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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 39.97
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.98

$q$-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+(1.86011 - 0.734843i) q^{2} -2.97071 q^{3} +(2.92001 - 2.73378i) q^{4} +(-0.632199 + 4.95987i) q^{5} +(-5.52585 + 2.18301i) q^{6} +5.57969 q^{7} +(3.42265 - 7.23087i) q^{8} -0.174871 q^{9} +(2.46877 + 9.69047i) q^{10} +6.76119i q^{11} +(-8.67452 + 8.12126i) q^{12} +13.4398i q^{13} +(10.3788 - 4.10019i) q^{14} +(1.87808 - 14.7343i) q^{15} +(1.05295 - 15.9653i) q^{16} +15.9028i q^{17} +(-0.325279 + 0.128503i) q^{18} +4.35890i q^{19} +(11.7131 + 16.2112i) q^{20} -16.5756 q^{21} +(4.96841 + 12.5766i) q^{22} +37.1354 q^{23} +(-10.1677 + 21.4808i) q^{24} +(-24.2006 - 6.27125i) q^{25} +(9.87616 + 24.9995i) q^{26} +27.2559 q^{27} +(16.2928 - 15.2536i) q^{28} +45.3295 q^{29} +(-7.33399 - 28.7876i) q^{30} +10.3171i q^{31} +(-9.77340 - 30.4710i) q^{32} -20.0856i q^{33} +(11.6861 + 29.5809i) q^{34} +(-3.52747 + 27.6745i) q^{35} +(-0.510626 + 0.478058i) q^{36} +73.5102i q^{37} +(3.20310 + 8.10803i) q^{38} -39.9258i q^{39} +(33.7004 + 21.5472i) q^{40} -73.2141 q^{41} +(-30.8325 + 12.1805i) q^{42} +69.3375 q^{43} +(18.4836 + 19.7428i) q^{44} +(0.110553 - 0.867338i) q^{45} +(69.0759 - 27.2887i) q^{46} -41.1514 q^{47} +(-3.12800 + 47.4284i) q^{48} -17.8671 q^{49} +(-49.6242 + 6.11846i) q^{50} -47.2426i q^{51} +(36.7415 + 39.2445i) q^{52} -75.3694i q^{53} +(50.6989 - 20.0288i) q^{54} +(-33.5346 - 4.27442i) q^{55} +(19.0973 - 40.3460i) q^{56} -12.9490i q^{57} +(84.3178 - 33.3100i) q^{58} -64.6396i q^{59} +(-34.7964 - 48.1587i) q^{60} +52.6371 q^{61} +(7.58142 + 19.1909i) q^{62} -0.975726 q^{63} +(-40.5710 - 49.4974i) q^{64} +(-66.6598 - 8.49664i) q^{65} +(-14.7597 - 37.3613i) q^{66} -40.0412 q^{67} +(43.4747 + 46.4364i) q^{68} -110.319 q^{69} +(13.7749 + 54.0698i) q^{70} -2.17424i q^{71} +(-0.598522 + 1.26447i) q^{72} +81.2412i q^{73} +(54.0184 + 136.737i) q^{74} +(71.8932 + 18.6301i) q^{75} +(11.9162 + 12.7280i) q^{76} +37.7254i q^{77} +(-29.3392 - 74.2664i) q^{78} -89.5872i q^{79} +(78.5202 + 15.3157i) q^{80} -79.3956 q^{81} +(-136.186 + 53.8008i) q^{82} -105.827 q^{83} +(-48.4011 + 45.3141i) q^{84} +(-78.8758 - 10.0537i) q^{85} +(128.975 - 50.9521i) q^{86} -134.661 q^{87} +(48.8893 + 23.1412i) q^{88} -9.02879 q^{89} +(-0.431716 - 1.69458i) q^{90} +74.9900i q^{91} +(108.436 - 101.520i) q^{92} -30.6490i q^{93} +(-76.5461 + 30.2398i) q^{94} +(-21.6196 - 2.75569i) q^{95} +(29.0339 + 90.5205i) q^{96} +36.2731i q^{97} +(-33.2347 + 13.1295i) q^{98} -1.18234i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86011 − 0.734843i 0.930055 − 0.367421i
\(3\) −2.97071 −0.990237 −0.495119 − 0.868825i \(-0.664875\pi\)
−0.495119 + 0.868825i \(0.664875\pi\)
\(4\) 2.92001 − 2.73378i 0.730003 − 0.683444i
\(5\) −0.632199 + 4.95987i −0.126440 + 0.991974i
\(6\) −5.52585 + 2.18301i −0.920975 + 0.363834i
\(7\) 5.57969 0.797098 0.398549 − 0.917147i \(-0.369514\pi\)
0.398549 + 0.917147i \(0.369514\pi\)
\(8\) 3.42265 − 7.23087i 0.427831 − 0.903859i
\(9\) −0.174871 −0.0194301
\(10\) 2.46877 + 9.69047i 0.246877 + 0.969047i
\(11\) 6.76119i 0.614654i 0.951604 + 0.307327i \(0.0994345\pi\)
−0.951604 + 0.307327i \(0.900565\pi\)
\(12\) −8.67452 + 8.12126i −0.722876 + 0.676772i
\(13\) 13.4398i 1.03383i 0.856036 + 0.516916i \(0.172920\pi\)
−0.856036 + 0.516916i \(0.827080\pi\)
\(14\) 10.3788 − 4.10019i 0.741345 − 0.292871i
\(15\) 1.87808 − 14.7343i 0.125205 − 0.982290i
\(16\) 1.05295 − 15.9653i 0.0658092 − 0.997832i
\(17\) 15.9028i 0.935459i 0.883872 + 0.467729i \(0.154928\pi\)
−0.883872 + 0.467729i \(0.845072\pi\)
\(18\) −0.325279 + 0.128503i −0.0180711 + 0.00713904i
\(19\) 4.35890i 0.229416i
\(20\) 11.7131 + 16.2112i 0.585657 + 0.810559i
\(21\) −16.5756 −0.789317
\(22\) 4.96841 + 12.5766i 0.225837 + 0.571662i
\(23\) 37.1354 1.61458 0.807291 − 0.590154i \(-0.200933\pi\)
0.807291 + 0.590154i \(0.200933\pi\)
\(24\) −10.1677 + 21.4808i −0.423654 + 0.895035i
\(25\) −24.2006 − 6.27125i −0.968026 − 0.250850i
\(26\) 9.87616 + 24.9995i 0.379852 + 0.961521i
\(27\) 27.2559 1.00948
\(28\) 16.2928 − 15.2536i 0.581884 − 0.544772i
\(29\) 45.3295 1.56309 0.781543 − 0.623851i \(-0.214433\pi\)
0.781543 + 0.623851i \(0.214433\pi\)
\(30\) −7.33399 − 28.7876i −0.244466 − 0.959586i
\(31\) 10.3171i 0.332808i 0.986058 + 0.166404i \(0.0532157\pi\)
−0.986058 + 0.166404i \(0.946784\pi\)
\(32\) −9.77340 − 30.4710i −0.305419 − 0.952218i
\(33\) − 20.0856i − 0.608653i
\(34\) 11.6861 + 29.5809i 0.343707 + 0.870028i
\(35\) −3.52747 + 27.6745i −0.100785 + 0.790701i
\(36\) −0.510626 + 0.478058i −0.0141841 + 0.0132794i
\(37\) 73.5102i 1.98676i 0.114864 + 0.993381i \(0.463357\pi\)
−0.114864 + 0.993381i \(0.536643\pi\)
\(38\) 3.20310 + 8.10803i 0.0842922 + 0.213369i
\(39\) − 39.9258i − 1.02374i
\(40\) 33.7004 + 21.5472i 0.842510 + 0.538681i
\(41\) −73.2141 −1.78571 −0.892855 − 0.450345i \(-0.851301\pi\)
−0.892855 + 0.450345i \(0.851301\pi\)
\(42\) −30.8325 + 12.1805i −0.734107 + 0.290012i
\(43\) 69.3375 1.61250 0.806250 − 0.591575i \(-0.201494\pi\)
0.806250 + 0.591575i \(0.201494\pi\)
\(44\) 18.4836 + 19.7428i 0.420081 + 0.448699i
\(45\) 0.110553 − 0.867338i 0.00245674 − 0.0192742i
\(46\) 69.0759 − 27.2887i 1.50165 − 0.593232i
\(47\) −41.1514 −0.875561 −0.437781 − 0.899082i \(-0.644235\pi\)
−0.437781 + 0.899082i \(0.644235\pi\)
\(48\) −3.12800 + 47.4284i −0.0651667 + 0.988091i
\(49\) −17.8671 −0.364634
\(50\) −49.6242 + 6.11846i −0.992485 + 0.122369i
\(51\) − 47.2426i − 0.926326i
\(52\) 36.7415 + 39.2445i 0.706567 + 0.754701i
\(53\) − 75.3694i − 1.42206i −0.703160 − 0.711032i \(-0.748228\pi\)
0.703160 − 0.711032i \(-0.251772\pi\)
\(54\) 50.6989 − 20.0288i 0.938869 − 0.370904i
\(55\) −33.5346 − 4.27442i −0.609721 − 0.0777167i
\(56\) 19.0973 − 40.3460i 0.341023 − 0.720464i
\(57\) − 12.9490i − 0.227176i
\(58\) 84.3178 − 33.3100i 1.45376 − 0.574311i
\(59\) − 64.6396i − 1.09559i −0.836614 − 0.547793i \(-0.815468\pi\)
0.836614 − 0.547793i \(-0.184532\pi\)
\(60\) −34.7964 − 48.1587i −0.579940 − 0.802646i
\(61\) 52.6371 0.862903 0.431452 − 0.902136i \(-0.358002\pi\)
0.431452 + 0.902136i \(0.358002\pi\)
\(62\) 7.58142 + 19.1909i 0.122281 + 0.309530i
\(63\) −0.975726 −0.0154877
\(64\) −40.5710 − 49.4974i −0.633921 − 0.773398i
\(65\) −66.6598 − 8.49664i −1.02554 − 0.130718i
\(66\) −14.7597 − 37.3613i −0.223632 − 0.566081i
\(67\) −40.0412 −0.597630 −0.298815 − 0.954311i \(-0.596591\pi\)
−0.298815 + 0.954311i \(0.596591\pi\)
\(68\) 43.4747 + 46.4364i 0.639333 + 0.682888i
\(69\) −110.319 −1.59882
\(70\) 13.7749 + 54.0698i 0.196785 + 0.772426i
\(71\) − 2.17424i − 0.0306230i −0.999883 − 0.0153115i \(-0.995126\pi\)
0.999883 − 0.0153115i \(-0.00487400\pi\)
\(72\) −0.598522 + 1.26447i −0.00831281 + 0.0175621i
\(73\) 81.2412i 1.11289i 0.830883 + 0.556446i \(0.187836\pi\)
−0.830883 + 0.556446i \(0.812164\pi\)
\(74\) 54.0184 + 136.737i 0.729979 + 1.84780i
\(75\) 71.8932 + 18.6301i 0.958575 + 0.248401i
\(76\) 11.9162 + 12.7280i 0.156793 + 0.167474i
\(77\) 37.7254i 0.489940i
\(78\) −29.3392 − 74.2664i −0.376144 − 0.952134i
\(79\) − 89.5872i − 1.13401i −0.823713 − 0.567007i \(-0.808101\pi\)
0.823713 − 0.567007i \(-0.191899\pi\)
\(80\) 78.5202 + 15.3157i 0.981503 + 0.191447i
\(81\) −79.3956 −0.980192
\(82\) −136.186 + 53.8008i −1.66081 + 0.656108i
\(83\) −105.827 −1.27502 −0.637512 − 0.770440i \(-0.720036\pi\)
−0.637512 + 0.770440i \(0.720036\pi\)
\(84\) −48.4011 + 45.3141i −0.576204 + 0.539453i
\(85\) −78.8758 − 10.0537i −0.927951 − 0.118279i
\(86\) 128.975 − 50.9521i 1.49971 − 0.592467i
\(87\) −134.661 −1.54783
\(88\) 48.8893 + 23.1412i 0.555560 + 0.262968i
\(89\) −9.02879 −0.101447 −0.0507235 − 0.998713i \(-0.516153\pi\)
−0.0507235 + 0.998713i \(0.516153\pi\)
\(90\) −0.431716 − 1.69458i −0.00479684 − 0.0188287i
\(91\) 74.9900i 0.824066i
\(92\) 108.436 − 101.520i 1.17865 − 1.10348i
\(93\) − 30.6490i − 0.329559i
\(94\) −76.5461 + 30.2398i −0.814320 + 0.321700i
\(95\) −21.6196 − 2.75569i −0.227575 − 0.0290073i
\(96\) 29.0339 + 90.5205i 0.302437 + 0.942922i
\(97\) 36.2731i 0.373949i 0.982365 + 0.186975i \(0.0598682\pi\)
−0.982365 + 0.186975i \(0.940132\pi\)
\(98\) −33.2347 + 13.1295i −0.339130 + 0.133974i
\(99\) − 1.18234i − 0.0119428i
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.39.97 yes 108
4.3 odd 2 inner 380.3.h.a.39.11 ✓ 108
5.4 even 2 inner 380.3.h.a.39.12 yes 108
20.19 odd 2 inner 380.3.h.a.39.98 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.11 ✓ 108 4.3 odd 2 inner
380.3.h.a.39.12 yes 108 5.4 even 2 inner
380.3.h.a.39.97 yes 108 1.1 even 1 trivial
380.3.h.a.39.98 yes 108 20.19 odd 2 inner