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 227.13
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.13

$q$-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} +(-37.9409 - 2.11789i) 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} +(-6.29885 + 11.3596i) 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} +(70.2747 + 28.9391i) 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} +(46.8797 + 82.6274i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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 \(-0.786347\pi\)
0.621934 + 0.783069i \(0.286347\pi\)
\(4\) 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 −0.579758 − 0.814789i \(-0.696853\pi\)
0.814789 + 0.579758i \(0.196853\pi\)
\(8\) −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 \(0.0783680\pi\)
−0.969845 + 0.243721i \(0.921632\pi\)
\(12\) −2.71628 + 0.315547i −0.226357 + 0.0262956i
\(13\) 9.72523 + 9.72523i 0.748094 + 0.748094i 0.974121 − 0.226027i \(-0.0725736\pi\)
−0.226027 + 0.974121i \(0.572574\pi\)
\(14\) −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 0.169824 − 0.985474i \(-0.445680\pi\)
−0.985474 + 0.169824i \(0.945680\pi\)
\(18\) 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 0.687264 − 0.726407i \(-0.258811\pi\)
−0.726407 + 0.687264i \(0.758811\pi\)
\(24\) 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 \(-0.662097\pi\)
−0.487517 + 0.873114i \(0.662097\pi\)
\(30\) 4.33835 + 5.28344i 0.144612 + 0.176115i
\(31\) −24.8851 −0.802744 −0.401372 − 0.915915i \(-0.631467\pi\)
−0.401372 + 0.915915i \(0.631467\pi\)
\(32\) 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 \(-0.834614\pi\)
0.496511 + 0.868030i \(0.334614\pi\)
\(38\) −37.9409 − 2.11789i −0.998446 − 0.0557340i
\(39\) −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 0.885215 − 0.465183i \(-0.154011\pi\)
−0.465183 + 0.885215i \(0.654011\pi\)
\(44\) −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 −0.530750 − 0.847529i \(-0.678089\pi\)
0.847529 + 0.530750i \(0.178089\pi\)
\(48\) −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 \(-0.0516767\pi\)
−0.161635 + 0.986851i \(0.551677\pi\)
\(54\) 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
\(57\) −6.29885 + 11.3596i −0.110506 + 0.199292i
\(58\) 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 \(-0.288649\pi\)
−0.787546 + 0.616256i \(0.788649\pi\)
\(68\) −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 \(-0.864800\pi\)
−0.911144 + 0.412088i \(0.864800\pi\)
\(72\) 57.6096 − 36.6158i 0.800133 − 0.508552i
\(73\) 17.7188 − 17.7188i 0.242724 − 0.242724i −0.575252 − 0.817976i \(-0.695096\pi\)
0.817976 + 0.575252i \(0.195096\pi\)
\(74\) 35.0006 − 16.9299i 0.472981 − 0.228782i
\(75\) 5.23765 − 16.2686i 0.0698353 − 0.216914i
\(76\) 70.2747 + 28.9391i 0.924667 + 0.380777i
\(77\) 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 −0.949677 − 0.313231i \(-0.898588\pi\)
−0.313231 − 0.949677i \(-0.601412\pi\)
\(84\) −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 \(-0.384255\pi\)
0.355664 + 0.934614i \(0.384255\pi\)
\(90\) 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
\(95\) 46.8797 + 82.6274i 0.493470 + 0.869763i
\(96\) 13.7867 + 16.9854i 0.143612 + 0.176931i
\(97\) −52.4495 + 52.4495i −0.540717 + 0.540717i −0.923739 − 0.383022i \(-0.874883\pi\)
0.383022 + 0.923739i \(0.374883\pi\)
\(98\) 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.227.13 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.71 yes 232
5.3 odd 4 inner 380.3.j.a.303.46 yes 232
19.18 odd 2 inner 380.3.j.a.227.104 yes 232
20.3 even 4 inner 380.3.j.a.303.104 yes 232
76.75 even 2 inner 380.3.j.a.227.46 yes 232
95.18 even 4 inner 380.3.j.a.303.71 yes 232
380.303 odd 4 inner 380.3.j.a.303.13 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.13 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.46 yes 232 76.75 even 2 inner
380.3.j.a.227.71 yes 232 4.3 odd 2 inner
380.3.j.a.227.104 yes 232 19.18 odd 2 inner
380.3.j.a.303.13 yes 232 380.303 odd 4 inner
380.3.j.a.303.46 yes 232 5.3 odd 4 inner
380.3.j.a.303.71 yes 232 95.18 even 4 inner
380.3.j.a.303.104 yes 232 20.3 even 4 inner