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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 239.16
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.16

$q$-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} +(2.00334 - 4.58111i) q^{5} +(0.0248009 - 0.230807i) q^{6} -7.81754 q^{7} +(-2.52509 + 7.59104i) q^{8} +(4.49326 + 7.78256i) q^{9} +(0.0430974 + 9.99991i) q^{10} -3.02550i q^{11} +(0.141415 + 0.442209i) q^{12} +(11.0968 - 6.40674i) q^{13} +(14.2981 - 6.32616i) q^{14} +(0.344221 + 0.467231i) 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} +(-8.17102 - 18.2547i) 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} +(-16.9732 - 18.3551i) q^{25} +(-15.1113 + 20.6976i) q^{26} -2.08765 q^{27} +(-21.0315 + 23.1408i) q^{28} +(-20.2682 - 35.1055i) q^{29} +(-1.00767 - 0.576001i) q^{30} -31.8484i q^{31} +(15.6771 + 27.8968i) q^{32} +(0.304116 + 0.175581i) q^{33} +(-39.7741 - 4.27385i) q^{34} +(-15.6612 + 35.8130i) q^{35} +(35.1255 + 7.63687i) q^{36} -47.3562i q^{37} +(-30.9084 - 22.1059i) q^{38} +1.48723i q^{39} +(29.7168 + 26.7752i) 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} +(45.8971 + 19.8358i) 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} +(-13.8602 - 6.06112i) 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} +(2.30911 + 0.238060i) 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} +(-0.336916 - 78.1746i) 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} +(-76.0185 - 24.9235i) q^{80} +(-40.3182 + 69.8332i) q^{81} +(-6.68057 + 62.1719i) q^{82} +38.3764 q^{83} +(-1.10552 - 3.45699i) q^{84} +(80.5163 - 59.3184i) q^{85} +(-7.00270 + 65.1698i) q^{86} +4.70496 q^{87} +(22.9667 + 7.63966i) q^{88} +(-22.1989 - 38.4497i) q^{89} +(-77.6312 + 45.2676i) 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} +(94.3039 - 11.4796i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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 \(-0.839491\pi\)
0.856191 + 0.516659i \(0.172825\pi\)
\(4\) 2.69030 − 2.96011i 0.672576 − 0.740028i
\(5\) 2.00334 − 4.58111i 0.400669 − 0.916223i
\(6\) 0.0248009 − 0.230807i 0.00413349 − 0.0384678i
\(7\) −7.81754 −1.11679 −0.558396 − 0.829575i \(-0.688583\pi\)
−0.558396 + 0.829575i \(0.688583\pi\)
\(8\) −2.52509 + 7.59104i −0.315636 + 0.948880i
\(9\) 4.49326 + 7.78256i 0.499252 + 0.864729i
\(10\) 0.0430974 + 9.99991i 0.00430974 + 0.999991i
\(11\) − 3.02550i − 0.275045i −0.990499 − 0.137523i \(-0.956086\pi\)
0.990499 − 0.137523i \(-0.0439140\pi\)
\(12\) 0.141415 + 0.442209i 0.0117846 + 0.0368508i
\(13\) 11.0968 − 6.40674i 0.853600 − 0.492826i −0.00826419 − 0.999966i \(-0.502631\pi\)
0.861864 + 0.507140i \(0.169297\pi\)
\(14\) 14.2981 − 6.32616i 1.02129 − 0.451869i
\(15\) 0.344221 + 0.467231i 0.0229481 + 0.0311487i
\(16\) −1.52455 − 15.9272i −0.0952843 − 0.995450i
\(17\) 17.3218 + 10.0008i 1.01893 + 0.588280i 0.913794 − 0.406177i \(-0.133138\pi\)
0.105137 + 0.994458i \(0.466472\pi\)
\(18\) −14.5159 − 10.5980i −0.806441 − 0.588780i
\(19\) 9.66050 + 16.3608i 0.508448 + 0.861093i
\(20\) −8.17102 − 18.2547i −0.408551 − 0.912735i
\(21\) 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 −0.937447 − 0.348129i \(-0.886817\pi\)
0.167235 − 0.985917i \(-0.446516\pi\)
\(24\) −0.616492 − 0.694353i −0.0256872 − 0.0289314i
\(25\) −16.9732 − 18.3551i −0.678929 − 0.734204i
\(26\) −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 −0.968848 − 0.247658i \(-0.920339\pi\)
0.269946 − 0.962876i \(-0.412994\pi\)
\(30\) −1.00767 − 0.576001i −0.0335889 − 0.0192000i
\(31\) − 31.8484i − 1.02737i −0.857980 − 0.513684i \(-0.828281\pi\)
0.857980 − 0.513684i \(-0.171719\pi\)
\(32\) 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
\(35\) −15.6612 + 35.8130i −0.447463 + 1.02323i
\(36\) 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
\(40\) 29.7168 + 26.7752i 0.742920 + 0.669380i
\(41\) 15.6324 − 27.0762i 0.381279 − 0.660395i −0.609966 − 0.792427i \(-0.708817\pi\)
0.991245 + 0.132032i \(0.0421503\pi\)
\(42\) −0.193882 + 1.80434i −0.00461624 + 0.0429605i
\(43\) 16.3862 − 28.3818i 0.381075 − 0.660041i −0.610141 − 0.792293i \(-0.708887\pi\)
0.991216 + 0.132251i \(0.0422206\pi\)
\(44\) −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 \(-0.0324052\pi\)
−0.585424 + 0.810727i \(0.699072\pi\)
\(48\) 1.68944 + 0.771073i 0.0351966 + 0.0160640i
\(49\) 12.1139 0.247222
\(50\) 45.8971 + 19.8358i 0.917941 + 0.396716i
\(51\) −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 \(-0.430290\pi\)
0.953968 + 0.299909i \(0.0969565\pi\)
\(54\) 3.81827 − 1.68939i 0.0707087 − 0.0312849i
\(55\) −13.8602 − 6.06112i −0.252003 − 0.110202i
\(56\) 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 0.998334 − 0.0576968i \(-0.0183757\pi\)
0.449200 + 0.893431i \(0.351709\pi\)
\(60\) 2.30911 + 0.238060i 0.0384852 + 0.00396766i
\(61\) 17.0850 + 29.5921i 0.280082 + 0.485116i 0.971405 − 0.237430i \(-0.0763050\pi\)
−0.691323 + 0.722546i \(0.742972\pi\)
\(62\) 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 −0.993083 − 0.117414i \(-0.962539\pi\)
0.394858 − 0.918742i \(-0.370794\pi\)
\(68\) 76.2044 − 24.3695i 1.12065 − 0.358375i
\(69\) 4.11225 0.0595978
\(70\) −0.336916 − 78.1746i −0.00481308 − 1.11678i
\(71\) −14.5276 − 8.38749i −0.204614 − 0.118134i 0.394192 − 0.919028i \(-0.371024\pi\)
−0.598806 + 0.800894i \(0.704358\pi\)
\(72\) −70.4237 + 14.4569i −0.978106 + 0.200790i
\(73\) −57.9890 − 33.4800i −0.794370 − 0.458630i 0.0471287 − 0.998889i \(-0.484993\pi\)
−0.841499 + 0.540259i \(0.818326\pi\)
\(74\) 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 −0.167452 − 0.985880i \(-0.553554\pi\)
−0.937523 + 0.347923i \(0.886887\pi\)
\(80\) −76.0185 − 24.9235i −0.950232 − 0.311544i
\(81\) −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 \(-0.425740\pi\)
0.231183 + 0.972910i \(0.425740\pi\)
\(84\) −1.10552 − 3.45699i −0.0131609 − 0.0411546i
\(85\) 80.5163 − 59.3184i 0.947250 − 0.697863i
\(86\) −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 0.713941 − 0.700206i \(-0.246909\pi\)
−0.963367 + 0.268187i \(0.913575\pi\)
\(90\) −77.6312 + 45.2676i −0.862569 + 0.502974i
\(91\) −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
\(95\) 94.3039 − 11.4796i 0.992672 − 0.120838i
\(96\) −3.71392 − 0.0431331i −0.0386866 − 0.000449303i
\(97\) 102.346 + 59.0898i 1.05512 + 0.609173i 0.924078 − 0.382204i \(-0.124835\pi\)
0.131040 + 0.991377i \(0.458168\pi\)
\(98\) −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.239.16 yes 232
4.3 odd 2 inner 380.3.p.a.239.56 yes 232
5.4 even 2 inner 380.3.p.a.239.101 yes 232
19.7 even 3 inner 380.3.p.a.159.61 yes 232
20.19 odd 2 inner 380.3.p.a.239.61 yes 232
76.7 odd 6 inner 380.3.p.a.159.101 yes 232
95.64 even 6 inner 380.3.p.a.159.56 yes 232
380.159 odd 6 inner 380.3.p.a.159.16 ✓ 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.16 ✓ 232 380.159 odd 6 inner
380.3.p.a.159.56 yes 232 95.64 even 6 inner
380.3.p.a.159.61 yes 232 19.7 even 3 inner
380.3.p.a.159.101 yes 232 76.7 odd 6 inner
380.3.p.a.239.16 yes 232 1.1 even 1 trivial
380.3.p.a.239.56 yes 232 4.3 odd 2 inner
380.3.p.a.239.61 yes 232 20.19 odd 2 inner
380.3.p.a.239.101 yes 232 5.4 even 2 inner