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 2.17
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.17

$q$-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+(0.913099 + 1.30404i) q^{2} +(0.393159 + 4.49382i) q^{3} +(1.86939 - 5.13610i) q^{4} +(10.1870 - 4.60717i) q^{5} +(-5.50113 + 4.61600i) q^{6} +(7.15577 - 26.7057i) q^{7} +(20.7062 - 5.54821i) q^{8} +(6.54994 - 1.15493i) q^{9} +(15.3096 + 9.07739i) q^{10} +(-30.6905 + 53.1574i) q^{11} +(23.8157 + 6.38140i) q^{12} +(0.0370636 + 0.00324264i) q^{13} +(41.3593 - 15.0535i) q^{14} +(24.7089 + 43.9670i) q^{15} +(-7.35404 - 6.17077i) q^{16} +(-67.7523 + 47.4407i) q^{17} +(7.48682 + 7.48682i) q^{18} +(79.8222 - 22.0775i) q^{19} +(-4.61955 - 60.9338i) q^{20} +(122.824 + 21.6572i) q^{21} +(-97.3429 + 8.51640i) q^{22} +(-106.298 + 49.5674i) q^{23} +(33.0735 + 90.8686i) q^{24} +(82.5479 - 93.8661i) q^{25} +(0.0296142 + 0.0512933i) q^{26} +(39.2885 + 146.627i) q^{27} +(-123.786 - 86.6762i) q^{28} +(-3.66084 - 20.7616i) q^{29} +(-34.7731 + 72.3676i) q^{30} +(103.876 - 59.9726i) q^{31} +(16.2786 - 186.065i) q^{32} +(-250.946 - 117.018i) q^{33} +(-123.729 - 45.0337i) q^{34} +(-50.1423 - 305.018i) q^{35} +(6.31254 - 35.8002i) q^{36} +(1.82101 - 1.82101i) q^{37} +(101.676 + 83.9325i) q^{38} +0.167832i q^{39} +(185.371 - 151.916i) q^{40} +(-167.868 + 200.058i) q^{41} +(83.9087 + 179.943i) q^{42} +(-86.6976 + 185.924i) q^{43} +(215.650 + 257.001i) q^{44} +(61.4029 - 41.9419i) q^{45} +(-161.698 - 93.3565i) q^{46} +(-245.973 + 351.286i) q^{47} +(24.8391 - 35.4738i) q^{48} +(-364.943 - 210.700i) q^{49} +(197.780 + 21.9368i) q^{50} +(-239.828 - 285.815i) q^{51} +(0.0859409 - 0.184301i) q^{52} +(-237.877 - 510.129i) q^{53} +(-155.333 + 185.119i) q^{54} +(-67.7367 + 682.908i) q^{55} -592.675i q^{56} +(130.595 + 350.027i) q^{57} +(23.7313 - 23.7313i) q^{58} +(2.47913 - 14.0598i) q^{59} +(272.010 - 44.7161i) q^{60} +(-140.112 - 50.9967i) q^{61} +(173.055 + 80.6971i) q^{62} +(16.0266 - 183.185i) q^{63} +(190.989 - 110.268i) q^{64} +(0.392504 - 0.137726i) q^{65} +(-76.5424 - 434.093i) q^{66} +(-553.884 - 387.834i) q^{67} +(117.005 + 436.668i) q^{68} +(-264.539 - 458.195i) q^{69} +(351.970 - 343.899i) q^{70} +(315.108 + 865.751i) q^{71} +(129.216 - 60.2546i) q^{72} +(395.795 - 34.6276i) q^{73} +(4.03744 + 0.711909i) q^{74} +(454.272 + 334.051i) q^{75} +(35.8265 - 451.247i) q^{76} +(1199.99 + 1199.99i) q^{77} +(-0.218860 + 0.153247i) q^{78} +(337.854 + 283.493i) q^{79} +(-103.345 - 28.9800i) q^{80} +(-474.721 + 172.784i) q^{81} +(-414.164 - 36.2346i) q^{82} +(-315.226 - 84.4645i) q^{83} +(340.840 - 590.352i) q^{84} +(-471.622 + 795.423i) q^{85} +(-321.616 + 56.7095i) q^{86} +(91.8598 - 24.6138i) q^{87} +(-340.554 + 1270.96i) q^{88} +(997.880 - 837.321i) q^{89} +(110.761 + 41.7748i) q^{90} +(0.351816 - 0.966606i) q^{91} +(55.8717 + 638.616i) q^{92} +(310.346 + 443.220i) q^{93} -682.688 q^{94} +(711.430 - 592.657i) q^{95} +842.543 q^{96} +(-252.945 - 361.242i) q^{97} +(-58.4678 - 668.290i) q^{98} +(-139.627 + 383.623i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.913099 + 1.30404i 0.322829 + 0.461048i 0.947461 − 0.319872i \(-0.103640\pi\)
−0.624632 + 0.780920i \(0.714751\pi\)
\(3\) 0.393159 + 4.49382i 0.0756634 + 0.864837i 0.935283 + 0.353902i \(0.115145\pi\)
−0.859619 + 0.510935i \(0.829299\pi\)
\(4\) 1.86939 − 5.13610i 0.233674 − 0.642013i
\(5\) 10.1870 − 4.60717i 0.911149 − 0.412078i
\(6\) −5.50113 + 4.61600i −0.374305 + 0.314079i
\(7\) 7.15577 − 26.7057i 0.386375 − 1.44197i −0.449612 − 0.893224i \(-0.648438\pi\)
0.835987 − 0.548749i \(-0.184896\pi\)
\(8\) 20.7062 − 5.54821i 0.915093 − 0.245198i
\(9\) 6.54994 − 1.15493i 0.242590 − 0.0427752i
\(10\) 15.3096 + 9.07739i 0.484133 + 0.287052i
\(11\) −30.6905 + 53.1574i −0.841229 + 1.45705i 0.0476271 + 0.998865i \(0.484834\pi\)
−0.888856 + 0.458186i \(0.848499\pi\)
\(12\) 23.8157 + 6.38140i 0.572917 + 0.153513i
\(13\) 0.0370636 + 0.00324264i 0.000790738 + 6.91806e-5i 0.0875511 − 0.996160i \(-0.472096\pi\)
−0.0867604 + 0.996229i \(0.527651\pi\)
\(14\) 41.3593 − 15.0535i 0.789552 − 0.287373i
\(15\) 24.7089 + 43.9670i 0.425321 + 0.756815i
\(16\) −7.35404 − 6.17077i −0.114907 − 0.0964183i
\(17\) −67.7523 + 47.4407i −0.966609 + 0.676827i −0.946410 − 0.322969i \(-0.895319\pi\)
−0.0201996 + 0.999796i \(0.506430\pi\)
\(18\) 7.48682 + 7.48682i 0.0980367 + 0.0980367i
\(19\) 79.8222 − 22.0775i 0.963814 − 0.266575i
\(20\) −4.61955 − 60.9338i −0.0516481 − 0.681261i
\(21\) 122.824 + 21.6572i 1.27631 + 0.225047i
\(22\) −97.3429 + 8.51640i −0.943344 + 0.0825319i
\(23\) −106.298 + 49.5674i −0.963678 + 0.449370i −0.839781 − 0.542925i \(-0.817317\pi\)
−0.123896 + 0.992295i \(0.539539\pi\)
\(24\) 33.0735 + 90.8686i 0.281296 + 0.772853i
\(25\) 82.5479 − 93.8661i 0.660383 − 0.750929i
\(26\) 0.0296142 + 0.0512933i 0.000223378 + 0.000386902i
\(27\) 39.2885 + 146.627i 0.280040 + 1.04512i
\(28\) −123.786 − 86.6762i −0.835480 − 0.585009i
\(29\) −3.66084 − 20.7616i −0.0234414 − 0.132943i 0.970841 − 0.239724i \(-0.0770568\pi\)
−0.994283 + 0.106781i \(0.965946\pi\)
\(30\) −34.7731 + 72.3676i −0.211622 + 0.440415i
\(31\) 103.876 − 59.9726i 0.601826 − 0.347465i −0.167933 − 0.985798i \(-0.553709\pi\)
0.769760 + 0.638334i \(0.220376\pi\)
\(32\) 16.2786 − 186.065i 0.0899272 − 1.02787i
\(33\) −250.946 − 117.018i −1.32376 − 0.617280i
\(34\) −123.729 − 45.0337i −0.624099 − 0.227154i
\(35\) −50.1423 − 305.018i −0.242160 − 1.47307i
\(36\) 6.31254 − 35.8002i 0.0292247 − 0.165742i
\(37\) 1.82101 − 1.82101i 0.00809115 − 0.00809115i −0.703050 − 0.711141i \(-0.748179\pi\)
0.711141 + 0.703050i \(0.248179\pi\)
\(38\) 101.676 + 83.9325i 0.434051 + 0.358306i
\(39\) 0.167832i 0.000689094i
\(40\) 185.371 − 151.916i 0.732745 − 0.600502i
\(41\) −167.868 + 200.058i −0.639430 + 0.762043i −0.984280 − 0.176615i \(-0.943485\pi\)
0.344850 + 0.938658i \(0.387930\pi\)
\(42\) 83.9087 + 179.943i 0.308271 + 0.661090i
\(43\) −86.6976 + 185.924i −0.307471 + 0.659374i −0.997951 − 0.0639771i \(-0.979622\pi\)
0.690480 + 0.723351i \(0.257399\pi\)
\(44\) 215.650 + 257.001i 0.738873 + 0.880555i
\(45\) 61.4029 − 41.9419i 0.203409 − 0.138941i
\(46\) −161.698 − 93.3565i −0.518285 − 0.299232i
\(47\) −245.973 + 351.286i −0.763379 + 1.09022i 0.229756 + 0.973248i \(0.426207\pi\)
−0.993135 + 0.116971i \(0.962682\pi\)
\(48\) 24.8391 − 35.4738i 0.0746918 − 0.106671i
\(49\) −364.943 − 210.700i −1.06397 − 0.614286i
\(50\) 197.780 + 21.9368i 0.559405 + 0.0620466i
\(51\) −239.828 − 285.815i −0.658482 − 0.784748i
\(52\) 0.0859409 − 0.184301i 0.000229189 − 0.000491498i
\(53\) −237.877 − 510.129i −0.616508 − 1.32211i −0.928256 − 0.371942i \(-0.878692\pi\)
0.311748 − 0.950165i \(-0.399086\pi\)
\(54\) −155.333 + 185.119i −0.391447 + 0.466508i
\(55\) −67.7367 + 682.908i −0.166066 + 1.67424i
\(56\) − 592.675i − 1.41428i
\(57\) 130.595 + 350.027i 0.303469 + 0.813372i
\(58\) 23.7313 − 23.7313i 0.0537254 − 0.0537254i
\(59\) 2.47913 − 14.0598i 0.00547042 − 0.0310243i −0.981950 − 0.189139i \(-0.939430\pi\)
0.987421 + 0.158114i \(0.0505415\pi\)
\(60\) 272.010 − 44.7161i 0.585272 − 0.0962137i
\(61\) −140.112 − 50.9967i −0.294091 − 0.107040i 0.190761 − 0.981636i \(-0.438904\pi\)
−0.484852 + 0.874596i \(0.661127\pi\)
\(62\) 173.055 + 80.6971i 0.354485 + 0.165299i
\(63\) 16.0266 − 183.185i 0.0320502 − 0.366336i
\(64\) 190.989 − 110.268i 0.373026 − 0.215367i
\(65\) 0.392504 − 0.137726i 0.000748988 − 0.000262812i
\(66\) −76.5424 − 434.093i −0.142753 − 0.809594i
\(67\) −553.884 − 387.834i −1.00997 − 0.707186i −0.0533616 − 0.998575i \(-0.516994\pi\)
−0.956605 + 0.291389i \(0.905882\pi\)
\(68\) 117.005 + 436.668i 0.208661 + 0.778732i
\(69\) −264.539 − 458.195i −0.461547 − 0.799423i
\(70\) 351.970 − 343.899i 0.600979 − 0.587197i
\(71\) 315.108 + 865.751i 0.526710 + 1.44712i 0.862922 + 0.505337i \(0.168632\pi\)
−0.336212 + 0.941786i \(0.609146\pi\)
\(72\) 129.216 − 60.2546i 0.211504 − 0.0986260i
\(73\) 395.795 − 34.6276i 0.634580 − 0.0555186i 0.234673 − 0.972074i \(-0.424598\pi\)
0.399907 + 0.916556i \(0.369042\pi\)
\(74\) 4.03744 + 0.711909i 0.00634247 + 0.00111835i
\(75\) 454.272 + 334.051i 0.699397 + 0.514306i
\(76\) 35.8265 − 451.247i 0.0540735 − 0.681073i
\(77\) 1199.99 + 1199.99i 1.77600 + 1.77600i
\(78\) −0.218860 + 0.153247i −0.000317705 + 0.000222460i
\(79\) 337.854 + 283.493i 0.481159 + 0.403740i 0.850845 − 0.525416i \(-0.176090\pi\)
−0.369686 + 0.929157i \(0.620535\pi\)
\(80\) −103.345 − 28.9800i −0.144429 − 0.0405008i
\(81\) −474.721 + 172.784i −0.651195 + 0.237016i
\(82\) −414.164 − 36.2346i −0.557765 − 0.0487981i
\(83\) −315.226 − 84.4645i −0.416874 − 0.111701i 0.0442844 − 0.999019i \(-0.485899\pi\)
−0.461158 + 0.887318i \(0.652566\pi\)
\(84\) 340.840 − 590.352i 0.442722 − 0.766817i
\(85\) −471.622 + 795.423i −0.601819 + 1.01501i
\(86\) −321.616 + 56.7095i −0.403264 + 0.0711063i
\(87\) 91.8598 − 24.6138i 0.113200 − 0.0303319i
\(88\) −340.554 + 1270.96i −0.412536 + 1.53961i
\(89\) 997.880 − 837.321i 1.18848 − 0.997256i 0.188600 − 0.982054i \(-0.439605\pi\)
0.999884 − 0.0152024i \(-0.00483926\pi\)
\(90\) 110.761 + 41.7748i 0.129725 + 0.0489272i
\(91\) 0.351816 − 0.966606i 0.000405278 − 0.00111349i
\(92\) 55.8717 + 638.616i 0.0633155 + 0.723700i
\(93\) 310.346 + 443.220i 0.346036 + 0.494191i
\(94\) −682.688 −0.749084
\(95\) 711.430 − 592.657i 0.768328 − 0.640056i
\(96\) 842.543 0.895747
\(97\) −252.945 − 361.242i −0.264769 − 0.378130i 0.664650 − 0.747155i \(-0.268581\pi\)
−0.929420 + 0.369025i \(0.879692\pi\)
\(98\) −58.4678 − 668.290i −0.0602668 − 0.688852i
\(99\) −139.627 + 383.623i −0.141748 + 0.389450i
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.2.17 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.17 yes 336
19.10 odd 18 inner 95.4.r.a.67.17 yes 336
95.48 even 36 inner 95.4.r.a.48.17 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.17 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.17 yes 336 95.48 even 36 inner
95.4.r.a.67.17 yes 336 19.10 odd 18 inner
95.4.r.a.78.17 yes 336 5.3 odd 4 inner