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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 239.55
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.55

$q$-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} +(-4.28911 + 2.56974i) q^{5} +(-3.17002 - 1.40155i) q^{6} +10.8203 q^{7} +(2.53123 + 7.58900i) q^{8} +(2.99833 + 5.19326i) q^{9} +(6.02869 + 7.97840i) q^{10} -20.5878i q^{11} +(-2.10791 + 6.60379i) q^{12} +(4.62800 - 2.67198i) q^{13} +(-2.31785 - 21.5162i) q^{14} +(0.140207 + 8.66393i) 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} +(14.5736 - 13.6971i) 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} +(11.7929 - 22.0438i) q^{25} +(-6.30459 - 8.63039i) q^{26} +25.9894 q^{27} +(-42.2883 + 9.21807i) q^{28} +(-20.7658 - 35.9674i) q^{29} +(17.1982 - 2.13472i) q^{30} +1.42469i q^{31} +(-16.3579 - 27.5031i) q^{32} +(-30.8989 - 17.8395i) q^{33} +(-1.74535 + 3.94762i) q^{34} +(-46.4096 + 27.8054i) q^{35} +(-16.1424 - 17.7421i) q^{36} +21.8400i q^{37} +(26.6065 - 27.1311i) q^{38} -9.26115i q^{39} +(-30.3585 - 26.0454i) 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} +(-46.3601 - 18.7281i) 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} +(52.9053 + 88.3033i) 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} +(-7.92895 - 33.7412i) 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} +(65.2324 + 86.3290i) 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} +(-45.2880 + 65.9469i) q^{80} +(-4.46499 + 7.73359i) q^{81} +(-52.6276 - 23.2680i) q^{82} +19.9327 q^{83} +(-22.8083 + 71.4552i) q^{84} +(10.7892 - 0.174599i) q^{85} +(-25.2611 - 11.1686i) q^{86} -71.9748 q^{87} +(156.241 - 52.1125i) q^{88} +(56.0002 + 96.9952i) q^{89} +(-23.3579 + 55.2305i) 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} +(-89.4609 - 31.9649i) 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 −0.684697 − 0.728828i \(-0.740065\pi\)
0.973532 + 0.228550i \(0.0733986\pi\)
\(4\) −3.90823 + 0.851921i −0.977056 + 0.212980i
\(5\) −4.28911 + 2.56974i −0.857822 + 0.513947i
\(6\) −3.17002 − 1.40155i −0.528336 − 0.233591i
\(7\) 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
\(10\) 6.02869 + 7.97840i 0.602869 + 0.797840i
\(11\) − 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 \(-0.600773\pi\)
0.667325 + 0.744767i \(0.267439\pi\)
\(14\) −2.31785 − 21.5162i −0.165561 − 1.53687i
\(15\) 0.140207 + 8.66393i 0.00934711 + 0.577595i
\(16\) 14.5485 − 6.65900i 0.909279 − 0.416188i
\(17\) −1.86899 − 1.07906i −0.109940 − 0.0634742i 0.444022 − 0.896016i \(-0.353551\pi\)
−0.553962 + 0.832542i \(0.686885\pi\)
\(18\) 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
\(22\) −40.9388 + 4.41017i −1.86085 + 0.200462i
\(23\) −12.9823 − 22.4860i −0.564448 − 0.977653i −0.997101 − 0.0760919i \(-0.975756\pi\)
0.432653 − 0.901561i \(-0.357578\pi\)
\(24\) 13.5831 + 2.77696i 0.565965 + 0.115707i
\(25\) 11.7929 − 22.0438i 0.471716 − 0.881750i
\(26\) −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 \(-0.420724\pi\)
\(30\) 17.1982 − 2.13472i 0.573272 − 0.0711574i
\(31\) 1.42469i 0.0459578i 0.999736 + 0.0229789i \(0.00731505\pi\)
−0.999736 + 0.0229789i \(0.992685\pi\)
\(32\) −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
\(35\) −46.4096 + 27.8054i −1.32599 + 0.794440i
\(36\) −16.1424 − 17.7421i −0.448400 − 0.492836i
\(37\) 21.8400i 0.590270i 0.955456 + 0.295135i \(0.0953647\pi\)
−0.955456 + 0.295135i \(0.904635\pi\)
\(38\) 26.6065 − 27.1311i 0.700171 − 0.713975i
\(39\) − 9.26115i − 0.237465i
\(40\) −30.3585 − 26.0454i −0.758961 − 0.651136i
\(41\) 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 −0.774496 − 0.632578i \(-0.781997\pi\)
0.935077 + 0.354444i \(0.115330\pi\)
\(44\) 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 \(0.0931130\pi\)
−0.229025 + 0.973421i \(0.573554\pi\)
\(48\) 2.61229 − 27.6049i 0.0544226 − 0.575102i
\(49\) 68.0796 1.38938
\(50\) −46.3601 − 18.7281i −0.927202 − 0.374562i
\(51\) −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 0.0779802 − 0.996955i \(-0.475153\pi\)
0.902378 + 0.430945i \(0.141820\pi\)
\(54\) −5.56726 − 51.6798i −0.103097 − 0.957034i
\(55\) 52.9053 + 88.3033i 0.961914 + 1.60552i
\(56\) 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 \(0.506792\pi\)
\(60\) −7.92895 − 33.7412i −0.132149 − 0.562353i
\(61\) −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 \(-0.983197\pi\)
0.453609 − 0.891201i \(-0.350136\pi\)
\(68\) 8.22370 + 2.62498i 0.120937 + 0.0386027i
\(69\) −44.9970 −0.652131
\(70\) 65.2324 + 86.3290i 0.931892 + 1.23327i
\(71\) −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 0.893560 − 0.448944i \(-0.148200\pi\)
0.0579826 + 0.998318i \(0.481533\pi\)
\(74\) 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 \(0.793807\pi\)
\(80\) −45.2880 + 65.9469i −0.566101 + 0.824336i
\(81\) −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 \(-0.461686\pi\)
0.120077 + 0.992765i \(0.461686\pi\)
\(84\) −22.8083 + 71.4552i −0.271527 + 0.850657i
\(85\) 10.7892 − 0.174599i 0.126932 − 0.00205411i
\(86\) −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 \(0.616709\pi\)
\(90\) −23.3579 + 55.2305i −0.259533 + 0.613672i
\(91\) 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
\(95\) −89.4609 − 31.9649i −0.941693 − 0.336473i
\(96\) −55.4518 + 0.718796i −0.577623 + 0.00748746i
\(97\) −2.63241 − 1.51982i −0.0271382 − 0.0156682i 0.486370 − 0.873753i \(-0.338321\pi\)
−0.513508 + 0.858085i \(0.671654\pi\)
\(98\) −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.239.55 yes 232
4.3 odd 2 inner 380.3.p.a.239.15 yes 232
5.4 even 2 inner 380.3.p.a.239.62 yes 232
19.7 even 3 inner 380.3.p.a.159.102 yes 232
20.19 odd 2 inner 380.3.p.a.239.102 yes 232
76.7 odd 6 inner 380.3.p.a.159.62 yes 232
95.64 even 6 inner 380.3.p.a.159.15 ✓ 232
380.159 odd 6 inner 380.3.p.a.159.55 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.15 ✓ 232 95.64 even 6 inner
380.3.p.a.159.55 yes 232 380.159 odd 6 inner
380.3.p.a.159.62 yes 232 76.7 odd 6 inner
380.3.p.a.159.102 yes 232 19.7 even 3 inner
380.3.p.a.239.15 yes 232 4.3 odd 2 inner
380.3.p.a.239.55 yes 232 1.1 even 1 trivial
380.3.p.a.239.62 yes 232 5.4 even 2 inner
380.3.p.a.239.102 yes 232 20.19 odd 2 inner