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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 39.90
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.89

$q$-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} +(-2.12407 + 9.77181i) 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} +(-14.0545 + 14.2292i) 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} +(12.6302 - 58.1051i) 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} +(-38.9793 + 8.97846i) 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} +(-49.5308 + 6.83397i) 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} +(83.5710 - 84.6097i) 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} +(-20.3810 + 93.7628i) 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} +(-75.4525 - 26.5880i) 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} +(-55.9848 + 257.558i) 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 \(-0.957341\pi\)
−0.991033 + 0.133617i \(0.957341\pi\)
\(4\) 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 \(-0.259639\pi\)
0.685374 + 0.728192i \(0.259639\pi\)
\(8\) −0.995461 + 7.93782i −0.124433 + 0.992228i
\(9\) 26.3573 2.92859
\(10\) −2.12407 + 9.77181i −0.212407 + 0.977181i
\(11\) − 5.60839i − 0.509854i −0.966960 − 0.254927i \(-0.917949\pi\)
0.966960 − 0.254927i \(-0.0820514\pi\)
\(12\) −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
\(20\) −14.0545 + 14.2292i −0.702726 + 0.711461i
\(21\) −57.0551 −2.71691
\(22\) 6.00738 − 9.47246i 0.273063 − 0.430566i
\(23\) −5.38208 −0.234004 −0.117002 − 0.993132i \(-0.537328\pi\)
−0.117002 + 0.993132i \(0.537328\pi\)
\(24\) 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 \(0.357953\pi\)
\(30\) 12.6302 − 58.1051i 0.421005 − 1.93684i
\(31\) 26.0540i 0.840453i 0.907419 + 0.420226i \(0.138049\pi\)
−0.907419 + 0.420226i \(0.861951\pi\)
\(32\) −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
\(40\) −38.9793 + 8.97846i −0.974483 + 0.224461i
\(41\) 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 \(-0.731317\pi\)
−0.664411 + 0.747368i \(0.731317\pi\)
\(44\) 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 \(-0.394450\pi\)
0.325553 + 0.945524i \(0.394450\pi\)
\(48\) 60.5552 − 73.3793i 1.26157 − 1.52874i
\(49\) 43.0685 0.878948
\(50\) −49.5308 + 6.83397i −0.990615 + 0.136679i
\(51\) − 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 \(0.141050\pi\)
−0.903418 + 0.428761i \(0.858950\pi\)
\(60\) 83.5710 − 84.6097i 1.39285 − 1.41016i
\(61\) −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 \(-0.468096\pi\)
0.100062 + 0.994981i \(0.468096\pi\)
\(68\) −110.267 + 51.9691i −1.62157 + 0.764252i
\(69\) 32.0029 0.463811
\(70\) −20.3810 + 93.7628i −0.291157 + 1.33947i
\(71\) − 23.8999i − 0.336618i −0.985734 − 0.168309i \(-0.946169\pi\)
0.985734 − 0.168309i \(-0.0538306\pi\)
\(72\) −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 \(0.117761\pi\)
−0.932343 + 0.361574i \(0.882239\pi\)
\(80\) −75.4525 − 26.5880i −0.943156 − 0.332350i
\(81\) 376.491 4.64803
\(82\) 5.37961 + 3.41172i 0.0656050 + 0.0416063i
\(83\) 71.3250 0.859338 0.429669 − 0.902987i \(-0.358630\pi\)
0.429669 + 0.902987i \(0.358630\pi\)
\(84\) −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 \(0.513503\pi\)
\(90\) −55.9848 + 257.558i −0.622053 + 2.86176i
\(91\) − 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.39.90 yes 108
4.3 odd 2 inner 380.3.h.a.39.20 yes 108
5.4 even 2 inner 380.3.h.a.39.19 ✓ 108
20.19 odd 2 inner 380.3.h.a.39.89 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.19 ✓ 108 5.4 even 2 inner
380.3.h.a.39.20 yes 108 4.3 odd 2 inner
380.3.h.a.39.89 yes 108 20.19 odd 2 inner
380.3.h.a.39.90 yes 108 1.1 even 1 trivial