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 4.10
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.10

$q$-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+(-0.811338 + 2.22913i) q^{2} +(-8.50321 + 1.49935i) q^{3} +(1.81760 + 1.52514i) q^{4} +(-9.66250 + 5.62459i) q^{5} +(3.55674 - 20.1713i) q^{6} +(-5.98741 - 3.45683i) q^{7} +(-21.3095 + 12.3030i) q^{8} +(44.6849 - 16.2640i) q^{9} +(-4.69840 - 26.1024i) q^{10} +(-3.49409 - 6.05194i) q^{11} +(-17.7421 - 10.2434i) q^{12} +(59.7409 + 10.5339i) q^{13} +(12.5635 - 10.5421i) q^{14} +(73.7291 - 62.3145i) q^{15} +(-6.83977 - 38.7902i) q^{16} +(19.0179 - 52.2513i) q^{17} +112.804i q^{18} +(-42.8048 - 70.8995i) q^{19} +(-26.1408 - 4.51348i) q^{20} +(56.0952 + 20.4170i) q^{21} +(16.3255 - 2.87862i) q^{22} +(-46.0941 + 54.9328i) q^{23} +(162.752 - 136.565i) q^{24} +(61.7280 - 108.695i) q^{25} +(-71.9515 + 124.624i) q^{26} +(-153.684 + 88.7298i) q^{27} +(-5.61053 - 15.4148i) q^{28} +(-195.497 + 71.1551i) q^{29} +(79.0881 + 214.910i) q^{30} +(-43.6469 + 75.5987i) q^{31} +(-101.840 - 17.9571i) q^{32} +(38.7849 + 46.2221i) q^{33} +(101.045 + 84.7869i) q^{34} +(77.2967 - 0.275066i) q^{35} +(106.024 + 38.5896i) q^{36} +90.2882i q^{37} +(192.774 - 37.8942i) q^{38} -523.783 q^{39} +(136.703 - 238.735i) q^{40} +(-13.9198 - 78.9434i) q^{41} +(-91.0243 + 108.479i) q^{42} +(-151.002 - 179.958i) q^{43} +(2.87924 - 16.3290i) q^{44} +(-340.290 + 408.485i) q^{45} +(-85.0546 - 147.319i) q^{46} +(-209.567 - 575.779i) q^{47} +(116.320 + 319.587i) q^{48} +(-147.601 - 255.652i) q^{49} +(192.214 + 225.788i) q^{50} +(-83.3707 + 472.819i) q^{51} +(92.5190 + 110.260i) q^{52} +(-74.1754 + 88.3988i) q^{53} +(-73.1004 - 414.573i) q^{54} +(67.8013 + 38.8241i) q^{55} +170.118 q^{56} +(470.282 + 538.695i) q^{57} -493.520i q^{58} +(700.002 + 254.780i) q^{59} +(229.048 - 0.815087i) q^{60} +(227.794 + 191.142i) q^{61} +(-133.107 - 158.631i) q^{62} +(-323.769 - 57.0891i) q^{63} +(280.210 - 485.338i) q^{64} +(-636.496 + 234.234i) q^{65} +(-134.503 + 48.9550i) q^{66} +(54.1306 + 148.723i) q^{67} +(114.258 - 65.9667i) q^{68} +(309.585 - 536.216i) q^{69} +(-62.1005 + 172.528i) q^{70} +(-569.897 + 478.200i) q^{71} +(-752.115 + 896.335i) q^{72} +(700.391 - 123.498i) q^{73} +(-201.264 - 73.2542i) q^{74} +(-361.914 + 1016.81i) q^{75} +(30.3301 - 194.150i) q^{76} +48.3139i q^{77} +(424.965 - 1167.58i) q^{78} +(-181.078 - 1026.94i) q^{79} +(284.269 + 336.340i) q^{80} +(190.234 - 159.625i) q^{81} +(187.269 + 33.0206i) q^{82} +(-972.154 - 561.273i) q^{83} +(70.8196 + 122.663i) q^{84} +(110.132 + 611.847i) q^{85} +(523.663 - 190.598i) q^{86} +(1555.67 - 898.165i) q^{87} +(148.914 + 85.9757i) q^{88} +(233.952 - 1326.81i) q^{89} +(-634.476 - 1089.97i) q^{90} +(-321.279 - 269.585i) q^{91} +(-167.561 + 29.5455i) q^{92} +(257.791 - 708.274i) q^{93} +1453.52 q^{94} +(812.383 + 444.307i) q^{95} +892.890 q^{96} +(168.557 - 463.105i) q^{97} +(689.635 - 121.601i) q^{98} +(-254.561 - 213.602i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.811338 + 2.22913i −0.286851 + 0.788117i 0.709651 + 0.704553i \(0.248853\pi\)
−0.996502 + 0.0835640i \(0.973370\pi\)
\(3\) −8.50321 + 1.49935i −1.63644 + 0.288549i −0.914858 − 0.403776i \(-0.867698\pi\)
−0.721586 + 0.692325i \(0.756587\pi\)
\(4\) 1.81760 + 1.52514i 0.227200 + 0.190643i
\(5\) −9.66250 + 5.62459i −0.864241 + 0.503079i
\(6\) 3.55674 − 20.1713i 0.242005 − 1.37248i
\(7\) −5.98741 − 3.45683i −0.323290 − 0.186651i 0.329568 − 0.944132i \(-0.393097\pi\)
−0.652858 + 0.757480i \(0.726430\pi\)
\(8\) −21.3095 + 12.3030i −0.941754 + 0.543722i
\(9\) 44.6849 − 16.2640i 1.65500 − 0.602369i
\(10\) −4.69840 − 26.1024i −0.148576 − 0.825431i
\(11\) −3.49409 − 6.05194i −0.0957734 − 0.165884i 0.814158 − 0.580644i \(-0.197199\pi\)
−0.909931 + 0.414759i \(0.863866\pi\)
\(12\) −17.7421 − 10.2434i −0.426809 − 0.246418i
\(13\) 59.7409 + 10.5339i 1.27455 + 0.224737i 0.769664 − 0.638449i \(-0.220424\pi\)
0.504885 + 0.863187i \(0.331535\pi\)
\(14\) 12.5635 − 10.5421i 0.239839 − 0.201249i
\(15\) 73.7291 − 62.3145i 1.26912 − 1.07264i
\(16\) −6.83977 − 38.7902i −0.106871 − 0.606098i
\(17\) 19.0179 − 52.2513i 0.271325 − 0.745459i −0.726947 − 0.686694i \(-0.759061\pi\)
0.998272 − 0.0587656i \(-0.0187165\pi\)
\(18\) 112.804i 1.47712i
\(19\) −42.8048 − 70.8995i −0.516848 − 0.856077i
\(20\) −26.1408 − 4.51348i −0.292264 − 0.0504622i
\(21\) 56.0952 + 20.4170i 0.582904 + 0.212160i
\(22\) 16.3255 − 2.87862i 0.158209 − 0.0278965i
\(23\) −46.0941 + 54.9328i −0.417882 + 0.498012i −0.933386 − 0.358875i \(-0.883160\pi\)
0.515504 + 0.856887i \(0.327605\pi\)
\(24\) 162.752 − 136.565i 1.38424 − 1.16151i
\(25\) 61.7280 − 108.695i 0.493824 − 0.869562i
\(26\) −71.9515 + 124.624i −0.542725 + 0.940028i
\(27\) −153.684 + 88.7298i −1.09543 + 0.632446i
\(28\) −5.61053 − 15.4148i −0.0378675 − 0.104040i
\(29\) −195.497 + 71.1551i −1.25182 + 0.455627i −0.881018 − 0.473083i \(-0.843141\pi\)
−0.370806 + 0.928710i \(0.620919\pi\)
\(30\) 79.0881 + 214.910i 0.481315 + 1.30790i
\(31\) −43.6469 + 75.5987i −0.252878 + 0.437998i −0.964317 − 0.264750i \(-0.914711\pi\)
0.711439 + 0.702748i \(0.248044\pi\)
\(32\) −101.840 − 17.9571i −0.562591 − 0.0992000i
\(33\) 38.7849 + 46.2221i 0.204594 + 0.243825i
\(34\) 101.045 + 84.7869i 0.509679 + 0.427672i
\(35\) 77.2967 − 0.275066i 0.373301 − 0.00132842i
\(36\) 106.024 + 38.5896i 0.490852 + 0.178655i
\(37\) 90.2882i 0.401170i 0.979676 + 0.200585i \(0.0642843\pi\)
−0.979676 + 0.200585i \(0.935716\pi\)
\(38\) 192.774 − 37.8942i 0.822948 − 0.161770i
\(39\) −523.783 −2.15058
\(40\) 136.703 − 238.735i 0.540367 − 0.943683i
\(41\) −13.9198 − 78.9434i −0.0530223 − 0.300704i 0.946752 − 0.321965i \(-0.104343\pi\)
−0.999774 + 0.0212603i \(0.993232\pi\)
\(42\) −91.0243 + 108.479i −0.334413 + 0.398538i
\(43\) −151.002 − 179.958i −0.535527 − 0.638216i 0.428652 − 0.903470i \(-0.358989\pi\)
−0.964179 + 0.265254i \(0.914544\pi\)
\(44\) 2.87924 − 16.3290i 0.00986503 − 0.0559474i
\(45\) −340.290 + 408.485i −1.12728 + 1.35318i
\(46\) −85.0546 − 147.319i −0.272622 − 0.472195i
\(47\) −209.567 − 575.779i −0.650392 − 1.78694i −0.616291 − 0.787519i \(-0.711365\pi\)
−0.0341012 − 0.999418i \(-0.510857\pi\)
\(48\) 116.320 + 319.587i 0.349778 + 0.961007i
\(49\) −147.601 − 255.652i −0.430322 − 0.745340i
\(50\) 192.214 + 225.788i 0.543663 + 0.638626i
\(51\) −83.3707 + 472.819i −0.228906 + 1.29819i
\(52\) 92.5190 + 110.260i 0.246732 + 0.294044i
\(53\) −74.1754 + 88.3988i −0.192241 + 0.229104i −0.853552 − 0.521008i \(-0.825556\pi\)
0.661311 + 0.750112i \(0.270001\pi\)
\(54\) −73.1004 − 414.573i −0.184217 − 1.04474i
\(55\) 67.8013 + 38.8241i 0.166224 + 0.0951825i
\(56\) 170.118 0.405946
\(57\) 470.282 + 538.695i 1.09281 + 1.25179i
\(58\) − 493.520i − 1.11728i
\(59\) 700.002 + 254.780i 1.54462 + 0.562195i 0.967147 − 0.254216i \(-0.0818174\pi\)
0.577471 + 0.816411i \(0.304040\pi\)
\(60\) 229.048 − 0.815087i 0.492834 − 0.00175379i
\(61\) 227.794 + 191.142i 0.478132 + 0.401200i 0.849750 − 0.527185i \(-0.176753\pi\)
−0.371618 + 0.928386i \(0.621197\pi\)
\(62\) −133.107 − 158.631i −0.272655 − 0.324938i
\(63\) −323.769 − 57.0891i −0.647476 − 0.114168i
\(64\) 280.210 − 485.338i 0.547285 − 0.947925i
\(65\) −636.496 + 234.234i −1.21458 + 0.446971i
\(66\) −134.503 + 48.9550i −0.250851 + 0.0913022i
\(67\) 54.1306 + 148.723i 0.0987030 + 0.271184i 0.979210 − 0.202849i \(-0.0650200\pi\)
−0.880507 + 0.474033i \(0.842798\pi\)
\(68\) 114.258 − 65.9667i 0.203762 − 0.117642i
\(69\) 309.585 − 536.216i 0.540139 − 0.935548i
\(70\) −62.1005 + 172.528i −0.106035 + 0.294586i
\(71\) −569.897 + 478.200i −0.952596 + 0.799323i −0.979733 − 0.200309i \(-0.935805\pi\)
0.0271369 + 0.999632i \(0.491361\pi\)
\(72\) −752.115 + 896.335i −1.23108 + 1.46714i
\(73\) 700.391 − 123.498i 1.12294 − 0.198004i 0.418808 − 0.908075i \(-0.362448\pi\)
0.704131 + 0.710070i \(0.251337\pi\)
\(74\) −201.264 − 73.2542i −0.316169 − 0.115076i
\(75\) −361.914 + 1016.81i −0.557203 + 1.56548i
\(76\) 30.3301 − 194.150i 0.0457777 − 0.293034i
\(77\) 48.3139i 0.0715050i
\(78\) 424.965 − 1167.58i 0.616895 − 1.69491i
\(79\) −181.078 − 1026.94i −0.257884 − 1.46253i −0.788560 − 0.614958i \(-0.789173\pi\)
0.530676 − 0.847575i \(-0.321938\pi\)
\(80\) 284.269 + 336.340i 0.397277 + 0.470050i
\(81\) 190.234 − 159.625i 0.260952 − 0.218965i
\(82\) 187.269 + 33.0206i 0.252200 + 0.0444696i
\(83\) −972.154 − 561.273i −1.28564 − 0.742262i −0.307763 − 0.951463i \(-0.599580\pi\)
−0.977873 + 0.209201i \(0.932914\pi\)
\(84\) 70.8196 + 122.663i 0.0919887 + 0.159329i
\(85\) 110.132 + 611.847i 0.140535 + 0.780754i
\(86\) 523.663 − 190.598i 0.656605 − 0.238985i
\(87\) 1555.67 − 898.165i 1.91707 − 1.10682i
\(88\) 148.914 + 85.9757i 0.180390 + 0.104148i
\(89\) 233.952 − 1326.81i 0.278639 − 1.58024i −0.448523 − 0.893771i \(-0.648050\pi\)
0.727161 − 0.686467i \(-0.240839\pi\)
\(90\) −634.476 − 1089.97i −0.743108 − 1.27659i
\(91\) −321.279 − 269.585i −0.370101 − 0.310552i
\(92\) −167.561 + 29.5455i −0.189885 + 0.0334819i
\(93\) 257.791 − 708.274i 0.287437 − 0.789727i
\(94\) 1453.52 1.59488
\(95\) 812.383 + 444.307i 0.877355 + 0.479842i
\(96\) 892.890 0.949273
\(97\) 168.557 − 463.105i 0.176436 − 0.484755i −0.819678 − 0.572825i \(-0.805848\pi\)
0.996114 + 0.0880697i \(0.0280698\pi\)
\(98\) 689.635 − 121.601i 0.710854 − 0.125343i
\(99\) −254.561 − 213.602i −0.258428 − 0.216847i
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.4.10 ✓ 168
5.4 even 2 inner 95.4.p.a.4.19 yes 168
19.5 even 9 inner 95.4.p.a.24.19 yes 168
95.24 even 18 inner 95.4.p.a.24.10 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.10 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.19 yes 168 5.4 even 2 inner
95.4.p.a.24.10 yes 168 95.24 even 18 inner
95.4.p.a.24.19 yes 168 19.5 even 9 inner