Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] 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("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), 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: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.96
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.95

$q$-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+(1.85933 + 0.736815i) q^{2} +2.04408 q^{3} +(2.91421 + 2.73996i) q^{4} +(-4.55181 + 2.06907i) q^{5} +(3.80062 + 1.50611i) q^{6} +11.4458 q^{7} +(3.39962 + 7.24172i) q^{8} -4.82174 q^{9} +(-9.98783 + 0.493235i) q^{10} -2.74195i q^{11} +(5.95687 + 5.60070i) q^{12} +23.9054i q^{13} +(21.2815 + 8.43345i) q^{14} +(-9.30426 + 4.22934i) q^{15} +(0.985208 + 15.9696i) q^{16} -18.6947i q^{17} +(-8.96520 - 3.55273i) q^{18} -4.35890i q^{19} +(-18.9341 - 6.44210i) q^{20} +23.3962 q^{21} +(2.02031 - 5.09819i) q^{22} +33.3470 q^{23} +(6.94910 + 14.8027i) q^{24} +(16.4379 - 18.8360i) q^{25} +(-17.6139 + 44.4480i) q^{26} -28.2527 q^{27} +(33.3555 + 31.3611i) q^{28} -14.3317 q^{29} +(-20.4159 + 1.00821i) q^{30} +6.39501i q^{31} +(-9.93485 + 30.4187i) q^{32} -5.60477i q^{33} +(13.7745 - 34.7596i) q^{34} +(-52.0992 + 23.6822i) q^{35} +(-14.0515 - 13.2114i) q^{36} -9.68409i q^{37} +(3.21170 - 8.10463i) q^{38} +48.8645i q^{39} +(-30.4580 - 25.9289i) q^{40} +8.81555 q^{41} +(43.5012 + 17.2386i) q^{42} -32.1849 q^{43} +(7.51285 - 7.99062i) q^{44} +(21.9476 - 9.97650i) q^{45} +(62.0030 + 24.5705i) q^{46} +11.9702 q^{47} +(2.01384 + 32.6432i) q^{48} +82.0068 q^{49} +(44.4421 - 22.9106i) q^{50} -38.2135i q^{51} +(-65.4999 + 69.6653i) q^{52} -101.668i q^{53} +(-52.5311 - 20.8170i) q^{54} +(5.67328 + 12.4808i) q^{55} +(38.9115 + 82.8875i) q^{56} -8.90994i q^{57} +(-26.6473 - 10.5598i) q^{58} -51.3607i q^{59} +(-38.7028 - 13.1682i) q^{60} +41.5193 q^{61} +(-4.71194 + 11.8904i) q^{62} -55.1887 q^{63} +(-40.8851 + 49.2383i) q^{64} +(-49.4619 - 108.813i) q^{65} +(4.12968 - 10.4211i) q^{66} -91.4221 q^{67} +(51.2228 - 54.4802i) q^{68} +68.1638 q^{69} +(-114.319 + 5.64548i) q^{70} -93.7587i q^{71} +(-16.3921 - 34.9177i) q^{72} +77.6994i q^{73} +(7.13539 - 18.0059i) q^{74} +(33.6004 - 38.5023i) q^{75} +(11.9432 - 12.7027i) q^{76} -31.3839i q^{77} +(-36.0041 + 90.8552i) q^{78} -59.7780i q^{79} +(-37.5267 - 70.6523i) q^{80} -14.3552 q^{81} +(16.3910 + 6.49543i) q^{82} +53.7068 q^{83} +(68.1813 + 64.1046i) q^{84} +(38.6806 + 85.0947i) q^{85} +(-59.8423 - 23.7143i) q^{86} -29.2951 q^{87} +(19.8565 - 9.32161i) q^{88} +16.6798 q^{89} +(48.1587 - 2.37825i) q^{90} +273.617i q^{91} +(97.1800 + 91.3694i) q^{92} +13.0719i q^{93} +(22.2566 + 8.81983i) q^{94} +(9.01885 + 19.8409i) q^{95} +(-20.3076 + 62.1783i) q^{96} -94.6828i q^{97} +(152.478 + 60.4238i) q^{98} +13.2210i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.85933 + 0.736815i 0.929664 + 0.368408i
\(3\) 2.04408 0.681360 0.340680 − 0.940179i \(-0.389343\pi\)
0.340680 + 0.940179i \(0.389343\pi\)
\(4\) 2.91421 + 2.73996i 0.728552 + 0.684991i
\(5\) −4.55181 + 2.06907i −0.910362 + 0.413813i
\(6\) 3.80062 + 1.50611i 0.633436 + 0.251018i
\(7\) 11.4458 1.63512 0.817558 − 0.575846i \(-0.195327\pi\)
0.817558 + 0.575846i \(0.195327\pi\)
\(8\) 3.39962 + 7.24172i 0.424953 + 0.905215i
\(9\) −4.82174 −0.535749
\(10\) −9.98783 + 0.493235i −0.998783 + 0.0493235i
\(11\) − 2.74195i − 0.249268i −0.992203 − 0.124634i \(-0.960224\pi\)
0.992203 − 0.124634i \(-0.0397757\pi\)
\(12\) 5.95687 + 5.60070i 0.496406 + 0.466725i
\(13\) 23.9054i 1.83888i 0.393235 + 0.919438i \(0.371356\pi\)
−0.393235 + 0.919438i \(0.628644\pi\)
\(14\) 21.2815 + 8.43345i 1.52011 + 0.602389i
\(15\) −9.30426 + 4.22934i −0.620284 + 0.281956i
\(16\) 0.985208 + 15.9696i 0.0615755 + 0.998102i
\(17\) − 18.6947i − 1.09969i −0.835267 − 0.549844i \(-0.814687\pi\)
0.835267 − 0.549844i \(-0.185313\pi\)
\(18\) −8.96520 − 3.55273i −0.498067 − 0.197374i
\(19\) − 4.35890i − 0.229416i
\(20\) −18.9341 − 6.44210i −0.946704 − 0.322105i
\(21\) 23.3962 1.11410
\(22\) 2.02031 − 5.09819i 0.0918324 − 0.231736i
\(23\) 33.3470 1.44987 0.724934 − 0.688818i \(-0.241870\pi\)
0.724934 + 0.688818i \(0.241870\pi\)
\(24\) 6.94910 + 14.8027i 0.289546 + 0.616777i
\(25\) 16.4379 − 18.8360i 0.657517 − 0.753440i
\(26\) −17.6139 + 44.4480i −0.677456 + 1.70954i
\(27\) −28.2527 −1.04640
\(28\) 33.3555 + 31.3611i 1.19127 + 1.12004i
\(29\) −14.3317 −0.494196 −0.247098 − 0.968991i \(-0.579477\pi\)
−0.247098 + 0.968991i \(0.579477\pi\)
\(30\) −20.4159 + 1.00821i −0.680531 + 0.0336071i
\(31\) 6.39501i 0.206291i 0.994666 + 0.103145i \(0.0328907\pi\)
−0.994666 + 0.103145i \(0.967109\pi\)
\(32\) −9.93485 + 30.4187i −0.310464 + 0.950585i
\(33\) − 5.60477i − 0.169841i
\(34\) 13.7745 − 34.7596i 0.405133 − 1.02234i
\(35\) −52.0992 + 23.6822i −1.48855 + 0.676633i
\(36\) −14.0515 − 13.2114i −0.390321 − 0.366983i
\(37\) − 9.68409i − 0.261732i −0.991400 − 0.130866i \(-0.958224\pi\)
0.991400 − 0.130866i \(-0.0417758\pi\)
\(38\) 3.21170 − 8.10463i 0.0845185 − 0.213280i
\(39\) 48.8645i 1.25294i
\(40\) −30.4580 − 25.9289i −0.761451 − 0.648222i
\(41\) 8.81555 0.215014 0.107507 − 0.994204i \(-0.465713\pi\)
0.107507 + 0.994204i \(0.465713\pi\)
\(42\) 43.5012 + 17.2386i 1.03574 + 0.410444i
\(43\) −32.1849 −0.748486 −0.374243 − 0.927331i \(-0.622097\pi\)
−0.374243 + 0.927331i \(0.622097\pi\)
\(44\) 7.51285 − 7.99062i 0.170747 − 0.181605i
\(45\) 21.9476 − 9.97650i 0.487725 − 0.221700i
\(46\) 62.0030 + 24.5705i 1.34789 + 0.534142i
\(47\) 11.9702 0.254685 0.127343 − 0.991859i \(-0.459355\pi\)
0.127343 + 0.991859i \(0.459355\pi\)
\(48\) 2.01384 + 32.6432i 0.0419551 + 0.680067i
\(49\) 82.0068 1.67361
\(50\) 44.4421 − 22.9106i 0.888843 − 0.458212i
\(51\) − 38.2135i − 0.749283i
\(52\) −65.4999 + 69.6653i −1.25961 + 1.33972i
\(53\) − 101.668i − 1.91826i −0.282958 − 0.959132i \(-0.591316\pi\)
0.282958 − 0.959132i \(-0.408684\pi\)
\(54\) −52.5311 − 20.8170i −0.972799 − 0.385501i
\(55\) 5.67328 + 12.4808i 0.103151 + 0.226924i
\(56\) 38.9115 + 82.8875i 0.694848 + 1.48013i
\(57\) − 8.90994i − 0.156315i
\(58\) −26.6473 − 10.5598i −0.459436 − 0.182065i
\(59\) − 51.3607i − 0.870520i −0.900305 − 0.435260i \(-0.856657\pi\)
0.900305 − 0.435260i \(-0.143343\pi\)
\(60\) −38.7028 − 13.1682i −0.645046 − 0.219469i
\(61\) 41.5193 0.680644 0.340322 − 0.940309i \(-0.389464\pi\)
0.340322 + 0.940309i \(0.389464\pi\)
\(62\) −4.71194 + 11.8904i −0.0759990 + 0.191781i
\(63\) −55.1887 −0.876012
\(64\) −40.8851 + 49.2383i −0.638830 + 0.769348i
\(65\) −49.4619 − 108.813i −0.760952 − 1.67404i
\(66\) 4.12968 − 10.4211i 0.0625709 − 0.157896i
\(67\) −91.4221 −1.36451 −0.682254 − 0.731115i \(-0.739000\pi\)
−0.682254 + 0.731115i \(0.739000\pi\)
\(68\) 51.2228 − 54.4802i 0.753276 − 0.801180i
\(69\) 68.1638 0.987882
\(70\) −114.319 + 5.64548i −1.63313 + 0.0806497i
\(71\) − 93.7587i − 1.32054i −0.751026 − 0.660272i \(-0.770441\pi\)
0.751026 − 0.660272i \(-0.229559\pi\)
\(72\) −16.3921 − 34.9177i −0.227668 − 0.484968i
\(73\) 77.6994i 1.06438i 0.846626 + 0.532188i \(0.178630\pi\)
−0.846626 + 0.532188i \(0.821370\pi\)
\(74\) 7.13539 − 18.0059i 0.0964241 − 0.243323i
\(75\) 33.6004 − 38.5023i 0.448006 − 0.513364i
\(76\) 11.9432 − 12.7027i 0.157148 − 0.167141i
\(77\) − 31.3839i − 0.407583i
\(78\) −36.0041 + 90.8552i −0.461591 + 1.16481i
\(79\) − 59.7780i − 0.756684i −0.925666 − 0.378342i \(-0.876494\pi\)
0.925666 − 0.378342i \(-0.123506\pi\)
\(80\) −37.5267 − 70.6523i −0.469084 − 0.883153i
\(81\) −14.3552 −0.177224
\(82\) 16.3910 + 6.49543i 0.199890 + 0.0792126i
\(83\) 53.7068 0.647070 0.323535 − 0.946216i \(-0.395129\pi\)
0.323535 + 0.946216i \(0.395129\pi\)
\(84\) 68.1813 + 64.1046i 0.811682 + 0.763150i
\(85\) 38.6806 + 85.0947i 0.455066 + 1.00111i
\(86\) −59.8423 − 23.7143i −0.695841 − 0.275748i
\(87\) −29.2951 −0.336725
\(88\) 19.8565 − 9.32161i 0.225642 − 0.105927i
\(89\) 16.6798 0.187413 0.0937067 − 0.995600i \(-0.470128\pi\)
0.0937067 + 0.995600i \(0.470128\pi\)
\(90\) 48.1587 − 2.37825i 0.535097 − 0.0264250i
\(91\) 273.617i 3.00678i
\(92\) 97.1800 + 91.3694i 1.05630 + 0.993146i
\(93\) 13.0719i 0.140558i
\(94\) 22.2566 + 8.81983i 0.236772 + 0.0938280i
\(95\) 9.01885 + 19.8409i 0.0949353 + 0.208851i
\(96\) −20.3076 + 62.1783i −0.211538 + 0.647691i
\(97\) − 94.6828i − 0.976112i −0.872812 − 0.488056i \(-0.837706\pi\)
0.872812 − 0.488056i \(-0.162294\pi\)
\(98\) 152.478 + 60.4238i 1.55589 + 0.616569i
\(99\) 13.2210i 0.133545i
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.h.a.39.96 yes 108
4.3 odd 2 inner 380.3.h.a.39.14 yes 108
5.4 even 2 inner 380.3.h.a.39.13 ✓ 108
20.19 odd 2 inner 380.3.h.a.39.95 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.13 ✓ 108 5.4 even 2 inner
380.3.h.a.39.14 yes 108 4.3 odd 2 inner
380.3.h.a.39.95 yes 108 20.19 odd 2 inner
380.3.h.a.39.96 yes 108 1.1 even 1 trivial