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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (of order \(6\), 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: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.1
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.1

$q$-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+(-1.99987 - 0.0230852i) q^{2} +(0.798339 + 0.460921i) q^{3} +(3.99893 + 0.0923347i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(-1.58593 - 0.940211i) q^{6} -13.2021i q^{7} +(-7.99520 - 0.276973i) q^{8} +(-4.07510 - 7.05829i) q^{9} +(2.28062 - 3.84692i) q^{10} +13.1700i q^{11} +(3.14995 + 1.91691i) q^{12} +(-2.74783 - 4.75939i) q^{13} +(-0.304772 + 26.4024i) q^{14} +(-1.78514 + 1.03065i) q^{15} +(15.9829 + 0.738481i) q^{16} +(-16.6580 + 28.8525i) q^{17} +(7.98672 + 14.2097i) q^{18} +(-15.7226 + 10.6677i) q^{19} +(-4.64975 + 7.64067i) q^{20} +(6.08511 - 10.5397i) q^{21} +(0.304033 - 26.3383i) q^{22} +(6.77660 - 3.91247i) q^{23} +(-6.25522 - 3.90628i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(5.38543 + 9.58158i) q^{26} -15.8098i q^{27} +(1.21901 - 52.7942i) q^{28} +(-6.67298 - 11.5579i) q^{29} +(3.59383 - 2.01995i) q^{30} +33.2302i q^{31} +(-31.9467 - 1.84583i) q^{32} +(-6.07034 + 10.5141i) q^{33} +(33.9798 - 57.3166i) q^{34} +(25.5657 + 14.7603i) q^{35} +(-15.6443 - 28.6019i) q^{36} -32.3591 q^{37} +(31.6894 - 20.9710i) q^{38} -5.06614i q^{39} +(9.47527 - 15.1730i) q^{40} +(-30.9152 + 53.5468i) q^{41} +(-12.4127 + 20.9375i) q^{42} +(0.396318 + 0.228814i) q^{43} +(-1.21605 + 52.6660i) q^{44} +18.2244 q^{45} +(-13.6426 + 7.66798i) q^{46} +(30.6683 - 17.7064i) q^{47} +(12.4194 + 7.95644i) q^{48} -125.294 q^{49} +(4.89970 + 8.71739i) q^{50} +(-26.5974 + 15.3560i) q^{51} +(-10.5490 - 19.2862i) q^{52} +(-22.1123 - 38.2997i) q^{53} +(-0.364972 + 31.6175i) q^{54} +(-25.5036 - 14.7245i) q^{55} +(-3.65662 + 105.553i) q^{56} +(-17.4689 + 1.26957i) q^{57} +(13.0783 + 23.2684i) q^{58} +(-58.2654 - 33.6396i) q^{59} +(-7.23382 + 3.95667i) q^{60} +(17.6931 + 30.6454i) q^{61} +(0.767126 - 66.4560i) q^{62} +(-93.1839 + 53.7997i) q^{63} +(63.8466 + 4.42892i) q^{64} +12.2887 q^{65} +(12.3826 - 20.8867i) q^{66} +(40.1271 - 23.1674i) q^{67} +(-69.2783 + 113.841i) q^{68} +7.21336 q^{69} +(-50.7872 - 30.1089i) q^{70} +(-32.0877 - 18.5259i) q^{71} +(30.6263 + 57.5611i) q^{72} +(-43.3661 + 75.1122i) q^{73} +(64.7139 + 0.747018i) q^{74} -4.60921i q^{75} +(-63.8586 + 41.2077i) q^{76} +173.871 q^{77} +(-0.116953 + 10.1316i) q^{78} +(-87.1619 - 50.3229i) q^{79} +(-19.2995 + 30.1252i) q^{80} +(-29.3889 + 50.9030i) q^{81} +(63.0625 - 106.373i) q^{82} -152.225i q^{83} +(25.3071 - 41.5858i) q^{84} +(-37.2484 - 64.5161i) q^{85} +(-0.787301 - 0.466747i) q^{86} -12.3029i q^{87} +(3.64774 - 105.297i) q^{88} +(41.1328 + 71.2440i) q^{89} +(-36.4464 - 0.420715i) q^{90} +(-62.8337 + 36.2771i) q^{91} +(27.4604 - 15.0200i) q^{92} +(-15.3165 + 26.5290i) q^{93} +(-61.7414 + 34.7024i) q^{94} +(-3.07953 - 42.3735i) q^{95} +(-24.6535 - 16.1985i) q^{96} +(29.5165 - 51.1241i) q^{97} +(250.572 + 2.89245i) q^{98} +(92.9577 - 53.6692i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99987 − 0.0230852i −0.999933 − 0.0115426i
\(3\) 0.798339 + 0.460921i 0.266113 + 0.153640i 0.627120 − 0.778923i \(-0.284234\pi\)
−0.361007 + 0.932563i \(0.617567\pi\)
\(4\) 3.99893 + 0.0923347i 0.999734 + 0.0230837i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) −1.58593 − 0.940211i −0.264322 − 0.156702i
\(7\) − 13.2021i − 1.88601i −0.332781 − 0.943004i \(-0.607987\pi\)
0.332781 − 0.943004i \(-0.392013\pi\)
\(8\) −7.99520 − 0.276973i −0.999400 − 0.0346217i
\(9\) −4.07510 − 7.05829i −0.452789 − 0.784254i
\(10\) 2.28062 − 3.84692i 0.228062 − 0.384692i
\(11\) 13.1700i 1.19727i 0.801020 + 0.598637i \(0.204291\pi\)
−0.801020 + 0.598637i \(0.795709\pi\)
\(12\) 3.14995 + 1.91691i 0.262495 + 0.159742i
\(13\) −2.74783 − 4.75939i −0.211372 − 0.366107i 0.740772 − 0.671756i \(-0.234460\pi\)
−0.952144 + 0.305649i \(0.901126\pi\)
\(14\) −0.304772 + 26.4024i −0.0217695 + 1.88588i
\(15\) −1.78514 + 1.03065i −0.119009 + 0.0687101i
\(16\) 15.9829 + 0.738481i 0.998934 + 0.0461551i
\(17\) −16.6580 + 28.8525i −0.979881 + 1.69720i −0.317099 + 0.948392i \(0.602709\pi\)
−0.662783 + 0.748812i \(0.730625\pi\)
\(18\) 7.98672 + 14.2097i 0.443707 + 0.789428i
\(19\) −15.7226 + 10.6677i −0.827505 + 0.561458i
\(20\) −4.64975 + 7.64067i −0.232487 + 0.382033i
\(21\) 6.08511 − 10.5397i 0.289767 − 0.501891i
\(22\) 0.304033 − 26.3383i 0.0138197 − 1.19719i
\(23\) 6.77660 − 3.91247i 0.294635 − 0.170107i −0.345395 − 0.938457i \(-0.612255\pi\)
0.640030 + 0.768350i \(0.278922\pi\)
\(24\) −6.25522 − 3.90628i −0.260634 − 0.162762i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 5.38543 + 9.58158i 0.207132 + 0.368522i
\(27\) − 15.8098i − 0.585548i
\(28\) 1.21901 − 52.7942i 0.0435360 − 1.88551i
\(29\) −6.67298 − 11.5579i −0.230103 − 0.398550i 0.727735 − 0.685858i \(-0.240573\pi\)
−0.957838 + 0.287308i \(0.907240\pi\)
\(30\) 3.59383 − 2.01995i 0.119794 − 0.0673318i
\(31\) 33.2302i 1.07194i 0.844236 + 0.535971i \(0.180054\pi\)
−0.844236 + 0.535971i \(0.819946\pi\)
\(32\) −31.9467 − 1.84583i −0.998335 − 0.0576823i
\(33\) −6.07034 + 10.5141i −0.183950 + 0.318610i
\(34\) 33.9798 − 57.3166i 0.999406 − 1.68578i
\(35\) 25.5657 + 14.7603i 0.730448 + 0.421724i
\(36\) −15.6443 − 28.6019i −0.434565 − 0.794497i
\(37\) −32.3591 −0.874571 −0.437285 − 0.899323i \(-0.644060\pi\)
−0.437285 + 0.899323i \(0.644060\pi\)
\(38\) 31.6894 − 20.9710i 0.833931 − 0.551869i
\(39\) − 5.06614i − 0.129901i
\(40\) 9.47527 − 15.1730i 0.236882 − 0.379325i
\(41\) −30.9152 + 53.5468i −0.754030 + 1.30602i 0.191825 + 0.981429i \(0.438559\pi\)
−0.945855 + 0.324589i \(0.894774\pi\)
\(42\) −12.4127 + 20.9375i −0.295541 + 0.498513i
\(43\) 0.396318 + 0.228814i 0.00921670 + 0.00532126i 0.504601 − 0.863353i \(-0.331640\pi\)
−0.495385 + 0.868674i \(0.664973\pi\)
\(44\) −1.21605 + 52.6660i −0.0276375 + 1.19696i
\(45\) 18.2244 0.404987
\(46\) −13.6426 + 7.66798i −0.296579 + 0.166695i
\(47\) 30.6683 − 17.7064i 0.652518 − 0.376731i −0.136902 − 0.990585i \(-0.543715\pi\)
0.789420 + 0.613853i \(0.210381\pi\)
\(48\) 12.4194 + 7.95644i 0.258738 + 0.165759i
\(49\) −125.294 −2.55703
\(50\) 4.89970 + 8.71739i 0.0979941 + 0.174348i
\(51\) −26.5974 + 15.3560i −0.521518 + 0.301099i
\(52\) −10.5490 − 19.2862i −0.202864 − 0.370888i
\(53\) −22.1123 − 38.2997i −0.417214 − 0.722636i 0.578444 − 0.815722i \(-0.303660\pi\)
−0.995658 + 0.0930861i \(0.970327\pi\)
\(54\) −0.364972 + 31.6175i −0.00675875 + 0.585509i
\(55\) −25.5036 − 14.7245i −0.463702 − 0.267719i
\(56\) −3.65662 + 105.553i −0.0652968 + 1.88488i
\(57\) −17.4689 + 1.26957i −0.306472 + 0.0222731i
\(58\) 13.0783 + 23.2684i 0.225487 + 0.401179i
\(59\) −58.2654 − 33.6396i −0.987549 − 0.570162i −0.0830085 − 0.996549i \(-0.526453\pi\)
−0.904541 + 0.426387i \(0.859786\pi\)
\(60\) −7.23382 + 3.95667i −0.120564 + 0.0659446i
\(61\) 17.6931 + 30.6454i 0.290051 + 0.502384i 0.973822 − 0.227314i \(-0.0729943\pi\)
−0.683770 + 0.729697i \(0.739661\pi\)
\(62\) 0.767126 − 66.4560i 0.0123730 − 1.07187i
\(63\) −93.1839 + 53.7997i −1.47911 + 0.853964i
\(64\) 63.8466 + 4.42892i 0.997603 + 0.0692018i
\(65\) 12.2887 0.189057
\(66\) 12.3826 − 20.8867i 0.187615 − 0.316466i
\(67\) 40.1271 − 23.1674i 0.598912 − 0.345782i −0.169702 − 0.985495i \(-0.554280\pi\)
0.768613 + 0.639714i \(0.220947\pi\)
\(68\) −69.2783 + 113.841i −1.01880 + 1.67413i
\(69\) 7.21336 0.104542
\(70\) −50.7872 − 30.1089i −0.725531 − 0.430127i
\(71\) −32.0877 − 18.5259i −0.451940 − 0.260928i 0.256709 − 0.966489i \(-0.417362\pi\)
−0.708649 + 0.705561i \(0.750695\pi\)
\(72\) 30.6263 + 57.5611i 0.425366 + 0.799460i
\(73\) −43.3661 + 75.1122i −0.594056 + 1.02893i 0.399624 + 0.916679i \(0.369141\pi\)
−0.993679 + 0.112255i \(0.964193\pi\)
\(74\) 64.7139 + 0.747018i 0.874513 + 0.0100948i
\(75\) − 4.60921i − 0.0614561i
\(76\) −63.8586 + 41.2077i −0.840245 + 0.542207i
\(77\) 173.871 2.25807
\(78\) −0.116953 + 10.1316i −0.00149940 + 0.129892i
\(79\) −87.1619 − 50.3229i −1.10331 − 0.636999i −0.166225 − 0.986088i \(-0.553158\pi\)
−0.937090 + 0.349089i \(0.886491\pi\)
\(80\) −19.2995 + 30.1252i −0.241244 + 0.376565i
\(81\) −29.3889 + 50.9030i −0.362826 + 0.628432i
\(82\) 63.0625 − 106.373i 0.769055 − 1.29723i
\(83\) − 152.225i − 1.83403i −0.398849 − 0.917017i \(-0.630590\pi\)
0.398849 − 0.917017i \(-0.369410\pi\)
\(84\) 25.3071 − 41.5858i 0.301275 − 0.495068i
\(85\) −37.2484 − 64.5161i −0.438216 − 0.759013i
\(86\) −0.787301 − 0.466747i −0.00915466 − 0.00542729i
\(87\) − 12.3029i − 0.141412i
\(88\) 3.64774 − 105.297i 0.0414516 − 1.19656i
\(89\) 41.1328 + 71.2440i 0.462166 + 0.800495i 0.999069 − 0.0431493i \(-0.0137391\pi\)
−0.536903 + 0.843644i \(0.680406\pi\)
\(90\) −36.4464 − 0.420715i −0.404960 − 0.00467461i
\(91\) −62.8337 + 36.2771i −0.690480 + 0.398649i
\(92\) 27.4604 − 15.0200i 0.298483 − 0.163261i
\(93\) −15.3165 + 26.5290i −0.164694 + 0.285258i
\(94\) −61.7414 + 34.7024i −0.656823 + 0.369175i
\(95\) −3.07953 − 42.3735i −0.0324161 − 0.446037i
\(96\) −24.6535 − 16.1985i −0.256808 − 0.168735i
\(97\) 29.5165 − 51.1241i 0.304294 − 0.527053i −0.672810 − 0.739815i \(-0.734913\pi\)
0.977104 + 0.212763i \(0.0682462\pi\)
\(98\) 250.572 + 2.89245i 2.55686 + 0.0295148i
\(99\) 92.9577 − 53.6692i 0.938967 − 0.542113i
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.q.a.11.1 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.55 yes 160
19.7 even 3 inner 380.3.q.a.311.55 yes 160
76.7 odd 6 inner 380.3.q.a.311.1 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.1 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.55 yes 160 4.3 odd 2 inner
380.3.q.a.311.1 yes 160 76.7 odd 6 inner
380.3.q.a.311.55 yes 160 19.7 even 3 inner