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(159,380)] 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("380.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (of order \(6\), degree \(2\), 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: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 239.62
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.62

$q$-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+(0.214213 + 1.98850i) q^{2} +(-0.866507 + 1.50083i) q^{3} +(-3.90823 + 0.851921i) q^{4} +(-0.0809034 + 4.99935i) q^{5} +(-3.17002 - 1.40155i) q^{6} -10.8203 q^{7} +(-2.53123 - 7.58900i) q^{8} +(2.99833 + 5.19326i) q^{9} +(-9.95850 + 0.910046i) q^{10} -20.5878i q^{11} +(2.10791 - 6.60379i) q^{12} +(-4.62800 + 2.67198i) q^{13} +(-2.31785 - 21.5162i) q^{14} +(-7.43308 - 4.45339i) q^{15} +(14.5485 - 6.65900i) q^{16} +(1.86899 + 1.07906i) q^{17} +(-9.68450 + 7.07463i) q^{18} +(12.0626 + 14.6797i) q^{19} +(-3.94286 - 19.6075i) q^{20} +(9.37589 - 16.2395i) q^{21} +(40.9388 - 4.41017i) q^{22} +(12.9823 + 22.4860i) q^{23} +(13.5831 + 2.77696i) q^{24} +(-24.9869 - 0.808928i) q^{25} +(-6.30459 - 8.63039i) q^{26} -25.9894 q^{27} +(42.2883 - 9.21807i) q^{28} +(-20.7658 - 35.9674i) q^{29} +(7.26328 - 15.7346i) q^{30} +1.42469i q^{31} +(16.3579 + 27.5031i) q^{32} +(30.8989 + 17.8395i) q^{33} +(-1.74535 + 3.94762i) q^{34} +(0.875402 - 54.0946i) q^{35} +(-16.1424 - 17.7421i) q^{36} -21.8400i q^{37} +(-26.6065 + 27.1311i) q^{38} -9.26115i q^{39} +(38.1448 - 12.0405i) q^{40} +(14.3855 - 24.9164i) q^{41} +(34.3006 + 15.1652i) q^{42} +(-6.90498 + 11.9598i) q^{43} +(17.5392 + 80.4618i) q^{44} +(-26.2055 + 14.5695i) q^{45} +(-41.9323 + 30.6320i) q^{46} +(-34.2392 - 59.3041i) q^{47} +(-2.61229 + 27.6049i) q^{48} +68.0796 q^{49} +(-3.74396 - 49.8596i) q^{50} +(-3.23898 + 1.87003i) q^{51} +(15.8110 - 14.3854i) q^{52} +(-51.9590 + 29.9985i) q^{53} +(-5.56726 - 51.6798i) q^{54} +(102.926 + 1.66562i) q^{55} +(27.3888 + 82.1155i) q^{56} +(-32.4841 + 5.38398i) q^{57} +(67.0727 - 48.9974i) q^{58} +(49.1957 + 28.4032i) q^{59} +(32.8441 + 11.0724i) q^{60} +(-45.8399 - 79.3970i) q^{61} +(-2.83299 + 0.305187i) q^{62} +(-32.4430 - 56.1929i) q^{63} +(-51.1857 + 38.4190i) q^{64} +(-12.9837 - 23.3532i) q^{65} +(-28.8548 + 65.2637i) q^{66} +(36.5149 + 63.2456i) q^{67} +(-8.22370 - 2.62498i) q^{68} -44.9970 q^{69} +(107.754 - 9.84701i) q^{70} +(-80.7206 - 46.6041i) q^{71} +(31.8222 - 35.8997i) q^{72} +(-69.4626 - 40.1042i) q^{73} +(43.4287 - 4.67840i) q^{74} +(22.8654 - 36.8002i) q^{75} +(-59.6494 - 47.0951i) q^{76} +222.767i q^{77} +(18.4157 - 1.98385i) q^{78} +(-53.2121 - 30.7220i) q^{79} +(32.1136 + 73.2715i) q^{80} +(-4.46499 + 7.73359i) q^{81} +(52.6276 + 23.2680i) q^{82} -19.9327 q^{83} +(-22.8083 + 71.4552i) q^{84} +(-5.54580 + 9.25642i) q^{85} +(-25.2611 - 11.1686i) q^{86} +71.9748 q^{87} +(-156.241 + 52.1125i) q^{88} +(56.0002 + 96.9952i) q^{89} +(-34.5850 - 48.9885i) q^{90} +(50.0765 - 28.9117i) q^{91} +(-69.8941 - 76.8205i) q^{92} +(-2.13822 - 1.23450i) q^{93} +(110.591 - 80.7883i) q^{94} +(-74.3647 + 59.1177i) q^{95} +(-55.4518 + 0.718796i) q^{96} +(2.63241 + 1.51982i) q^{97} +(14.5835 + 135.376i) q^{98} +(106.918 - 61.7291i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.214213 + 1.98850i 0.107106 + 0.994248i
\(3\) −0.866507 + 1.50083i −0.288836 + 0.500278i −0.973532 − 0.228550i \(-0.926601\pi\)
0.684697 + 0.728828i \(0.259935\pi\)
\(4\) −3.90823 + 0.851921i −0.977056 + 0.212980i
\(5\) −0.0809034 + 4.99935i −0.0161807 + 0.999869i
\(6\) −3.17002 − 1.40155i −0.528336 − 0.233591i
\(7\) −10.8203 −1.54576 −0.772881 − 0.634551i \(-0.781185\pi\)
−0.772881 + 0.634551i \(0.781185\pi\)
\(8\) −2.53123 − 7.58900i −0.316404 − 0.948625i
\(9\) 2.99833 + 5.19326i 0.333148 + 0.577029i
\(10\) −9.95850 + 0.910046i −0.995850 + 0.0910046i
\(11\) − 20.5878i − 1.87162i −0.352506 − 0.935809i \(-0.614670\pi\)
0.352506 − 0.935809i \(-0.385330\pi\)
\(12\) 2.10791 − 6.60379i 0.175659 − 0.550316i
\(13\) −4.62800 + 2.67198i −0.356000 + 0.205537i −0.667325 − 0.744767i \(-0.732561\pi\)
0.311325 + 0.950304i \(0.399227\pi\)
\(14\) −2.31785 − 21.5162i −0.165561 − 1.53687i
\(15\) −7.43308 − 4.45339i −0.495539 − 0.296893i
\(16\) 14.5485 − 6.65900i 0.909279 − 0.416188i
\(17\) 1.86899 + 1.07906i 0.109940 + 0.0634742i 0.553962 − 0.832542i \(-0.313115\pi\)
−0.444022 + 0.896016i \(0.646449\pi\)
\(18\) −9.68450 + 7.07463i −0.538028 + 0.393035i
\(19\) 12.0626 + 14.6797i 0.634876 + 0.772614i
\(20\) −3.94286 − 19.6075i −0.197143 − 0.980375i
\(21\) 9.37589 − 16.2395i 0.446471 − 0.773310i
\(22\) 40.9388 − 4.41017i 1.86085 − 0.200462i
\(23\) 12.9823 + 22.4860i 0.564448 + 0.977653i 0.997101 + 0.0760919i \(0.0242443\pi\)
−0.432653 + 0.901561i \(0.642422\pi\)
\(24\) 13.5831 + 2.77696i 0.565965 + 0.115707i
\(25\) −24.9869 − 0.808928i −0.999476 − 0.0323571i
\(26\) −6.30459 − 8.63039i −0.242484 − 0.331938i
\(27\) −25.9894 −0.962571
\(28\) 42.2883 − 9.21807i 1.51030 − 0.329217i
\(29\) −20.7658 − 35.9674i −0.716062 − 1.24026i −0.962549 − 0.271109i \(-0.912610\pi\)
0.246487 − 0.969146i \(-0.420724\pi\)
\(30\) 7.26328 − 15.7346i 0.242109 − 0.524487i
\(31\) 1.42469i 0.0459578i 0.999736 + 0.0229789i \(0.00731505\pi\)
−0.999736 + 0.0229789i \(0.992685\pi\)
\(32\) 16.3579 + 27.5031i 0.511183 + 0.859472i
\(33\) 30.8989 + 17.8395i 0.936329 + 0.540590i
\(34\) −1.74535 + 3.94762i −0.0513337 + 0.116107i
\(35\) 0.875402 − 54.0946i 0.0250115 − 1.54556i
\(36\) −16.1424 − 17.7421i −0.448400 − 0.492836i
\(37\) − 21.8400i − 0.590270i −0.955456 − 0.295135i \(-0.904635\pi\)
0.955456 − 0.295135i \(-0.0953647\pi\)
\(38\) −26.6065 + 27.1311i −0.700171 + 0.713975i
\(39\) − 9.26115i − 0.237465i
\(40\) 38.1448 − 12.0405i 0.953620 − 0.301013i
\(41\) 14.3855 − 24.9164i 0.350865 − 0.607716i −0.635536 − 0.772071i \(-0.719221\pi\)
0.986401 + 0.164355i \(0.0525543\pi\)
\(42\) 34.3006 + 15.1652i 0.816682 + 0.361076i
\(43\) −6.90498 + 11.9598i −0.160581 + 0.278134i −0.935077 − 0.354444i \(-0.884670\pi\)
0.774496 + 0.632578i \(0.218003\pi\)
\(44\) 17.5392 + 80.4618i 0.398618 + 1.82868i
\(45\) −26.2055 + 14.5695i −0.582344 + 0.323768i
\(46\) −41.9323 + 30.6320i −0.911573 + 0.665914i
\(47\) −34.2392 − 59.3041i −0.728495 − 1.26179i −0.957519 − 0.288369i \(-0.906887\pi\)
0.229025 − 0.973421i \(-0.426446\pi\)
\(48\) −2.61229 + 27.6049i −0.0544226 + 0.575102i
\(49\) 68.0796 1.38938
\(50\) −3.74396 − 49.8596i −0.0748792 − 0.997193i
\(51\) −3.23898 + 1.87003i −0.0635094 + 0.0366672i
\(52\) 15.8110 − 14.3854i 0.304057 − 0.276642i
\(53\) −51.9590 + 29.9985i −0.980359 + 0.566010i −0.902378 − 0.430945i \(-0.858180\pi\)
−0.0779802 + 0.996955i \(0.524847\pi\)
\(54\) −5.56726 − 51.6798i −0.103097 − 0.957034i
\(55\) 102.926 + 1.66562i 1.87137 + 0.0302841i
\(56\) 27.3888 + 82.1155i 0.489085 + 1.46635i
\(57\) −32.4841 + 5.38398i −0.569896 + 0.0944558i
\(58\) 67.0727 − 48.9974i 1.15643 − 0.844782i
\(59\) 49.1957 + 28.4032i 0.833826 + 0.481410i 0.855161 − 0.518363i \(-0.173458\pi\)
−0.0213348 + 0.999772i \(0.506792\pi\)
\(60\) 32.8441 + 11.0724i 0.547402 + 0.184541i
\(61\) −45.8399 − 79.3970i −0.751474 − 1.30159i −0.947108 − 0.320914i \(-0.896010\pi\)
0.195635 − 0.980677i \(-0.437323\pi\)
\(62\) −2.83299 + 0.305187i −0.0456934 + 0.00492237i
\(63\) −32.4430 − 56.1929i −0.514968 − 0.891950i
\(64\) −51.1857 + 38.4190i −0.799777 + 0.600297i
\(65\) −12.9837 − 23.3532i −0.199750 − 0.359279i
\(66\) −28.8548 + 65.2637i −0.437194 + 0.988844i
\(67\) 36.5149 + 63.2456i 0.544998 + 0.943965i 0.998607 + 0.0527637i \(0.0168030\pi\)
−0.453609 + 0.891201i \(0.649864\pi\)
\(68\) −8.22370 − 2.62498i −0.120937 − 0.0386027i
\(69\) −44.9970 −0.652131
\(70\) 107.754 − 9.84701i 1.53935 − 0.140672i
\(71\) −80.7206 − 46.6041i −1.13691 − 0.656395i −0.191247 − 0.981542i \(-0.561253\pi\)
−0.945664 + 0.325147i \(0.894586\pi\)
\(72\) 31.8222 − 35.8997i 0.441975 − 0.498607i
\(73\) −69.4626 − 40.1042i −0.951542 − 0.549373i −0.0579826 − 0.998318i \(-0.518467\pi\)
−0.893560 + 0.448944i \(0.851800\pi\)
\(74\) 43.4287 − 4.67840i 0.586875 − 0.0632216i
\(75\) 22.8654 − 36.8002i 0.304872 − 0.490670i
\(76\) −59.6494 − 47.0951i −0.784861 − 0.619672i
\(77\) 222.767i 2.89308i
\(78\) 18.4157 − 1.98385i 0.236099 − 0.0254340i
\(79\) −53.2121 − 30.7220i −0.673571 − 0.388886i 0.123858 − 0.992300i \(-0.460473\pi\)
−0.797428 + 0.603414i \(0.793807\pi\)
\(80\) 32.1136 + 73.2715i 0.401420 + 0.915894i
\(81\) −4.46499 + 7.73359i −0.0551233 + 0.0954764i
\(82\) 52.6276 + 23.2680i 0.641800 + 0.283757i
\(83\) −19.9327 −0.240153 −0.120077 − 0.992765i \(-0.538314\pi\)
−0.120077 + 0.992765i \(0.538314\pi\)
\(84\) −22.8083 + 71.4552i −0.271527 + 0.850657i
\(85\) −5.54580 + 9.25642i −0.0652448 + 0.108899i
\(86\) −25.2611 − 11.1686i −0.293733 − 0.129867i
\(87\) 71.9748 0.827296
\(88\) −156.241 + 52.1125i −1.77546 + 0.592188i
\(89\) 56.0002 + 96.9952i 0.629216 + 1.08983i 0.987709 + 0.156302i \(0.0499572\pi\)
−0.358493 + 0.933532i \(0.616709\pi\)
\(90\) −34.5850 − 48.9885i −0.384278 − 0.544317i
\(91\) 50.0765 − 28.9117i 0.550291 − 0.317711i
\(92\) −69.8941 − 76.8205i −0.759718 − 0.835005i
\(93\) −2.13822 − 1.23450i −0.0229917 − 0.0132742i
\(94\) 110.591 − 80.7883i 1.17650 − 0.859450i
\(95\) −74.3647 + 59.1177i −0.782786 + 0.622291i
\(96\) −55.4518 + 0.718796i −0.577623 + 0.00748746i
\(97\) 2.63241 + 1.51982i 0.0271382 + 0.0156682i 0.513508 − 0.858085i \(-0.328346\pi\)
−0.486370 + 0.873753i \(0.661679\pi\)
\(98\) 14.5835 + 135.376i 0.148811 + 1.38139i
\(99\) 106.918 − 61.7291i 1.07998 − 0.623526i
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.p.a.239.62 yes 232
4.3 odd 2 inner 380.3.p.a.239.102 yes 232
5.4 even 2 inner 380.3.p.a.239.55 yes 232
19.7 even 3 inner 380.3.p.a.159.15 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.15 yes 232
76.7 odd 6 inner 380.3.p.a.159.55 yes 232
95.64 even 6 inner 380.3.p.a.159.102 yes 232
380.159 odd 6 inner 380.3.p.a.159.62 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.15 ✓ 232 19.7 even 3 inner
380.3.p.a.159.55 yes 232 76.7 odd 6 inner
380.3.p.a.159.62 yes 232 380.159 odd 6 inner
380.3.p.a.159.102 yes 232 95.64 even 6 inner
380.3.p.a.239.15 yes 232 20.19 odd 2 inner
380.3.p.a.239.55 yes 232 5.4 even 2 inner
380.3.p.a.239.62 yes 232 1.1 even 1 trivial
380.3.p.a.239.102 yes 232 4.3 odd 2 inner