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 2.7
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.7

$q$-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+(-1.99889 - 2.85471i) q^{2} +(-0.595193 - 6.80308i) q^{3} +(-1.41766 + 3.89499i) q^{4} +(0.600619 + 11.1642i) q^{5} +(-18.2311 + 15.2977i) q^{6} +(-8.87294 + 33.1143i) q^{7} +(-12.9769 + 3.47715i) q^{8} +(-19.3379 + 3.40979i) q^{9} +(30.6700 - 24.0306i) q^{10} +(8.50209 - 14.7261i) q^{11} +(27.3417 + 7.32619i) q^{12} +(-45.0618 - 3.94239i) q^{13} +(112.268 - 40.8621i) q^{14} +(75.5935 - 10.7309i) q^{15} +(61.2675 + 51.4095i) q^{16} +(-63.4188 + 44.4063i) q^{17} +(48.3883 + 48.3883i) q^{18} +(-77.7848 - 28.4345i) q^{19} +(-44.3359 - 13.4876i) q^{20} +(230.560 + 40.6540i) q^{21} +(-59.0334 + 5.16476i) q^{22} +(32.2329 - 15.0305i) q^{23} +(31.3791 + 86.2133i) q^{24} +(-124.279 + 13.4109i) q^{25} +(78.8192 + 136.519i) q^{26} +(-13.0155 - 48.5744i) q^{27} +(-116.401 - 81.5048i) q^{28} +(10.8978 + 61.8048i) q^{29} +(-181.737 - 194.348i) q^{30} +(38.0836 - 21.9876i) q^{31} +(14.9251 - 170.595i) q^{32} +(-105.243 - 49.0756i) q^{33} +(253.535 + 92.2790i) q^{34} +(-375.023 - 79.1702i) q^{35} +(14.1334 - 80.1547i) q^{36} +(-83.9599 + 83.9599i) q^{37} +(74.3113 + 278.891i) q^{38} +308.905i q^{39} +(-46.6137 - 142.788i) q^{40} +(103.536 - 123.389i) q^{41} +(-344.809 - 739.446i) q^{42} +(-209.826 + 449.973i) q^{43} +(45.3048 + 53.9921i) q^{44} +(-49.6822 - 213.844i) q^{45} +(-107.338 - 61.9715i) q^{46} +(260.403 - 371.895i) q^{47} +(313.277 - 447.406i) q^{48} +(-720.778 - 416.142i) q^{49} +(286.703 + 327.973i) q^{50} +(339.846 + 405.013i) q^{51} +(79.2378 - 169.926i) q^{52} +(56.0438 + 120.186i) q^{53} +(-112.650 + 134.251i) q^{54} +(169.511 + 86.0743i) q^{55} -460.573i q^{56} +(-147.145 + 546.101i) q^{57} +(154.651 - 154.651i) q^{58} +(-110.159 + 624.740i) q^{59} +(-65.3691 + 309.648i) q^{60} +(-365.052 - 132.868i) q^{61} +(-138.893 - 64.7669i) q^{62} +(58.6711 - 670.614i) q^{63} +(37.2778 - 21.5223i) q^{64} +(16.9487 - 505.446i) q^{65} +(70.2725 + 398.535i) q^{66} +(-150.960 - 105.704i) q^{67} +(-83.0558 - 309.968i) q^{68} +(-121.438 - 210.337i) q^{69} +(523.623 + 1228.84i) q^{70} +(230.498 + 633.287i) q^{71} +(239.089 - 111.489i) q^{72} +(82.8189 - 7.24572i) q^{73} +(407.508 + 71.8547i) q^{74} +(165.205 + 837.495i) q^{75} +(221.024 - 262.661i) q^{76} +(412.204 + 412.204i) q^{77} +(881.837 - 617.469i) q^{78} +(4.73346 + 3.97185i) q^{79} +(-537.147 + 714.879i) q^{80} +(-820.913 + 298.788i) q^{81} +(-559.197 - 48.9234i) q^{82} +(891.384 + 238.846i) q^{83} +(-485.203 + 840.396i) q^{84} +(-533.851 - 681.348i) q^{85} +(1703.96 - 300.455i) q^{86} +(413.977 - 110.925i) q^{87} +(-59.1261 + 220.661i) q^{88} +(1130.61 - 948.696i) q^{89} +(-511.154 + 569.279i) q^{90} +(530.380 - 1457.21i) q^{91} +(12.8481 + 146.855i) q^{92} +(-172.250 - 245.999i) q^{93} -1582.17 q^{94} +(270.729 - 885.483i) q^{95} -1169.45 q^{96} +(-626.779 - 895.133i) q^{97} +(252.793 + 2889.44i) q^{98} +(-114.200 + 313.761i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99889 − 2.85471i −0.706715 − 1.00929i −0.998603 − 0.0528412i \(-0.983172\pi\)
0.291888 − 0.956453i \(-0.405717\pi\)
\(3\) −0.595193 − 6.80308i −0.114545 − 1.30925i −0.808183 − 0.588931i \(-0.799549\pi\)
0.693639 − 0.720323i \(-0.256006\pi\)
\(4\) −1.41766 + 3.89499i −0.177207 + 0.486874i
\(5\) 0.600619 + 11.1642i 0.0537210 + 0.998556i
\(6\) −18.2311 + 15.2977i −1.24047 + 1.04088i
\(7\) −8.87294 + 33.1143i −0.479094 + 1.78800i 0.126206 + 0.992004i \(0.459720\pi\)
−0.605299 + 0.795998i \(0.706947\pi\)
\(8\) −12.9769 + 3.47715i −0.573503 + 0.153670i
\(9\) −19.3379 + 3.40979i −0.716218 + 0.126288i
\(10\) 30.6700 − 24.0306i 0.969871 − 0.759915i
\(11\) 8.50209 − 14.7261i 0.233043 − 0.403643i −0.725659 − 0.688055i \(-0.758465\pi\)
0.958702 + 0.284412i \(0.0917982\pi\)
\(12\) 27.3417 + 7.32619i 0.657739 + 0.176241i
\(13\) −45.0618 − 3.94239i −0.961376 − 0.0841095i −0.404350 − 0.914604i \(-0.632502\pi\)
−0.557026 + 0.830495i \(0.688058\pi\)
\(14\) 112.268 − 40.8621i 2.14320 − 0.780062i
\(15\) 75.5935 − 10.7309i 1.30121 − 0.184714i
\(16\) 61.2675 + 51.4095i 0.957304 + 0.803273i
\(17\) −63.4188 + 44.4063i −0.904783 + 0.633536i −0.930563 − 0.366132i \(-0.880682\pi\)
0.0257800 + 0.999668i \(0.491793\pi\)
\(18\) 48.3883 + 48.3883i 0.633624 + 0.633624i
\(19\) −77.7848 − 28.4345i −0.939214 − 0.343332i
\(20\) −44.3359 − 13.4876i −0.495690 − 0.150796i
\(21\) 230.560 + 40.6540i 2.39583 + 0.422449i
\(22\) −59.0334 + 5.16476i −0.572090 + 0.0500514i
\(23\) 32.2329 − 15.0305i 0.292219 − 0.136264i −0.270981 − 0.962585i \(-0.587348\pi\)
0.563200 + 0.826321i \(0.309570\pi\)
\(24\) 31.3791 + 86.2133i 0.266884 + 0.733259i
\(25\) −124.279 + 13.4109i −0.994228 + 0.107287i
\(26\) 78.8192 + 136.519i 0.594528 + 1.02975i
\(27\) −13.0155 − 48.5744i −0.0927715 − 0.346228i
\(28\) −116.401 − 81.5048i −0.785632 − 0.550105i
\(29\) 10.8978 + 61.8048i 0.0697821 + 0.395754i 0.999614 + 0.0277701i \(0.00884063\pi\)
−0.929832 + 0.367984i \(0.880048\pi\)
\(30\) −181.737 − 194.348i −1.10602 − 1.18276i
\(31\) 38.0836 − 21.9876i 0.220645 − 0.127390i −0.385604 − 0.922665i \(-0.626007\pi\)
0.606249 + 0.795275i \(0.292673\pi\)
\(32\) 14.9251 − 170.595i 0.0824503 − 0.942412i
\(33\) −105.243 − 49.0756i −0.555165 − 0.258878i
\(34\) 253.535 + 92.2790i 1.27885 + 0.465462i
\(35\) −375.023 − 79.1702i −1.81116 − 0.382349i
\(36\) 14.1334 − 80.1547i 0.0654326 − 0.371087i
\(37\) −83.9599 + 83.9599i −0.373052 + 0.373052i −0.868588 − 0.495536i \(-0.834972\pi\)
0.495536 + 0.868588i \(0.334972\pi\)
\(38\) 74.3113 + 278.891i 0.317234 + 1.19058i
\(39\) 308.905i 1.26832i
\(40\) −46.6137 − 142.788i −0.184257 − 0.564420i
\(41\) 103.536 − 123.389i 0.394379 − 0.470003i −0.531918 − 0.846796i \(-0.678529\pi\)
0.926298 + 0.376793i \(0.122973\pi\)
\(42\) −344.809 − 739.446i −1.26679 − 2.71664i
\(43\) −209.826 + 449.973i −0.744143 + 1.59582i 0.0577353 + 0.998332i \(0.481612\pi\)
−0.801878 + 0.597488i \(0.796166\pi\)
\(44\) 45.3048 + 53.9921i 0.155226 + 0.184991i
\(45\) −49.6822 − 213.844i −0.164582 − 0.708399i
\(46\) −107.338 − 61.9715i −0.344046 − 0.198635i
\(47\) 260.403 − 371.895i 0.808165 − 1.15418i −0.177276 − 0.984161i \(-0.556729\pi\)
0.985441 − 0.170018i \(-0.0543825\pi\)
\(48\) 313.277 − 447.406i 0.942035 − 1.34536i
\(49\) −720.778 − 416.142i −2.10139 − 1.21324i
\(50\) 286.703 + 327.973i 0.810920 + 0.927647i
\(51\) 339.846 + 405.013i 0.933098 + 1.11202i
\(52\) 79.2378 − 169.926i 0.211314 − 0.453164i
\(53\) 56.0438 + 120.186i 0.145249 + 0.311488i 0.965450 − 0.260587i \(-0.0839162\pi\)
−0.820201 + 0.572075i \(0.806138\pi\)
\(54\) −112.650 + 134.251i −0.283883 + 0.338318i
\(55\) 169.511 + 86.0743i 0.415580 + 0.211023i
\(56\) − 460.573i − 1.09905i
\(57\) −147.145 + 546.101i −0.341927 + 1.26900i
\(58\) 154.651 − 154.651i 0.350116 − 0.350116i
\(59\) −110.159 + 624.740i −0.243075 + 1.37855i 0.581846 + 0.813299i \(0.302331\pi\)
−0.824921 + 0.565248i \(0.808781\pi\)
\(60\) −65.3691 + 309.648i −0.140652 + 0.666257i
\(61\) −365.052 − 132.868i −0.766232 − 0.278886i −0.0708127 − 0.997490i \(-0.522559\pi\)
−0.695420 + 0.718604i \(0.744781\pi\)
\(62\) −138.893 − 64.7669i −0.284507 − 0.132668i
\(63\) 58.6711 − 670.614i 0.117331 − 1.34110i
\(64\) 37.2778 − 21.5223i 0.0728082 − 0.0420358i
\(65\) 16.9487 − 505.446i 0.0323420 − 0.964506i
\(66\) 70.2725 + 398.535i 0.131060 + 0.743278i
\(67\) −150.960 − 105.704i −0.275265 − 0.192742i 0.427790 − 0.903878i \(-0.359292\pi\)
−0.703055 + 0.711136i \(0.748181\pi\)
\(68\) −83.0558 − 309.968i −0.148118 − 0.552782i
\(69\) −121.438 − 210.337i −0.211876 − 0.366980i
\(70\) 523.623 + 1228.84i 0.894070 + 2.09820i
\(71\) 230.498 + 633.287i 0.385282 + 1.05855i 0.969100 + 0.246669i \(0.0793361\pi\)
−0.583818 + 0.811885i \(0.698442\pi\)
\(72\) 239.089 − 111.489i 0.391346 − 0.182488i
\(73\) 82.8189 − 7.24572i 0.132784 − 0.0116171i −0.0205700 − 0.999788i \(-0.506548\pi\)
0.153354 + 0.988171i \(0.450993\pi\)
\(74\) 407.508 + 71.8547i 0.640160 + 0.112878i
\(75\) 165.205 + 837.495i 0.254349 + 1.28941i
\(76\) 221.024 − 262.661i 0.333595 − 0.396437i
\(77\) 412.204 + 412.204i 0.610065 + 0.610065i
\(78\) 881.837 − 617.469i 1.28011 − 0.896340i
\(79\) 4.73346 + 3.97185i 0.00674122 + 0.00565655i 0.646152 − 0.763209i \(-0.276377\pi\)
−0.639411 + 0.768865i \(0.720822\pi\)
\(80\) −537.147 + 714.879i −0.750686 + 0.999074i
\(81\) −820.913 + 298.788i −1.12608 + 0.409860i
\(82\) −559.197 − 48.9234i −0.753085 − 0.0658864i
\(83\) 891.384 + 238.846i 1.17882 + 0.315864i 0.794459 − 0.607318i \(-0.207755\pi\)
0.384362 + 0.923182i \(0.374421\pi\)
\(84\) −485.203 + 840.396i −0.630238 + 1.09160i
\(85\) −533.851 − 681.348i −0.681227 − 0.869442i
\(86\) 1703.96 − 300.455i 2.13655 − 0.376731i
\(87\) 413.977 − 110.925i 0.510149 − 0.136694i
\(88\) −59.1261 + 220.661i −0.0716234 + 0.267302i
\(89\) 1130.61 − 948.696i 1.34657 − 1.12991i 0.366683 − 0.930346i \(-0.380493\pi\)
0.979886 − 0.199560i \(-0.0639512\pi\)
\(90\) −511.154 + 569.279i −0.598670 + 0.666748i
\(91\) 530.380 − 1457.21i 0.610977 − 1.67865i
\(92\) 12.8481 + 146.855i 0.0145599 + 0.166421i
\(93\) −172.250 − 245.999i −0.192059 − 0.274289i
\(94\) −1582.17 −1.73605
\(95\) 270.729 − 885.483i 0.292381 − 0.956302i
\(96\) −1169.45 −1.24330
\(97\) −626.779 − 895.133i −0.656080 − 0.936980i 0.343918 − 0.939000i \(-0.388246\pi\)
−0.999998 + 0.00202023i \(0.999357\pi\)
\(98\) 252.793 + 2889.44i 0.260571 + 2.97834i
\(99\) −114.200 + 313.761i −0.115934 + 0.318527i
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.2.7 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.7 yes 336
19.10 odd 18 inner 95.4.r.a.67.7 yes 336
95.48 even 36 inner 95.4.r.a.48.7 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.7 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.7 yes 336 95.48 even 36 inner
95.4.r.a.67.7 yes 336 19.10 odd 18 inner
95.4.r.a.78.7 yes 336 5.3 odd 4 inner