Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), 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 \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 303.104
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.104

$q$-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+(1.88890 - 0.657300i) q^{2} +(0.483404 + 0.483404i) q^{3} +(3.13591 - 2.48315i) q^{4} +(1.17376 - 4.86028i) q^{5} +(1.23085 + 0.595363i) q^{6} +(1.64522 + 1.64522i) q^{7} +(4.29126 - 6.75167i) q^{8} -8.53264i q^{9} +(-0.977532 - 9.95211i) q^{10} -5.36186i q^{11} +(2.71628 + 0.315547i) q^{12} +(-9.72523 + 9.72523i) q^{13} +(4.18906 + 2.02626i) q^{14} +(2.91688 - 1.78208i) q^{15} +(3.66790 - 15.5739i) q^{16} +(-13.8661 + 13.8661i) q^{17} +(-5.60851 - 16.1173i) q^{18} +(18.2647 - 5.23452i) q^{19} +(-8.38799 - 18.1560i) q^{20} +1.59061i q^{21} +(-3.52435 - 10.1280i) q^{22} +(-0.900290 + 0.900290i) q^{23} +(5.33820 - 1.18937i) q^{24} +(-22.2446 - 11.4096i) q^{25} +(-11.9776 + 24.7624i) q^{26} +(8.47536 - 8.47536i) q^{27} +(9.24459 + 1.07393i) q^{28} +28.2760 q^{29} +(4.33835 - 5.28344i) q^{30} +24.8851 q^{31} +(-3.30842 - 31.8285i) q^{32} +(2.59195 - 2.59195i) q^{33} +(-17.0775 + 35.3058i) q^{34} +(9.92731 - 6.06511i) q^{35} +(-21.1879 - 26.7576i) q^{36} +(13.7462 + 13.7462i) q^{37} +(31.0596 - 21.8929i) q^{38} -9.40244 q^{39} +(-27.7781 - 28.7816i) q^{40} +33.6063i q^{41} +(1.04551 + 3.00451i) q^{42} +(18.0614 - 18.0614i) q^{43} +(-13.3143 - 16.8143i) q^{44} +(-41.4710 - 10.0153i) q^{45} +(-1.10880 + 2.29232i) q^{46} +(14.8886 + 14.8886i) q^{47} +(9.30157 - 5.75542i) q^{48} -43.5865i q^{49} +(-49.5174 - 6.93036i) q^{50} -13.4058 q^{51} +(-6.34824 + 54.6467i) q^{52} +(-43.7364 + 43.7364i) q^{53} +(10.4383 - 21.5800i) q^{54} +(-26.0601 - 6.29356i) q^{55} +(18.1680 - 4.04792i) q^{56} +(11.3596 + 6.29885i) q^{57} +(53.4106 - 18.5858i) q^{58} +91.5416i q^{59} +(4.72192 - 12.8315i) q^{60} +3.74309 q^{61} +(47.0055 - 16.3570i) q^{62} +(14.0381 - 14.0381i) q^{63} +(-27.1702 - 57.9464i) q^{64} +(35.8522 + 58.6824i) q^{65} +(3.19225 - 6.59962i) q^{66} +(11.4764 - 11.4764i) q^{67} +(-9.05120 + 77.9143i) q^{68} -0.870408 q^{69} +(14.7651 - 17.9816i) q^{70} +129.382 q^{71} +(-57.6096 - 36.6158i) q^{72} +(17.7188 + 17.7188i) q^{73} +(35.0006 + 16.9299i) q^{74} +(-5.23765 - 16.2686i) q^{75} +(44.2784 - 61.7691i) q^{76} +(8.82142 - 8.82142i) q^{77} +(-17.7603 + 6.18022i) q^{78} +36.2202i q^{79} +(-71.3882 - 36.1071i) q^{80} -68.5997 q^{81} +(22.0895 + 63.4792i) q^{82} +(-104.821 + 104.821i) q^{83} +(3.94973 + 4.98802i) q^{84} +(51.1174 + 83.6683i) q^{85} +(22.2444 - 45.9879i) q^{86} +(13.6687 + 13.6687i) q^{87} +(-36.2015 - 23.0091i) q^{88} -63.3083 q^{89} +(-84.9177 + 8.34093i) q^{90} -32.0002 q^{91} +(-0.587673 + 5.05879i) q^{92} +(12.0295 + 12.0295i) q^{93} +(37.9095 + 18.3369i) q^{94} +(-4.00274 - 94.9156i) q^{95} +(13.7867 - 16.9854i) q^{96} +(52.4495 + 52.4495i) q^{97} +(-28.6494 - 82.3307i) q^{98} -45.7508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.88890 − 0.657300i 0.944452 − 0.328650i
\(3\) 0.483404 + 0.483404i 0.161135 + 0.161135i 0.783069 − 0.621934i \(-0.213653\pi\)
−0.621934 + 0.783069i \(0.713653\pi\)
\(4\) 3.13591 − 2.48315i 0.783978 − 0.620788i
\(5\) 1.17376 − 4.86028i 0.234753 − 0.972055i
\(6\) 1.23085 + 0.595363i 0.205141 + 0.0992271i
\(7\) 1.64522 + 1.64522i 0.235031 + 0.235031i 0.814789 − 0.579758i \(-0.196853\pi\)
−0.579758 + 0.814789i \(0.696853\pi\)
\(8\) 4.29126 − 6.75167i 0.536407 − 0.843959i
\(9\) − 8.53264i − 0.948071i
\(10\) −0.977532 − 9.95211i −0.0977532 − 0.995211i
\(11\) − 5.36186i − 0.487441i −0.969845 − 0.243721i \(-0.921632\pi\)
0.969845 − 0.243721i \(-0.0783680\pi\)
\(12\) 2.71628 + 0.315547i 0.226357 + 0.0262956i
\(13\) −9.72523 + 9.72523i −0.748094 + 0.748094i −0.974121 − 0.226027i \(-0.927426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(14\) 4.18906 + 2.02626i 0.299219 + 0.144733i
\(15\) 2.91688 − 1.78208i 0.194459 − 0.118805i
\(16\) 3.66790 − 15.5739i 0.229244 − 0.973369i
\(17\) −13.8661 + 13.8661i −0.815650 + 0.815650i −0.985474 − 0.169824i \(-0.945680\pi\)
0.169824 + 0.985474i \(0.445680\pi\)
\(18\) −5.60851 − 16.1173i −0.311584 − 0.895407i
\(19\) 18.2647 − 5.23452i 0.961301 − 0.275501i
\(20\) −8.38799 − 18.1560i −0.419399 − 0.907802i
\(21\) 1.59061i 0.0757434i
\(22\) −3.52435 − 10.1280i −0.160198 − 0.460365i
\(23\) −0.900290 + 0.900290i −0.0391430 + 0.0391430i −0.726407 − 0.687264i \(-0.758811\pi\)
0.687264 + 0.726407i \(0.258811\pi\)
\(24\) 5.33820 − 1.18937i 0.222425 − 0.0495573i
\(25\) −22.2446 − 11.4096i −0.889782 − 0.456386i
\(26\) −11.9776 + 24.7624i −0.460678 + 0.952400i
\(27\) 8.47536 − 8.47536i 0.313902 − 0.313902i
\(28\) 9.24459 + 1.07393i 0.330164 + 0.0383547i
\(29\) 28.2760 0.975034 0.487517 − 0.873114i \(-0.337903\pi\)
0.487517 + 0.873114i \(0.337903\pi\)
\(30\) 4.33835 − 5.28344i 0.144612 − 0.176115i
\(31\) 24.8851 0.802744 0.401372 − 0.915915i \(-0.368533\pi\)
0.401372 + 0.915915i \(0.368533\pi\)
\(32\) −3.30842 − 31.8285i −0.103388 − 0.994641i
\(33\) 2.59195 − 2.59195i 0.0785438 − 0.0785438i
\(34\) −17.0775 + 35.3058i −0.502279 + 1.03841i
\(35\) 9.92731 − 6.06511i 0.283638 − 0.173289i
\(36\) −21.1879 − 26.7576i −0.588551 − 0.743267i
\(37\) 13.7462 + 13.7462i 0.371519 + 0.371519i 0.868030 − 0.496511i \(-0.165386\pi\)
−0.496511 + 0.868030i \(0.665386\pi\)
\(38\) 31.0596 − 21.8929i 0.817359 − 0.576129i
\(39\) −9.40244 −0.241088
\(40\) −27.7781 − 28.7816i −0.694452 − 0.719539i
\(41\) 33.6063i 0.819667i 0.912160 + 0.409834i \(0.134413\pi\)
−0.912160 + 0.409834i \(0.865587\pi\)
\(42\) 1.04551 + 3.00451i 0.0248931 + 0.0715360i
\(43\) 18.0614 − 18.0614i 0.420032 − 0.420032i −0.465183 − 0.885215i \(-0.654011\pi\)
0.885215 + 0.465183i \(0.154011\pi\)
\(44\) −13.3143 − 16.8143i −0.302598 − 0.382144i
\(45\) −41.4710 − 10.0153i −0.921577 − 0.222562i
\(46\) −1.10880 + 2.29232i −0.0241043 + 0.0498331i
\(47\) 14.8886 + 14.8886i 0.316779 + 0.316779i 0.847529 − 0.530750i \(-0.178089\pi\)
−0.530750 + 0.847529i \(0.678089\pi\)
\(48\) 9.30157 − 5.75542i 0.193783 − 0.119905i
\(49\) − 43.5865i − 0.889521i
\(50\) −49.5174 − 6.93036i −0.990347 − 0.138607i
\(51\) −13.4058 −0.262859
\(52\) −6.34824 + 54.6467i −0.122081 + 1.05090i
\(53\) −43.7364 + 43.7364i −0.825216 + 0.825216i −0.986851 − 0.161635i \(-0.948323\pi\)
0.161635 + 0.986851i \(0.448323\pi\)
\(54\) 10.4383 − 21.5800i 0.193301 − 0.399629i
\(55\) −26.0601 − 6.29356i −0.473820 − 0.114428i
\(56\) 18.1680 − 4.04792i 0.324429 − 0.0722842i
\(57\) 11.3596 + 6.29885i 0.199292 + 0.110506i
\(58\) 53.4106 − 18.5858i 0.920872 − 0.320445i
\(59\) 91.5416i 1.55155i 0.631008 + 0.775776i \(0.282641\pi\)
−0.631008 + 0.775776i \(0.717359\pi\)
\(60\) 4.72192 − 12.8315i 0.0786987 − 0.213858i
\(61\) 3.74309 0.0613621 0.0306811 − 0.999529i \(-0.490232\pi\)
0.0306811 + 0.999529i \(0.490232\pi\)
\(62\) 47.0055 − 16.3570i 0.758153 − 0.263822i
\(63\) 14.0381 − 14.0381i 0.222826 − 0.222826i
\(64\) −27.1702 − 57.9464i −0.424534 − 0.905412i
\(65\) 35.8522 + 58.6824i 0.551572 + 0.902806i
\(66\) 3.19225 − 6.59962i 0.0483674 − 0.0999943i
\(67\) 11.4764 − 11.4764i 0.171290 − 0.171290i −0.616256 − 0.787546i \(-0.711351\pi\)
0.787546 + 0.616256i \(0.211351\pi\)
\(68\) −9.05120 + 77.9143i −0.133106 + 1.14580i
\(69\) −0.870408 −0.0126146
\(70\) 14.7651 − 17.9816i 0.210930 − 0.256881i
\(71\) 129.382 1.82229 0.911144 − 0.412088i \(-0.135200\pi\)
0.911144 + 0.412088i \(0.135200\pi\)
\(72\) −57.6096 − 36.6158i −0.800133 − 0.508552i
\(73\) 17.7188 + 17.7188i 0.242724 + 0.242724i 0.817976 − 0.575252i \(-0.195096\pi\)
−0.575252 + 0.817976i \(0.695096\pi\)
\(74\) 35.0006 + 16.9299i 0.472981 + 0.228782i
\(75\) −5.23765 − 16.2686i −0.0698353 − 0.216914i
\(76\) 44.2784 − 61.7691i 0.582611 − 0.812751i
\(77\) 8.82142 − 8.82142i 0.114564 − 0.114564i
\(78\) −17.7603 + 6.18022i −0.227696 + 0.0792336i
\(79\) 36.2202i 0.458483i 0.973370 + 0.229242i \(0.0736246\pi\)
−0.973370 + 0.229242i \(0.926375\pi\)
\(80\) −71.3882 − 36.1071i −0.892353 − 0.451339i
\(81\) −68.5997 −0.846910
\(82\) 22.0895 + 63.4792i 0.269384 + 0.774136i
\(83\) −104.821 + 104.821i −1.26291 + 1.26291i −0.313231 + 0.949677i \(0.601412\pi\)
−0.949677 + 0.313231i \(0.898588\pi\)
\(84\) 3.94973 + 4.98802i 0.0470206 + 0.0593812i
\(85\) 51.1174 + 83.6683i 0.601381 + 0.984333i
\(86\) 22.2444 − 45.9879i 0.258656 − 0.534743i
\(87\) 13.6687 + 13.6687i 0.157112 + 0.157112i
\(88\) −36.2015 − 23.0091i −0.411381 − 0.261467i
\(89\) −63.3083 −0.711329 −0.355664 − 0.934614i \(-0.615745\pi\)
−0.355664 + 0.934614i \(0.615745\pi\)
\(90\) −84.9177 + 8.34093i −0.943531 + 0.0926770i
\(91\) −32.0002 −0.351651
\(92\) −0.587673 + 5.05879i −0.00638775 + 0.0549868i
\(93\) 12.0295 + 12.0295i 0.129350 + 0.129350i
\(94\) 37.9095 + 18.3369i 0.403292 + 0.195073i
\(95\) −4.00274 − 94.9156i −0.0421341 − 0.999112i
\(96\) 13.7867 − 16.9854i 0.143612 − 0.176931i
\(97\) 52.4495 + 52.4495i 0.540717 + 0.540717i 0.923739 − 0.383022i \(-0.125117\pi\)
−0.383022 + 0.923739i \(0.625117\pi\)
\(98\) −28.6494 − 82.3307i −0.292341 − 0.840109i
\(99\) −45.7508 −0.462129
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.j.a.303.104 yes 232
4.3 odd 2 inner 380.3.j.a.303.46 yes 232
5.2 odd 4 inner 380.3.j.a.227.71 yes 232
19.18 odd 2 inner 380.3.j.a.303.13 yes 232
20.7 even 4 inner 380.3.j.a.227.13 ✓ 232
76.75 even 2 inner 380.3.j.a.303.71 yes 232
95.37 even 4 inner 380.3.j.a.227.46 yes 232
380.227 odd 4 inner 380.3.j.a.227.104 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.13 ✓ 232 20.7 even 4 inner
380.3.j.a.227.46 yes 232 95.37 even 4 inner
380.3.j.a.227.71 yes 232 5.2 odd 4 inner
380.3.j.a.227.104 yes 232 380.227 odd 4 inner
380.3.j.a.303.13 yes 232 19.18 odd 2 inner
380.3.j.a.303.46 yes 232 4.3 odd 2 inner
380.3.j.a.303.71 yes 232 76.75 even 2 inner
380.3.j.a.303.104 yes 232 1.1 even 1 trivial