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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 239.56
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.56

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.213676 − 1.98855i −0.106838 − 0.994276i
\(3\) 0.0580338 − 0.100518i 0.0193446 − 0.0335059i −0.856191 − 0.516659i \(-0.827175\pi\)
0.875536 + 0.483154i \(0.160509\pi\)
\(4\) −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 \(0.0439140\pi\)
−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 \(-0.502631\pi\)
0.861864 + 0.507140i \(0.169297\pi\)
\(14\) −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 0.913794 − 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
\(20\) −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 0.937447 + 0.348129i \(0.113183\pi\)
−0.167235 + 0.985917i \(0.553484\pi\)
\(24\) 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 −0.968848 − 0.247658i \(-0.920339\pi\)
0.269946 − 0.962876i \(-0.412994\pi\)
\(30\) −0.855562 + 0.784337i −0.0285187 + 0.0261446i
\(31\) 31.8484i 1.02737i 0.857980 + 0.513684i \(0.171719\pi\)
−0.857980 + 0.513684i \(0.828281\pi\)
\(32\) −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 \(-0.778956\pi\)
0.768418 − 0.639948i \(-0.221044\pi\)
\(38\) −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 −0.609966 − 0.792427i \(-0.708817\pi\)
0.991245 + 0.132032i \(0.0421503\pi\)
\(42\) −1.65955 − 0.734263i −0.0395130 − 0.0174825i
\(43\) −16.3862 + 28.3818i −0.381075 + 0.660041i −0.991216 − 0.132251i \(-0.957779\pi\)
0.610141 + 0.792293i \(0.291113\pi\)
\(44\) −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 0.585424 − 0.810727i \(-0.300928\pi\)
−0.994822 + 0.101628i \(0.967595\pi\)
\(48\) 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 0.217255 − 0.976115i \(-0.430290\pi\)
0.953968 + 0.299909i \(0.0969565\pi\)
\(54\) −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 −0.449200 − 0.893431i \(-0.648291\pi\)
−0.998334 + 0.0576968i \(0.981624\pi\)
\(60\) 1.74251 + 1.53374i 0.0290418 + 0.0255623i
\(61\) 17.0850 + 29.5921i 0.280082 + 0.485116i 0.971405 − 0.237430i \(-0.0763050\pi\)
−0.691323 + 0.722546i \(0.742972\pi\)
\(62\) 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 0.993083 + 0.117414i \(0.0374606\pi\)
−0.394858 + 0.918742i \(0.629206\pi\)
\(68\) −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 0.598806 − 0.800894i \(-0.295642\pi\)
−0.394192 + 0.919028i \(0.628976\pi\)
\(72\) −47.7318 + 53.7602i −0.662942 + 0.746670i
\(73\) −57.9890 − 33.4800i −0.794370 − 0.458630i 0.0471287 − 0.998889i \(-0.484993\pi\)
−0.841499 + 0.540259i \(0.818326\pi\)
\(74\) −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 0.937523 − 0.347923i \(-0.113113\pi\)
0.167452 + 0.985880i \(0.446446\pi\)
\(80\) −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 0.713941 − 0.700206i \(-0.246909\pi\)
−0.963367 + 0.268187i \(0.913575\pi\)
\(90\) −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 0.924078 − 0.382204i \(-0.124835\pi\)
0.131040 + 0.991377i \(0.458168\pi\)
\(98\) −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.239.56 yes 232
4.3 odd 2 inner 380.3.p.a.239.16 yes 232
5.4 even 2 inner 380.3.p.a.239.61 yes 232
19.7 even 3 inner 380.3.p.a.159.101 yes 232
20.19 odd 2 inner 380.3.p.a.239.101 yes 232
76.7 odd 6 inner 380.3.p.a.159.61 yes 232
95.64 even 6 inner 380.3.p.a.159.16 ✓ 232
380.159 odd 6 inner 380.3.p.a.159.56 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.16 ✓ 232 95.64 even 6 inner
380.3.p.a.159.56 yes 232 380.159 odd 6 inner
380.3.p.a.159.61 yes 232 76.7 odd 6 inner
380.3.p.a.159.101 yes 232 19.7 even 3 inner
380.3.p.a.239.16 yes 232 4.3 odd 2 inner
380.3.p.a.239.56 yes 232 1.1 even 1 trivial
380.3.p.a.239.61 yes 232 5.4 even 2 inner
380.3.p.a.239.101 yes 232 20.19 odd 2 inner