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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 303.46
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.46

$q$-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} +(2.11789 - 37.9409i) 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} +(11.3596 + 6.29885i) 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} +(70.2747 + 28.9391i) 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} +(4.00274 + 94.9156i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.657300 + 1.88890i −0.328650 + 0.944452i
\(3\) −0.483404 − 0.483404i −0.161135 − 0.161135i 0.621934 − 0.783069i \(-0.286347\pi\)
−0.783069 + 0.621934i \(0.786347\pi\)
\(4\) −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 \(-0.303147\pi\)
−0.814789 + 0.579758i \(0.803147\pi\)
\(8\) 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 \(0.0783680\pi\)
−0.969845 + 0.243721i \(0.921632\pi\)
\(12\) 0.315547 + 2.71628i 0.0262956 + 0.226357i
\(13\) −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 \(0.445680\pi\)
\(18\) 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 −0.687264 − 0.726407i \(-0.741189\pi\)
0.726407 + 0.687264i \(0.241189\pi\)
\(24\) −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 \(-0.337903\pi\)
0.487517 + 0.873114i \(0.337903\pi\)
\(30\) −1.44890 − 6.68107i −0.0482967 − 0.222702i
\(31\) −24.8851 −0.802744 −0.401372 − 0.915915i \(-0.631467\pi\)
−0.401372 + 0.915915i \(0.631467\pi\)
\(32\) −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 \(-0.165386\pi\)
−0.496511 + 0.868030i \(0.665386\pi\)
\(38\) 2.11789 − 37.9409i 0.0557340 − 0.998446i
\(39\) 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 −0.885215 − 0.465183i \(-0.845989\pi\)
0.465183 + 0.885215i \(0.345989\pi\)
\(44\) 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 \(-0.321911\pi\)
−0.847529 + 0.530750i \(0.821911\pi\)
\(48\) 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 \(-0.948323\pi\)
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
\(58\) −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 \(-0.788649\pi\)
0.616256 + 0.787546i \(0.288649\pi\)
\(68\) 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 \(-0.864800\pi\)
−0.911144 + 0.412088i \(0.864800\pi\)
\(72\) −36.6158 − 57.6096i −0.508552 − 0.800133i
\(73\) 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
\(76\) 70.2747 + 28.9391i 0.924667 + 0.380777i
\(77\) 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 0.313231 − 0.949677i \(-0.398588\pi\)
0.949677 − 0.313231i \(-0.101412\pi\)
\(84\) 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 \(-0.615745\pi\)
−0.355664 + 0.934614i \(0.615745\pi\)
\(90\) 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
\(95\) 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 \(0.625117\pi\)
\(98\) 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.303.46 yes 232
4.3 odd 2 inner 380.3.j.a.303.104 yes 232
5.2 odd 4 inner 380.3.j.a.227.13 ✓ 232
19.18 odd 2 inner 380.3.j.a.303.71 yes 232
20.7 even 4 inner 380.3.j.a.227.71 yes 232
76.75 even 2 inner 380.3.j.a.303.13 yes 232
95.37 even 4 inner 380.3.j.a.227.104 yes 232
380.227 odd 4 inner 380.3.j.a.227.46 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.13 ✓ 232 5.2 odd 4 inner
380.3.j.a.227.46 yes 232 380.227 odd 4 inner
380.3.j.a.227.71 yes 232 20.7 even 4 inner
380.3.j.a.227.104 yes 232 95.37 even 4 inner
380.3.j.a.303.13 yes 232 76.75 even 2 inner
380.3.j.a.303.46 yes 232 1.1 even 1 trivial
380.3.j.a.303.71 yes 232 19.18 odd 2 inner
380.3.j.a.303.104 yes 232 4.3 odd 2 inner