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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 227.75
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.75

$q$-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+(0.912666 - 1.77962i) q^{2} +(3.66846 - 3.66846i) q^{3} +(-2.33408 - 3.24839i) q^{4} +(4.37305 - 2.42414i) q^{5} +(-3.18038 - 9.87655i) q^{6} +(2.51789 - 2.51789i) q^{7} +(-7.91114 + 1.18908i) q^{8} -17.9153i q^{9} +(-0.322920 - 9.99478i) q^{10} +14.1899i q^{11} +(-20.4791 - 3.35412i) q^{12} +(0.192716 + 0.192716i) q^{13} +(-2.18289 - 6.77886i) q^{14} +(7.14948 - 24.9352i) q^{15} +(-5.10412 + 15.1640i) q^{16} +(18.2406 + 18.2406i) q^{17} +(-31.8823 - 16.3506i) q^{18} +(-17.8506 + 6.50817i) q^{19} +(-18.0816 - 8.54722i) q^{20} -18.4735i q^{21} +(25.2527 + 12.9507i) q^{22} +(18.5273 + 18.5273i) q^{23} +(-24.6596 + 33.3838i) q^{24} +(13.2471 - 21.2018i) q^{25} +(0.518847 - 0.167076i) q^{26} +(-32.7053 - 32.7053i) q^{27} +(-14.0560 - 2.30213i) q^{28} +31.6301 q^{29} +(-37.8501 - 35.4809i) q^{30} -35.7449 q^{31} +(22.3278 + 22.9231i) q^{32} +(52.0552 + 52.0552i) q^{33} +(49.1088 - 15.8137i) q^{34} +(4.90711 - 17.1145i) q^{35} +(-58.1958 + 41.8157i) q^{36} +(-17.7701 + 17.7701i) q^{37} +(-4.70956 + 37.7070i) q^{38} +1.41395 q^{39} +(-31.7133 + 24.3776i) q^{40} -49.7840i q^{41} +(-32.8759 - 16.8602i) q^{42} +(-13.5259 - 13.5259i) q^{43} +(46.0945 - 33.1205i) q^{44} +(-43.4291 - 78.3442i) q^{45} +(49.8808 - 16.0623i) q^{46} +(-37.4331 + 37.4331i) q^{47} +(36.9044 + 74.3530i) q^{48} +36.3205i q^{49} +(-25.6409 - 42.9248i) q^{50} +133.830 q^{51} +(0.176203 - 1.07583i) q^{52} +(-20.7721 - 20.7721i) q^{53} +(-88.0519 + 28.3539i) q^{54} +(34.3984 + 62.0532i) q^{55} +(-16.9254 + 22.9133i) q^{56} +(-41.6092 + 89.3593i) q^{57} +(28.8677 - 56.2895i) q^{58} +17.7674i q^{59} +(-97.6869 + 34.9766i) q^{60} -85.5197 q^{61} +(-32.6232 + 63.6124i) q^{62} +(-45.1086 - 45.1086i) q^{63} +(61.1722 - 18.8139i) q^{64} +(1.30993 + 0.375585i) q^{65} +(140.147 - 45.1294i) q^{66} +(-28.1260 - 28.1260i) q^{67} +(16.6775 - 101.827i) q^{68} +135.934 q^{69} +(-25.9788 - 24.3527i) q^{70} +62.1745 q^{71} +(21.3026 + 141.730i) q^{72} +(-24.6006 + 24.6006i) q^{73} +(15.4058 + 47.8421i) q^{74} +(-29.1816 - 126.374i) q^{75} +(62.8059 + 42.7951i) q^{76} +(35.7286 + 35.7286i) q^{77} +(1.29046 - 2.51628i) q^{78} -116.222i q^{79} +(14.4392 + 78.6861i) q^{80} -78.7189 q^{81} +(-88.5965 - 45.4362i) q^{82} +(-26.7366 - 26.7366i) q^{83} +(-60.0093 + 43.1188i) q^{84} +(123.985 + 35.5491i) q^{85} +(-36.4155 + 11.7263i) q^{86} +(116.034 - 116.034i) q^{87} +(-16.8729 - 112.258i) q^{88} -149.178 q^{89} +(-179.059 + 5.78519i) q^{90} +0.970476 q^{91} +(16.9397 - 103.428i) q^{92} +(-131.129 + 131.129i) q^{93} +(32.4527 + 100.781i) q^{94} +(-62.2847 + 71.7329i) q^{95} +(166.001 + 2.18362i) q^{96} +(9.47531 - 9.47531i) q^{97} +(64.6366 + 33.1485i) q^{98} +254.216 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.912666 − 1.77962i 0.456333 − 0.889809i
\(3\) 3.66846 − 3.66846i 1.22282 − 1.22282i 0.256196 − 0.966625i \(-0.417531\pi\)
0.966625 − 0.256196i \(-0.0824694\pi\)
\(4\) −2.33408 − 3.24839i −0.583521 − 0.812098i
\(5\) 4.37305 − 2.42414i 0.874609 − 0.484829i
\(6\) −3.18038 − 9.87655i −0.530064 − 1.64609i
\(7\) 2.51789 − 2.51789i 0.359698 − 0.359698i −0.504004 − 0.863702i \(-0.668140\pi\)
0.863702 + 0.504004i \(0.168140\pi\)
\(8\) −7.91114 + 1.18908i −0.988892 + 0.148635i
\(9\) − 17.9153i − 1.99058i
\(10\) −0.322920 − 9.99478i −0.0322920 − 0.999478i
\(11\) 14.1899i 1.28999i 0.764185 + 0.644997i \(0.223141\pi\)
−0.764185 + 0.644997i \(0.776859\pi\)
\(12\) −20.4791 − 3.35412i −1.70659 − 0.279510i
\(13\) 0.192716 + 0.192716i 0.0148243 + 0.0148243i 0.714480 − 0.699656i \(-0.246663\pi\)
−0.699656 + 0.714480i \(0.746663\pi\)
\(14\) −2.18289 − 6.77886i −0.155921 − 0.484205i
\(15\) 7.14948 − 24.9352i 0.476632 − 1.66235i
\(16\) −5.10412 + 15.1640i −0.319007 + 0.947752i
\(17\) 18.2406 + 18.2406i 1.07297 + 1.07297i 0.997119 + 0.0758551i \(0.0241686\pi\)
0.0758551 + 0.997119i \(0.475831\pi\)
\(18\) −31.8823 − 16.3506i −1.77124 − 0.908369i
\(19\) −17.8506 + 6.50817i −0.939505 + 0.342536i
\(20\) −18.0816 − 8.54722i −0.904081 − 0.427361i
\(21\) − 18.4735i − 0.879693i
\(22\) 25.2527 + 12.9507i 1.14785 + 0.588666i
\(23\) 18.5273 + 18.5273i 0.805535 + 0.805535i 0.983955 − 0.178419i \(-0.0570983\pi\)
−0.178419 + 0.983955i \(0.557098\pi\)
\(24\) −24.6596 + 33.3838i −1.02748 + 1.39099i
\(25\) 13.2471 − 21.2018i 0.529882 − 0.848071i
\(26\) 0.518847 − 0.167076i 0.0199557 − 0.00642600i
\(27\) −32.7053 − 32.7053i −1.21131 − 1.21131i
\(28\) −14.0560 − 2.30213i −0.502001 − 0.0822190i
\(29\) 31.6301 1.09069 0.545347 − 0.838211i \(-0.316398\pi\)
0.545347 + 0.838211i \(0.316398\pi\)
\(30\) −37.8501 − 35.4809i −1.26167 − 1.18270i
\(31\) −35.7449 −1.15306 −0.576531 − 0.817075i \(-0.695594\pi\)
−0.576531 + 0.817075i \(0.695594\pi\)
\(32\) 22.3278 + 22.9231i 0.697745 + 0.716346i
\(33\) 52.0552 + 52.0552i 1.57743 + 1.57743i
\(34\) 49.1088 − 15.8137i 1.44438 − 0.465109i
\(35\) 4.90711 − 17.1145i 0.140203 − 0.488987i
\(36\) −58.1958 + 41.8157i −1.61655 + 1.16155i
\(37\) −17.7701 + 17.7701i −0.480272 + 0.480272i −0.905219 − 0.424946i \(-0.860293\pi\)
0.424946 + 0.905219i \(0.360293\pi\)
\(38\) −4.70956 + 37.7070i −0.123936 + 0.992290i
\(39\) 1.41395 0.0362550
\(40\) −31.7133 + 24.3776i −0.792832 + 0.609441i
\(41\) − 49.7840i − 1.21424i −0.794609 − 0.607122i \(-0.792324\pi\)
0.794609 − 0.607122i \(-0.207676\pi\)
\(42\) −32.8759 − 16.8602i −0.782759 − 0.401433i
\(43\) −13.5259 − 13.5259i −0.314556 − 0.314556i 0.532116 − 0.846672i \(-0.321397\pi\)
−0.846672 + 0.532116i \(0.821397\pi\)
\(44\) 46.0945 − 33.1205i 1.04760 − 0.752738i
\(45\) −43.4291 − 78.3442i −0.965092 − 1.74098i
\(46\) 49.8808 − 16.0623i 1.08436 − 0.349180i
\(47\) −37.4331 + 37.4331i −0.796449 + 0.796449i −0.982534 − 0.186084i \(-0.940420\pi\)
0.186084 + 0.982534i \(0.440420\pi\)
\(48\) 36.9044 + 74.3530i 0.768842 + 1.54902i
\(49\) 36.3205i 0.741235i
\(50\) −25.6409 − 42.9248i −0.512819 − 0.858497i
\(51\) 133.830 2.62411
\(52\) 0.176203 − 1.07583i 0.00338851 − 0.0206891i
\(53\) −20.7721 − 20.7721i −0.391926 − 0.391926i 0.483447 − 0.875373i \(-0.339384\pi\)
−0.875373 + 0.483447i \(0.839384\pi\)
\(54\) −88.0519 + 28.3539i −1.63059 + 0.525073i
\(55\) 34.3984 + 62.0532i 0.625426 + 1.12824i
\(56\) −16.9254 + 22.9133i −0.302239 + 0.409166i
\(57\) −41.6092 + 89.3593i −0.729987 + 1.56771i
\(58\) 28.8677 − 56.2895i 0.497719 − 0.970509i
\(59\) 17.7674i 0.301143i 0.988599 + 0.150571i \(0.0481114\pi\)
−0.988599 + 0.150571i \(0.951889\pi\)
\(60\) −97.6869 + 34.9766i −1.62812 + 0.582943i
\(61\) −85.5197 −1.40196 −0.700981 − 0.713180i \(-0.747254\pi\)
−0.700981 + 0.713180i \(0.747254\pi\)
\(62\) −32.6232 + 63.6124i −0.526180 + 1.02601i
\(63\) −45.1086 − 45.1086i −0.716009 − 0.716009i
\(64\) 61.1722 − 18.8139i 0.955815 − 0.293968i
\(65\) 1.30993 + 0.375585i 0.0201528 + 0.00577824i
\(66\) 140.147 − 45.1294i 2.12345 − 0.683779i
\(67\) −28.1260 − 28.1260i −0.419791 − 0.419791i 0.465341 − 0.885132i \(-0.345932\pi\)
−0.885132 + 0.465341i \(0.845932\pi\)
\(68\) 16.6775 − 101.827i 0.245258 − 1.49746i
\(69\) 135.934 1.97005
\(70\) −25.9788 − 24.3527i −0.371126 − 0.347895i
\(71\) 62.1745 0.875697 0.437848 − 0.899049i \(-0.355741\pi\)
0.437848 + 0.899049i \(0.355741\pi\)
\(72\) 21.3026 + 141.730i 0.295870 + 1.96847i
\(73\) −24.6006 + 24.6006i −0.336995 + 0.336995i −0.855235 − 0.518240i \(-0.826587\pi\)
0.518240 + 0.855235i \(0.326587\pi\)
\(74\) 15.4058 + 47.8421i 0.208187 + 0.646515i
\(75\) −29.1816 − 126.374i −0.389088 − 1.68499i
\(76\) 62.8059 + 42.7951i 0.826393 + 0.563094i
\(77\) 35.7286 + 35.7286i 0.464008 + 0.464008i
\(78\) 1.29046 − 2.51628i 0.0165444 − 0.0322601i
\(79\) − 116.222i − 1.47117i −0.677435 − 0.735583i \(-0.736908\pi\)
0.677435 − 0.735583i \(-0.263092\pi\)
\(80\) 14.4392 + 78.6861i 0.180491 + 0.983577i
\(81\) −78.7189 −0.971839
\(82\) −88.5965 − 45.4362i −1.08045 − 0.554100i
\(83\) −26.7366 − 26.7366i −0.322127 − 0.322127i 0.527455 − 0.849583i \(-0.323146\pi\)
−0.849583 + 0.527455i \(0.823146\pi\)
\(84\) −60.0093 + 43.1188i −0.714397 + 0.513319i
\(85\) 123.985 + 35.5491i 1.45864 + 0.418224i
\(86\) −36.4155 + 11.7263i −0.423436 + 0.136352i
\(87\) 116.034 − 116.034i 1.33372 − 1.33372i
\(88\) −16.8729 − 112.258i −0.191738 − 1.27566i
\(89\) −149.178 −1.67616 −0.838081 − 0.545546i \(-0.816323\pi\)
−0.838081 + 0.545546i \(0.816323\pi\)
\(90\) −179.059 + 5.78519i −1.98955 + 0.0642799i
\(91\) 0.970476 0.0106646
\(92\) 16.9397 − 103.428i 0.184127 − 1.12422i
\(93\) −131.129 + 131.129i −1.40999 + 1.40999i
\(94\) 32.4527 + 100.781i 0.345242 + 1.07213i
\(95\) −62.2847 + 71.7329i −0.655629 + 0.755084i
\(96\) 166.001 + 2.18362i 1.72918 + 0.0227460i
\(97\) 9.47531 − 9.47531i 0.0976837 − 0.0976837i −0.656576 − 0.754260i \(-0.727996\pi\)
0.754260 + 0.656576i \(0.227996\pi\)
\(98\) 64.6366 + 33.1485i 0.659557 + 0.338250i
\(99\) 254.216 2.56784
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.227.75 yes 232
4.3 odd 2 inner 380.3.j.a.227.100 yes 232
5.3 odd 4 inner 380.3.j.a.303.17 yes 232
19.18 odd 2 inner 380.3.j.a.227.42 yes 232
20.3 even 4 inner 380.3.j.a.303.42 yes 232
76.75 even 2 inner 380.3.j.a.227.17 ✓ 232
95.18 even 4 inner 380.3.j.a.303.100 yes 232
380.303 odd 4 inner 380.3.j.a.303.75 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.17 ✓ 232 76.75 even 2 inner
380.3.j.a.227.42 yes 232 19.18 odd 2 inner
380.3.j.a.227.75 yes 232 1.1 even 1 trivial
380.3.j.a.227.100 yes 232 4.3 odd 2 inner
380.3.j.a.303.17 yes 232 5.3 odd 4 inner
380.3.j.a.303.42 yes 232 20.3 even 4 inner
380.3.j.a.303.75 yes 232 380.303 odd 4 inner
380.3.j.a.303.100 yes 232 95.18 even 4 inner