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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 159.102
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.102

$q$-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+(1.82919 + 0.808734i) q^{2} +(0.866507 + 1.50083i) q^{3} +(2.69190 + 2.95866i) q^{4} +(-0.0809034 - 4.99935i) q^{5} +(0.371233 + 3.44609i) q^{6} +10.8203 q^{7} +(2.53123 + 7.58900i) q^{8} +(2.99833 - 5.19326i) q^{9} +(3.89515 - 9.21020i) q^{10} -20.5878i q^{11} +(-2.10791 + 6.60379i) q^{12} +(-4.62800 - 2.67198i) q^{13} +(19.7925 + 8.75077i) q^{14} +(7.43308 - 4.45339i) q^{15} +(-1.50737 + 15.9288i) q^{16} +(1.86899 - 1.07906i) q^{17} +(9.68450 - 7.07463i) q^{18} +(-12.0626 + 14.6797i) q^{19} +(14.5736 - 13.6971i) q^{20} +(9.37589 + 16.2395i) q^{21} +(16.6501 - 37.6591i) q^{22} +(-12.9823 + 22.4860i) q^{23} +(-9.19649 + 10.3749i) q^{24} +(-24.9869 + 0.808928i) q^{25} +(-6.30459 - 8.63039i) q^{26} +25.9894 q^{27} +(29.1272 + 32.0137i) q^{28} +(-20.7658 + 35.9674i) q^{29} +(17.1982 - 2.13472i) q^{30} +1.42469i q^{31} +(-15.6395 + 27.9179i) q^{32} +(30.8989 - 17.8395i) q^{33} +(4.29141 - 0.462297i) q^{34} +(-0.875402 - 54.0946i) q^{35} +(23.4363 - 5.10869i) q^{36} +21.8400i q^{37} +(-33.9369 + 17.0965i) q^{38} -9.26115i q^{39} +(37.7352 - 13.2685i) q^{40} +(14.3855 + 24.9164i) q^{41} +(4.01687 + 37.2878i) q^{42} +(6.90498 + 11.9598i) q^{43} +(60.9124 - 55.4203i) q^{44} +(-26.2055 - 14.5695i) q^{45} +(-41.9323 + 30.6320i) q^{46} +(34.2392 - 59.3041i) q^{47} +(-25.2127 + 11.5401i) q^{48} +68.0796 q^{49} +(-46.3601 - 18.7281i) q^{50} +(3.23898 + 1.87003i) q^{51} +(-4.55263 - 20.8854i) q^{52} +(-51.9590 - 29.9985i) q^{53} +(47.5397 + 21.0185i) 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} +(33.1852 + 10.0039i) q^{60} +(-45.8399 + 79.3970i) q^{61} +(-1.15220 + 2.60604i) q^{62} +(32.4430 - 56.1929i) q^{63} +(-51.1857 + 38.4190i) q^{64} +(-12.9837 + 23.3532i) q^{65} +(70.9474 - 7.64288i) q^{66} +(-36.5149 + 63.2456i) q^{67} +(8.22370 + 2.62498i) q^{68} -44.9970 q^{69} +(42.1469 - 99.6574i) q^{70} +(80.7206 - 46.6041i) q^{71} +(47.0011 + 9.60898i) q^{72} +(-69.4626 + 40.1042i) q^{73} +(-17.6628 + 39.9496i) q^{74} +(-22.8654 - 36.8002i) q^{75} +(-75.9036 + 3.82691i) q^{76} -222.767i q^{77} +(7.48980 - 16.9404i) q^{78} +(53.2121 - 30.7220i) q^{79} +(79.7557 + 6.24715i) q^{80} +(-4.46499 - 7.73359i) q^{81} +(6.16309 + 57.2109i) q^{82} +19.9327 q^{83} +(-22.8083 + 71.4552i) q^{84} +(-5.54580 - 9.25642i) q^{85} +(2.95826 + 27.4610i) q^{86} -71.9748 q^{87} +(156.241 - 52.1125i) q^{88} +(56.0002 - 96.9952i) q^{89} +(-36.1520 - 47.8438i) q^{90} +(-50.0765 - 28.9117i) q^{91} +(-101.476 + 22.1198i) 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} +(124.531 + 55.0583i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82919 + 0.808734i 0.914597 + 0.404367i
\(3\) 0.866507 + 1.50083i 0.288836 + 0.500278i 0.973532 − 0.228550i \(-0.0733986\pi\)
−0.684697 + 0.728828i \(0.740065\pi\)
\(4\) 2.69190 + 2.95866i 0.672975 + 0.739666i
\(5\) −0.0809034 − 4.99935i −0.0161807 − 0.999869i
\(6\) 0.371233 + 3.44609i 0.0618722 + 0.574348i
\(7\) 10.8203 1.54576 0.772881 − 0.634551i \(-0.218815\pi\)
0.772881 + 0.634551i \(0.218815\pi\)
\(8\) 2.53123 + 7.58900i 0.316404 + 0.948625i
\(9\) 2.99833 − 5.19326i 0.333148 − 0.577029i
\(10\) 3.89515 − 9.21020i 0.389515 − 0.921020i
\(11\) − 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 0.311325 − 0.950304i \(-0.399227\pi\)
−0.667325 + 0.744767i \(0.732561\pi\)
\(14\) 19.7925 + 8.75077i 1.41375 + 0.625055i
\(15\) 7.43308 − 4.45339i 0.495539 − 0.296893i
\(16\) −1.50737 + 15.9288i −0.0942104 + 0.995552i
\(17\) 1.86899 − 1.07906i 0.109940 − 0.0634742i −0.444022 − 0.896016i \(-0.646449\pi\)
0.553962 + 0.832542i \(0.313115\pi\)
\(18\) 9.68450 − 7.07463i 0.538028 − 0.393035i
\(19\) −12.0626 + 14.6797i −0.634876 + 0.772614i
\(20\) 14.5736 − 13.6971i 0.728680 − 0.684855i
\(21\) 9.37589 + 16.2395i 0.446471 + 0.773310i
\(22\) 16.6501 − 37.6591i 0.756821 − 1.71178i
\(23\) −12.9823 + 22.4860i −0.564448 + 0.977653i 0.432653 + 0.901561i \(0.357578\pi\)
−0.997101 + 0.0760919i \(0.975756\pi\)
\(24\) −9.19649 + 10.3749i −0.383187 + 0.432286i
\(25\) −24.9869 + 0.808928i −0.999476 + 0.0323571i
\(26\) −6.30459 − 8.63039i −0.242484 − 0.331938i
\(27\) 25.9894 0.962571
\(28\) 29.1272 + 32.0137i 1.04026 + 1.14335i
\(29\) −20.7658 + 35.9674i −0.716062 + 1.24026i 0.246487 + 0.969146i \(0.420724\pi\)
−0.962549 + 0.271109i \(0.912610\pi\)
\(30\) 17.1982 − 2.13472i 0.573272 − 0.0711574i
\(31\) 1.42469i 0.0459578i 0.999736 + 0.0229789i \(0.00731505\pi\)
−0.999736 + 0.0229789i \(0.992685\pi\)
\(32\) −15.6395 + 27.9179i −0.488733 + 0.872433i
\(33\) 30.8989 − 17.8395i 0.936329 − 0.540590i
\(34\) 4.29141 − 0.462297i 0.126218 − 0.0135970i
\(35\) −0.875402 − 54.0946i −0.0250115 − 1.54556i
\(36\) 23.4363 − 5.10869i 0.651009 − 0.141908i
\(37\) 21.8400i 0.590270i 0.955456 + 0.295135i \(0.0953647\pi\)
−0.955456 + 0.295135i \(0.904635\pi\)
\(38\) −33.9369 + 17.0965i −0.893075 + 0.449908i
\(39\) − 9.26115i − 0.237465i
\(40\) 37.7352 − 13.2685i 0.943381 − 0.331712i
\(41\) 14.3855 + 24.9164i 0.350865 + 0.607716i 0.986401 − 0.164355i \(-0.0525543\pi\)
−0.635536 + 0.772071i \(0.719221\pi\)
\(42\) 4.01687 + 37.2878i 0.0956397 + 0.887805i
\(43\) 6.90498 + 11.9598i 0.160581 + 0.278134i 0.935077 − 0.354444i \(-0.115330\pi\)
−0.774496 + 0.632578i \(0.781997\pi\)
\(44\) 60.9124 − 55.4203i 1.38437 − 1.25955i
\(45\) −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 −0.229025 − 0.973421i \(-0.573554\pi\)
0.957519 − 0.288369i \(-0.0931130\pi\)
\(48\) −25.2127 + 11.5401i −0.525264 + 0.240420i
\(49\) 68.0796 1.38938
\(50\) −46.3601 − 18.7281i −0.927202 − 0.374562i
\(51\) 3.23898 + 1.87003i 0.0635094 + 0.0366672i
\(52\) −4.55263 − 20.8854i −0.0875506 − 0.401642i
\(53\) −51.9590 − 29.9985i −0.980359 − 0.566010i −0.0779802 − 0.996955i \(-0.524847\pi\)
−0.902378 + 0.430945i \(0.858180\pi\)
\(54\) 47.5397 + 21.0185i 0.880364 + 0.389232i
\(55\) −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 \(-0.826542\pi\)
0.0213348 + 0.999772i \(0.493208\pi\)
\(60\) 33.1852 + 10.0039i 0.553086 + 0.166732i
\(61\) −45.8399 + 79.3970i −0.751474 + 1.30159i 0.195635 + 0.980677i \(0.437323\pi\)
−0.947108 + 0.320914i \(0.896010\pi\)
\(62\) −1.15220 + 2.60604i −0.0185838 + 0.0420328i
\(63\) 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
\(66\) 70.9474 − 7.64288i 1.07496 − 0.115801i
\(67\) −36.5149 + 63.2456i −0.544998 + 0.943965i 0.453609 + 0.891201i \(0.350136\pi\)
−0.998607 + 0.0527637i \(0.983197\pi\)
\(68\) 8.22370 + 2.62498i 0.120937 + 0.0386027i
\(69\) −44.9970 −0.652131
\(70\) 42.1469 − 99.6574i 0.602098 − 1.42368i
\(71\) 80.7206 − 46.6041i 1.13691 − 0.656395i 0.191247 − 0.981542i \(-0.438747\pi\)
0.945664 + 0.325147i \(0.105414\pi\)
\(72\) 47.0011 + 9.60898i 0.652794 + 0.133458i
\(73\) −69.4626 + 40.1042i −0.951542 + 0.549373i −0.893560 − 0.448944i \(-0.851800\pi\)
−0.0579826 + 0.998318i \(0.518467\pi\)
\(74\) −17.6628 + 39.9496i −0.238686 + 0.539859i
\(75\) −22.8654 − 36.8002i −0.304872 − 0.490670i
\(76\) −75.9036 + 3.82691i −0.998731 + 0.0503541i
\(77\) − 222.767i − 2.89308i
\(78\) 7.48980 − 16.9404i 0.0960231 − 0.217185i
\(79\) 53.2121 − 30.7220i 0.673571 − 0.388886i −0.123858 − 0.992300i \(-0.539527\pi\)
0.797428 + 0.603414i \(0.206193\pi\)
\(80\) 79.7557 + 6.24715i 0.996946 + 0.0780893i
\(81\) −4.46499 − 7.73359i −0.0551233 − 0.0954764i
\(82\) 6.16309 + 57.2109i 0.0751597 + 0.697693i
\(83\) 19.9327 0.240153 0.120077 − 0.992765i \(-0.461686\pi\)
0.120077 + 0.992765i \(0.461686\pi\)
\(84\) −22.8083 + 71.4552i −0.271527 + 0.850657i
\(85\) −5.54580 − 9.25642i −0.0652448 − 0.108899i
\(86\) 2.95826 + 27.4610i 0.0343984 + 0.319314i
\(87\) −71.9748 −0.827296
\(88\) 156.241 − 52.1125i 1.77546 − 0.592188i
\(89\) 56.0002 − 96.9952i 0.629216 − 1.08983i −0.358493 − 0.933532i \(-0.616709\pi\)
0.987709 − 0.156302i \(-0.0499572\pi\)
\(90\) −36.1520 − 47.8438i −0.401689 − 0.531598i
\(91\) −50.0765 − 28.9117i −0.550291 − 0.317711i
\(92\) −101.476 + 22.1198i −1.10300 + 0.240433i
\(93\) −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 −0.486370 − 0.873753i \(-0.661679\pi\)
0.513508 + 0.858085i \(0.328346\pi\)
\(98\) 124.531 + 55.0583i 1.27072 + 0.561819i
\(99\) −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.159.102 yes 232
4.3 odd 2 inner 380.3.p.a.159.62 yes 232
5.4 even 2 inner 380.3.p.a.159.15 ✓ 232
19.11 even 3 inner 380.3.p.a.239.55 yes 232
20.19 odd 2 inner 380.3.p.a.159.55 yes 232
76.11 odd 6 inner 380.3.p.a.239.15 yes 232
95.49 even 6 inner 380.3.p.a.239.62 yes 232
380.239 odd 6 inner 380.3.p.a.239.102 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.15 ✓ 232 5.4 even 2 inner
380.3.p.a.159.55 yes 232 20.19 odd 2 inner
380.3.p.a.159.62 yes 232 4.3 odd 2 inner
380.3.p.a.159.102 yes 232 1.1 even 1 trivial
380.3.p.a.239.15 yes 232 76.11 odd 6 inner
380.3.p.a.239.55 yes 232 19.11 even 3 inner
380.3.p.a.239.62 yes 232 95.49 even 6 inner
380.3.p.a.239.102 yes 232 380.239 odd 6 inner