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(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 2.5
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.5

$q$-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+(-2.47264 - 3.53129i) q^{2} +(0.343809 + 3.92975i) q^{3} +(-3.61992 + 9.94565i) q^{4} +(-9.75843 + 5.45647i) q^{5} +(13.0270 - 10.9309i) q^{6} +(1.83448 - 6.84636i) q^{7} +(10.7596 - 2.88302i) q^{8} +(11.2651 - 1.98633i) q^{9} +(43.3974 + 20.9680i) q^{10} +(31.6293 - 54.7835i) q^{11} +(-40.3285 - 10.8060i) q^{12} +(70.1456 + 6.13695i) q^{13} +(-28.7125 + 10.4505i) q^{14} +(-24.7976 - 36.4722i) q^{15} +(28.0769 + 23.5593i) q^{16} +(-18.0368 + 12.6295i) q^{17} +(-34.8687 - 34.8687i) q^{18} +(-7.85092 - 82.4461i) q^{19} +(-18.9434 - 116.806i) q^{20} +(27.5352 + 4.85520i) q^{21} +(-271.664 + 23.7675i) q^{22} +(-35.5390 + 16.5721i) q^{23} +(15.0288 + 41.2913i) q^{24} +(65.4539 - 106.493i) q^{25} +(-151.773 - 262.879i) q^{26} +(39.2453 + 146.465i) q^{27} +(61.4509 + 43.0284i) q^{28} +(26.0418 + 147.691i) q^{29} +(-67.4786 + 177.750i) q^{30} +(226.226 - 130.612i) q^{31} +(21.5375 - 246.175i) q^{32} +(226.160 + 105.460i) q^{33} +(89.1967 + 32.4650i) q^{34} +(19.4553 + 76.8195i) q^{35} +(-21.0232 + 119.229i) q^{36} +(-3.11739 + 3.11739i) q^{37} +(-271.729 + 231.583i) q^{38} +277.765i q^{39} +(-89.2654 + 86.8430i) q^{40} +(115.565 - 137.724i) q^{41} +(-50.9394 - 109.240i) q^{42} +(-105.490 + 226.223i) q^{43} +(430.362 + 512.885i) q^{44} +(-99.0909 + 80.8509i) q^{45} +(146.396 + 84.5217i) q^{46} +(10.0714 - 14.3835i) q^{47} +(-82.9291 + 118.435i) q^{48} +(253.539 + 146.381i) q^{49} +(-537.902 + 32.1820i) q^{50} +(-55.8319 - 66.5379i) q^{51} +(-314.958 + 675.429i) q^{52} +(-305.648 - 655.465i) q^{53} +(420.173 - 500.742i) q^{54} +(-9.72754 + 707.185i) q^{55} -78.9528i q^{56} +(321.294 - 59.1979i) q^{57} +(457.146 - 457.146i) q^{58} +(119.052 - 675.175i) q^{59} +(452.505 - 114.602i) q^{60} +(117.557 + 42.7873i) q^{61} +(-1020.60 - 475.914i) q^{62} +(7.06633 - 80.7686i) q^{63} +(-668.639 + 386.039i) q^{64} +(-717.997 + 322.861i) q^{65} +(-186.801 - 1059.40i) q^{66} +(358.928 + 251.324i) q^{67} +(-60.3167 - 225.105i) q^{68} +(-77.3428 - 133.962i) q^{69} +(223.166 - 258.649i) q^{70} +(160.916 + 442.112i) q^{71} +(115.481 - 53.8495i) q^{72} +(-421.501 + 36.8765i) q^{73} +(18.7166 + 3.30024i) q^{74} +(440.995 + 220.604i) q^{75} +(848.400 + 220.366i) q^{76} +(-317.044 - 317.044i) q^{77} +(980.869 - 686.812i) q^{78} +(-452.320 - 379.541i) q^{79} +(-402.537 - 76.7011i) q^{80} +(-271.857 + 98.9479i) q^{81} +(-772.094 - 67.5495i) q^{82} +(-631.959 - 169.333i) q^{83} +(-147.963 + 256.280i) q^{84} +(107.098 - 221.661i) q^{85} +(1059.70 - 186.853i) q^{86} +(-571.434 + 153.115i) q^{87} +(182.375 - 680.635i) q^{88} +(-54.0230 + 45.3307i) q^{89} +(530.524 + 150.004i) q^{90} +(170.696 - 468.984i) q^{91} +(-36.1720 - 413.448i) q^{92} +(591.050 + 844.106i) q^{93} -75.6952 q^{94} +(526.477 + 761.706i) q^{95} +974.812 q^{96} +(374.599 + 534.983i) q^{97} +(-109.997 - 1257.27i) q^{98} +(247.487 - 679.965i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.47264 − 3.53129i −0.874209 − 1.24850i −0.967503 − 0.252861i \(-0.918628\pi\)
0.0932937 − 0.995639i \(-0.470260\pi\)
\(3\) 0.343809 + 3.92975i 0.0661660 + 0.756281i 0.954719 + 0.297508i \(0.0961555\pi\)
−0.888553 + 0.458773i \(0.848289\pi\)
\(4\) −3.61992 + 9.94565i −0.452490 + 1.24321i
\(5\) −9.75843 + 5.45647i −0.872820 + 0.488041i
\(6\) 13.0270 − 10.9309i 0.886374 − 0.743756i
\(7\) 1.83448 − 6.84636i 0.0990524 − 0.369669i −0.898550 − 0.438871i \(-0.855379\pi\)
0.997603 + 0.0692019i \(0.0220453\pi\)
\(8\) 10.7596 − 2.88302i 0.475511 − 0.127413i
\(9\) 11.2651 − 1.98633i 0.417224 − 0.0735679i
\(10\) 43.3974 + 20.9680i 1.37235 + 0.663066i
\(11\) 31.6293 − 54.7835i 0.866962 − 1.50162i 0.00187548 − 0.999998i \(-0.499403\pi\)
0.865086 − 0.501623i \(-0.167264\pi\)
\(12\) −40.3285 − 10.8060i −0.970153 − 0.259952i
\(13\) 70.1456 + 6.13695i 1.49653 + 0.130929i 0.805802 − 0.592185i \(-0.201734\pi\)
0.690729 + 0.723114i \(0.257290\pi\)
\(14\) −28.7125 + 10.4505i −0.548124 + 0.199501i
\(15\) −24.7976 − 36.4722i −0.426848 − 0.627806i
\(16\) 28.0769 + 23.5593i 0.438701 + 0.368114i
\(17\) −18.0368 + 12.6295i −0.257327 + 0.180182i −0.695120 − 0.718893i \(-0.744649\pi\)
0.437794 + 0.899076i \(0.355760\pi\)
\(18\) −34.8687 − 34.8687i −0.456591 − 0.456591i
\(19\) −7.85092 − 82.4461i −0.0947960 − 0.995497i
\(20\) −18.9434 − 116.806i −0.211794 − 1.30593i
\(21\) 27.5352 + 4.85520i 0.286127 + 0.0504520i
\(22\) −271.664 + 23.7675i −2.63268 + 0.230330i
\(23\) −35.5390 + 16.5721i −0.322191 + 0.150240i −0.576984 − 0.816756i \(-0.695770\pi\)
0.254793 + 0.966996i \(0.417993\pi\)
\(24\) 15.0288 + 41.2913i 0.127822 + 0.351189i
\(25\) 65.4539 − 106.493i 0.523631 − 0.851945i
\(26\) −151.773 − 262.879i −1.14482 − 1.98288i
\(27\) 39.2453 + 146.465i 0.279732 + 1.04397i
\(28\) 61.4509 + 43.0284i 0.414754 + 0.290414i
\(29\) 26.0418 + 147.691i 0.166753 + 0.945705i 0.947238 + 0.320530i \(0.103861\pi\)
−0.780485 + 0.625175i \(0.785028\pi\)
\(30\) −67.4786 + 177.750i −0.410662 + 1.08175i
\(31\) 226.226 − 130.612i 1.31069 − 0.756727i 0.328479 − 0.944511i \(-0.393464\pi\)
0.982210 + 0.187784i \(0.0601306\pi\)
\(32\) 21.5375 − 246.175i 0.118979 − 1.35994i
\(33\) 226.160 + 105.460i 1.19301 + 0.556310i
\(34\) 89.1967 + 32.4650i 0.449915 + 0.163756i
\(35\) 19.4553 + 76.8195i 0.0939587 + 0.370996i
\(36\) −21.0232 + 119.229i −0.0973298 + 0.551985i
\(37\) −3.11739 + 3.11739i −0.0138512 + 0.0138512i −0.713998 − 0.700147i \(-0.753118\pi\)
0.700147 + 0.713998i \(0.253118\pi\)
\(38\) −271.729 + 231.583i −1.16001 + 0.988625i
\(39\) 277.765i 1.14046i
\(40\) −89.2654 + 86.8430i −0.352853 + 0.343277i
\(41\) 115.565 − 137.724i 0.440199 − 0.524608i −0.499637 − 0.866235i \(-0.666533\pi\)
0.939836 + 0.341626i \(0.110978\pi\)
\(42\) −50.9394 − 109.240i −0.187146 − 0.401336i
\(43\) −105.490 + 226.223i −0.374117 + 0.802296i 0.625637 + 0.780114i \(0.284839\pi\)
−0.999754 + 0.0221820i \(0.992939\pi\)
\(44\) 430.362 + 512.885i 1.47453 + 1.75728i
\(45\) −99.0909 + 80.8509i −0.328258 + 0.267834i
\(46\) 146.396 + 84.5217i 0.469237 + 0.270914i
\(47\) 10.0714 − 14.3835i 0.0312568 − 0.0446393i −0.803220 − 0.595683i \(-0.796881\pi\)
0.834476 + 0.551044i \(0.185770\pi\)
\(48\) −82.9291 + 118.435i −0.249370 + 0.356138i
\(49\) 253.539 + 146.381i 0.739182 + 0.426767i
\(50\) −537.902 + 32.1820i −1.52142 + 0.0910246i
\(51\) −55.8319 − 66.5379i −0.153295 − 0.182690i
\(52\) −314.958 + 675.429i −0.839938 + 1.80125i
\(53\) −305.648 − 655.465i −0.792152 − 1.69878i −0.712200 − 0.701976i \(-0.752301\pi\)
−0.0799517 − 0.996799i \(-0.525477\pi\)
\(54\) 420.173 − 500.742i 1.05886 − 1.26190i
\(55\) −9.72754 + 707.185i −0.0238484 + 1.73376i
\(56\) − 78.9528i − 0.188402i
\(57\) 321.294 − 59.1979i 0.746603 − 0.137561i
\(58\) 457.146 − 457.146i 1.03494 − 1.03494i
\(59\) 119.052 − 675.175i 0.262698 − 1.48984i −0.512812 − 0.858501i \(-0.671396\pi\)
0.775510 − 0.631335i \(-0.217493\pi\)
\(60\) 452.505 − 114.602i 0.973637 − 0.246584i
\(61\) 117.557 + 42.7873i 0.246748 + 0.0898090i 0.462434 − 0.886654i \(-0.346976\pi\)
−0.215685 + 0.976463i \(0.569199\pi\)
\(62\) −1020.60 − 475.914i −2.09059 − 0.974858i
\(63\) 7.06633 − 80.7686i 0.0141313 − 0.161522i
\(64\) −668.639 + 386.039i −1.30594 + 0.753982i
\(65\) −717.997 + 322.861i −1.37010 + 0.616091i
\(66\) −186.801 − 1059.40i −0.348388 − 1.97581i
\(67\) 358.928 + 251.324i 0.654477 + 0.458270i 0.853059 − 0.521814i \(-0.174745\pi\)
−0.198582 + 0.980084i \(0.563634\pi\)
\(68\) −60.3167 − 225.105i −0.107566 − 0.401441i
\(69\) −77.3428 − 133.962i −0.134942 − 0.233726i
\(70\) 223.166 − 258.649i 0.381049 − 0.441635i
\(71\) 160.916 + 442.112i 0.268974 + 0.739000i 0.998485 + 0.0550306i \(0.0175256\pi\)
−0.729511 + 0.683969i \(0.760252\pi\)
\(72\) 115.481 − 53.8495i 0.189021 − 0.0881420i
\(73\) −421.501 + 36.8765i −0.675793 + 0.0591243i −0.419886 − 0.907577i \(-0.637930\pi\)
−0.255908 + 0.966701i \(0.582374\pi\)
\(74\) 18.7166 + 3.30024i 0.0294021 + 0.00518439i
\(75\) 440.995 + 220.604i 0.678957 + 0.339643i
\(76\) 848.400 + 220.366i 1.28050 + 0.332601i
\(77\) −317.044 − 317.044i −0.469228 − 0.469228i
\(78\) 980.869 − 686.812i 1.42387 − 0.997001i
\(79\) −452.320 − 379.541i −0.644176 − 0.540528i 0.261121 − 0.965306i \(-0.415908\pi\)
−0.905298 + 0.424778i \(0.860352\pi\)
\(80\) −402.537 − 76.7011i −0.562562 − 0.107193i
\(81\) −271.857 + 98.9479i −0.372918 + 0.135731i
\(82\) −772.094 − 67.5495i −1.03980 − 0.0909706i
\(83\) −631.959 − 169.333i −0.835741 − 0.223936i −0.184524 − 0.982828i \(-0.559074\pi\)
−0.651217 + 0.758892i \(0.725741\pi\)
\(84\) −147.963 + 256.280i −0.192192 + 0.332886i
\(85\) 107.098 − 221.661i 0.136664 − 0.282853i
\(86\) 1059.70 − 186.853i 1.32872 − 0.234290i
\(87\) −571.434 + 153.115i −0.704186 + 0.188686i
\(88\) 182.375 − 680.635i 0.220924 − 0.824499i
\(89\) −54.0230 + 45.3307i −0.0643419 + 0.0539893i −0.674391 − 0.738374i \(-0.735594\pi\)
0.610050 + 0.792363i \(0.291150\pi\)
\(90\) 530.524 + 150.004i 0.621357 + 0.175687i
\(91\) 170.696 − 468.984i 0.196636 − 0.540252i
\(92\) −36.1720 − 413.448i −0.0409912 − 0.468532i
\(93\) 591.050 + 844.106i 0.659021 + 0.941180i
\(94\) −75.6952 −0.0830571
\(95\) 526.477 + 761.706i 0.568584 + 0.822626i
\(96\) 974.812 1.03637
\(97\) 374.599 + 534.983i 0.392111 + 0.559993i 0.965869 − 0.259032i \(-0.0834036\pi\)
−0.573757 + 0.819025i \(0.694515\pi\)
\(98\) −109.997 − 1257.27i −0.113381 − 1.29595i
\(99\) 247.487 − 679.965i 0.251246 − 0.690294i
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.2.5 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.5 yes 336
19.10 odd 18 inner 95.4.r.a.67.5 yes 336
95.48 even 36 inner 95.4.r.a.48.5 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.5 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.5 yes 336 95.48 even 36 inner
95.4.r.a.67.5 yes 336 19.10 odd 18 inner
95.4.r.a.78.5 yes 336 5.3 odd 4 inner