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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 303.112
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.112

$q$-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+(1.99311 + 0.165836i) 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} +(7.75320 + 1.97178i) q^{8} +2.16219i q^{9} +(6.26591 - 7.79348i) q^{10} +14.7134i q^{11} +(-7.75809 - 10.8815i) 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} +(-0.358568 + 4.30948i) q^{18} +(-12.7707 - 14.0680i) q^{19} +(13.7811 - 14.4942i) q^{20} -39.3498i q^{21} +(-2.44001 + 29.3256i) 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} +(27.3494 + 38.3603i) q^{28} -25.8430 q^{29} +(-33.2144 + 3.60878i) q^{30} +39.2318 q^{31} +(29.2829 + 12.9040i) 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} +(-23.1205 - 30.1570i) q^{38} -21.8384 q^{39} +(29.8709 - 26.6031i) q^{40} +2.81998i q^{41} +(6.52559 - 78.4285i) 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} +(-23.4123 - 48.0560i) q^{48} +89.7187i q^{49} +(-14.7505 - 47.7747i) q^{50} -12.6685 q^{51} +(21.2892 - 15.1784i) 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} +(-3.06485 + 63.4047i) q^{57} +(-51.5080 - 4.28569i) q^{58} -14.4806i q^{59} +(-66.7985 + 1.68458i) q^{60} +73.3218 q^{61} +(78.1934 + 6.50603i) 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} +(12.3499 - 8.80503i) q^{68} -90.0793 q^{69} +(117.090 - 12.7219i) q^{70} -90.3599 q^{71} +(-4.26337 + 16.7639i) 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} +(-43.5263 - 3.62158i) q^{78} -123.283i q^{79} +(63.9479 - 48.0694i) q^{80} +95.7846 q^{81} +(-0.467653 + 5.62054i) 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} +(-29.0117 + 114.076i) q^{88} +56.2222 q^{89} +(16.8510 + 13.5481i) q^{90} +76.9862 q^{91} +(87.8142 - 62.6081i) q^{92} +(-92.6825 - 92.6825i) q^{93} +(-37.8739 - 44.7485i) q^{94} +(-94.0311 + 13.5330i) q^{95} +(-38.6940 - 99.6637i) q^{96} +(99.3888 + 99.3888i) q^{97} +(-14.8785 + 178.819i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.99311 + 0.165836i 0.996556 + 0.0829178i
\(3\) −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 \(0.0681961\pi\)
0.212609 + 0.977137i \(0.431804\pi\)
\(8\) 7.75320 + 1.97178i 0.969150 + 0.246473i
\(9\) 2.16219i 0.240243i
\(10\) 6.26591 − 7.79348i 0.626591 − 0.779348i
\(11\) 14.7134i 1.33759i 0.743449 + 0.668793i \(0.233189\pi\)
−0.743449 + 0.668793i \(0.766811\pi\)
\(12\) −7.75809 − 10.8815i −0.646507 − 0.906792i
\(13\) 4.62200 − 4.62200i 0.355539 − 0.355539i −0.506627 − 0.862166i \(-0.669108\pi\)
0.862166 + 0.506627i \(0.169108\pi\)
\(14\) 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 \(0.214427\pi\)
\(18\) −0.358568 + 4.30948i −0.0199204 + 0.239416i
\(19\) −12.7707 − 14.0680i −0.672142 − 0.740422i
\(20\) 13.7811 − 14.4942i 0.689055 − 0.724709i
\(21\) − 39.3498i − 1.87380i
\(22\) −2.44001 + 29.3256i −0.110910 + 1.33298i
\(23\) 19.0649 − 19.0649i 0.828911 − 0.828911i −0.158456 − 0.987366i \(-0.550651\pi\)
0.987366 + 0.158456i \(0.0506515\pi\)
\(24\) −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
\(28\) 27.3494 + 38.3603i 0.976764 + 1.37001i
\(29\) −25.8430 −0.891138 −0.445569 − 0.895248i \(-0.646999\pi\)
−0.445569 + 0.895248i \(0.646999\pi\)
\(30\) −33.2144 + 3.60878i −1.10715 + 0.120293i
\(31\) 39.2318 1.26554 0.632771 − 0.774339i \(-0.281917\pi\)
0.632771 + 0.774339i \(0.281917\pi\)
\(32\) 29.2829 + 12.9040i 0.915090 + 0.403250i
\(33\) 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 \(-0.376275\pi\)
−0.925405 + 0.378979i \(0.876275\pi\)
\(38\) −23.1205 − 30.1570i −0.608433 − 0.793605i
\(39\) −21.8384 −0.559958
\(40\) 29.8709 − 26.6031i 0.746774 − 0.665078i
\(41\) 2.81998i 0.0687800i 0.999408 + 0.0343900i \(0.0109488\pi\)
−0.999408 + 0.0343900i \(0.989051\pi\)
\(42\) 6.52559 − 78.4285i 0.155371 − 1.86735i
\(43\) −32.2611 + 32.2611i −0.750259 + 0.750259i −0.974527 − 0.224269i \(-0.928001\pi\)
0.224269 + 0.974527i \(0.428001\pi\)
\(44\) −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 \(-0.350943\pi\)
−0.892348 + 0.451348i \(0.850943\pi\)
\(48\) −23.4123 − 48.0560i −0.487757 − 1.00117i
\(49\) 89.7187i 1.83099i
\(50\) −14.7505 − 47.7747i −0.295010 − 0.955494i
\(51\) −12.6685 −0.248402
\(52\) 21.2892 − 15.1784i 0.409408 − 0.291892i
\(53\) −6.53842 + 6.53842i −0.123366 + 0.123366i −0.766094 − 0.642728i \(-0.777803\pi\)
0.642728 + 0.766094i \(0.277803\pi\)
\(54\) −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
\(57\) −3.06485 + 63.4047i −0.0537693 + 1.11236i
\(58\) −51.5080 − 4.28569i −0.888069 − 0.0738912i
\(59\) − 14.4806i − 0.245434i −0.992442 − 0.122717i \(-0.960839\pi\)
0.992442 − 0.122717i \(-0.0391608\pi\)
\(60\) −66.7985 + 1.68458i −1.11331 + 0.0280763i
\(61\) 73.3218 1.20200 0.600999 − 0.799250i \(-0.294770\pi\)
0.600999 + 0.799250i \(0.294770\pi\)
\(62\) 78.1934 + 6.50603i 1.26118 + 0.104936i
\(63\) −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 \(0.979775\pi\)
\(68\) 12.3499 − 8.80503i 0.181617 − 0.129486i
\(69\) −90.0793 −1.30550
\(70\) 117.090 − 12.7219i 1.67271 − 0.181742i
\(71\) −90.3599 −1.27267 −0.636337 − 0.771411i \(-0.719551\pi\)
−0.636337 + 0.771411i \(0.719551\pi\)
\(72\) −4.26337 + 16.7639i −0.0592134 + 0.232831i
\(73\) −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
\(78\) −43.5263 − 3.62158i −0.558030 − 0.0464305i
\(79\) − 123.283i − 1.56054i −0.625441 − 0.780271i \(-0.715081\pi\)
0.625441 − 0.780271i \(-0.284919\pi\)
\(80\) 63.9479 − 48.0694i 0.799349 − 0.600867i
\(81\) 95.7846 1.18253
\(82\) −0.467653 + 5.62054i −0.00570309 + 0.0685432i
\(83\) −75.5185 + 75.5185i −0.909861 + 0.909861i −0.996261 − 0.0863994i \(-0.972464\pi\)
0.0863994 + 0.996261i \(0.472464\pi\)
\(84\) 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
\(88\) −29.0117 + 114.076i −0.329679 + 1.29632i
\(89\) 56.2222 0.631710 0.315855 − 0.948808i \(-0.397709\pi\)
0.315855 + 0.948808i \(0.397709\pi\)
\(90\) 16.8510 + 13.5481i 0.187233 + 0.150534i
\(91\) 76.9862 0.846002
\(92\) 87.8142 − 62.6081i 0.954502 − 0.680523i
\(93\) −92.6825 − 92.6825i −0.996586 − 0.996586i
\(94\) −37.8739 − 44.7485i −0.402914 − 0.476047i
\(95\) −94.0311 + 13.5330i −0.989802 + 0.142453i
\(96\) −38.6940 − 99.6637i −0.403063 − 1.03816i
\(97\) 99.3888 + 99.3888i 1.02463 + 1.02463i 0.999689 + 0.0249374i \(0.00793866\pi\)
0.0249374 + 0.999689i \(0.492061\pi\)
\(98\) −14.8785 + 178.819i −0.151822 + 1.82469i
\(99\) −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.303.112 yes 232
4.3 odd 2 inner 380.3.j.a.303.63 yes 232
5.2 odd 4 inner 380.3.j.a.227.54 yes 232
19.18 odd 2 inner 380.3.j.a.303.5 yes 232
20.7 even 4 inner 380.3.j.a.227.5 ✓ 232
76.75 even 2 inner 380.3.j.a.303.54 yes 232
95.37 even 4 inner 380.3.j.a.227.63 yes 232
380.227 odd 4 inner 380.3.j.a.227.112 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.5 ✓ 232 20.7 even 4 inner
380.3.j.a.227.54 yes 232 5.2 odd 4 inner
380.3.j.a.227.63 yes 232 95.37 even 4 inner
380.3.j.a.227.112 yes 232 380.227 odd 4 inner
380.3.j.a.303.5 yes 232 19.18 odd 2 inner
380.3.j.a.303.54 yes 232 76.75 even 2 inner
380.3.j.a.303.63 yes 232 4.3 odd 2 inner
380.3.j.a.303.112 yes 232 1.1 even 1 trivial