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Properties

Label 380.3.p.a.159.3
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 159.3
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.3

$q$-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+(-1.99798 - 0.0898151i) q^{2} +(0.192588 + 0.333573i) q^{3} +(3.98387 + 0.358898i) q^{4} +(0.758288 + 4.94217i) q^{5} +(-0.354828 - 0.683770i) q^{6} +5.98368 q^{7} +(-7.92746 - 1.07488i) q^{8} +(4.42582 - 7.66574i) q^{9} +(-1.07117 - 9.94246i) q^{10} -17.6878i q^{11} +(0.647528 + 1.39803i) q^{12} +(-17.5165 - 10.1131i) q^{13} +(-11.9553 - 0.537425i) q^{14} +(-1.50254 + 1.20475i) q^{15} +(15.7424 + 2.85960i) q^{16} +(7.16606 - 4.13733i) q^{17} +(-9.53121 + 14.9185i) q^{18} +(9.14229 + 16.6559i) q^{19} +(1.24719 + 19.9611i) q^{20} +(1.15239 + 1.99599i) q^{21} +(-1.58863 + 35.3399i) q^{22} +(11.2176 - 19.4294i) q^{23} +(-1.16819 - 2.85140i) q^{24} +(-23.8500 + 7.49517i) q^{25} +(34.0893 + 21.7791i) q^{26} +6.87604 q^{27} +(23.8382 + 2.14753i) q^{28} +(17.6496 - 30.5700i) q^{29} +(3.11024 - 2.27212i) q^{30} +21.3190i q^{31} +(-31.1962 - 7.12734i) q^{32} +(5.90018 - 3.40647i) q^{33} +(-14.6893 + 7.62268i) q^{34} +(4.53735 + 29.5723i) q^{35} +(20.3831 - 28.9509i) q^{36} -26.7915i q^{37} +(-16.7702 - 34.0993i) q^{38} -7.79069i q^{39} +(-0.699049 - 39.9939i) q^{40} +(-29.5321 - 51.1511i) q^{41} +(-2.12318 - 4.09146i) q^{42} +(33.4293 + 57.9012i) q^{43} +(6.34812 - 70.4659i) q^{44} +(41.2414 + 16.0603i) q^{45} +(-24.1575 + 37.8121i) q^{46} +(22.1016 - 38.2810i) q^{47} +(2.07792 + 5.80196i) q^{48} -13.1956 q^{49} +(48.3251 - 12.8331i) q^{50} +(2.76020 + 1.59360i) q^{51} +(-66.1536 - 46.5760i) q^{52} +(55.2217 + 31.8822i) q^{53} +(-13.7382 - 0.617572i) q^{54} +(87.4161 - 13.4125i) q^{55} +(-47.4354 - 6.43176i) q^{56} +(-3.79525 + 6.25735i) q^{57} +(-38.0093 + 59.4932i) q^{58} +(-12.0246 + 6.94238i) q^{59} +(-6.41828 + 4.26030i) q^{60} +(8.51902 - 14.7554i) q^{61} +(1.91477 - 42.5949i) q^{62} +(26.4827 - 45.8694i) q^{63} +(61.6893 + 17.0422i) q^{64} +(36.6982 - 94.2379i) q^{65} +(-12.0944 + 6.27614i) q^{66} +(-24.8498 + 43.0411i) q^{67} +(30.0335 - 13.9107i) q^{68} +8.64149 q^{69} +(-6.40951 - 59.4925i) q^{70} +(58.9583 - 34.0396i) q^{71} +(-43.3253 + 56.0126i) q^{72} +(85.6728 - 49.4632i) q^{73} +(-2.40628 + 53.5289i) q^{74} +(-7.09342 - 6.51223i) q^{75} +(30.4439 + 69.6360i) q^{76} -105.838i q^{77} +(-0.699721 + 15.5657i) q^{78} +(22.8122 - 13.1707i) q^{79} +(-2.19537 + 79.9699i) q^{80} +(-38.5081 - 66.6980i) q^{81} +(54.4105 + 104.852i) q^{82} +50.3251 q^{83} +(3.87460 + 8.36536i) q^{84} +(25.8813 + 32.2786i) q^{85} +(-61.5907 - 118.688i) q^{86} +13.5964 q^{87} +(-19.0123 + 140.219i) q^{88} +(-55.7536 + 96.5681i) q^{89} +(-80.9572 - 35.7923i) q^{90} +(-104.813 - 60.5137i) q^{91} +(51.6624 - 73.3781i) q^{92} +(-7.11144 + 4.10579i) q^{93} +(-47.5967 + 74.4997i) q^{94} +(-75.3836 + 57.8127i) q^{95} +(-3.63053 - 11.7788i) q^{96} +(-44.3122 + 25.5836i) q^{97} +(26.3645 + 1.18516i) q^{98} +(-135.590 - 78.2831i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99798 − 0.0898151i −0.998991 − 0.0449075i
\(3\) 0.192588 + 0.333573i 0.0641961 + 0.111191i 0.896337 − 0.443373i \(-0.146218\pi\)
−0.832141 + 0.554564i \(0.812885\pi\)
\(4\) 3.98387 + 0.358898i 0.995967 + 0.0897245i
\(5\) 0.758288 + 4.94217i 0.151658 + 0.988433i
\(6\) −0.354828 − 0.683770i −0.0591381 − 0.113962i
\(7\) 5.98368 0.854812 0.427406 − 0.904060i \(-0.359428\pi\)
0.427406 + 0.904060i \(0.359428\pi\)
\(8\) −7.92746 − 1.07488i −0.990933 − 0.134360i
\(9\) 4.42582 − 7.66574i 0.491758 − 0.851749i
\(10\) −1.07117 − 9.94246i −0.107117 − 0.994246i
\(11\) − 17.6878i − 1.60798i −0.594640 − 0.803992i \(-0.702706\pi\)
0.594640 − 0.803992i \(-0.297294\pi\)
\(12\) 0.647528 + 1.39803i 0.0539607 + 0.116502i
\(13\) −17.5165 − 10.1131i −1.34742 − 0.777933i −0.359536 − 0.933131i \(-0.617065\pi\)
−0.987883 + 0.155198i \(0.950398\pi\)
\(14\) −11.9553 − 0.537425i −0.853949 − 0.0383875i
\(15\) −1.50254 + 1.20475i −0.100169 + 0.0803166i
\(16\) 15.7424 + 2.85960i 0.983899 + 0.178725i
\(17\) 7.16606 − 4.13733i 0.421533 − 0.243372i −0.274200 − 0.961673i \(-0.588413\pi\)
0.695733 + 0.718301i \(0.255080\pi\)
\(18\) −9.53121 + 14.9185i −0.529512 + 0.828806i
\(19\) 9.14229 + 16.6559i 0.481173 + 0.876626i
\(20\) 1.24719 + 19.9611i 0.0623593 + 0.998054i
\(21\) 1.15239 + 1.99599i 0.0548756 + 0.0950473i
\(22\) −1.58863 + 35.3399i −0.0722106 + 1.60636i
\(23\) 11.2176 − 19.4294i 0.487720 − 0.844756i −0.512180 − 0.858878i \(-0.671162\pi\)
0.999900 + 0.0141220i \(0.00449533\pi\)
\(24\) −1.16819 − 2.85140i −0.0486744 − 0.118808i
\(25\) −23.8500 + 7.49517i −0.954000 + 0.299807i
\(26\) 34.0893 + 21.7791i 1.31113 + 0.837658i
\(27\) 6.87604 0.254668
\(28\) 23.8382 + 2.14753i 0.851364 + 0.0766975i
\(29\) 17.6496 − 30.5700i 0.608607 − 1.05414i −0.382863 − 0.923805i \(-0.625062\pi\)
0.991470 − 0.130334i \(-0.0416049\pi\)
\(30\) 3.11024 − 2.27212i 0.103675 − 0.0757372i
\(31\) 21.3190i 0.687709i 0.939023 + 0.343855i \(0.111733\pi\)
−0.939023 + 0.343855i \(0.888267\pi\)
\(32\) −31.1962 − 7.12734i −0.974880 − 0.222729i
\(33\) 5.90018 − 3.40647i 0.178793 − 0.103226i
\(34\) −14.6893 + 7.62268i −0.432037 + 0.224197i
\(35\) 4.53735 + 29.5723i 0.129639 + 0.844924i
\(36\) 20.3831 − 28.9509i 0.566197 − 0.804191i
\(37\) − 26.7915i − 0.724094i −0.932160 − 0.362047i \(-0.882078\pi\)
0.932160 − 0.362047i \(-0.117922\pi\)
\(38\) −16.7702 − 34.0993i −0.441321 − 0.897349i
\(39\) − 7.79069i − 0.199761i
\(40\) −0.699049 − 39.9939i −0.0174762 − 0.999847i
\(41\) −29.5321 − 51.1511i −0.720296 − 1.24759i −0.960881 − 0.276961i \(-0.910673\pi\)
0.240585 − 0.970628i \(-0.422661\pi\)
\(42\) −2.12318 − 4.09146i −0.0505519 − 0.0974158i
\(43\) 33.4293 + 57.9012i 0.777425 + 1.34654i 0.933422 + 0.358781i \(0.116808\pi\)
−0.155997 + 0.987758i \(0.549859\pi\)
\(44\) 6.34812 − 70.4659i 0.144275 − 1.60150i
\(45\) 41.2414 + 16.0603i 0.916476 + 0.356895i
\(46\) −24.1575 + 37.8121i −0.525164 + 0.822001i
\(47\) 22.1016 − 38.2810i 0.470246 − 0.814490i −0.529175 − 0.848513i \(-0.677499\pi\)
0.999421 + 0.0340229i \(0.0108319\pi\)
\(48\) 2.07792 + 5.80196i 0.0432899 + 0.120874i
\(49\) −13.1956 −0.269297
\(50\) 48.3251 − 12.8331i 0.966501 − 0.256663i
\(51\) 2.76020 + 1.59360i 0.0541216 + 0.0312471i
\(52\) −66.1536 − 46.5760i −1.27219 − 0.895692i
\(53\) 55.2217 + 31.8822i 1.04192 + 0.601552i 0.920376 − 0.391035i \(-0.127883\pi\)
0.121542 + 0.992586i \(0.461216\pi\)
\(54\) −13.7382 − 0.617572i −0.254411 − 0.0114365i
\(55\) 87.4161 − 13.4125i 1.58938 − 0.243863i
\(56\) −47.4354 − 6.43176i −0.847061 − 0.114853i
\(57\) −3.79525 + 6.25735i −0.0665834 + 0.109778i
\(58\) −38.0093 + 59.4932i −0.655332 + 1.02574i
\(59\) −12.0246 + 6.94238i −0.203806 + 0.117668i −0.598430 − 0.801175i \(-0.704208\pi\)
0.394623 + 0.918843i \(0.370875\pi\)
\(60\) −6.41828 + 4.26030i −0.106971 + 0.0710050i
\(61\) 8.51902 − 14.7554i 0.139656 − 0.241891i −0.787710 − 0.616046i \(-0.788734\pi\)
0.927366 + 0.374154i \(0.122067\pi\)
\(62\) 1.91477 − 42.5949i 0.0308833 − 0.687015i
\(63\) 26.4827 − 45.8694i 0.420360 − 0.728085i
\(64\) 61.6893 + 17.0422i 0.963895 + 0.266284i
\(65\) 36.6982 − 94.2379i 0.564588 − 1.44981i
\(66\) −12.0944 + 6.27614i −0.183249 + 0.0950930i
\(67\) −24.8498 + 43.0411i −0.370892 + 0.642404i −0.989703 − 0.143136i \(-0.954281\pi\)
0.618811 + 0.785540i \(0.287615\pi\)
\(68\) 30.0335 − 13.9107i 0.441669 − 0.204569i
\(69\) 8.64149 0.125239
\(70\) −6.40951 − 59.4925i −0.0915644 − 0.849893i
\(71\) 58.9583 − 34.0396i 0.830399 − 0.479431i −0.0235903 − 0.999722i \(-0.507510\pi\)
0.853989 + 0.520291i \(0.174176\pi\)
\(72\) −43.3253 + 56.0126i −0.601740 + 0.777953i
\(73\) 85.6728 − 49.4632i 1.17360 − 0.677578i 0.219075 − 0.975708i \(-0.429696\pi\)
0.954525 + 0.298130i \(0.0963629\pi\)
\(74\) −2.40628 + 53.5289i −0.0325173 + 0.723364i
\(75\) −7.09342 − 6.51223i −0.0945789 − 0.0868298i
\(76\) 30.4439 + 69.6360i 0.400578 + 0.916263i
\(77\) − 105.838i − 1.37452i
\(78\) −0.699721 + 15.5657i −0.00897079 + 0.199560i
\(79\) 22.8122 − 13.1707i 0.288763 − 0.166717i −0.348621 − 0.937264i \(-0.613350\pi\)
0.637384 + 0.770547i \(0.280017\pi\)
\(80\) −2.19537 + 79.9699i −0.0274421 + 0.999623i
\(81\) −38.5081 − 66.6980i −0.475409 − 0.823433i
\(82\) 54.4105 + 104.852i 0.663543 + 1.27868i
\(83\) 50.3251 0.606327 0.303163 − 0.952939i \(-0.401957\pi\)
0.303163 + 0.952939i \(0.401957\pi\)
\(84\) 3.87460 + 8.36536i 0.0461262 + 0.0995877i
\(85\) 25.8813 + 32.2786i 0.304486 + 0.379748i
\(86\) −61.5907 − 118.688i −0.716171 − 1.38009i
\(87\) 13.5964 0.156281
\(88\) −19.0123 + 140.219i −0.216049 + 1.59340i
\(89\) −55.7536 + 96.5681i −0.626445 + 1.08504i 0.361814 + 0.932250i \(0.382158\pi\)
−0.988259 + 0.152785i \(0.951176\pi\)
\(90\) −80.9572 − 35.7923i −0.899524 − 0.397692i
\(91\) −104.813 − 60.5137i −1.15179 − 0.664986i
\(92\) 51.6624 − 73.3781i 0.561548 − 0.797588i
\(93\) −7.11144 + 4.10579i −0.0764670 + 0.0441483i
\(94\) −47.5967 + 74.4997i −0.506348 + 0.792550i
\(95\) −75.3836 + 57.8127i −0.793512 + 0.608554i
\(96\) −3.63053 − 11.7788i −0.0378181 − 0.122696i
\(97\) −44.3122 + 25.5836i −0.456827 + 0.263749i −0.710709 − 0.703486i \(-0.751626\pi\)
0.253882 + 0.967235i \(0.418292\pi\)
\(98\) 26.3645 + 1.18516i 0.269026 + 0.0120935i
\(99\) −135.590 − 78.2831i −1.36960 − 0.790738i
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.159.3 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.40 yes 232
5.4 even 2 inner 380.3.p.a.159.114 yes 232
19.11 even 3 inner 380.3.p.a.239.77 yes 232
20.19 odd 2 inner 380.3.p.a.159.77 yes 232
76.11 odd 6 inner 380.3.p.a.239.114 yes 232
95.49 even 6 inner 380.3.p.a.239.40 yes 232
380.239 odd 6 inner 380.3.p.a.239.3 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.3 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.40 yes 232 4.3 odd 2 inner
380.3.p.a.159.77 yes 232 20.19 odd 2 inner
380.3.p.a.159.114 yes 232 5.4 even 2 inner
380.3.p.a.239.3 yes 232 380.239 odd 6 inner
380.3.p.a.239.40 yes 232 95.49 even 6 inner
380.3.p.a.239.77 yes 232 19.11 even 3 inner
380.3.p.a.239.114 yes 232 76.11 odd 6 inner