 0.895460i \(0.646847\pi\)
\(32\) 15.0415 − 171.925i 0.0830933 − 0.949761i
\(33\) −0.680439 − 0.317294i −0.00358937 − 0.00167375i
\(34\) −64.3459 − 23.4200i −0.324566 − 0.118132i
\(35\) −84.3052 + 243.474i −0.407148 + 1.17585i
\(36\) 21.0955 − 119.639i 0.0976645 − 0.553883i
\(37\) −97.1259 + 97.1259i −0.431551 + 0.431551i −0.889156 − 0.457605i \(-0.848708\pi\)
0.457605 + 0.889156i \(0.348708\pi\)
\(38\) −51.4985 − 146.134i −0.219846 − 0.623843i
\(39\) 0.0533629i 0.000219100i
\(40\) −256.740 + 49.4453i −1.01485 + 0.195450i
\(41\) −0.359759 + 0.428744i −0.00137037 + 0.00163314i −0.766729 − 0.641971i \(-0.778117\pi\)
0.765359 + 0.643604i \(0.222562\pi\)
\(42\) −0.984048 − 2.11030i −0.00361528 − 0.00775300i
\(43\) 16.1811 − 34.7005i 0.0573860 − 0.123065i −0.875524 − 0.483175i \(-0.839483\pi\)
0.932910 + 0.360111i \(0.117261\pi\)
\(44\) −40.2108 − 47.9213i −0.137773 − 0.164191i
\(45\) −300.237 + 31.0322i −0.994593 + 0.102800i
\(46\) 336.461 + 194.256i 1.07845 + 0.622641i
\(47\) 120.076 − 171.486i 0.372657 − 0.532209i −0.588396 − 0.808573i \(-0.700241\pi\)
0.961053 + 0.276364i \(0.0891294\pi\)
\(48\) 0.240118 − 0.342924i 0.000722043 − 0.00103118i
\(49\) −162.895 − 94.0476i −0.474913 − 0.274191i
\(50\) 105.449 + 208.733i 0.298254 + 0.590387i
\(51\) 1.27059 + 1.51423i 0.00348859 + 0.00415754i
\(52\) −1.87909 + 4.02973i −0.00501122 + 0.0107466i
\(53\) 195.037 + 418.257i 0.505478 + 1.08400i 0.979120 + 0.203284i \(0.0651613\pi\)
−0.473642 + 0.880718i \(0.657061\pi\)
\(54\) 3.50687 − 4.17933i 0.00883749 − 0.0105321i
\(55\) −87.1360 + 128.705i −0.213626 + 0.315537i
\(56\) − 538.931i − 1.28603i
\(57\) −0.821783 + 4.39659i −0.00190961 + 0.0102165i
\(58\) −187.019 + 187.019i −0.423393 + 0.423393i
\(59\) −102.303 + 580.192i −0.225742 + 1.28025i 0.635520 + 0.772085i \(0.280786\pi\)
−0.861262 + 0.508162i \(0.830325\pi\)
\(60\) 2.56751 + 0.889025i 0.00552440 + 0.00191288i
\(61\) 601.536 + 218.941i 1.26260 + 0.459550i 0.884642 − 0.466271i \(-0.154403\pi\)
0.377962 + 0.925821i \(0.376625\pi\)
\(62\) −487.703 − 227.419i −0.999005 − 0.465844i
\(63\) 54.2249 − 619.794i 0.108440 − 1.23947i
\(64\) 333.323 − 192.444i 0.651022 − 0.375867i
\(65\) 10.9082 + 1.74700i 0.0208153 + 0.00333367i
\(66\) −0.243907 − 1.38326i −0.000454891 − 0.00257982i
\(67\) −503.491 − 352.548i −0.918077 − 0.642845i 0.0160168 − 0.999872i \(-0.494901\pi\)
−0.934094 + 0.357027i \(0.883790\pi\)
\(68\) 42.6280 + 159.090i 0.0760206 + 0.283713i
\(69\) −5.60760 − 9.71264i −0.00978370 − 0.0169459i
\(70\) −463.593 + 132.067i −0.791571 + 0.225501i
\(71\) 172.689 + 474.458i 0.288653 + 0.793068i 0.996256 + 0.0864573i \(0.0275546\pi\)
−0.707602 + 0.706611i \(0.750223\pi\)
\(72\) 572.189 − 266.816i 0.936570 − 0.436730i
\(73\) 370.409 − 32.4066i 0.593878 − 0.0519576i 0.213745 − 0.976889i \(-0.431434\pi\)
0.380133 + 0.924932i \(0.375878\pi\)
\(74\) −253.070 − 44.6231i −0.397552 − 0.0700991i
\(75\) 0.376582 − 6.74024i 0.000579786 − 0.0103773i
\(76\) −216.877 + 303.073i −0.327335 + 0.457433i
\(77\) −226.539 − 226.539i −0.335279 − 0.335279i
\(78\) −0.0817793 + 0.0572625i −0.000118714 + 8.31244e-5i
\(79\) 275.324 + 231.024i 0.392106 + 0.329016i 0.817433 − 0.576024i \(-0.195397\pi\)
−0.425327 + 0.905040i \(0.639841\pi\)
\(80\) −62.2380 − 60.3105i −0.0869802 − 0.0842865i
\(81\) 684.814 − 249.252i 0.939388 − 0.341909i
\(82\) −1.04311 − 0.0912600i −0.00140478 − 0.000122902i
\(83\) 1075.90 + 288.287i 1.42284 + 0.381248i 0.886489 − 0.462749i \(-0.153137\pi\)
0.536348 + 0.843997i \(0.319804\pi\)
\(84\) −2.80028 + 4.85023i −0.00363733 + 0.00630004i
\(85\) 351.128 − 210.155i 0.448061 − 0.268170i
\(86\) 70.5426 − 12.4386i 0.0884512 − 0.0155963i
\(87\) 7.37473 − 1.97605i 0.00908798 − 0.00243512i
\(88\) 84.1423 − 314.023i 0.101927 − 0.380398i
\(89\) −123.514 + 103.641i −0.147106 + 0.123437i −0.713371 − 0.700787i \(-0.752832\pi\)
0.566265 + 0.824224i \(0.308388\pi\)
\(90\) −369.735 − 426.818i −0.433039 − 0.499895i
\(91\) −7.78816 + 21.3978i −0.00897166 + 0.0246494i
\(92\) −81.4449 − 930.920i −0.0922959 − 1.05495i
\(93\) 8.90990 + 12.7247i 0.00993456 + 0.0141880i
\(94\) 391.655 0.429747
\(95\) 871.826 + 311.921i 0.941552 + 0.336867i
\(96\) −9.32046 −0.00990901
\(97\) −862.977 − 1232.46i −0.903320 − 1.29007i −0.956573 − 0.291493i \(-0.905848\pi\)
0.0532531 − 0.998581i \(-0.483041\pi\)
\(98\) −30.6700 − 350.559i −0.0316136 − 0.361345i
\(99\) 128.363 − 352.675i 0.130313 − 0.358032i
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.2.20 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.20 yes 336
19.10 odd 18 inner 95.4.r.a.67.20 yes 336
95.48 even 36 inner 95.4.r.a.48.20 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.20 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.20 yes 336 95.48 even 36 inner
95.4.r.a.67.20 yes 336 19.10 odd 18 inner
95.4.r.a.78.20 yes 336 5.3 odd 4 inner