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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 227.71
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.71

$q$-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+(0.657300 + 1.88890i) 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} +(-6.75167 - 4.29126i) q^{8} +8.53264i q^{9} +(-8.40908 + 5.41179i) q^{10} -5.36186i q^{11} +(-0.315547 + 2.71628i) 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} +(-16.1173 + 5.60851i) q^{18} +(-18.2647 + 5.23452i) q^{19} +(-15.7496 - 12.3268i) q^{20} +1.59061i q^{21} +(10.1280 - 3.52435i) 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} +(1.07393 - 9.24459i) q^{28} -28.2760 q^{29} +(-1.44890 + 6.68107i) q^{30} +24.8851 q^{31} +(31.8285 - 3.30842i) 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} +(-21.8929 - 31.0596i) q^{38} +9.40244 q^{39} +(12.9318 - 37.8519i) q^{40} +33.6063i q^{41} +(-3.00451 + 1.04551i) 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} +(-5.75542 - 9.30157i) q^{48} +43.5865i q^{49} +(-36.1731 - 34.5183i) q^{50} -13.4058 q^{51} +(-54.6467 - 6.34824i) 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} +(-18.5858 - 53.4106i) q^{58} -91.5416i q^{59} +(-13.5723 + 1.65463i) q^{60} +3.74309 q^{61} +(16.3570 + 47.0055i) 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} +(77.9143 + 9.05120i) q^{68} +0.870408 q^{69} +(4.93119 - 22.7383i) q^{70} +129.382 q^{71} +(36.6158 - 57.6096i) 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} +(6.18022 + 17.7603i) q^{78} -36.2202i q^{79} +(79.9987 - 0.453098i) q^{80} -68.5997 q^{81} +(-63.4792 + 22.0895i) 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} +(-23.0091 + 36.2015i) q^{88} +63.3083 q^{89} +(-46.1768 - 71.7516i) q^{90} -32.0002 q^{91} +(-5.05879 - 0.587673i) 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} +(-82.3307 + 28.6494i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.657300 + 1.88890i 0.328650 + 0.944452i
\(3\) 0.483404 − 0.483404i 0.161135 − 0.161135i −0.621934 − 0.783069i \(-0.713653\pi\)
0.783069 + 0.621934i \(0.213653\pi\)
\(4\) −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 \(-0.803147\pi\)
0.579758 + 0.814789i \(0.303147\pi\)
\(8\) −6.75167 − 4.29126i −0.843959 − 0.536407i
\(9\) 8.53264i 0.948071i
\(10\) −8.40908 + 5.41179i −0.840908 + 0.541179i
\(11\) − 5.36186i − 0.487441i −0.969845 − 0.243721i \(-0.921632\pi\)
0.969845 − 0.243721i \(-0.0783680\pi\)
\(12\) −0.315547 + 2.71628i −0.0262956 + 0.226357i
\(13\) 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 \(0.945680\pi\)
\(18\) −16.1173 + 5.60851i −0.895407 + 0.311584i
\(19\) −18.2647 + 5.23452i −0.961301 + 0.275501i
\(20\) −15.7496 − 12.3268i −0.787482 − 0.616338i
\(21\) 1.59061i 0.0757434i
\(22\) 10.1280 − 3.52435i 0.460365 − 0.160198i
\(23\) 0.900290 + 0.900290i 0.0391430 + 0.0391430i 0.726407 − 0.687264i \(-0.241189\pi\)
−0.687264 + 0.726407i \(0.741189\pi\)
\(24\) −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
\(28\) 1.07393 − 9.24459i 0.0383547 − 0.330164i
\(29\) −28.2760 −0.975034 −0.487517 − 0.873114i \(-0.662097\pi\)
−0.487517 + 0.873114i \(0.662097\pi\)
\(30\) −1.44890 + 6.68107i −0.0482967 + 0.222702i
\(31\) 24.8851 0.802744 0.401372 − 0.915915i \(-0.368533\pi\)
0.401372 + 0.915915i \(0.368533\pi\)
\(32\) 31.8285 − 3.30842i 0.994641 − 0.103388i
\(33\) −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 \(0.334614\pi\)
\(38\) −21.8929 − 31.0596i −0.576129 − 0.817359i
\(39\) 9.40244 0.241088
\(40\) 12.9318 − 37.8519i 0.323296 − 0.946298i
\(41\) 33.6063i 0.819667i 0.912160 + 0.409834i \(0.134413\pi\)
−0.912160 + 0.409834i \(0.865587\pi\)
\(42\) −3.00451 + 1.04551i −0.0715360 + 0.0248931i
\(43\) −18.0614 − 18.0614i −0.420032 − 0.420032i 0.465183 − 0.885215i \(-0.345989\pi\)
−0.885215 + 0.465183i \(0.845989\pi\)
\(44\) 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 \(-0.821911\pi\)
0.530750 + 0.847529i \(0.321911\pi\)
\(48\) −5.75542 − 9.30157i −0.119905 − 0.193783i
\(49\) 43.5865i 0.889521i
\(50\) −36.1731 − 34.5183i −0.723461 − 0.690365i
\(51\) −13.4058 −0.262859
\(52\) −54.6467 − 6.34824i −1.05090 − 0.122081i
\(53\) 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
\(58\) −18.5858 − 53.4106i −0.320445 − 0.920872i
\(59\) − 91.5416i − 1.55155i −0.631008 − 0.775776i \(-0.717359\pi\)
0.631008 − 0.775776i \(-0.282641\pi\)
\(60\) −13.5723 + 1.65463i −0.226204 + 0.0275772i
\(61\) 3.74309 0.0613621 0.0306811 − 0.999529i \(-0.490232\pi\)
0.0306811 + 0.999529i \(0.490232\pi\)
\(62\) 16.3570 + 47.0055i 0.263822 + 0.758153i
\(63\) −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 0.787546 − 0.616256i \(-0.211351\pi\)
−0.616256 + 0.787546i \(0.711351\pi\)
\(68\) 77.9143 + 9.05120i 1.14580 + 0.133106i
\(69\) 0.870408 0.0126146
\(70\) 4.93119 − 22.7383i 0.0704456 − 0.324833i
\(71\) 129.382 1.82229 0.911144 − 0.412088i \(-0.135200\pi\)
0.911144 + 0.412088i \(0.135200\pi\)
\(72\) 36.6158 − 57.6096i 0.508552 − 0.800133i
\(73\) 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
\(76\) 44.2784 − 61.7691i 0.582611 − 0.812751i
\(77\) 8.82142 + 8.82142i 0.114564 + 0.114564i
\(78\) 6.18022 + 17.7603i 0.0792336 + 0.227696i
\(79\) − 36.2202i − 0.458483i −0.973370 − 0.229242i \(-0.926375\pi\)
0.973370 − 0.229242i \(-0.0736246\pi\)
\(80\) 79.9987 − 0.453098i 0.999984 − 0.00566373i
\(81\) −68.5997 −0.846910
\(82\) −63.4792 + 22.0895i −0.774136 + 0.269384i
\(83\) 104.821 + 104.821i 1.26291 + 1.26291i 0.949677 + 0.313231i \(0.101412\pi\)
0.313231 + 0.949677i \(0.398588\pi\)
\(84\) −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
\(88\) −23.0091 + 36.2015i −0.261467 + 0.411381i
\(89\) 63.3083 0.711329 0.355664 − 0.934614i \(-0.384255\pi\)
0.355664 + 0.934614i \(0.384255\pi\)
\(90\) −46.1768 − 71.7516i −0.513076 − 0.797240i
\(91\) −32.0002 −0.351651
\(92\) −5.05879 − 0.587673i −0.0549868 − 0.00638775i
\(93\) 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 \(0.374883\pi\)
\(98\) −82.3307 + 28.6494i −0.840109 + 0.292341i
\(99\) 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.227.71 yes 232
4.3 odd 2 inner 380.3.j.a.227.13 ✓ 232
5.3 odd 4 inner 380.3.j.a.303.104 yes 232
19.18 odd 2 inner 380.3.j.a.227.46 yes 232
20.3 even 4 inner 380.3.j.a.303.46 yes 232
76.75 even 2 inner 380.3.j.a.227.104 yes 232
95.18 even 4 inner 380.3.j.a.303.13 yes 232
380.303 odd 4 inner 380.3.j.a.303.71 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.13 ✓ 232 4.3 odd 2 inner
380.3.j.a.227.46 yes 232 19.18 odd 2 inner
380.3.j.a.227.71 yes 232 1.1 even 1 trivial
380.3.j.a.227.104 yes 232 76.75 even 2 inner
380.3.j.a.303.13 yes 232 95.18 even 4 inner
380.3.j.a.303.46 yes 232 20.3 even 4 inner
380.3.j.a.303.71 yes 232 380.303 odd 4 inner
380.3.j.a.303.104 yes 232 5.3 odd 4 inner