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(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), 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 \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 303.48
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.48

$q$-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+(-0.617236 + 1.90237i) q^{2} +(1.61266 + 1.61266i) q^{3} +(-3.23804 - 2.34842i) q^{4} +(1.65685 + 4.71750i) q^{5} +(-4.06328 + 2.07249i) q^{6} +(5.55382 + 5.55382i) q^{7} +(6.46621 - 4.71042i) q^{8} -3.79864i q^{9} +(-9.99712 + 0.240130i) q^{10} +8.72018i q^{11} +(-1.43465 - 9.00908i) q^{12} +(-12.4280 + 12.4280i) q^{13} +(-13.9934 + 7.13741i) q^{14} +(-4.93580 + 10.2797i) q^{15} +(4.96980 + 15.2086i) q^{16} +(21.6646 - 21.6646i) q^{17} +(7.22643 + 2.34466i) q^{18} +(11.2885 + 15.2830i) q^{19} +(5.71376 - 19.1665i) q^{20} +17.9129i q^{21} +(-16.5890 - 5.38241i) q^{22} +(-15.1312 + 15.1312i) q^{23} +(18.0241 + 2.83149i) q^{24} +(-19.5097 + 15.6324i) q^{25} +(-15.9717 - 31.3138i) q^{26} +(20.6399 - 20.6399i) q^{27} +(-4.94076 - 31.0262i) q^{28} +6.81256 q^{29} +(-16.5092 - 15.7347i) q^{30} -30.7465 q^{31} +(-31.9999 + 0.0671278i) q^{32} +(-14.0627 + 14.0627i) q^{33} +(27.8420 + 54.5864i) q^{34} +(-16.9983 + 35.4020i) q^{35} +(-8.92083 + 12.3002i) q^{36} +(-33.1548 - 33.1548i) q^{37} +(-36.0416 + 12.0417i) q^{38} -40.0844 q^{39} +(32.9350 + 22.6999i) q^{40} +32.6770i q^{41} +(-34.0769 - 11.0565i) q^{42} +(30.2479 - 30.2479i) q^{43} +(20.4787 - 28.2363i) q^{44} +(17.9201 - 6.29378i) q^{45} +(-19.4456 - 38.1246i) q^{46} +(-30.9152 - 30.9152i) q^{47} +(-16.5117 + 32.5409i) q^{48} +12.6897i q^{49} +(-17.6965 - 46.7636i) q^{50} +69.8754 q^{51} +(69.4288 - 11.0562i) q^{52} +(-21.5207 + 21.5207i) q^{53} +(26.5251 + 52.0044i) q^{54} +(-41.1375 + 14.4480i) q^{55} +(62.0730 + 9.75132i) q^{56} +(-6.44178 + 42.8508i) q^{57} +(-4.20496 + 12.9600i) q^{58} +26.9423i q^{59} +(40.1234 - 21.6946i) q^{60} +72.3852 q^{61} +(18.9778 - 58.4912i) q^{62} +(21.0970 - 21.0970i) q^{63} +(19.6238 - 60.9172i) q^{64} +(-79.2207 - 38.0379i) q^{65} +(-18.0725 - 35.4325i) q^{66} +(44.8566 - 44.8566i) q^{67} +(-121.029 + 19.2732i) q^{68} -48.8029 q^{69} +(-56.8558 - 54.1885i) q^{70} -102.438 q^{71} +(-17.8932 - 24.5628i) q^{72} +(31.3053 + 31.3053i) q^{73} +(83.5371 - 42.6084i) q^{74} +(-56.6723 - 6.25280i) q^{75} +(-0.661628 - 75.9971i) q^{76} +(-48.4303 + 48.4303i) q^{77} +(24.7416 - 76.2555i) q^{78} -46.5275i q^{79} +(-63.5124 + 48.6434i) q^{80} +32.3825 q^{81} +(-62.1638 - 20.1694i) q^{82} +(-56.4992 + 56.4992i) q^{83} +(42.0670 - 58.0025i) q^{84} +(138.098 + 66.3079i) q^{85} +(38.8726 + 76.2128i) q^{86} +(10.9864 + 10.9864i) q^{87} +(41.0758 + 56.3865i) q^{88} +33.8164 q^{89} +(0.912168 + 37.9755i) q^{90} -138.046 q^{91} +(84.5296 - 13.4609i) q^{92} +(-49.5837 - 49.5837i) q^{93} +(77.8941 - 39.7302i) q^{94} +(-53.3943 + 78.5751i) q^{95} +(-51.7133 - 51.4968i) q^{96} +(111.020 + 111.020i) q^{97} +(-24.1406 - 7.83257i) q^{98} +33.1249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.617236 + 1.90237i −0.308618 + 0.951186i
\(3\) 1.61266 + 1.61266i 0.537554 + 0.537554i 0.922810 − 0.385256i \(-0.125887\pi\)
−0.385256 + 0.922810i \(0.625887\pi\)
\(4\) −3.23804 − 2.34842i −0.809510 − 0.587106i
\(5\) 1.65685 + 4.71750i 0.331370 + 0.943501i
\(6\) −4.06328 + 2.07249i −0.677213 + 0.345415i
\(7\) 5.55382 + 5.55382i 0.793402 + 0.793402i 0.982046 − 0.188643i \(-0.0604090\pi\)
−0.188643 + 0.982046i \(0.560409\pi\)
\(8\) 6.46621 − 4.71042i 0.808277 − 0.588803i
\(9\) − 3.79864i − 0.422071i
\(10\) −9.99712 + 0.240130i −0.999712 + 0.0240130i
\(11\) 8.72018i 0.792744i 0.918090 + 0.396372i \(0.129731\pi\)
−0.918090 + 0.396372i \(0.870269\pi\)
\(12\) −1.43465 − 9.00908i −0.119554 − 0.750757i
\(13\) −12.4280 + 12.4280i −0.956003 + 0.956003i −0.999072 − 0.0430694i \(-0.986286\pi\)
0.0430694 + 0.999072i \(0.486286\pi\)
\(14\) −13.9934 + 7.13741i −0.999531 + 0.509815i
\(15\) −4.93580 + 10.2797i −0.329054 + 0.685312i
\(16\) 4.96980 + 15.2086i 0.310613 + 0.950537i
\(17\) 21.6646 − 21.6646i 1.27439 − 1.27439i 0.330628 − 0.943761i \(-0.392739\pi\)
0.943761 − 0.330628i \(-0.107261\pi\)
\(18\) 7.22643 + 2.34466i 0.401468 + 0.130259i
\(19\) 11.2885 + 15.2830i 0.594131 + 0.804368i
\(20\) 5.71376 − 19.1665i 0.285688 − 0.958323i
\(21\) 17.9129i 0.852993i
\(22\) −16.5890 − 5.38241i −0.754047 − 0.244655i
\(23\) −15.1312 + 15.1312i −0.657876 + 0.657876i −0.954877 − 0.297001i \(-0.904014\pi\)
0.297001 + 0.954877i \(0.404014\pi\)
\(24\) 18.0241 + 2.83149i 0.751006 + 0.117979i
\(25\) −19.5097 + 15.6324i −0.780388 + 0.625295i
\(26\) −15.9717 − 31.3138i −0.614297 − 1.20438i
\(27\) 20.6399 − 20.6399i 0.764440 − 0.764440i
\(28\) −4.94076 − 31.0262i −0.176456 − 1.10808i
\(29\) 6.81256 0.234916 0.117458 − 0.993078i \(-0.462525\pi\)
0.117458 + 0.993078i \(0.462525\pi\)
\(30\) −16.5092 − 15.7347i −0.550307 − 0.524491i
\(31\) −30.7465 −0.991822 −0.495911 − 0.868373i \(-0.665166\pi\)
−0.495911 + 0.868373i \(0.665166\pi\)
\(32\) −31.9999 + 0.0671278i −0.999998 + 0.00209774i
\(33\) −14.0627 + 14.0627i −0.426143 + 0.426143i
\(34\) 27.8420 + 54.5864i 0.818882 + 1.60548i
\(35\) −16.9983 + 35.4020i −0.485666 + 1.01149i
\(36\) −8.92083 + 12.3002i −0.247801 + 0.341671i
\(37\) −33.1548 − 33.1548i −0.896076 − 0.896076i 0.0990107 − 0.995086i \(-0.468432\pi\)
−0.995086 + 0.0990107i \(0.968432\pi\)
\(38\) −36.0416 + 12.0417i −0.948463 + 0.316887i
\(39\) −40.0844 −1.02781
\(40\) 32.9350 + 22.6999i 0.823375 + 0.567498i
\(41\) 32.6770i 0.797000i 0.917168 + 0.398500i \(0.130469\pi\)
−0.917168 + 0.398500i \(0.869531\pi\)
\(42\) −34.0769 − 11.0565i −0.811355 − 0.263249i
\(43\) 30.2479 − 30.2479i 0.703439 − 0.703439i −0.261708 − 0.965147i \(-0.584286\pi\)
0.965147 + 0.261708i \(0.0842858\pi\)
\(44\) 20.4787 − 28.2363i 0.465425 − 0.641734i
\(45\) 17.9201 − 6.29378i 0.398225 − 0.139862i
\(46\) −19.4456 − 38.1246i −0.422730 − 0.828795i
\(47\) −30.9152 − 30.9152i −0.657770 − 0.657770i 0.297082 − 0.954852i \(-0.403986\pi\)
−0.954852 + 0.297082i \(0.903986\pi\)
\(48\) −16.5117 + 32.5409i −0.343994 + 0.677936i
\(49\) 12.6897i 0.258974i
\(50\) −17.6965 − 46.7636i −0.353931 − 0.935272i
\(51\) 69.8754 1.37011
\(52\) 69.4288 − 11.0562i 1.33517 − 0.212619i
\(53\) −21.5207 + 21.5207i −0.406050 + 0.406050i −0.880359 − 0.474309i \(-0.842698\pi\)
0.474309 + 0.880359i \(0.342698\pi\)
\(54\) 26.5251 + 52.0044i 0.491205 + 0.963045i
\(55\) −41.1375 + 14.4480i −0.747955 + 0.262691i
\(56\) 62.0730 + 9.75132i 1.10845 + 0.174131i
\(57\) −6.44178 + 42.8508i −0.113014 + 0.751769i
\(58\) −4.20496 + 12.9600i −0.0724993 + 0.223449i
\(59\) 26.9423i 0.456649i 0.973585 + 0.228325i \(0.0733247\pi\)
−0.973585 + 0.228325i \(0.926675\pi\)
\(60\) 40.1234 − 21.6946i 0.668723 − 0.361577i
\(61\) 72.3852 1.18664 0.593321 − 0.804966i \(-0.297816\pi\)
0.593321 + 0.804966i \(0.297816\pi\)
\(62\) 18.9778 − 58.4912i 0.306094 − 0.943407i
\(63\) 21.0970 − 21.0970i 0.334872 − 0.334872i
\(64\) 19.6238 − 60.9172i 0.306622 − 0.951831i
\(65\) −79.2207 − 38.0379i −1.21878 − 0.585199i
\(66\) −18.0725 − 35.4325i −0.273826 − 0.536856i
\(67\) 44.8566 − 44.8566i 0.669502 − 0.669502i −0.288099 − 0.957601i \(-0.593023\pi\)
0.957601 + 0.288099i \(0.0930232\pi\)
\(68\) −121.029 + 19.2732i −1.77983 + 0.283429i
\(69\) −48.8029 −0.707288
\(70\) −56.8558 − 54.1885i −0.812225 − 0.774122i
\(71\) −102.438 −1.44279 −0.721397 − 0.692522i \(-0.756500\pi\)
−0.721397 + 0.692522i \(0.756500\pi\)
\(72\) −17.8932 − 24.5628i −0.248517 − 0.341150i
\(73\) 31.3053 + 31.3053i 0.428840 + 0.428840i 0.888233 − 0.459393i \(-0.151933\pi\)
−0.459393 + 0.888233i \(0.651933\pi\)
\(74\) 83.5371 − 42.6084i 1.12888 − 0.575790i
\(75\) −56.6723 − 6.25280i −0.755631 − 0.0833707i
\(76\) −0.661628 − 75.9971i −0.00870564 − 0.999962i
\(77\) −48.4303 + 48.4303i −0.628965 + 0.628965i
\(78\) 24.7416 − 76.2555i 0.317199 − 0.977635i
\(79\) − 46.5275i − 0.588955i −0.955658 − 0.294478i \(-0.904854\pi\)
0.955658 − 0.294478i \(-0.0951457\pi\)
\(80\) −63.5124 + 48.6434i −0.793905 + 0.608042i
\(81\) 32.3825 0.399784
\(82\) −62.1638 − 20.1694i −0.758095 − 0.245969i
\(83\) −56.4992 + 56.4992i −0.680714 + 0.680714i −0.960161 − 0.279447i \(-0.909849\pi\)
0.279447 + 0.960161i \(0.409849\pi\)
\(84\) 42.0670 − 58.0025i 0.500798 − 0.690506i
\(85\) 138.098 + 66.3079i 1.62468 + 0.780093i
\(86\) 38.8726 + 76.2128i 0.452007 + 0.886195i
\(87\) 10.9864 + 10.9864i 0.126280 + 0.126280i
\(88\) 41.0758 + 56.3865i 0.466770 + 0.640756i
\(89\) 33.8164 0.379959 0.189980 − 0.981788i \(-0.439158\pi\)
0.189980 + 0.981788i \(0.439158\pi\)
\(90\) 0.912168 + 37.9755i 0.0101352 + 0.421950i
\(91\) −138.046 −1.51699
\(92\) 84.5296 − 13.4609i 0.918800 − 0.146314i
\(93\) −49.5837 − 49.5837i −0.533158 − 0.533158i
\(94\) 77.8941 − 39.7302i 0.828661 − 0.422662i
\(95\) −53.3943 + 78.5751i −0.562045 + 0.827107i
\(96\) −51.7133 − 51.4968i −0.538680 − 0.536425i
\(97\) 111.020 + 111.020i 1.14454 + 1.14454i 0.987610 + 0.156927i \(0.0501587\pi\)
0.156927 + 0.987610i \(0.449841\pi\)
\(98\) −24.1406 − 7.83257i −0.246333 − 0.0799241i
\(99\) 33.1249 0.334594
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.j.a.303.48 yes 232
4.3 odd 2 inner 380.3.j.a.303.106 yes 232
5.2 odd 4 inner 380.3.j.a.227.11 ✓ 232
19.18 odd 2 inner 380.3.j.a.303.69 yes 232
20.7 even 4 inner 380.3.j.a.227.69 yes 232
76.75 even 2 inner 380.3.j.a.303.11 yes 232
95.37 even 4 inner 380.3.j.a.227.106 yes 232
380.227 odd 4 inner 380.3.j.a.227.48 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.11 ✓ 232 5.2 odd 4 inner
380.3.j.a.227.48 yes 232 380.227 odd 4 inner
380.3.j.a.227.69 yes 232 20.7 even 4 inner
380.3.j.a.227.106 yes 232 95.37 even 4 inner
380.3.j.a.303.11 yes 232 76.75 even 2 inner
380.3.j.a.303.48 yes 232 1.1 even 1 trivial
380.3.j.a.303.69 yes 232 19.18 odd 2 inner
380.3.j.a.303.106 yes 232 4.3 odd 2 inner