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Properties

Label 380.3.h.a.39.14
Level $380$
Weight $3$
Character 380.39
Analytic conductor $10.354$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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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 39.14
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.13

$q$-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} +(6.93879 - 7.20092i) 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} +(-7.59575 + 18.5015i) 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} +(-14.1834 + 14.7192i) 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} +(0.490825 - 39.9970i) 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} +(-16.6849 + 47.1340i) 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} +(15.5263 - 37.8185i) 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} +(-79.4201 + 82.4204i) 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} +(28.5578 + 74.7292i) 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} +(-33.4570 + 34.7209i) 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 \(-0.610657\pi\)
−0.340680 + 0.940179i \(0.610657\pi\)
\(4\) 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 \(-0.804673\pi\)
−0.817558 + 0.575846i \(0.804673\pi\)
\(8\) −3.39962 + 7.24172i −0.424953 + 0.905215i
\(9\) −4.82174 −0.535749
\(10\) 6.93879 − 7.20092i 0.693879 − 0.720092i
\(11\) 2.74195i 0.249268i 0.992203 + 0.124634i \(0.0397757\pi\)
−0.992203 + 0.124634i \(0.960224\pi\)
\(12\) −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
\(20\) −7.59575 + 18.5015i −0.379787 + 0.925074i
\(21\) 23.3962 1.11410
\(22\) −2.02031 − 5.09819i −0.0918324 − 0.231736i
\(23\) −33.3470 −1.44987 −0.724934 − 0.688818i \(-0.758130\pi\)
−0.724934 + 0.688818i \(0.758130\pi\)
\(24\) 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 \(0.579477\pi\)
\(30\) −14.1834 + 14.7192i −0.472781 + 0.490642i
\(31\) − 6.39501i − 0.206291i −0.994666 − 0.103145i \(-0.967109\pi\)
0.994666 − 0.103145i \(-0.0328907\pi\)
\(32\) 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
\(40\) 0.490825 − 39.9970i 0.0122706 − 0.999925i
\(41\) 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 \(-0.377903\pi\)
0.374243 + 0.927331i \(0.377903\pi\)
\(44\) 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 \(-0.540645\pi\)
−0.127343 + 0.991859i \(0.540645\pi\)
\(48\) −2.01384 + 32.6432i −0.0419551 + 0.680067i
\(49\) 82.0068 1.67361
\(50\) −16.6849 + 47.1340i −0.333697 + 0.942680i
\(51\) 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 \(0.143343\pi\)
−0.900305 + 0.435260i \(0.856657\pi\)
\(60\) 15.5263 − 37.8185i 0.258772 − 0.630308i
\(61\) 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 \(-0.261000\pi\)
0.682254 + 0.731115i \(0.261000\pi\)
\(68\) −51.2228 − 54.4802i −0.753276 − 0.801180i
\(69\) 68.1638 0.987882
\(70\) −79.4201 + 82.4204i −1.13457 + 1.17743i
\(71\) 93.7587i 1.32054i 0.751026 + 0.660272i \(0.229559\pi\)
−0.751026 + 0.660272i \(0.770441\pi\)
\(72\) 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 \(0.123506\pi\)
−0.925666 + 0.378342i \(0.876494\pi\)
\(80\) 28.5578 + 74.7292i 0.356972 + 0.934115i
\(81\) −14.3552 −0.177224
\(82\) −16.3910 + 6.49543i −0.199890 + 0.0792126i
\(83\) −53.7068 −0.647070 −0.323535 − 0.946216i \(-0.604871\pi\)
−0.323535 + 0.946216i \(0.604871\pi\)
\(84\) 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 \(0.470128\pi\)
\(90\) −33.4570 + 34.7209i −0.371745 + 0.385788i
\(91\) − 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.39.14 yes 108
4.3 odd 2 inner 380.3.h.a.39.96 yes 108
5.4 even 2 inner 380.3.h.a.39.95 yes 108
20.19 odd 2 inner 380.3.h.a.39.13 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.13 ✓ 108 20.19 odd 2 inner
380.3.h.a.39.14 yes 108 1.1 even 1 trivial
380.3.h.a.39.95 yes 108 5.4 even 2 inner
380.3.h.a.39.96 yes 108 4.3 odd 2 inner