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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 4.7
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.7

$q$-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+(-1.15144 + 3.16356i) q^{2} +(8.44720 - 1.48947i) q^{3} +(-2.55395 - 2.14302i) q^{4} +(-1.08681 + 11.1274i) q^{5} +(-5.01443 + 28.4383i) q^{6} +(-14.7094 - 8.49249i) q^{7} +(-13.6041 + 7.85435i) q^{8} +(43.7649 - 15.9291i) q^{9} +(-33.9508 - 16.2507i) q^{10} +(32.5985 + 56.4623i) q^{11} +(-24.7657 - 14.2985i) q^{12} +(23.4919 + 4.14225i) q^{13} +(43.8036 - 36.7556i) q^{14} +(7.39341 + 95.6140i) q^{15} +(-13.8148 - 78.3477i) q^{16} +(35.2563 - 96.8658i) q^{17} +156.794i q^{18} +(-39.9137 + 72.5665i) q^{19} +(26.6219 - 26.0898i) q^{20} +(-136.903 - 49.8285i) q^{21} +(-216.157 + 38.1144i) q^{22} +(34.3036 - 40.8814i) q^{23} +(-103.218 + 86.6101i) q^{24} +(-122.638 - 24.1867i) q^{25} +(-40.1538 + 69.5484i) q^{26} +(145.399 - 83.9464i) q^{27} +(19.3676 + 53.2120i) q^{28} +(26.4045 - 9.61046i) q^{29} +(-310.994 - 86.7045i) q^{30} +(112.584 - 195.001i) q^{31} +(140.005 + 24.6866i) q^{32} +(359.465 + 428.394i) q^{33} +(265.845 + 223.071i) q^{34} +(110.486 - 154.448i) q^{35} +(-145.910 - 53.1069i) q^{36} -304.917i q^{37} +(-183.610 - 209.826i) q^{38} +204.610 q^{39} +(-72.6133 - 159.915i) q^{40} +(50.0971 + 284.115i) q^{41} +(315.271 - 375.725i) q^{42} +(148.028 + 176.413i) q^{43} +(37.7448 - 214.061i) q^{44} +(129.685 + 504.301i) q^{45} +(89.8323 + 155.594i) q^{46} +(-143.393 - 393.970i) q^{47} +(-233.393 - 641.242i) q^{48} +(-27.2552 - 47.2075i) q^{49} +(217.726 - 360.122i) q^{50} +(153.538 - 870.757i) q^{51} +(-51.1202 - 60.9226i) q^{52} +(206.909 - 246.585i) q^{53} +(98.1506 + 556.640i) q^{54} +(-663.707 + 301.373i) q^{55} +266.812 q^{56} +(-229.073 + 672.434i) q^{57} +94.5982i q^{58} +(-149.352 - 54.3596i) q^{59} +(186.020 - 260.038i) q^{60} +(134.517 + 112.873i) q^{61} +(487.263 + 580.697i) q^{62} +(-779.034 - 137.365i) q^{63} +(78.9207 - 136.695i) q^{64} +(-71.6236 + 256.901i) q^{65} +(-1769.15 + 643.919i) q^{66} +(156.455 + 429.857i) q^{67} +(-297.628 + 171.836i) q^{68} +(228.877 - 396.427i) q^{69} +(361.387 + 527.366i) q^{70} +(258.412 - 216.833i) q^{71} +(-470.270 + 560.446i) q^{72} +(-276.610 + 48.7739i) q^{73} +(964.623 + 351.094i) q^{74} +(-1071.97 - 21.6447i) q^{75} +(257.449 - 99.7955i) q^{76} -1107.37i q^{77} +(-235.597 + 647.297i) q^{78} +(-19.4955 - 110.564i) q^{79} +(886.820 - 68.5739i) q^{80} +(139.891 - 117.382i) q^{81} +(-956.500 - 168.657i) q^{82} +(156.771 + 90.5119i) q^{83} +(242.859 + 420.645i) q^{84} +(1039.55 + 497.585i) q^{85} +(-728.538 + 265.166i) q^{86} +(208.730 - 120.510i) q^{87} +(-886.949 - 512.080i) q^{88} +(64.0689 - 363.353i) q^{89} +(-1744.71 - 170.406i) q^{90} +(-310.374 - 260.434i) q^{91} +(-175.219 + 30.8959i) q^{92} +(660.569 - 1814.90i) q^{93} +1411.46 q^{94} +(-764.097 - 523.001i) q^{95} +1219.42 q^{96} +(-382.511 + 1050.94i) q^{97} +(180.727 - 31.8670i) q^{98} +(2326.07 + 1951.80i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15144 + 3.16356i −0.407096 + 1.11849i 0.551613 + 0.834100i \(0.314013\pi\)
−0.958709 + 0.284388i \(0.908210\pi\)
\(3\) 8.44720 − 1.48947i 1.62566 − 0.286648i 0.714792 − 0.699338i \(-0.246522\pi\)
0.910872 + 0.412689i \(0.135410\pi\)
\(4\) −2.55395 − 2.14302i −0.319244 − 0.267878i
\(5\) −1.08681 + 11.1274i −0.0972071 + 0.995264i
\(6\) −5.01443 + 28.4383i −0.341189 + 1.93498i
\(7\) −14.7094 − 8.49249i −0.794234 − 0.458551i 0.0472169 − 0.998885i \(-0.484965\pi\)
−0.841451 + 0.540333i \(0.818298\pi\)
\(8\) −13.6041 + 7.85435i −0.601223 + 0.347116i
\(9\) 43.7649 − 15.9291i 1.62092 − 0.589967i
\(10\) −33.9508 − 16.2507i −1.07362 − 0.513894i
\(11\) 32.5985 + 56.4623i 0.893530 + 1.54764i 0.835614 + 0.549318i \(0.185112\pi\)
0.0579162 + 0.998321i \(0.481554\pi\)
\(12\) −24.7657 − 14.2985i −0.595770 − 0.343968i
\(13\) 23.4919 + 4.14225i 0.501190 + 0.0883733i 0.418527 − 0.908204i \(-0.362546\pi\)
0.0826627 + 0.996578i \(0.473658\pi\)
\(14\) 43.8036 − 36.7556i 0.836214 − 0.701667i
\(15\) 7.39341 + 95.6140i 0.127265 + 1.64583i
\(16\) −13.8148 − 78.3477i −0.215857 − 1.22418i
\(17\) 35.2563 − 96.8658i 0.502994 − 1.38197i −0.385343 − 0.922773i \(-0.625917\pi\)
0.888337 − 0.459192i \(-0.151861\pi\)
\(18\) 156.794i 2.05316i
\(19\) −39.9137 + 72.5665i −0.481938 + 0.876205i
\(20\) 26.6219 − 26.0898i 0.297642 − 0.291693i
\(21\) −136.903 − 49.8285i −1.42260 − 0.517784i
\(22\) −216.157 + 38.1144i −2.09477 + 0.369364i
\(23\) 34.3036 − 40.8814i 0.310991 − 0.370624i −0.587797 − 0.809008i \(-0.700005\pi\)
0.898788 + 0.438384i \(0.144449\pi\)
\(24\) −103.218 + 86.6101i −0.877886 + 0.736634i
\(25\) −122.638 − 24.1867i −0.981102 − 0.193494i
\(26\) −40.1538 + 69.5484i −0.302877 + 0.524599i
\(27\) 145.399 − 83.9464i 1.03638 − 0.598352i
\(28\) 19.3676 + 53.2120i 0.130719 + 0.359147i
\(29\) 26.4045 − 9.61046i 0.169076 − 0.0615385i −0.256095 − 0.966652i \(-0.582436\pi\)
0.425171 + 0.905113i \(0.360214\pi\)
\(30\) −310.994 − 86.7045i −1.89265 − 0.527667i
\(31\) 112.584 − 195.001i 0.652278 − 1.12978i −0.330291 − 0.943879i \(-0.607147\pi\)
0.982569 − 0.185900i \(-0.0595199\pi\)
\(32\) 140.005 + 24.6866i 0.773423 + 0.136375i
\(33\) 359.465 + 428.394i 1.89621 + 2.25981i
\(34\) 265.845 + 223.071i 1.34094 + 1.12519i
\(35\) 110.486 − 154.448i 0.533585 − 0.745898i
\(36\) −145.910 − 53.1069i −0.675509 − 0.245865i
\(37\) − 304.917i − 1.35481i −0.735610 − 0.677405i \(-0.763104\pi\)
0.735610 − 0.677405i \(-0.236896\pi\)
\(38\) −183.610 − 209.826i −0.783830 − 0.895743i
\(39\) 204.610 0.840098
\(40\) −72.6133 − 159.915i −0.287029 − 0.632118i
\(41\) 50.0971 + 284.115i 0.190826 + 1.08223i 0.918239 + 0.396027i \(0.129611\pi\)
−0.727413 + 0.686200i \(0.759277\pi\)
\(42\) 315.271 − 375.725i 1.15827 − 1.38037i
\(43\) 148.028 + 176.413i 0.524977 + 0.625644i 0.961750 − 0.273928i \(-0.0883231\pi\)
−0.436773 + 0.899572i \(0.643879\pi\)
\(44\) 37.7448 − 214.061i 0.129324 − 0.733431i
\(45\) 129.685 + 504.301i 0.429608 + 1.67059i
\(46\) 89.8323 + 155.594i 0.287936 + 0.498719i
\(47\) −143.393 − 393.970i −0.445023 − 1.22269i −0.936149 − 0.351603i \(-0.885637\pi\)
0.491126 − 0.871089i \(-0.336585\pi\)
\(48\) −233.393 − 641.242i −0.701820 − 1.92824i
\(49\) −27.2552 − 47.2075i −0.0794614 − 0.137631i
\(50\) 217.726 − 360.122i 0.615823 − 1.01858i
\(51\) 153.538 − 870.757i 0.421561 − 2.39079i
\(52\) −51.1202 − 60.9226i −0.136329 − 0.162470i
\(53\) 206.909 − 246.585i 0.536248 − 0.639076i −0.428094 − 0.903734i \(-0.640815\pi\)
0.964342 + 0.264658i \(0.0852592\pi\)
\(54\) 98.1506 + 556.640i 0.247345 + 1.40276i
\(55\) −663.707 + 301.373i −1.62717 + 0.738857i
\(56\) 266.812 0.636683
\(57\) −229.073 + 672.434i −0.532307 + 1.56256i
\(58\) 94.5982i 0.214161i
\(59\) −149.352 − 54.3596i −0.329558 − 0.119949i 0.171942 − 0.985107i \(-0.444996\pi\)
−0.501500 + 0.865158i \(0.667218\pi\)
\(60\) 186.020 − 260.038i 0.400252 − 0.559513i
\(61\) 134.517 + 112.873i 0.282346 + 0.236916i 0.772951 − 0.634466i \(-0.218780\pi\)
−0.490605 + 0.871382i \(0.663224\pi\)
\(62\) 487.263 + 580.697i 0.998104 + 1.18949i
\(63\) −779.034 − 137.365i −1.55792 − 0.274704i
\(64\) 78.9207 − 136.695i 0.154142 − 0.266982i
\(65\) −71.6236 + 256.901i −0.136674 + 0.490226i
\(66\) −1769.15 + 643.919i −3.29951 + 1.20092i
\(67\) 156.455 + 429.857i 0.285284 + 0.783811i 0.996710 + 0.0810504i \(0.0258275\pi\)
−0.711426 + 0.702761i \(0.751950\pi\)
\(68\) −297.628 + 171.836i −0.530775 + 0.306443i
\(69\) 228.877 − 396.427i 0.399327 − 0.691655i
\(70\) 361.387 + 527.366i 0.617058 + 0.900461i
\(71\) 258.412 − 216.833i 0.431941 − 0.362441i −0.400743 − 0.916191i \(-0.631248\pi\)
0.832683 + 0.553749i \(0.186803\pi\)
\(72\) −470.270 + 560.446i −0.769749 + 0.917351i
\(73\) −276.610 + 48.7739i −0.443490 + 0.0781993i −0.390934 − 0.920419i \(-0.627848\pi\)
−0.0525562 + 0.998618i \(0.516737\pi\)
\(74\) 964.623 + 351.094i 1.51534 + 0.551538i
\(75\) −1071.97 − 21.6447i −1.65041 − 0.0333242i
\(76\) 257.449 − 99.7955i 0.388572 − 0.150623i
\(77\) − 1107.37i − 1.63892i
\(78\) −235.597 + 647.297i −0.342001 + 0.939640i
\(79\) −19.4955 − 110.564i −0.0277647 − 0.157461i 0.967773 − 0.251823i \(-0.0810301\pi\)
−0.995538 + 0.0943616i \(0.969919\pi\)
\(80\) 886.820 − 68.5739i 1.23937 − 0.0958350i
\(81\) 139.891 − 117.382i 0.191894 − 0.161018i
\(82\) −956.500 − 168.657i −1.28814 − 0.227134i
\(83\) 156.771 + 90.5119i 0.207324 + 0.119698i 0.600067 − 0.799950i \(-0.295141\pi\)
−0.392743 + 0.919648i \(0.628474\pi\)
\(84\) 242.859 + 420.645i 0.315454 + 0.546382i
\(85\) 1039.55 + 497.585i 1.32653 + 0.634949i
\(86\) −728.538 + 265.166i −0.913491 + 0.332484i
\(87\) 208.730 − 120.510i 0.257220 − 0.148506i
\(88\) −886.949 − 512.080i −1.07442 − 0.620318i
\(89\) 64.0689 − 363.353i 0.0763067 − 0.432757i −0.922589 − 0.385783i \(-0.873931\pi\)
0.998896 − 0.0469734i \(-0.0149576\pi\)
\(90\) −1744.71 − 170.406i −2.04343 − 0.199581i
\(91\) −310.374 − 260.434i −0.357538 − 0.300010i
\(92\) −175.219 + 30.8959i −0.198564 + 0.0350122i
\(93\) 660.569 − 1814.90i 0.736535 − 2.02361i
\(94\) 1411.46 1.54873
\(95\) −764.097 − 523.001i −0.825208 − 0.564829i
\(96\) 1219.42 1.29642
\(97\) −382.511 + 1050.94i −0.400392 + 1.10007i 0.561699 + 0.827342i \(0.310148\pi\)
−0.962091 + 0.272727i \(0.912074\pi\)
\(98\) 180.727 − 31.8670i 0.186287 − 0.0328475i
\(99\) 2326.07 + 1951.80i 2.36140 + 1.98145i
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.4.7 ✓ 168
5.4 even 2 inner 95.4.p.a.4.22 yes 168
19.5 even 9 inner 95.4.p.a.24.22 yes 168
95.24 even 18 inner 95.4.p.a.24.7 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.7 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.22 yes 168 5.4 even 2 inner
95.4.p.a.24.7 yes 168 95.24 even 18 inner
95.4.p.a.24.22 yes 168 19.5 even 9 inner