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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(159,380)] 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("380.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (of order \(6\), degree \(2\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 159.101
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.101

$q$-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+(1.82898 + 0.809227i) q^{2} +(0.0580338 + 0.100518i) q^{3} +(2.69030 + 2.96011i) q^{4} +(2.96569 + 4.02550i) q^{5} +(0.0248009 + 0.230807i) q^{6} +7.81754 q^{7} +(2.52509 + 7.59104i) q^{8} +(4.49326 - 7.78256i) q^{9} +(2.16663 + 9.76247i) q^{10} +3.02550i q^{11} +(-0.141415 + 0.442209i) q^{12} +(-11.0968 - 6.40674i) q^{13} +(14.2981 + 6.32616i) q^{14} +(-0.232523 + 0.531719i) q^{15} +(-1.52455 + 15.9272i) q^{16} +(-17.3218 + 10.0008i) q^{17} +(14.5159 - 10.5980i) q^{18} +(9.66050 - 16.3608i) q^{19} +(-3.93735 + 19.6086i) q^{20} +(0.453682 + 0.785800i) q^{21} +(-2.44832 + 5.53356i) q^{22} +(17.7149 - 30.6831i) q^{23} +(-0.616492 + 0.694353i) q^{24} +(-7.40936 + 23.8768i) q^{25} +(-15.1113 - 20.6976i) q^{26} +2.08765 q^{27} +(21.0315 + 23.1408i) q^{28} +(-20.2682 + 35.1055i) q^{29} +(-0.855562 + 0.784337i) q^{30} +31.8484i q^{31} +(-15.6771 + 27.8968i) q^{32} +(-0.304116 + 0.175581i) q^{33} +(-39.7741 + 4.27385i) q^{34} +(23.1844 + 31.4695i) q^{35} +(35.1255 - 7.63687i) q^{36} -47.3562i q^{37} +(30.9084 - 22.1059i) q^{38} -1.48723i q^{39} +(-23.0691 + 32.6774i) q^{40} +(15.6324 + 27.0762i) q^{41} +(0.193882 + 1.80434i) q^{42} +(-16.3862 - 28.3818i) q^{43} +(-8.95582 + 8.13951i) q^{44} +(44.6544 - 4.99301i) q^{45} +(57.2296 - 41.7832i) q^{46} +(-19.2417 + 33.3277i) q^{47} +(-1.68944 + 0.771073i) q^{48} +12.1139 q^{49} +(-32.8733 + 37.6742i) q^{50} +(-2.01051 - 1.16077i) q^{51} +(-10.8891 - 50.0838i) q^{52} +(-62.0748 - 35.8389i) q^{53} +(3.81827 + 1.68939i) q^{54} +(-12.1792 + 8.97270i) q^{55} +(19.7400 + 59.3433i) q^{56} +(2.20518 + 0.0215721i) q^{57} +(-65.4783 + 47.8055i) q^{58} +(85.4045 - 49.3083i) q^{59} +(-2.19951 + 0.742190i) q^{60} +(17.0850 - 29.5921i) q^{61} +(-25.7726 + 58.2499i) q^{62} +(35.1263 - 60.8405i) q^{63} +(-51.2478 + 38.3361i) q^{64} +(-7.11931 - 63.6706i) q^{65} +(-0.698306 + 0.0750352i) q^{66} +(40.0811 - 69.4225i) q^{67} +(-76.2044 - 24.3695i) q^{68} +4.11225 q^{69} +(16.9377 + 76.3184i) q^{70} +(-14.5276 + 8.38749i) q^{71} +(70.4237 + 14.4569i) q^{72} +(57.9890 - 33.4800i) q^{73} +(38.3219 - 86.6133i) q^{74} +(-2.83003 + 0.640891i) q^{75} +(74.4194 - 15.4192i) q^{76} +23.6520i q^{77} +(1.20351 - 2.72011i) q^{78} +(-87.2930 + 50.3986i) q^{79} +(-68.6364 + 41.0981i) q^{80} +(-40.3182 - 69.8332i) q^{81} +(6.68057 + 62.1719i) q^{82} -38.3764 q^{83} +(-1.10552 + 3.45699i) q^{84} +(-91.6293 - 40.0699i) q^{85} +(-7.00270 - 65.1698i) q^{86} -4.70496 q^{87} +(-22.9667 + 7.63966i) q^{88} +(-22.1989 + 38.4497i) q^{89} +(85.7122 + 27.0034i) q^{90} +(-86.7496 - 50.0849i) q^{91} +(138.484 - 30.1087i) q^{92} +(-3.20132 + 1.84828i) q^{93} +(-62.1623 + 45.3845i) q^{94} +(94.5104 - 9.63257i) q^{95} +(-3.71392 + 0.0431331i) q^{96} +(-102.346 + 59.0898i) q^{97} +(22.1560 + 9.80289i) q^{98} +(23.5461 + 13.5944i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82898 + 0.809227i 0.914488 + 0.404614i
\(3\) 0.0580338 + 0.100518i 0.0193446 + 0.0335059i 0.875536 − 0.483154i \(-0.160509\pi\)
−0.856191 + 0.516659i \(0.827175\pi\)
\(4\) 2.69030 + 2.96011i 0.672576 + 0.740028i
\(5\) 2.96569 + 4.02550i 0.593138 + 0.805101i
\(6\) 0.0248009 + 0.230807i 0.00413349 + 0.0384678i
\(7\) 7.81754 1.11679 0.558396 − 0.829575i \(-0.311417\pi\)
0.558396 + 0.829575i \(0.311417\pi\)
\(8\) 2.52509 + 7.59104i 0.315636 + 0.948880i
\(9\) 4.49326 − 7.78256i 0.499252 − 0.864729i
\(10\) 2.16663 + 9.76247i 0.216663 + 0.976247i
\(11\) 3.02550i 0.275045i 0.990499 + 0.137523i \(0.0439140\pi\)
−0.990499 + 0.137523i \(0.956086\pi\)
\(12\) −0.141415 + 0.442209i −0.0117846 + 0.0368508i
\(13\) −11.0968 − 6.40674i −0.853600 − 0.492826i 0.00826419 − 0.999966i \(-0.497369\pi\)
−0.861864 + 0.507140i \(0.830703\pi\)
\(14\) 14.2981 + 6.32616i 1.02129 + 0.451869i
\(15\) −0.232523 + 0.531719i −0.0155016 + 0.0354480i
\(16\) −1.52455 + 15.9272i −0.0952843 + 0.995450i
\(17\) −17.3218 + 10.0008i −1.01893 + 0.588280i −0.913794 − 0.406177i \(-0.866862\pi\)
−0.105137 + 0.994458i \(0.533528\pi\)
\(18\) 14.5159 − 10.5980i 0.806441 − 0.588780i
\(19\) 9.66050 − 16.3608i 0.508448 − 0.861093i
\(20\) −3.93735 + 19.6086i −0.196867 + 0.980430i
\(21\) 0.453682 + 0.785800i 0.0216039 + 0.0374190i
\(22\) −2.44832 + 5.53356i −0.111287 + 0.251526i
\(23\) 17.7149 − 30.6831i 0.770212 − 1.33405i −0.167235 − 0.985917i \(-0.553484\pi\)
0.937447 − 0.348129i \(-0.113183\pi\)
\(24\) −0.616492 + 0.694353i −0.0256872 + 0.0289314i
\(25\) −7.40936 + 23.8768i −0.296375 + 0.955072i
\(26\) −15.1113 − 20.6976i −0.581202 − 0.796061i
\(27\) 2.08765 0.0773205
\(28\) 21.0315 + 23.1408i 0.751126 + 0.826457i
\(29\) −20.2682 + 35.1055i −0.698902 + 1.21053i 0.269946 + 0.962876i \(0.412994\pi\)
−0.968848 + 0.247658i \(0.920339\pi\)
\(30\) −0.855562 + 0.784337i −0.0285187 + 0.0261446i
\(31\) 31.8484i 1.02737i 0.857980 + 0.513684i \(0.171719\pi\)
−0.857980 + 0.513684i \(0.828281\pi\)
\(32\) −15.6771 + 27.8968i −0.489909 + 0.871774i
\(33\) −0.304116 + 0.175581i −0.00921563 + 0.00532065i
\(34\) −39.7741 + 4.27385i −1.16983 + 0.125702i
\(35\) 23.1844 + 31.4695i 0.662411 + 0.899129i
\(36\) 35.1255 − 7.63687i 0.975709 − 0.212135i
\(37\) − 47.3562i − 1.27990i −0.768418 − 0.639948i \(-0.778956\pi\)
0.768418 − 0.639948i \(-0.221044\pi\)
\(38\) 30.9084 − 22.1059i 0.813379 − 0.581734i
\(39\) − 1.48723i − 0.0381341i
\(40\) −23.0691 + 32.6774i −0.576728 + 0.816936i
\(41\) 15.6324 + 27.0762i 0.381279 + 0.660395i 0.991245 − 0.132032i \(-0.0421503\pi\)
−0.609966 + 0.792427i \(0.708817\pi\)
\(42\) 0.193882 + 1.80434i 0.00461624 + 0.0429605i
\(43\) −16.3862 − 28.3818i −0.381075 − 0.660041i 0.610141 − 0.792293i \(-0.291113\pi\)
−0.991216 + 0.132251i \(0.957779\pi\)
\(44\) −8.95582 + 8.13951i −0.203541 + 0.184989i
\(45\) 44.6544 − 4.99301i 0.992319 − 0.110956i
\(46\) 57.2296 − 41.7832i 1.24412 − 0.908331i
\(47\) −19.2417 + 33.3277i −0.409399 + 0.709099i −0.994822 − 0.101628i \(-0.967595\pi\)
0.585424 + 0.810727i \(0.300928\pi\)
\(48\) −1.68944 + 0.771073i −0.0351966 + 0.0160640i
\(49\) 12.1139 0.247222
\(50\) −32.8733 + 37.6742i −0.657466 + 0.753484i
\(51\) −2.01051 − 1.16077i −0.0394217 − 0.0227601i
\(52\) −10.8891 − 50.0838i −0.209405 − 0.963151i
\(53\) −62.0748 − 35.8389i −1.17122 − 0.676205i −0.217255 − 0.976115i \(-0.569710\pi\)
−0.953968 + 0.299909i \(0.903044\pi\)
\(54\) 3.81827 + 1.68939i 0.0707087 + 0.0312849i
\(55\) −12.1792 + 8.97270i −0.221439 + 0.163140i
\(56\) 19.7400 + 59.3433i 0.352500 + 1.05970i
\(57\) 2.20518 + 0.0215721i 0.0386874 + 0.000378457i
\(58\) −65.4783 + 47.8055i −1.12894 + 0.824233i
\(59\) 85.4045 − 49.3083i 1.44753 − 0.835734i 0.449200 − 0.893431i \(-0.351709\pi\)
0.998334 + 0.0576968i \(0.0183757\pi\)
\(60\) −2.19951 + 0.742190i −0.0366585 + 0.0123698i
\(61\) 17.0850 − 29.5921i 0.280082 − 0.485116i −0.691323 − 0.722546i \(-0.742972\pi\)
0.971405 + 0.237430i \(0.0763050\pi\)
\(62\) −25.7726 + 58.2499i −0.415687 + 0.939515i
\(63\) 35.1263 − 60.8405i 0.557560 − 0.965722i
\(64\) −51.2478 + 38.3361i −0.800747 + 0.599002i
\(65\) −7.11931 − 63.6706i −0.109528 − 0.979547i
\(66\) −0.698306 + 0.0750352i −0.0105804 + 0.00113690i
\(67\) 40.0811 − 69.4225i 0.598225 − 1.03616i −0.394858 − 0.918742i \(-0.629206\pi\)
0.993083 − 0.117414i \(-0.0374606\pi\)
\(68\) −76.2044 − 24.3695i −1.12065 − 0.358375i
\(69\) 4.11225 0.0595978
\(70\) 16.9377 + 76.3184i 0.241967 + 1.09026i
\(71\) −14.5276 + 8.38749i −0.204614 + 0.118134i −0.598806 − 0.800894i \(-0.704358\pi\)
0.394192 + 0.919028i \(0.371024\pi\)
\(72\) 70.4237 + 14.4569i 0.978106 + 0.200790i
\(73\) 57.9890 − 33.4800i 0.794370 − 0.458630i −0.0471287 − 0.998889i \(-0.515007\pi\)
0.841499 + 0.540259i \(0.181674\pi\)
\(74\) 38.3219 − 86.6133i 0.517864 − 1.17045i
\(75\) −2.83003 + 0.640891i −0.0377337 + 0.00854521i
\(76\) 74.4194 − 15.4192i 0.979203 − 0.202884i
\(77\) 23.6520i 0.307168i
\(78\) 1.20351 − 2.72011i 0.0154296 − 0.0348732i
\(79\) −87.2930 + 50.3986i −1.10497 + 0.637957i −0.937523 − 0.347923i \(-0.886887\pi\)
−0.167452 + 0.985880i \(0.553554\pi\)
\(80\) −68.6364 + 41.0981i −0.857954 + 0.513726i
\(81\) −40.3182 − 69.8332i −0.497756 − 0.862138i
\(82\) 6.68057 + 62.1719i 0.0814703 + 0.758194i
\(83\) −38.3764 −0.462366 −0.231183 − 0.972910i \(-0.574260\pi\)
−0.231183 + 0.972910i \(0.574260\pi\)
\(84\) −1.10552 + 3.45699i −0.0131609 + 0.0411546i
\(85\) −91.6293 − 40.0699i −1.07799 − 0.471411i
\(86\) −7.00270 − 65.1698i −0.0814267 − 0.757788i
\(87\) −4.70496 −0.0540799
\(88\) −22.9667 + 7.63966i −0.260985 + 0.0868144i
\(89\) −22.1989 + 38.4497i −0.249426 + 0.432019i −0.963367 − 0.268187i \(-0.913575\pi\)
0.713941 + 0.700206i \(0.246909\pi\)
\(90\) 85.7122 + 27.0034i 0.952358 + 0.300038i
\(91\) −86.7496 − 50.0849i −0.953292 − 0.550384i
\(92\) 138.484 − 30.1087i 1.50526 − 0.327268i
\(93\) −3.20132 + 1.84828i −0.0344228 + 0.0198740i
\(94\) −62.1623 + 45.3845i −0.661301 + 0.482814i
\(95\) 94.5104 − 9.63257i 0.994846 − 0.101396i
\(96\) −3.71392 + 0.0431331i −0.0386866 + 0.000449303i
\(97\) −102.346 + 59.0898i −1.05512 + 0.609173i −0.924078 − 0.382204i \(-0.875165\pi\)
−0.131040 + 0.991377i \(0.541832\pi\)
\(98\) 22.1560 + 9.80289i 0.226082 + 0.100029i
\(99\) 23.5461 + 13.5944i 0.237840 + 0.137317i
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 380.3.p.a.159.101 yes 232
4.3 odd 2 inner 380.3.p.a.159.61 yes 232
5.4 even 2 inner 380.3.p.a.159.16 ✓ 232
19.11 even 3 inner 380.3.p.a.239.56 yes 232
20.19 odd 2 inner 380.3.p.a.159.56 yes 232
76.11 odd 6 inner 380.3.p.a.239.16 yes 232
95.49 even 6 inner 380.3.p.a.239.61 yes 232
380.239 odd 6 inner 380.3.p.a.239.101 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.16 ✓ 232 5.4 even 2 inner
380.3.p.a.159.56 yes 232 20.19 odd 2 inner
380.3.p.a.159.61 yes 232 4.3 odd 2 inner
380.3.p.a.159.101 yes 232 1.1 even 1 trivial
380.3.p.a.239.16 yes 232 76.11 odd 6 inner
380.3.p.a.239.56 yes 232 19.11 even 3 inner
380.3.p.a.239.61 yes 232 95.49 even 6 inner
380.3.p.a.239.101 yes 232 380.239 odd 6 inner