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 4.2
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.2

$q$-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+(-1.83075 + 5.02994i) q^{2} +(-6.67395 + 1.17680i) q^{3} +(-15.8203 - 13.2748i) q^{4} +(6.58809 + 9.03311i) q^{5} +(6.29910 - 35.7240i) q^{6} +(-23.1341 - 13.3565i) q^{7} +(58.6494 - 33.8613i) q^{8} +(17.7851 - 6.47325i) q^{9} +(-57.4971 + 16.6004i) q^{10} +(18.1835 + 31.4948i) q^{11} +(121.206 + 69.9781i) q^{12} +(-13.6659 - 2.40968i) q^{13} +(109.535 - 91.9109i) q^{14} +(-54.5988 - 52.5337i) q^{15} +(34.2584 + 194.289i) q^{16} +(-19.5168 + 53.6220i) q^{17} +101.309i q^{18} +(-6.98094 - 82.5243i) q^{19} +(15.6871 - 230.362i) q^{20} +(170.114 + 61.9165i) q^{21} +(-191.707 + 33.8030i) q^{22} +(54.9156 - 65.4458i) q^{23} +(-351.576 + 295.007i) q^{24} +(-38.1940 + 119.022i) q^{25} +(37.1394 - 64.3274i) q^{26} +(47.3831 - 27.3567i) q^{27} +(188.684 + 518.405i) q^{28} +(-12.9054 + 4.69719i) q^{29} +(364.198 - 178.453i) q^{30} +(127.370 - 220.611i) q^{31} +(-506.431 - 89.2975i) q^{32} +(-158.419 - 188.797i) q^{33} +(-233.985 - 196.337i) q^{34} +(-31.7592 - 296.967i) q^{35} +(-367.296 - 133.685i) q^{36} -363.545i q^{37} +(427.873 + 115.968i) q^{38} +94.0416 q^{39} +(692.260 + 306.705i) q^{40} +(6.40024 + 36.2976i) q^{41} +(-622.872 + 742.310i) q^{42} +(-200.017 - 238.370i) q^{43} +(130.419 - 739.640i) q^{44} +(175.643 + 118.008i) q^{45} +(228.652 + 396.037i) q^{46} +(51.6187 + 141.821i) q^{47} +(-457.278 - 1256.36i) q^{48} +(185.293 + 320.936i) q^{49} +(-528.749 - 410.013i) q^{50} +(67.1520 - 380.838i) q^{51} +(184.211 + 219.534i) q^{52} +(-248.195 + 295.787i) q^{53} +(50.8558 + 288.417i) q^{54} +(-164.701 + 371.745i) q^{55} -1809.07 q^{56} +(143.705 + 542.548i) q^{57} -73.5128i q^{58} +(-734.898 - 267.481i) q^{59} +(166.394 + 1555.89i) q^{60} +(189.191 + 158.750i) q^{61} +(876.479 + 1044.55i) q^{62} +(-497.903 - 87.7937i) q^{63} +(587.164 - 1017.00i) q^{64} +(-68.2657 - 139.321i) q^{65} +(1239.66 - 451.200i) q^{66} +(-231.830 - 636.948i) q^{67} +(1020.58 - 589.233i) q^{68} +(-289.487 + 501.407i) q^{69} +(1551.87 + 383.925i) q^{70} +(-341.242 + 286.336i) q^{71} +(823.894 - 981.878i) q^{72} +(133.904 - 23.6109i) q^{73} +(1828.61 + 665.560i) q^{74} +(114.841 - 839.293i) q^{75} +(-985.053 + 1398.23i) q^{76} -971.475i q^{77} +(-172.166 + 473.023i) q^{78} +(-17.0147 - 96.4954i) q^{79} +(-1529.34 + 1589.45i) q^{80} +(-675.500 + 566.812i) q^{81} +(-194.292 - 34.2589i) q^{82} +(-158.734 - 91.6451i) q^{83} +(-1869.33 - 3237.77i) q^{84} +(-612.951 + 176.969i) q^{85} +(1565.17 - 569.675i) q^{86} +(80.6025 - 46.5359i) q^{87} +(2132.91 + 1231.44i) q^{88} +(10.0724 - 57.1232i) q^{89} +(-915.134 + 667.432i) q^{90} +(283.965 + 238.275i) q^{91} +(-1737.56 + 306.379i) q^{92} +(-590.446 + 1622.24i) q^{93} -807.852 q^{94} +(699.460 - 606.738i) q^{95} +3484.98 q^{96} +(572.622 - 1573.27i) q^{97} +(-1953.51 + 344.457i) q^{98} +(527.270 + 442.432i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.83075 + 5.02994i −0.647267 + 1.77835i −0.0196788 + 0.999806i \(0.506264\pi\)
−0.627588 + 0.778545i \(0.715958\pi\)
\(3\) −6.67395 + 1.17680i −1.28440 + 0.226475i −0.773849 − 0.633371i \(-0.781671\pi\)
−0.510554 + 0.859846i \(0.670560\pi\)
\(4\) −15.8203 − 13.2748i −1.97754 − 1.65935i
\(5\) 6.58809 + 9.03311i 0.589257 + 0.807946i
\(6\) 6.29910 − 35.7240i 0.428600 − 2.43071i
\(7\) −23.1341 − 13.3565i −1.24913 − 0.721184i −0.278191 − 0.960526i \(-0.589735\pi\)
−0.970935 + 0.239342i \(0.923068\pi\)
\(8\) 58.6494 − 33.8613i 2.59196 − 1.49647i
\(9\) 17.7851 − 6.47325i 0.658707 − 0.239750i
\(10\) −57.4971 + 16.6004i −1.81822 + 0.524950i
\(11\) 18.1835 + 31.4948i 0.498413 + 0.863277i 0.999998 − 0.00183117i \(-0.000582881\pi\)
−0.501585 + 0.865108i \(0.667250\pi\)
\(12\) 121.206 + 69.9781i 2.91575 + 1.68341i
\(13\) −13.6659 − 2.40968i −0.291558 − 0.0514095i 0.0259562 − 0.999663i \(-0.491737\pi\)
−0.317514 + 0.948254i \(0.602848\pi\)
\(14\) 109.535 − 91.9109i 2.09104 − 1.75459i
\(15\) −54.5988 − 52.5337i −0.939823 − 0.904276i
\(16\) 34.2584 + 194.289i 0.535288 + 3.03577i
\(17\) −19.5168 + 53.6220i −0.278442 + 0.765014i 0.719097 + 0.694909i \(0.244556\pi\)
−0.997540 + 0.0701046i \(0.977667\pi\)
\(18\) 101.309i 1.32660i
\(19\) −6.98094 − 82.5243i −0.0842914 − 0.996441i
\(20\) 15.6871 − 230.362i 0.175387 − 2.57552i
\(21\) 170.114 + 61.9165i 1.76771 + 0.643394i
\(22\) −191.707 + 33.8030i −1.85782 + 0.327583i
\(23\) 54.9156 − 65.4458i 0.497856 − 0.593322i −0.457341 − 0.889291i \(-0.651198\pi\)
0.955197 + 0.295970i \(0.0956427\pi\)
\(24\) −351.576 + 295.007i −2.99021 + 2.50909i
\(25\) −38.1940 + 119.022i −0.305552 + 0.952175i
\(26\) 37.1394 − 64.3274i 0.280140 − 0.485217i
\(27\) 47.3831 − 27.3567i 0.337737 − 0.194992i
\(28\) 188.684 + 518.405i 1.27350 + 3.49890i
\(29\) −12.9054 + 4.69719i −0.0826371 + 0.0300775i −0.383008 − 0.923745i \(-0.625112\pi\)
0.300371 + 0.953823i \(0.402890\pi\)
\(30\) 364.198 − 178.453i 2.21644 − 1.08603i
\(31\) 127.370 − 220.611i 0.737946 − 1.27816i −0.215473 − 0.976510i \(-0.569129\pi\)
0.953419 − 0.301650i \(-0.0975375\pi\)
\(32\) −506.431 − 89.2975i −2.79766 − 0.493304i
\(33\) −158.419 − 188.797i −0.835674 − 0.995918i
\(34\) −233.985 − 196.337i −1.18024 − 0.990337i
\(35\) −31.7592 − 296.967i −0.153379 − 1.43419i
\(36\) −367.296 − 133.685i −1.70045 − 0.618912i
\(37\) − 363.545i − 1.61531i −0.589655 − 0.807655i \(-0.700736\pi\)
0.589655 − 0.807655i \(-0.299264\pi\)
\(38\) 427.873 + 115.968i 1.82658 + 0.495064i
\(39\) 94.0416 0.386121
\(40\) 692.260 + 306.705i 2.73640 + 1.21236i
\(41\) 6.40024 + 36.2976i 0.0243793 + 0.138262i 0.994568 − 0.104089i \(-0.0331927\pi\)
−0.970189 + 0.242351i \(0.922082\pi\)
\(42\) −622.872 + 742.310i −2.28836 + 2.72717i
\(43\) −200.017 − 238.370i −0.709354 − 0.845375i 0.284196 − 0.958766i \(-0.408273\pi\)
−0.993550 + 0.113391i \(0.963829\pi\)
\(44\) 130.419 − 739.640i 0.446848 − 2.53420i
\(45\) 175.643 + 118.008i 0.581853 + 0.390925i
\(46\) 228.652 + 396.037i 0.732889 + 1.26940i
\(47\) 51.6187 + 141.821i 0.160199 + 0.440143i 0.993659 − 0.112437i \(-0.0358655\pi\)
−0.833460 + 0.552580i \(0.813643\pi\)
\(48\) −457.278 − 1256.36i −1.37505 − 3.77792i
\(49\) 185.293 + 320.936i 0.540211 + 0.935674i
\(50\) −528.749 − 410.013i −1.49553 − 1.15969i
\(51\) 67.1520 − 380.838i 0.184376 − 1.04565i
\(52\) 184.211 + 219.534i 0.491260 + 0.585461i
\(53\) −248.195 + 295.787i −0.643248 + 0.766593i −0.984880 − 0.173241i \(-0.944576\pi\)
0.341631 + 0.939834i \(0.389021\pi\)
\(54\) 50.8558 + 288.417i 0.128159 + 0.726827i
\(55\) −164.701 + 371.745i −0.403788 + 0.911383i
\(56\) −1809.07 −4.31692
\(57\) 143.705 + 542.548i 0.333933 + 1.26074i
\(58\) − 73.5128i − 0.166426i
\(59\) −734.898 − 267.481i −1.62162 − 0.590221i −0.637930 − 0.770094i \(-0.720209\pi\)
−0.983690 + 0.179873i \(0.942431\pi\)
\(60\) 166.394 + 1555.89i 0.358024 + 3.34773i
\(61\) 189.191 + 158.750i 0.397106 + 0.333211i 0.819374 − 0.573260i \(-0.194321\pi\)
−0.422268 + 0.906471i \(0.638766\pi\)
\(62\) 876.479 + 1044.55i 1.79537 + 2.13964i
\(63\) −497.903 − 87.7937i −0.995713 − 0.175571i
\(64\) 587.164 − 1017.00i 1.14680 − 1.98632i
\(65\) −68.2657 − 139.321i −0.130266 − 0.265856i
\(66\) 1239.66 − 451.200i 2.31200 − 0.841498i
\(67\) −231.830 − 636.948i −0.422725 − 1.16143i −0.950141 − 0.311820i \(-0.899062\pi\)
0.527417 − 0.849607i \(-0.323161\pi\)
\(68\) 1020.58 − 589.233i 1.82006 − 1.05081i
\(69\) −289.487 + 501.407i −0.505075 + 0.874816i
\(70\) 1551.87 + 383.925i 2.64977 + 0.655541i
\(71\) −341.242 + 286.336i −0.570394 + 0.478617i −0.881776 − 0.471667i \(-0.843652\pi\)
0.311383 + 0.950285i \(0.399208\pi\)
\(72\) 823.894 − 981.878i 1.34857 − 1.60716i
\(73\) 133.904 − 23.6109i 0.214689 − 0.0378554i −0.0652691 − 0.997868i \(-0.520791\pi\)
0.279958 + 0.960012i \(0.409679\pi\)
\(74\) 1828.61 + 665.560i 2.87259 + 1.04554i
\(75\) 114.841 − 839.293i 0.176809 − 1.29218i
\(76\) −985.053 + 1398.23i −1.48675 + 2.11037i
\(77\) − 971.475i − 1.43779i
\(78\) −172.166 + 473.023i −0.249923 + 0.686659i
\(79\) −17.0147 − 96.4954i −0.0242318 − 0.137425i 0.970292 − 0.241938i \(-0.0777829\pi\)
−0.994524 + 0.104512i \(0.966672\pi\)
\(80\) −1529.34 + 1589.45i −2.13731 + 2.22133i
\(81\) −675.500 + 566.812i −0.926612 + 0.777520i
\(82\) −194.292 − 34.2589i −0.261658 − 0.0461373i
\(83\) −158.734 − 91.6451i −0.209919 − 0.121197i 0.391355 − 0.920240i \(-0.372007\pi\)
−0.601274 + 0.799043i \(0.705340\pi\)
\(84\) −1869.33 − 3237.77i −2.42810 − 4.20559i
\(85\) −612.951 + 176.969i −0.782164 + 0.225824i
\(86\) 1565.17 − 569.675i 1.96252 − 0.714298i
\(87\) 80.6025 − 46.5359i 0.0993276 − 0.0573468i
\(88\) 2132.91 + 1231.44i 2.58374 + 1.49172i
\(89\) 10.0724 − 57.1232i 0.0119963 − 0.0680342i −0.978222 − 0.207562i \(-0.933447\pi\)
0.990218 + 0.139527i \(0.0445583\pi\)
\(90\) −915.134 + 667.432i −1.07182 + 0.781706i
\(91\) 283.965 + 238.275i 0.327117 + 0.274484i
\(92\) −1737.56 + 306.379i −1.96906 + 0.347198i
\(93\) −590.446 + 1622.24i −0.658349 + 1.80880i
\(94\) −807.852 −0.886421
\(95\) 699.460 − 606.738i 0.755401 − 0.655263i
\(96\) 3484.98 3.70505
\(97\) 572.622 − 1573.27i 0.599391 − 1.64681i −0.153100 − 0.988211i \(-0.548926\pi\)
0.752491 − 0.658603i \(-0.228852\pi\)
\(98\) −1953.51 + 344.457i −2.01362 + 0.355055i
\(99\) 527.270 + 442.432i 0.535279 + 0.449153i
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.4.2 ✓ 168
5.4 even 2 inner 95.4.p.a.4.27 yes 168
19.5 even 9 inner 95.4.p.a.24.27 yes 168
95.24 even 18 inner 95.4.p.a.24.2 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.2 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.27 yes 168 5.4 even 2 inner
95.4.p.a.24.2 yes 168 95.24 even 18 inner
95.4.p.a.24.27 yes 168 19.5 even 9 inner