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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 303.54
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.54

$q$-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+(-0.165836 - 1.99311i) q^{2} +(-2.36243 - 2.36243i) q^{3} +(-3.94500 + 0.661058i) q^{4} +(2.79906 - 4.14310i) q^{5} +(-4.31682 + 5.10037i) q^{6} +(-8.32823 - 8.32823i) q^{7} +(1.97178 + 7.75320i) q^{8} +2.16219i q^{9} +(-8.72185 - 4.89177i) q^{10} -14.7134i q^{11} +(10.8815 + 7.75809i) q^{12} +(-4.62200 + 4.62200i) q^{13} +(-15.2180 + 17.9802i) q^{14} +(-16.4004 + 3.17521i) q^{15} +(15.1260 - 5.21574i) q^{16} +(2.68124 - 2.68124i) q^{17} +(4.30948 - 0.358568i) q^{18} +(12.7707 - 14.0680i) q^{19} +(-8.30345 + 18.1949i) q^{20} +39.3498i q^{21} +(-29.3256 + 2.44001i) q^{22} +(-19.0649 + 19.0649i) q^{23} +(13.6582 - 22.9746i) q^{24} +(-9.33054 - 23.1936i) q^{25} +(9.97867 + 8.44568i) q^{26} +(-16.1539 + 16.1539i) q^{27} +(38.3603 + 27.3494i) q^{28} +25.8430 q^{29} +(9.04831 + 32.1613i) q^{30} +39.2318 q^{31} +(-12.9040 - 29.2829i) q^{32} +(-34.7596 + 34.7596i) q^{33} +(-5.78866 - 4.89937i) q^{34} +(-57.8159 + 11.1935i) q^{35} +(-1.42933 - 8.52982i) q^{36} +(20.2178 + 20.2178i) q^{37} +(-30.1570 - 23.1205i) q^{38} +21.8384 q^{39} +(37.6414 + 13.5324i) q^{40} -2.81998i q^{41} +(78.4285 - 6.52559i) q^{42} +(32.2611 - 32.2611i) q^{43} +(9.72644 + 58.0445i) q^{44} +(8.95816 + 6.05209i) q^{45} +(41.1602 + 34.8369i) q^{46} +(20.7270 + 20.7270i) q^{47} +(-48.0560 - 23.4123i) q^{48} +89.7187i q^{49} +(-44.6800 + 22.4431i) q^{50} -12.6685 q^{51} +(15.1784 - 21.2892i) q^{52} +(6.53842 - 6.53842i) q^{53} +(34.8754 + 29.5176i) q^{54} +(-60.9593 - 41.1838i) q^{55} +(48.1489 - 80.9918i) q^{56} +(-63.4047 + 3.06485i) q^{57} +(-4.28569 - 51.5080i) q^{58} -14.4806i q^{59} +(62.6005 - 23.3678i) q^{60} +73.3218 q^{61} +(-6.50603 - 78.1934i) q^{62} +(18.0072 - 18.0072i) q^{63} +(-56.2241 + 30.5753i) q^{64} +(6.21216 + 32.0867i) q^{65} +(75.0441 + 63.5153i) q^{66} +(-71.1189 + 71.1189i) q^{67} +(-8.80503 + 12.3499i) q^{68} +90.0793 q^{69} +(31.8978 + 113.377i) q^{70} -90.3599 q^{71} +(-16.7639 + 4.26337i) q^{72} +(-30.6686 - 30.6686i) q^{73} +(36.9435 - 43.6491i) q^{74} +(-32.7504 + 76.8360i) q^{75} +(-41.0806 + 63.9405i) q^{76} +(-122.537 + 122.537i) q^{77} +(-3.62158 - 43.5263i) q^{78} -123.283i q^{79} +(20.7292 - 77.2677i) q^{80} +95.7846 q^{81} +(-5.62054 + 0.467653i) q^{82} +(75.5185 - 75.5185i) q^{83} +(-26.0125 - 155.235i) q^{84} +(-3.60370 - 18.6136i) q^{85} +(-69.6501 - 58.9500i) q^{86} +(-61.0524 - 61.0524i) q^{87} +(114.076 - 29.0117i) q^{88} -56.2222 q^{89} +(10.5769 - 18.8583i) q^{90} +76.9862 q^{91} +(62.6081 - 87.8142i) q^{92} +(-92.6825 - 92.6825i) q^{93} +(37.8739 - 44.7485i) q^{94} +(-22.5393 - 92.2875i) q^{95} +(-38.6940 + 99.6637i) q^{96} +(-99.3888 - 99.3888i) q^{97} +(178.819 - 14.8785i) q^{98} +31.8132 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.165836 − 1.99311i −0.0829178 − 0.996556i
\(3\) −2.36243 − 2.36243i −0.787478 − 0.787478i 0.193602 − 0.981080i \(-0.437983\pi\)
−0.981080 + 0.193602i \(0.937983\pi\)
\(4\) −3.94500 + 0.661058i −0.986249 + 0.165265i
\(5\) 2.79906 − 4.14310i 0.559812 − 0.828620i
\(6\) −4.31682 + 5.10037i −0.719470 + 0.850062i
\(7\) −8.32823 − 8.32823i −1.18975 − 1.18975i −0.977137 − 0.212609i \(-0.931804\pi\)
−0.212609 − 0.977137i \(-0.568196\pi\)
\(8\) 1.97178 + 7.75320i 0.246473 + 0.969150i
\(9\) 2.16219i 0.240243i
\(10\) −8.72185 − 4.89177i −0.872185 − 0.489177i
\(11\) − 14.7134i − 1.33759i −0.743449 − 0.668793i \(-0.766811\pi\)
0.743449 − 0.668793i \(-0.233189\pi\)
\(12\) 10.8815 + 7.75809i 0.906792 + 0.646507i
\(13\) −4.62200 + 4.62200i −0.355539 + 0.355539i −0.862166 − 0.506627i \(-0.830892\pi\)
0.506627 + 0.862166i \(0.330892\pi\)
\(14\) −15.2180 + 17.9802i −1.08700 + 1.28430i
\(15\) −16.4004 + 3.17521i −1.09336 + 0.211680i
\(16\) 15.1260 − 5.21574i 0.945375 − 0.325984i
\(17\) 2.68124 − 2.68124i 0.157720 − 0.157720i −0.623836 − 0.781556i \(-0.714427\pi\)
0.781556 + 0.623836i \(0.214427\pi\)
\(18\) 4.30948 − 0.358568i 0.239416 − 0.0199204i
\(19\) 12.7707 − 14.0680i 0.672142 − 0.740422i
\(20\) −8.30345 + 18.1949i −0.415172 + 0.909743i
\(21\) 39.3498i 1.87380i
\(22\) −29.3256 + 2.44001i −1.33298 + 0.110910i
\(23\) −19.0649 + 19.0649i −0.828911 + 0.828911i −0.987366 − 0.158456i \(-0.949349\pi\)
0.158456 + 0.987366i \(0.449349\pi\)
\(24\) 13.6582 − 22.9746i 0.569092 − 0.957276i
\(25\) −9.33054 − 23.1936i −0.373222 − 0.927742i
\(26\) 9.97867 + 8.44568i 0.383795 + 0.324834i
\(27\) −16.1539 + 16.1539i −0.598292 + 0.598292i
\(28\) 38.3603 + 27.3494i 1.37001 + 0.976764i
\(29\) 25.8430 0.891138 0.445569 − 0.895248i \(-0.353001\pi\)
0.445569 + 0.895248i \(0.353001\pi\)
\(30\) 9.04831 + 32.1613i 0.301610 + 1.07204i
\(31\) 39.2318 1.26554 0.632771 − 0.774339i \(-0.281917\pi\)
0.632771 + 0.774339i \(0.281917\pi\)
\(32\) −12.9040 − 29.2829i −0.403250 − 0.915090i
\(33\) −34.7596 + 34.7596i −1.05332 + 1.05332i
\(34\) −5.78866 − 4.89937i −0.170255 − 0.144099i
\(35\) −57.8159 + 11.1935i −1.65188 + 0.319814i
\(36\) −1.42933 − 8.52982i −0.0397037 − 0.236940i
\(37\) 20.2178 + 20.2178i 0.546426 + 0.546426i 0.925405 − 0.378979i \(-0.123725\pi\)
−0.378979 + 0.925405i \(0.623725\pi\)
\(38\) −30.1570 − 23.1205i −0.793605 − 0.608433i
\(39\) 21.8384 0.559958
\(40\) 37.6414 + 13.5324i 0.941035 + 0.338309i
\(41\) − 2.81998i − 0.0687800i −0.999408 − 0.0343900i \(-0.989051\pi\)
0.999408 − 0.0343900i \(-0.0109488\pi\)
\(42\) 78.4285 − 6.52559i 1.86735 − 0.155371i
\(43\) 32.2611 − 32.2611i 0.750259 − 0.750259i −0.224269 − 0.974527i \(-0.571999\pi\)
0.974527 + 0.224269i \(0.0719993\pi\)
\(44\) 9.72644 + 58.0445i 0.221056 + 1.31919i
\(45\) 8.95816 + 6.05209i 0.199070 + 0.134491i
\(46\) 41.1602 + 34.8369i 0.894788 + 0.757325i
\(47\) 20.7270 + 20.7270i 0.440999 + 0.440999i 0.892348 − 0.451348i \(-0.149057\pi\)
−0.451348 + 0.892348i \(0.649057\pi\)
\(48\) −48.0560 − 23.4123i −1.00117 − 0.487757i
\(49\) 89.7187i 1.83099i
\(50\) −44.6800 + 22.4431i −0.893601 + 0.448863i
\(51\) −12.6685 −0.248402
\(52\) 15.1784 − 21.2892i 0.291892 − 0.409408i
\(53\) 6.53842 − 6.53842i 0.123366 − 0.123366i −0.642728 − 0.766094i \(-0.722197\pi\)
0.766094 + 0.642728i \(0.222197\pi\)
\(54\) 34.8754 + 29.5176i 0.645841 + 0.546623i
\(55\) −60.9593 − 41.1838i −1.10835 − 0.748796i
\(56\) 48.1489 − 80.9918i 0.859802 − 1.44628i
\(57\) −63.4047 + 3.06485i −1.11236 + 0.0537693i
\(58\) −4.28569 − 51.5080i −0.0738912 − 0.888069i
\(59\) − 14.4806i − 0.245434i −0.992442 − 0.122717i \(-0.960839\pi\)
0.992442 − 0.122717i \(-0.0391608\pi\)
\(60\) 62.6005 − 23.3678i 1.04334 − 0.389463i
\(61\) 73.3218 1.20200 0.600999 − 0.799250i \(-0.294770\pi\)
0.600999 + 0.799250i \(0.294770\pi\)
\(62\) −6.50603 − 78.1934i −0.104936 − 1.26118i
\(63\) 18.0072 − 18.0072i 0.285828 − 0.285828i
\(64\) −56.2241 + 30.5753i −0.878502 + 0.477738i
\(65\) 6.21216 + 32.0867i 0.0955717 + 0.493641i
\(66\) 75.0441 + 63.5153i 1.13703 + 0.962354i
\(67\) −71.1189 + 71.1189i −1.06148 + 1.06148i −0.0634947 + 0.997982i \(0.520225\pi\)
−0.997982 + 0.0634947i \(0.979775\pi\)
\(68\) −8.80503 + 12.3499i −0.129486 + 0.181617i
\(69\) 90.0793 1.30550
\(70\) 31.8978 + 113.377i 0.455683 + 1.61967i
\(71\) −90.3599 −1.27267 −0.636337 − 0.771411i \(-0.719551\pi\)
−0.636337 + 0.771411i \(0.719551\pi\)
\(72\) −16.7639 + 4.26337i −0.232831 + 0.0592134i
\(73\) −30.6686 − 30.6686i −0.420117 − 0.420117i 0.465127 − 0.885244i \(-0.346009\pi\)
−0.885244 + 0.465127i \(0.846009\pi\)
\(74\) 36.9435 − 43.6491i 0.499236 − 0.589853i
\(75\) −32.7504 + 76.8360i −0.436673 + 1.02448i
\(76\) −41.0806 + 63.9405i −0.540534 + 0.841322i
\(77\) −122.537 + 122.537i −1.59139 + 1.59139i
\(78\) −3.62158 − 43.5263i −0.0464305 − 0.558030i
\(79\) − 123.283i − 1.56054i −0.625441 − 0.780271i \(-0.715081\pi\)
0.625441 − 0.780271i \(-0.284919\pi\)
\(80\) 20.7292 − 77.2677i 0.259115 − 0.965846i
\(81\) 95.7846 1.18253
\(82\) −5.62054 + 0.467653i −0.0685432 + 0.00570309i
\(83\) 75.5185 − 75.5185i 0.909861 − 0.909861i −0.0863994 − 0.996261i \(-0.527536\pi\)
0.996261 + 0.0863994i \(0.0275361\pi\)
\(84\) −26.0125 − 155.235i −0.309672 − 1.84803i
\(85\) −3.60370 − 18.6136i −0.0423964 − 0.218984i
\(86\) −69.6501 − 58.9500i −0.809885 − 0.685465i
\(87\) −61.0524 − 61.0524i −0.701751 − 0.701751i
\(88\) 114.076 − 29.0117i 1.29632 − 0.329679i
\(89\) −56.2222 −0.631710 −0.315855 − 0.948808i \(-0.602291\pi\)
−0.315855 + 0.948808i \(0.602291\pi\)
\(90\) 10.5769 − 18.8583i 0.117521 − 0.209536i
\(91\) 76.9862 0.846002
\(92\) 62.6081 − 87.8142i 0.680523 − 0.954502i
\(93\) −92.6825 − 92.6825i −0.996586 − 0.996586i
\(94\) 37.8739 − 44.7485i 0.402914 − 0.476047i
\(95\) −22.5393 − 92.2875i −0.237256 − 0.971447i
\(96\) −38.6940 + 99.6637i −0.403063 + 1.03816i
\(97\) −99.3888 − 99.3888i −1.02463 − 1.02463i −0.999689 − 0.0249374i \(-0.992061\pi\)
−0.0249374 − 0.999689i \(-0.507939\pi\)
\(98\) 178.819 − 14.8785i 1.82469 − 0.151822i
\(99\) 31.8132 0.321346
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.303.54 yes 232
4.3 odd 2 inner 380.3.j.a.303.5 yes 232
5.2 odd 4 inner 380.3.j.a.227.112 yes 232
19.18 odd 2 inner 380.3.j.a.303.63 yes 232
20.7 even 4 inner 380.3.j.a.227.63 yes 232
76.75 even 2 inner 380.3.j.a.303.112 yes 232
95.37 even 4 inner 380.3.j.a.227.5 ✓ 232
380.227 odd 4 inner 380.3.j.a.227.54 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.5 ✓ 232 95.37 even 4 inner
380.3.j.a.227.54 yes 232 380.227 odd 4 inner
380.3.j.a.227.63 yes 232 20.7 even 4 inner
380.3.j.a.227.112 yes 232 5.2 odd 4 inner
380.3.j.a.303.5 yes 232 4.3 odd 2 inner
380.3.j.a.303.54 yes 232 1.1 even 1 trivial
380.3.j.a.303.63 yes 232 19.18 odd 2 inner
380.3.j.a.303.112 yes 232 76.75 even 2 inner