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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 227.54
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.54

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.165836 + 1.99311i −0.0829178 + 0.996556i
\(3\) −2.36243 + 2.36243i −0.787478 + 0.787478i −0.981080 − 0.193602i \(-0.937983\pi\)
0.193602 + 0.981080i \(0.437983\pi\)
\(4\) −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 −0.212609 + 0.977137i \(0.568196\pi\)
−0.977137 + 0.212609i \(0.931804\pi\)
\(8\) 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 \(0.233189\pi\)
−0.743449 + 0.668793i \(0.766811\pi\)
\(12\) 10.8815 − 7.75809i 0.906792 − 0.646507i
\(13\) −4.62200 − 4.62200i −0.355539 − 0.355539i 0.506627 − 0.862166i \(-0.330892\pi\)
−0.862166 + 0.506627i \(0.830892\pi\)
\(14\) −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 0.781556 − 0.623836i \(-0.214427\pi\)
−0.623836 + 0.781556i \(0.714427\pi\)
\(18\) 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 0.158456 − 0.987366i \(-0.449349\pi\)
−0.987366 + 0.158456i \(0.949349\pi\)
\(24\) 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 −0.378979 − 0.925405i \(-0.623725\pi\)
0.925405 + 0.378979i \(0.123725\pi\)
\(38\) −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 \(0.0109488\pi\)
−0.999408 + 0.0343900i \(0.989051\pi\)
\(42\) 78.4285 + 6.52559i 1.86735 + 0.155371i
\(43\) 32.2611 + 32.2611i 0.750259 + 0.750259i 0.974527 − 0.224269i \(-0.0719993\pi\)
−0.224269 + 0.974527i \(0.571999\pi\)
\(44\) 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 −0.451348 − 0.892348i \(-0.649057\pi\)
0.892348 + 0.451348i \(0.149057\pi\)
\(48\) −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 0.766094 − 0.642728i \(-0.222197\pi\)
−0.642728 + 0.766094i \(0.722197\pi\)
\(54\) 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 \(0.0391608\pi\)
−0.992442 + 0.122717i \(0.960839\pi\)
\(60\) 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 −0.997982 − 0.0634947i \(-0.979775\pi\)
−0.0634947 − 0.997982i \(-0.520225\pi\)
\(68\) −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 −0.885244 − 0.465127i \(-0.846009\pi\)
0.465127 + 0.885244i \(0.346009\pi\)
\(74\) 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 \(0.284919\pi\)
−0.625441 + 0.780271i \(0.715081\pi\)
\(80\) 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 0.996261 − 0.0863994i \(-0.0275361\pi\)
−0.0863994 + 0.996261i \(0.527536\pi\)
\(84\) −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 −0.0249374 + 0.999689i \(0.507939\pi\)
−0.999689 + 0.0249374i \(0.992061\pi\)
\(98\) 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.227.54 yes 232
4.3 odd 2 inner 380.3.j.a.227.5 ✓ 232
5.3 odd 4 inner 380.3.j.a.303.112 yes 232
19.18 odd 2 inner 380.3.j.a.227.63 yes 232
20.3 even 4 inner 380.3.j.a.303.63 yes 232
76.75 even 2 inner 380.3.j.a.227.112 yes 232
95.18 even 4 inner 380.3.j.a.303.5 yes 232
380.303 odd 4 inner 380.3.j.a.303.54 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.5 ✓ 232 4.3 odd 2 inner
380.3.j.a.227.54 yes 232 1.1 even 1 trivial
380.3.j.a.227.63 yes 232 19.18 odd 2 inner
380.3.j.a.227.112 yes 232 76.75 even 2 inner
380.3.j.a.303.5 yes 232 95.18 even 4 inner
380.3.j.a.303.54 yes 232 380.303 odd 4 inner
380.3.j.a.303.63 yes 232 20.3 even 4 inner
380.3.j.a.303.112 yes 232 5.3 odd 4 inner