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Properties

Label 380.3.p.a.159.15
Level $380$
Weight $3$
Character 380.159
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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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 159.15
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.15

$q$-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} +(-4.28911 - 2.56974i) q^{5} +(0.371233 + 3.44609i) q^{6} -10.8203 q^{7} +(-2.53123 - 7.58900i) q^{8} +(2.99833 - 5.19326i) q^{9} +(5.76738 + 8.16929i) q^{10} -20.5878i q^{11} +(2.10791 - 6.60379i) q^{12} +(4.62800 + 2.67198i) q^{13} +(19.7925 + 8.75077i) q^{14} +(-0.140207 + 8.66393i) 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} +(-3.94286 - 19.6075i) 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} +(11.7929 + 22.0438i) q^{25} +(-6.30459 - 8.63039i) q^{26} -25.9894 q^{27} +(-29.1272 - 32.0137i) q^{28} +(-20.7658 + 35.9674i) q^{29} +(7.26328 - 15.7346i) q^{30} +1.42469i q^{31} +(15.6395 - 27.9179i) q^{32} +(-30.8989 + 17.8395i) q^{33} +(4.29141 - 0.462297i) q^{34} +(46.4096 + 27.8054i) q^{35} +(23.4363 - 5.10869i) q^{36} -21.8400i q^{37} +(33.9369 - 17.0965i) q^{38} -9.26115i q^{39} +(-8.64500 + 39.0546i) 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} +(-3.74396 - 49.8596i) 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} +(-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} +(-26.0111 + 22.9076i) 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} +(-62.4049 - 88.3945i) 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} +(47.3982 - 64.4470i) q^{80} +(-4.46499 - 7.73359i) q^{81} +(-6.16309 - 57.2109i) q^{82} -19.9327 q^{83} +(-22.8083 + 71.4552i) q^{84} +(10.7892 + 0.174599i) q^{85} +(2.95826 + 27.4610i) q^{86} +71.9748 q^{87} +(-156.241 + 52.1125i) q^{88} +(56.0002 - 96.9952i) q^{89} +(59.7178 - 5.45724i) 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} +(89.4609 - 31.9649i) 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 0.684697 − 0.728828i \(-0.259935\pi\)
−0.973532 + 0.228550i \(0.926601\pi\)
\(4\) 2.69190 + 2.95866i 0.672975 + 0.739666i
\(5\) −4.28911 − 2.56974i −0.857822 − 0.513947i
\(6\) 0.371233 + 3.44609i 0.0618722 + 0.574348i
\(7\) −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
\(10\) 5.76738 + 8.16929i 0.576738 + 0.816929i
\(11\) − 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 \(-0.267439\pi\)
−0.311325 + 0.950304i \(0.600773\pi\)
\(14\) 19.7925 + 8.75077i 1.41375 + 0.625055i
\(15\) −0.140207 + 8.66393i −0.00934711 + 0.577595i
\(16\) −1.50737 + 15.9288i −0.0942104 + 0.995552i
\(17\) −1.86899 + 1.07906i −0.109940 + 0.0634742i −0.553962 − 0.832542i \(-0.686885\pi\)
0.444022 + 0.896016i \(0.353551\pi\)
\(18\) −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
\(22\) −16.6501 + 37.6591i −0.756821 + 1.71178i
\(23\) 12.9823 − 22.4860i 0.564448 − 0.977653i −0.432653 − 0.901561i \(-0.642422\pi\)
0.997101 − 0.0760919i \(-0.0242443\pi\)
\(24\) −9.19649 + 10.3749i −0.383187 + 0.432286i
\(25\) 11.7929 + 22.0438i 0.471716 + 0.881750i
\(26\) −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 \(0.912610\pi\)
\(30\) 7.26328 − 15.7346i 0.242109 − 0.524487i
\(31\) 1.42469i 0.0459578i 0.999736 + 0.0229789i \(0.00731505\pi\)
−0.999736 + 0.0229789i \(0.992685\pi\)
\(32\) 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
\(35\) 46.4096 + 27.8054i 1.32599 + 0.794440i
\(36\) 23.4363 − 5.10869i 0.651009 − 0.141908i
\(37\) − 21.8400i − 0.590270i −0.955456 − 0.295135i \(-0.904635\pi\)
0.955456 − 0.295135i \(-0.0953647\pi\)
\(38\) 33.9369 − 17.0965i 0.893075 − 0.449908i
\(39\) − 9.26115i − 0.237465i
\(40\) −8.64500 + 39.0546i −0.216125 + 0.976366i
\(41\) 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 0.774496 − 0.632578i \(-0.218003\pi\)
−0.935077 + 0.354444i \(0.884670\pi\)
\(44\) 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 \(0.426446\pi\)
−0.957519 + 0.288369i \(0.906887\pi\)
\(48\) 25.2127 − 11.5401i 0.525264 − 0.240420i
\(49\) 68.0796 1.38938
\(50\) −3.74396 − 49.8596i −0.0748792 − 0.997193i
\(51\) 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 0.902378 − 0.430945i \(-0.141820\pi\)
0.0779802 + 0.996955i \(0.475153\pi\)
\(54\) 47.5397 + 21.0185i 0.880364 + 0.389232i
\(55\) −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 \(-0.826542\pi\)
0.0213348 + 0.999772i \(0.493208\pi\)
\(60\) −26.0111 + 22.9076i −0.433518 + 0.381793i
\(61\) −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 \(-0.649864\pi\)
0.998607 − 0.0527637i \(-0.0168030\pi\)
\(68\) −8.22370 − 2.62498i −0.120937 − 0.0386027i
\(69\) −44.9970 −0.652131
\(70\) −62.4049 − 88.3945i −0.891499 − 1.26278i
\(71\) 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 0.0579826 − 0.998318i \(-0.481533\pi\)
0.893560 + 0.448944i \(0.148200\pi\)
\(74\) −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 \(0.206193\pi\)
\(80\) 47.3982 − 64.4470i 0.592477 − 0.805587i
\(81\) −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 \(-0.538314\pi\)
−0.120077 + 0.992765i \(0.538314\pi\)
\(84\) −22.8083 + 71.4552i −0.271527 + 0.850657i
\(85\) 10.7892 + 0.174599i 0.126932 + 0.00205411i
\(86\) 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 \(-0.0499572\pi\)
\(90\) 59.7178 − 5.45724i 0.663531 − 0.0606360i
\(91\) −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
\(95\) 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 −0.513508 − 0.858085i \(-0.671654\pi\)
0.486370 + 0.873753i \(0.338321\pi\)
\(98\) −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.159.15 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.55 yes 232
5.4 even 2 inner 380.3.p.a.159.102 yes 232
19.11 even 3 inner 380.3.p.a.239.62 yes 232
20.19 odd 2 inner 380.3.p.a.159.62 yes 232
76.11 odd 6 inner 380.3.p.a.239.102 yes 232
95.49 even 6 inner 380.3.p.a.239.55 yes 232
380.239 odd 6 inner 380.3.p.a.239.15 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.15 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.55 yes 232 4.3 odd 2 inner
380.3.p.a.159.62 yes 232 20.19 odd 2 inner
380.3.p.a.159.102 yes 232 5.4 even 2 inner
380.3.p.a.239.15 yes 232 380.239 odd 6 inner
380.3.p.a.239.55 yes 232 95.49 even 6 inner
380.3.p.a.239.62 yes 232 19.11 even 3 inner
380.3.p.a.239.102 yes 232 76.11 odd 6 inner