Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(4,95)] 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("95.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(18)) 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.p (of order \(18\), degree \(6\), 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: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.20
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.20

$q$-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+(0.854996 - 2.34908i) q^{2} +(-9.69690 + 1.70982i) q^{3} +(1.34119 + 1.12539i) q^{4} +(10.8371 + 2.74907i) q^{5} +(-4.27429 + 24.2407i) q^{6} +(-12.3819 - 7.14868i) q^{7} +(21.1097 - 12.1877i) q^{8} +(65.7346 - 23.9254i) q^{9} +(15.7235 - 23.1068i) q^{10} +(-20.5820 - 35.6491i) q^{11} +(-14.9296 - 8.61960i) q^{12} +(32.8276 + 5.78839i) q^{13} +(-27.3793 + 22.9739i) q^{14} +(-109.787 - 8.12792i) q^{15} +(-8.14902 - 46.2154i) q^{16} +(21.7055 - 59.6353i) q^{17} -174.872i q^{18} +(74.0181 + 37.1526i) q^{19} +(11.4408 + 15.8830i) q^{20} +(132.289 + 48.1492i) q^{21} +(-101.340 + 17.8690i) q^{22} +(119.659 - 142.605i) q^{23} +(-183.860 + 154.277i) q^{24} +(109.885 + 59.5839i) q^{25} +(41.6649 - 72.1657i) q^{26} +(-366.276 + 211.470i) q^{27} +(-8.56137 - 23.5222i) q^{28} +(-35.0814 + 12.7686i) q^{29} +(-112.960 + 250.948i) q^{30} +(-52.6733 + 91.2329i) q^{31} +(76.5099 + 13.4908i) q^{32} +(260.536 + 310.494i) q^{33} +(-121.530 - 101.976i) q^{34} +(-114.531 - 111.510i) q^{35} +(115.088 + 41.8886i) q^{36} +42.7376i q^{37} +(150.560 - 142.109i) q^{38} -328.223 q^{39} +(262.273 - 74.0473i) q^{40} +(-24.8730 - 141.062i) q^{41} +(226.213 - 269.590i) q^{42} +(-138.213 - 164.716i) q^{43} +(12.5148 - 70.9750i) q^{44} +(778.145 - 78.5732i) q^{45} +(-232.681 - 403.016i) q^{46} +(130.027 + 357.246i) q^{47} +(158.040 + 434.212i) q^{48} +(-69.2928 - 120.019i) q^{49} +(233.919 - 207.185i) q^{50} +(-108.510 + 615.390i) q^{51} +(37.5138 + 44.7072i) q^{52} +(-350.669 + 417.912i) q^{53} +(183.595 + 1041.22i) q^{54} +(-125.047 - 442.914i) q^{55} -348.504 q^{56} +(-781.271 - 233.707i) q^{57} +93.3263i q^{58} +(125.160 + 45.5545i) q^{59} +(-138.097 - 134.454i) q^{60} +(-269.946 - 226.512i) q^{61} +(169.278 + 201.738i) q^{62} +(-984.953 - 173.674i) q^{63} +(284.820 - 493.322i) q^{64} +(339.843 + 152.975i) q^{65} +(952.133 - 346.548i) q^{66} +(-204.138 - 560.865i) q^{67} +(96.2242 - 55.5550i) q^{68} +(-916.496 + 1587.42i) q^{69} +(-359.869 + 173.703i) q^{70} +(91.1304 - 76.4675i) q^{71} +(1096.04 - 1306.22i) q^{72} +(-212.875 + 37.5357i) q^{73} +(100.394 + 36.5405i) q^{74} +(-1167.42 - 389.894i) q^{75} +(57.4611 + 133.128i) q^{76} +588.537i q^{77} +(-280.629 + 771.023i) q^{78} +(147.888 + 838.715i) q^{79} +(38.7376 - 523.243i) q^{80} +(1743.31 - 1462.81i) q^{81} +(-352.632 - 62.1785i) q^{82} +(795.498 + 459.281i) q^{83} +(123.238 + 213.454i) q^{84} +(399.166 - 586.603i) q^{85} +(-505.103 + 183.842i) q^{86} +(318.349 - 183.799i) q^{87} +(-868.963 - 501.696i) q^{88} +(164.097 - 930.643i) q^{89} +(480.736 - 1895.11i) q^{90} +(-365.088 - 306.345i) q^{91} +(320.972 - 56.5960i) q^{92} +(354.776 - 974.738i) q^{93} +950.373 q^{94} +(700.006 + 606.107i) q^{95} -764.975 q^{96} +(-300.768 + 826.353i) q^{97} +(-341.179 + 60.1590i) q^{98} +(-2205.87 - 1850.95i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q + 6 q^{4} - 6 q^{5} + 6 q^{6} - 12 q^{9} + 3 q^{10} - 90 q^{11} + 162 q^{14} - 72 q^{15} + 426 q^{16} - 444 q^{19} - 966 q^{20} + 114 q^{21} + 108 q^{24} + 270 q^{25} - 642 q^{26} - 60 q^{29} + 396 q^{30}+ \cdots + 22692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.854996 − 2.34908i 0.302287 − 0.830526i −0.691815 − 0.722075i \(-0.743189\pi\)
0.994102 − 0.108451i \(-0.0345891\pi\)
\(3\) −9.69690 + 1.70982i −1.86617 + 0.329056i −0.988620 − 0.150432i \(-0.951933\pi\)
−0.877548 + 0.479488i \(0.840822\pi\)
\(4\) 1.34119 + 1.12539i 0.167649 + 0.140674i
\(5\) 10.8371 + 2.74907i 0.969299 + 0.245884i
\(6\) −4.27429 + 24.2407i −0.290829 + 1.64937i
\(7\) −12.3819 − 7.14868i −0.668558 − 0.385992i 0.126972 − 0.991906i \(-0.459474\pi\)
−0.795530 + 0.605914i \(0.792808\pi\)
\(8\) 21.1097 − 12.1877i 0.932928 − 0.538626i
\(9\) 65.7346 − 23.9254i 2.43462 − 0.886127i
\(10\) 15.7235 − 23.1068i 0.497219 − 0.730700i
\(11\) −20.5820 − 35.6491i −0.564156 − 0.977147i −0.997128 − 0.0757392i \(-0.975868\pi\)
0.432972 − 0.901407i \(-0.357465\pi\)
\(12\) −14.9296 − 8.61960i −0.359150 − 0.207355i
\(13\) 32.8276 + 5.78839i 0.700365 + 0.123493i 0.512482 − 0.858698i \(-0.328726\pi\)
0.187883 + 0.982191i \(0.439837\pi\)
\(14\) −27.3793 + 22.9739i −0.522673 + 0.438575i
\(15\) −109.787 − 8.12792i −1.88979 − 0.139908i
\(16\) −8.14902 − 46.2154i −0.127328 − 0.722115i
\(17\) 21.7055 − 59.6353i 0.309668 − 0.850805i −0.683053 − 0.730368i \(-0.739348\pi\)
0.992721 − 0.120436i \(-0.0384294\pi\)
\(18\) − 174.872i − 2.28988i
\(19\) 74.0181 + 37.1526i 0.893733 + 0.448600i
\(20\) 11.4408 + 15.8830i 0.127912 + 0.177577i
\(21\) 132.289 + 48.1492i 1.37466 + 0.500334i
\(22\) −101.340 + 17.8690i −0.982082 + 0.173168i
\(23\) 119.659 − 142.605i 1.08481 − 1.29283i 0.131345 − 0.991337i \(-0.458070\pi\)
0.953468 − 0.301494i \(-0.0974852\pi\)
\(24\) −183.860 + 154.277i −1.56376 + 1.31215i
\(25\) 109.885 + 59.5839i 0.879082 + 0.476671i
\(26\) 41.6649 − 72.1657i 0.314275 − 0.544341i
\(27\) −366.276 + 211.470i −2.61074 + 1.50731i
\(28\) −8.56137 − 23.5222i −0.0577838 − 0.158760i
\(29\) −35.0814 + 12.7686i −0.224637 + 0.0817610i −0.451886 − 0.892075i \(-0.649249\pi\)
0.227250 + 0.973836i \(0.427027\pi\)
\(30\) −112.960 + 250.948i −0.687454 + 1.52722i
\(31\) −52.6733 + 91.2329i −0.305175 + 0.528578i −0.977300 − 0.211859i \(-0.932048\pi\)
0.672126 + 0.740437i \(0.265381\pi\)
\(32\) 76.5099 + 13.4908i 0.422661 + 0.0745266i
\(33\) 260.536 + 310.494i 1.37435 + 1.63788i
\(34\) −121.530 − 101.976i −0.613007 − 0.514374i
\(35\) −114.531 − 111.510i −0.553124 − 0.538530i
\(36\) 115.088 + 41.8886i 0.532815 + 0.193929i
\(37\) 42.7376i 0.189892i 0.995482 + 0.0949462i \(0.0302679\pi\)
−0.995482 + 0.0949462i \(0.969732\pi\)
\(38\) 150.560 − 142.109i 0.642737 − 0.606662i
\(39\) −328.223 −1.34763
\(40\) 262.273 − 74.0473i 1.03673 − 0.292698i
\(41\) −24.8730 − 141.062i −0.0947441 − 0.537320i −0.994825 − 0.101599i \(-0.967604\pi\)
0.900081 − 0.435722i \(-0.143507\pi\)
\(42\) 226.213 − 269.590i 0.831080 − 0.990443i
\(43\) −138.213 − 164.716i −0.490170 − 0.584162i 0.463091 − 0.886311i \(-0.346740\pi\)
−0.953261 + 0.302149i \(0.902296\pi\)
\(44\) 12.5148 − 70.9750i 0.0428791 − 0.243179i
\(45\) 778.145 − 78.5732i 2.57776 − 0.260289i
\(46\) −232.681 − 403.016i −0.745804 − 1.29177i
\(47\) 130.027 + 357.246i 0.403540 + 1.10872i 0.960525 + 0.278195i \(0.0897360\pi\)
−0.556985 + 0.830523i \(0.688042\pi\)
\(48\) 158.040 + 434.212i 0.475233 + 1.30569i
\(49\) −69.2928 − 120.019i −0.202020 − 0.349909i
\(50\) 233.919 − 207.185i 0.661622 − 0.586009i
\(51\) −108.510 + 615.390i −0.297930 + 1.68964i
\(52\) 37.5138 + 44.7072i 0.100043 + 0.119226i
\(53\) −350.669 + 417.912i −0.908833 + 1.08310i 0.0873814 + 0.996175i \(0.472150\pi\)
−0.996214 + 0.0869300i \(0.972294\pi\)
\(54\) 183.595 + 1041.22i 0.462668 + 2.62392i
\(55\) −125.047 − 442.914i −0.306571 − 1.08586i
\(56\) −348.504 −0.831622
\(57\) −781.271 − 233.707i −1.81547 − 0.543075i
\(58\) 93.3263i 0.211282i
\(59\) 125.160 + 45.5545i 0.276177 + 0.100520i 0.476396 − 0.879231i \(-0.341943\pi\)
−0.200219 + 0.979751i \(0.564165\pi\)
\(60\) −138.097 − 134.454i −0.297138 − 0.289299i
\(61\) −269.946 − 226.512i −0.566608 − 0.475441i 0.313910 − 0.949453i \(-0.398361\pi\)
−0.880518 + 0.474012i \(0.842805\pi\)
\(62\) 169.278 + 201.738i 0.346747 + 0.413237i
\(63\) −984.953 − 173.674i −1.96972 − 0.347315i
\(64\) 284.820 − 493.322i 0.556289 − 0.963520i
\(65\) 339.843 + 152.975i 0.648498 + 0.291910i
\(66\) 952.133 − 346.548i 1.77575 − 0.646320i
\(67\) −204.138 − 560.865i −0.372231 − 1.02270i −0.974497 − 0.224400i \(-0.927958\pi\)
0.602266 − 0.798295i \(-0.294265\pi\)
\(68\) 96.2242 − 55.5550i 0.171601 − 0.0990741i
\(69\) −916.496 + 1587.42i −1.59903 + 2.76960i
\(70\) −359.869 + 173.703i −0.614465 + 0.296593i
\(71\) 91.1304 − 76.4675i 0.152327 − 0.127817i −0.563439 − 0.826157i \(-0.690522\pi\)
0.715766 + 0.698340i \(0.246078\pi\)
\(72\) 1096.04 − 1306.22i 1.79403 − 2.13804i
\(73\) −212.875 + 37.5357i −0.341304 + 0.0601811i −0.341673 − 0.939819i \(-0.610994\pi\)
0.000369249 1.00000i \(0.499882\pi\)
\(74\) 100.394 + 36.5405i 0.157710 + 0.0574019i
\(75\) −1167.42 − 389.894i −1.79737 − 0.600281i
\(76\) 57.4611 + 133.128i 0.0867268 + 0.200932i
\(77\) 588.537i 0.871039i
\(78\) −280.629 + 771.023i −0.407372 + 1.11925i
\(79\) 147.888 + 838.715i 0.210617 + 1.19447i 0.888353 + 0.459161i \(0.151850\pi\)
−0.677736 + 0.735305i \(0.737039\pi\)
\(80\) 38.7376 − 523.243i 0.0541375 − 0.731254i
\(81\) 1743.31 − 1462.81i 2.39137 − 2.00660i
\(82\) −352.632 − 62.1785i −0.474898 − 0.0837374i
\(83\) 795.498 + 459.281i 1.05202 + 0.607381i 0.923213 − 0.384289i \(-0.125553\pi\)
0.128803 + 0.991670i \(0.458887\pi\)
\(84\) 123.238 + 213.454i 0.160075 + 0.277258i
\(85\) 399.166 − 586.603i 0.509360 − 0.748542i
\(86\) −505.103 + 183.842i −0.633333 + 0.230514i
\(87\) 318.349 − 183.799i 0.392306 − 0.226498i
\(88\) −868.963 − 501.696i −1.05263 − 0.607738i
\(89\) 164.097 − 930.643i 0.195442 − 1.10840i −0.716347 − 0.697744i \(-0.754187\pi\)
0.911789 − 0.410660i \(-0.134702\pi\)
\(90\) 480.736 − 1895.11i 0.563044 − 2.21957i
\(91\) −365.088 − 306.345i −0.420567 − 0.352898i
\(92\) 320.972 − 56.5960i 0.363735 − 0.0641363i
\(93\) 354.776 − 974.738i 0.395576 − 1.08683i
\(94\) 950.373 1.04280
\(95\) 700.006 + 606.107i 0.755991 + 0.654582i
\(96\) −764.975 −0.813281
\(97\) −300.768 + 826.353i −0.314828 + 0.864984i 0.676836 + 0.736134i \(0.263351\pi\)
−0.991664 + 0.128850i \(0.958871\pi\)
\(98\) −341.179 + 60.1590i −0.351676 + 0.0620100i
\(99\) −2205.87 − 1850.95i −2.23938 − 1.87906i
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.p.a.4.20 yes 168
5.4 even 2 inner 95.4.p.a.4.9 ✓ 168
19.5 even 9 inner 95.4.p.a.24.9 yes 168
95.24 even 18 inner 95.4.p.a.24.20 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.9 ✓ 168 5.4 even 2 inner
95.4.p.a.4.20 yes 168 1.1 even 1 trivial
95.4.p.a.24.9 yes 168 19.5 even 9 inner
95.4.p.a.24.20 yes 168 95.24 even 18 inner