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Properties

Label 380.3.h.a.39.20
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 39.20
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.19

$q$-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+(-1.68898 + 1.07114i) q^{2} +5.94620 q^{3} +(1.70531 - 3.61828i) q^{4} +(1.71987 + 4.69490i) q^{5} +(-10.0430 + 6.36923i) q^{6} -9.59523 q^{7} +(0.995461 + 7.93782i) q^{8} +26.3573 q^{9} +(-7.93373 - 6.08736i) q^{10} +5.60839i q^{11} +(10.1401 - 21.5150i) q^{12} -1.46557i q^{13} +(16.2062 - 10.2779i) q^{14} +(10.2267 + 27.9168i) q^{15} +(-10.1839 - 12.3405i) q^{16} +30.4749i q^{17} +(-44.5169 + 28.2324i) q^{18} +4.35890i q^{19} +(19.9203 + 1.78326i) q^{20} -57.0551 q^{21} +(-6.00738 - 9.47246i) q^{22} +5.38208 q^{23} +(5.91921 + 47.1999i) q^{24} +(-19.0841 + 16.1492i) q^{25} +(1.56984 + 2.47533i) q^{26} +103.210 q^{27} +(-16.3628 + 34.7182i) q^{28} +25.0321 q^{29} +(-47.1755 - 36.1966i) q^{30} -26.0540i q^{31} +(30.4188 + 9.93458i) q^{32} +33.3486i q^{33} +(-32.6430 - 51.4716i) q^{34} +(-16.5026 - 45.0486i) q^{35} +(44.9473 - 95.3679i) q^{36} -35.8792i q^{37} +(-4.66900 - 7.36209i) q^{38} -8.71460i q^{39} +(-35.5552 + 18.3256i) q^{40} +3.18512 q^{41} +(96.3650 - 61.1142i) q^{42} +57.1393 q^{43} +(20.2927 + 9.56403i) q^{44} +(45.3311 + 123.745i) q^{45} +(-9.09023 + 5.76498i) q^{46} -30.6020 q^{47} +(-60.5552 - 73.3793i) q^{48} +43.0685 q^{49} +(14.9345 - 47.7175i) q^{50} +181.210i q^{51} +(-5.30285 - 2.49925i) q^{52} -61.6843i q^{53} +(-174.319 + 110.552i) q^{54} +(-26.3308 + 9.64571i) q^{55} +(-9.55168 - 76.1653i) q^{56} +25.9189i q^{57} +(-42.2788 + 26.8130i) q^{58} -50.5937i q^{59} +(118.450 + 10.6036i) q^{60} -31.5158 q^{61} +(27.9076 + 44.0047i) q^{62} -252.904 q^{63} +(-62.0181 + 15.8036i) q^{64} +(6.88072 - 2.52060i) q^{65} +(-35.7211 - 56.3251i) q^{66} -13.4084 q^{67} +(110.267 + 51.9691i) q^{68} +32.0029 q^{69} +(76.1260 + 58.4096i) q^{70} +23.8999i q^{71} +(26.2376 + 209.219i) q^{72} +63.7064i q^{73} +(38.4317 + 60.5992i) q^{74} +(-113.478 + 96.0266i) q^{75} +(15.7717 + 7.43326i) q^{76} -53.8138i q^{77} +(9.33458 + 14.7188i) q^{78} -57.1288i q^{79} +(40.4226 - 69.0363i) q^{80} +376.491 q^{81} +(-5.37961 + 3.41172i) q^{82} -71.3250 q^{83} +(-97.2965 + 206.441i) q^{84} +(-143.077 + 52.4130i) q^{85} +(-96.5072 + 61.2044i) q^{86} +148.846 q^{87} +(-44.5184 + 5.58293i) q^{88} -7.54874 q^{89} +(-209.112 - 160.446i) q^{90} +14.0625i q^{91} +(9.17811 - 19.4739i) q^{92} -154.922i q^{93} +(51.6861 - 32.7791i) q^{94} +(-20.4646 + 7.49675i) q^{95} +(180.876 + 59.0730i) q^{96} -118.660i q^{97} +(-72.7418 + 46.1325i) q^{98} +147.822i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.68898 + 1.07114i −0.844490 + 0.535571i
\(3\) 5.94620 1.98207 0.991033 − 0.133617i \(-0.0426591\pi\)
0.991033 + 0.133617i \(0.0426591\pi\)
\(4\) 1.70531 − 3.61828i 0.426327 − 0.904569i
\(5\) 1.71987 + 4.69490i 0.343974 + 0.938979i
\(6\) −10.0430 + 6.36923i −1.67384 + 1.06154i
\(7\) −9.59523 −1.37075 −0.685374 − 0.728192i \(-0.740361\pi\)
−0.685374 + 0.728192i \(0.740361\pi\)
\(8\) 0.995461 + 7.93782i 0.124433 + 0.992228i
\(9\) 26.3573 2.92859
\(10\) −7.93373 − 6.08736i −0.793373 − 0.608736i
\(11\) 5.60839i 0.509854i 0.966960 + 0.254927i \(0.0820514\pi\)
−0.966960 + 0.254927i \(0.917949\pi\)
\(12\) 10.1401 − 21.5150i 0.845008 − 1.79292i
\(13\) − 1.46557i − 0.112736i −0.998410 − 0.0563682i \(-0.982048\pi\)
0.998410 − 0.0563682i \(-0.0179521\pi\)
\(14\) 16.2062 − 10.2779i 1.15758 − 0.734133i
\(15\) 10.2267 + 27.9168i 0.681780 + 1.86112i
\(16\) −10.1839 − 12.3405i −0.636491 − 0.771284i
\(17\) 30.4749i 1.79264i 0.443404 + 0.896322i \(0.353771\pi\)
−0.443404 + 0.896322i \(0.646229\pi\)
\(18\) −44.5169 + 28.2324i −2.47316 + 1.56847i
\(19\) 4.35890i 0.229416i
\(20\) 19.9203 + 1.78326i 0.996017 + 0.0891632i
\(21\) −57.0551 −2.71691
\(22\) −6.00738 − 9.47246i −0.273063 − 0.430566i
\(23\) 5.38208 0.234004 0.117002 − 0.993132i \(-0.462672\pi\)
0.117002 + 0.993132i \(0.462672\pi\)
\(24\) 5.91921 + 47.1999i 0.246634 + 1.96666i
\(25\) −19.0841 + 16.1492i −0.763363 + 0.645970i
\(26\) 1.56984 + 2.47533i 0.0603784 + 0.0952048i
\(27\) 103.210 3.82259
\(28\) −16.3628 + 34.7182i −0.584386 + 1.23994i
\(29\) 25.0321 0.863177 0.431589 − 0.902071i \(-0.357953\pi\)
0.431589 + 0.902071i \(0.357953\pi\)
\(30\) −47.1755 − 36.1966i −1.57252 − 1.20655i
\(31\) − 26.0540i − 0.840453i −0.907419 − 0.420226i \(-0.861951\pi\)
0.907419 − 0.420226i \(-0.138049\pi\)
\(32\) 30.4188 + 9.93458i 0.950588 + 0.310456i
\(33\) 33.3486i 1.01056i
\(34\) −32.6430 − 51.4716i −0.960088 − 1.51387i
\(35\) −16.5026 − 45.0486i −0.471502 − 1.28710i
\(36\) 44.9473 − 95.3679i 1.24853 − 2.64911i
\(37\) − 35.8792i − 0.969708i −0.874595 − 0.484854i \(-0.838873\pi\)
0.874595 − 0.484854i \(-0.161127\pi\)
\(38\) −4.66900 − 7.36209i −0.122868 − 0.193739i
\(39\) − 8.71460i − 0.223451i
\(40\) −35.5552 + 18.3256i −0.888880 + 0.458141i
\(41\) 3.18512 0.0776859 0.0388430 − 0.999245i \(-0.487633\pi\)
0.0388430 + 0.999245i \(0.487633\pi\)
\(42\) 96.3650 − 61.1142i 2.29440 − 1.45510i
\(43\) 57.1393 1.32882 0.664411 − 0.747368i \(-0.268683\pi\)
0.664411 + 0.747368i \(0.268683\pi\)
\(44\) 20.2927 + 9.56403i 0.461198 + 0.217364i
\(45\) 45.3311 + 123.745i 1.00736 + 2.74988i
\(46\) −9.09023 + 5.76498i −0.197614 + 0.125326i
\(47\) −30.6020 −0.651106 −0.325553 − 0.945524i \(-0.605550\pi\)
−0.325553 + 0.945524i \(0.605550\pi\)
\(48\) −60.5552 − 73.3793i −1.26157 − 1.52874i
\(49\) 43.0685 0.878948
\(50\) 14.9345 − 47.7175i 0.298690 − 0.954350i
\(51\) 181.210i 3.55314i
\(52\) −5.30285 − 2.49925i −0.101978 − 0.0480626i
\(53\) − 61.6843i − 1.16385i −0.813241 − 0.581927i \(-0.802299\pi\)
0.813241 − 0.581927i \(-0.197701\pi\)
\(54\) −174.319 + 110.552i −3.22814 + 2.04727i
\(55\) −26.3308 + 9.64571i −0.478742 + 0.175377i
\(56\) −9.55168 − 76.1653i −0.170566 − 1.36009i
\(57\) 25.9189i 0.454717i
\(58\) −42.2788 + 26.8130i −0.728944 + 0.462293i
\(59\) − 50.5937i − 0.857521i −0.903418 − 0.428761i \(-0.858950\pi\)
0.903418 − 0.428761i \(-0.141050\pi\)
\(60\) 118.450 + 10.6036i 1.97417 + 0.176727i
\(61\) −31.5158 −0.516652 −0.258326 − 0.966058i \(-0.583171\pi\)
−0.258326 + 0.966058i \(0.583171\pi\)
\(62\) 27.9076 + 44.0047i 0.450122 + 0.709754i
\(63\) −252.904 −4.01435
\(64\) −62.0181 + 15.8036i −0.969033 + 0.246931i
\(65\) 6.88072 − 2.52060i 0.105857 − 0.0387785i
\(66\) −35.7211 − 56.3251i −0.541229 − 0.853411i
\(67\) −13.4084 −0.200125 −0.100062 − 0.994981i \(-0.531904\pi\)
−0.100062 + 0.994981i \(0.531904\pi\)
\(68\) 110.267 + 51.9691i 1.62157 + 0.764252i
\(69\) 32.0029 0.463811
\(70\) 76.1260 + 58.4096i 1.08751 + 0.834423i
\(71\) 23.8999i 0.336618i 0.985734 + 0.168309i \(0.0538306\pi\)
−0.985734 + 0.168309i \(0.946169\pi\)
\(72\) 26.2376 + 209.219i 0.364412 + 2.90583i
\(73\) 63.7064i 0.872690i 0.899779 + 0.436345i \(0.143727\pi\)
−0.899779 + 0.436345i \(0.856273\pi\)
\(74\) 38.4317 + 60.5992i 0.519348 + 0.818908i
\(75\) −113.478 + 96.0266i −1.51304 + 1.28035i
\(76\) 15.7717 + 7.43326i 0.207522 + 0.0978061i
\(77\) − 53.8138i − 0.698880i
\(78\) 9.33458 + 14.7188i 0.119674 + 0.188702i
\(79\) − 57.1288i − 0.723149i −0.932343 − 0.361574i \(-0.882239\pi\)
0.932343 − 0.361574i \(-0.117761\pi\)
\(80\) 40.4226 − 69.0363i 0.505283 − 0.862954i
\(81\) 376.491 4.64803
\(82\) −5.37961 + 3.41172i −0.0656050 + 0.0416063i
\(83\) −71.3250 −0.859338 −0.429669 − 0.902987i \(-0.641370\pi\)
−0.429669 + 0.902987i \(0.641370\pi\)
\(84\) −97.2965 + 206.441i −1.15829 + 2.45763i
\(85\) −143.077 + 52.4130i −1.68325 + 0.616623i
\(86\) −96.5072 + 61.2044i −1.12218 + 0.711679i
\(87\) 148.846 1.71087
\(88\) −44.5184 + 5.58293i −0.505891 + 0.0634424i
\(89\) −7.54874 −0.0848173 −0.0424086 − 0.999100i \(-0.513503\pi\)
−0.0424086 + 0.999100i \(0.513503\pi\)
\(90\) −209.112 − 160.446i −2.32346 − 1.78273i
\(91\) 14.0625i 0.154533i
\(92\) 9.17811 − 19.4739i 0.0997620 − 0.211673i
\(93\) − 154.922i − 1.66583i
\(94\) 51.6861 − 32.7791i 0.549852 − 0.348714i
\(95\) −20.4646 + 7.49675i −0.215417 + 0.0789131i
\(96\) 180.876 + 59.0730i 1.88413 + 0.615343i
\(97\) − 118.660i − 1.22330i −0.791128 − 0.611650i \(-0.790506\pi\)
0.791128 − 0.611650i \(-0.209494\pi\)
\(98\) −72.7418 + 46.1325i −0.742263 + 0.470739i
\(99\) 147.822i 1.49315i
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.39.20 yes 108
4.3 odd 2 inner 380.3.h.a.39.90 yes 108
5.4 even 2 inner 380.3.h.a.39.89 yes 108
20.19 odd 2 inner 380.3.h.a.39.19 ✓ 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.19 ✓ 108 20.19 odd 2 inner
380.3.h.a.39.20 yes 108 1.1 even 1 trivial
380.3.h.a.39.89 yes 108 5.4 even 2 inner
380.3.h.a.39.90 yes 108 4.3 odd 2 inner