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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 159.56
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.56

$q$-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+(-0.213676 + 1.98855i) q^{2} +(0.0580338 + 0.100518i) q^{3} +(-3.90868 - 0.849813i) q^{4} +(2.00334 + 4.58111i) q^{5} +(-0.212285 + 0.0939251i) q^{6} +7.81754 q^{7} +(2.52509 - 7.59104i) q^{8} +(4.49326 - 7.78256i) q^{9} +(-9.53786 + 3.00488i) q^{10} -3.02550i q^{11} +(-0.141415 - 0.442209i) q^{12} +(11.0968 + 6.40674i) q^{13} +(-1.67042 + 15.5456i) q^{14} +(-0.344221 + 0.467231i) q^{15} +(14.5556 + 6.64330i) 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} +(6.01637 + 0.646478i) q^{22} +(17.7149 - 30.6831i) q^{23} +(0.909574 - 0.186721i) q^{24} +(-16.9732 + 18.3551i) q^{25} +(-15.1113 + 20.6976i) q^{26} +2.08765 q^{27} +(-30.5563 - 6.64345i) q^{28} +(-20.2682 + 35.1055i) q^{29} +(-0.855562 - 0.784337i) q^{30} -31.8484i q^{31} +(-16.3208 + 27.5251i) q^{32} +(0.304116 - 0.175581i) q^{33} +(16.1858 + 36.5823i) q^{34} +(15.6612 + 35.8130i) q^{35} +(-24.1765 + 26.6011i) q^{36} +47.3562i q^{37} +(-30.4700 - 22.7063i) q^{38} +1.48723i q^{39} +(39.8341 - 3.63973i) q^{40} +(15.6324 + 27.0762i) q^{41} +(-1.65955 + 0.734263i) q^{42} +(-16.3862 - 28.3818i) q^{43} +(-2.57111 + 11.8257i) q^{44} +(44.6544 + 4.99301i) q^{45} +(57.2296 + 41.7832i) q^{46} +(-19.2417 + 33.3277i) q^{47} +(0.176951 + 1.84863i) q^{48} +12.1139 q^{49} +(-32.8733 - 37.6742i) q^{50} +(2.01051 + 1.16077i) q^{51} +(-37.9293 - 34.4721i) q^{52} +(62.0748 + 35.8389i) q^{53} +(-0.446082 + 4.15141i) 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} +(1.74251 - 1.53374i) q^{60} +(17.0850 - 29.5921i) q^{61} +(63.3322 + 6.80524i) q^{62} +(35.1263 - 60.8405i) q^{63} +(-51.2478 - 38.3361i) q^{64} +(-7.11931 + 63.6706i) q^{65} +(0.284171 + 0.642268i) q^{66} +(40.0811 - 69.4225i) q^{67} +(-76.2044 + 24.3695i) q^{68} +4.11225 q^{69} +(-74.5625 + 23.4908i) q^{70} +(14.5276 - 8.38749i) q^{71} +(-47.7318 - 53.7602i) q^{72} +(-57.9890 + 33.4800i) q^{73} +(-94.1702 - 10.1189i) q^{74} +(-2.83003 - 0.640891i) q^{75} +(51.6635 - 55.7395i) q^{76} -23.6520i q^{77} +(-2.95744 - 0.317786i) q^{78} +(87.2930 - 50.3986i) q^{79} +(-1.27380 + 79.9899i) q^{80} +(-40.3182 - 69.8332i) q^{81} +(-57.1827 + 25.3004i) q^{82} -38.3764 q^{83} +(-1.10552 - 3.45699i) q^{84} +(80.5163 + 59.3184i) q^{85} +(59.9400 - 26.5204i) q^{86} -4.70496 q^{87} +(-22.9667 - 7.63966i) q^{88} +(-22.1989 + 38.4497i) q^{89} +(-19.4705 + 87.7307i) q^{90} +(86.7496 + 50.0849i) q^{91} +(-95.3167 + 104.876i) 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} +(-2.58845 + 24.0891i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.213676 + 1.98855i −0.106838 + 0.994276i
\(3\) 0.0580338 + 0.100518i 0.0193446 + 0.0335059i 0.875536 − 0.483154i \(-0.160509\pi\)
−0.856191 + 0.516659i \(0.827175\pi\)
\(4\) −3.90868 − 0.849813i −0.977171 − 0.212453i
\(5\) 2.00334 + 4.58111i 0.400669 + 0.916223i
\(6\) −0.212285 + 0.0939251i −0.0353808 + 0.0156542i
\(7\) 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
\(10\) −9.53786 + 3.00488i −0.953786 + 0.300488i
\(11\) − 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 0.861864 − 0.507140i \(-0.169297\pi\)
−0.00826419 + 0.999966i \(0.502631\pi\)
\(14\) −1.67042 + 15.5456i −0.119316 + 1.11040i
\(15\) −0.344221 + 0.467231i −0.0229481 + 0.0311487i
\(16\) 14.5556 + 6.64330i 0.909727 + 0.415206i
\(17\) 17.3218 − 10.0008i 1.01893 − 0.588280i 0.105137 − 0.994458i \(-0.466472\pi\)
0.913794 + 0.406177i \(0.133138\pi\)
\(18\) 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
\(22\) 6.01637 + 0.646478i 0.273471 + 0.0293853i
\(23\) 17.7149 − 30.6831i 0.770212 − 1.33405i −0.167235 − 0.985917i \(-0.553484\pi\)
0.937447 − 0.348129i \(-0.113183\pi\)
\(24\) 0.909574 − 0.186721i 0.0378989 − 0.00778005i
\(25\) −16.9732 + 18.3551i −0.678929 + 0.734204i
\(26\) −15.1113 + 20.6976i −0.581202 + 0.796061i
\(27\) 2.08765 0.0773205
\(28\) −30.5563 − 6.64345i −1.09130 − 0.237266i
\(29\) −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 \(-0.828281\pi\)
0.857980 − 0.513684i \(-0.171719\pi\)
\(32\) −16.3208 + 27.5251i −0.510024 + 0.860160i
\(33\) 0.304116 − 0.175581i 0.00921563 − 0.00532065i
\(34\) 16.1858 + 36.5823i 0.476053 + 1.07595i
\(35\) 15.6612 + 35.8130i 0.447463 + 1.02323i
\(36\) −24.1765 + 26.6011i −0.671569 + 0.738921i
\(37\) 47.3562i 1.27990i 0.768418 + 0.639948i \(0.221044\pi\)
−0.768418 + 0.639948i \(0.778956\pi\)
\(38\) −30.4700 − 22.7063i −0.801843 − 0.597535i
\(39\) 1.48723i 0.0381341i
\(40\) 39.8341 − 3.63973i 0.995852 − 0.0909932i
\(41\) 15.6324 + 27.0762i 0.381279 + 0.660395i 0.991245 − 0.132032i \(-0.0421503\pi\)
−0.609966 + 0.792427i \(0.708817\pi\)
\(42\) −1.65955 + 0.734263i −0.0395130 + 0.0174825i
\(43\) −16.3862 − 28.3818i −0.381075 − 0.660041i 0.610141 − 0.792293i \(-0.291113\pi\)
−0.991216 + 0.132251i \(0.957779\pi\)
\(44\) −2.57111 + 11.8257i −0.0584343 + 0.268766i
\(45\) 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 \(0.300928\pi\)
\(48\) 0.176951 + 1.84863i 0.00368647 + 0.0385132i
\(49\) 12.1139 0.247222
\(50\) −32.8733 − 37.6742i −0.657466 − 0.753484i
\(51\) 2.01051 + 1.16077i 0.0394217 + 0.0227601i
\(52\) −37.9293 − 34.4721i −0.729410 − 0.662925i
\(53\) 62.0748 + 35.8389i 1.17122 + 0.676205i 0.953968 − 0.299909i \(-0.0969565\pi\)
0.217255 + 0.976115i \(0.430290\pi\)
\(54\) −0.446082 + 4.15141i −0.00826078 + 0.0768780i
\(55\) 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 \(-0.981624\pi\)
−0.449200 + 0.893431i \(0.648291\pi\)
\(60\) 1.74251 − 1.53374i 0.0290418 − 0.0255623i
\(61\) 17.0850 − 29.5921i 0.280082 − 0.485116i −0.691323 − 0.722546i \(-0.742972\pi\)
0.971405 + 0.237430i \(0.0763050\pi\)
\(62\) 63.3322 + 6.80524i 1.02149 + 0.109762i
\(63\) 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
\(66\) 0.284171 + 0.642268i 0.00430561 + 0.00973133i
\(67\) 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
\(70\) −74.5625 + 23.4908i −1.06518 + 0.335582i
\(71\) 14.5276 − 8.38749i 0.204614 − 0.118134i −0.394192 − 0.919028i \(-0.628976\pi\)
0.598806 + 0.800894i \(0.295642\pi\)
\(72\) −47.7318 − 53.7602i −0.662942 − 0.746670i
\(73\) −57.9890 + 33.4800i −0.794370 + 0.458630i −0.841499 − 0.540259i \(-0.818326\pi\)
0.0471287 + 0.998889i \(0.484993\pi\)
\(74\) −94.1702 − 10.1189i −1.27257 − 0.136742i
\(75\) −2.83003 − 0.640891i −0.0377337 − 0.00854521i
\(76\) 51.6635 − 55.7395i 0.679782 − 0.733414i
\(77\) − 23.6520i − 0.307168i
\(78\) −2.95744 − 0.317786i −0.0379158 − 0.00407418i
\(79\) 87.2930 − 50.3986i 1.10497 − 0.637957i 0.167452 − 0.985880i \(-0.446446\pi\)
0.937523 + 0.347923i \(0.113113\pi\)
\(80\) −1.27380 + 79.9899i −0.0159225 + 0.999873i
\(81\) −40.3182 − 69.8332i −0.497756 − 0.862138i
\(82\) −57.1827 + 25.3004i −0.697350 + 0.308542i
\(83\) −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
\(85\) 80.5163 + 59.3184i 0.947250 + 0.697863i
\(86\) 59.9400 − 26.5204i 0.696977 − 0.308376i
\(87\) −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 \(0.246909\pi\)
\(90\) −19.4705 + 87.7307i −0.216338 + 0.974785i
\(91\) 86.7496 + 50.0849i 0.953292 + 0.550384i
\(92\) −95.3167 + 104.876i −1.03605 + 1.13996i
\(93\) 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 0.131040 − 0.991377i \(-0.458168\pi\)
0.924078 + 0.382204i \(0.124835\pi\)
\(98\) −2.58845 + 24.0891i −0.0264128 + 0.245807i
\(99\) −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.159.56 yes 232
4.3 odd 2 inner 380.3.p.a.159.16 ✓ 232
5.4 even 2 inner 380.3.p.a.159.61 yes 232
19.11 even 3 inner 380.3.p.a.239.101 yes 232
20.19 odd 2 inner 380.3.p.a.159.101 yes 232
76.11 odd 6 inner 380.3.p.a.239.61 yes 232
95.49 even 6 inner 380.3.p.a.239.16 yes 232
380.239 odd 6 inner 380.3.p.a.239.56 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.16 ✓ 232 4.3 odd 2 inner
380.3.p.a.159.56 yes 232 1.1 even 1 trivial
380.3.p.a.159.61 yes 232 5.4 even 2 inner
380.3.p.a.159.101 yes 232 20.19 odd 2 inner
380.3.p.a.239.16 yes 232 95.49 even 6 inner
380.3.p.a.239.56 yes 232 380.239 odd 6 inner
380.3.p.a.239.61 yes 232 76.11 odd 6 inner
380.3.p.a.239.101 yes 232 19.11 even 3 inner