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 227.11
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.11

$q$-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+(-1.90237 - 0.617236i) 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} +(-4.71042 - 6.46621i) q^{8} +3.79864i q^{9} +(-6.06376 + 7.95178i) q^{10} +8.72018i q^{11} +(9.00908 - 1.43465i) 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} +(2.34466 - 7.22643i) q^{18} +(-11.2885 - 15.2830i) q^{19} +(16.4436 - 11.3845i) q^{20} +17.9129i q^{21} +(5.38241 - 16.5890i) 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} +(-31.0262 + 4.94076i) q^{28} -6.81256 q^{29} +(3.04475 + 22.6023i) q^{30} -30.7465 q^{31} +(-0.0671278 - 31.9999i) 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} +(12.0417 + 36.0416i) q^{38} +40.0844 q^{39} +(-38.3088 + 11.5079i) q^{40} +32.6770i q^{41} +(11.0565 - 34.0769i) 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} +(32.5409 + 16.5117i) q^{48} -12.6897i q^{49} +(27.4658 + 41.7807i) q^{50} +69.8754 q^{51} +(11.0562 + 69.4288i) 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} +(-42.8508 - 6.44178i) q^{57} +(12.9600 + 4.20496i) q^{58} -26.9423i q^{59} +(8.15872 - 44.8774i) q^{60} +72.3852 q^{61} +(58.4912 + 18.9778i) 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} +(19.2732 + 121.029i) q^{68} +48.8029 q^{69} +(-10.4858 - 77.8397i) q^{70} -102.438 q^{71} +(24.5628 - 17.8932i) 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} +(-76.2555 - 24.7416i) q^{78} +46.5275i q^{79} +(79.9808 + 1.75326i) q^{80} +32.3825 q^{81} +(20.1694 - 62.1638i) 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} +(56.3865 - 41.0758i) q^{88} -33.8164 q^{89} +(-30.2060 - 23.0340i) q^{90} -138.046 q^{91} +(13.4609 + 84.5296i) q^{92} +(-49.5837 + 49.5837i) q^{93} +(-77.8941 + 39.7302i) q^{94} +(-90.8009 + 27.9319i) q^{95} +(-51.7133 - 51.4968i) q^{96} +(-111.020 + 111.020i) q^{97} +(-7.83257 + 24.1406i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90237 − 0.617236i −0.951186 − 0.308618i
\(3\) 1.61266 − 1.61266i 0.537554 − 0.537554i −0.385256 − 0.922810i \(-0.625887\pi\)
0.922810 + 0.385256i \(0.125887\pi\)
\(4\) 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 \(-0.939591\pi\)
0.188643 + 0.982046i \(0.439591\pi\)
\(8\) −4.71042 − 6.46621i −0.588803 − 0.808277i
\(9\) 3.79864i 0.422071i
\(10\) −6.06376 + 7.95178i −0.606376 + 0.795178i
\(11\) 8.72018i 0.792744i 0.918090 + 0.396372i \(0.129731\pi\)
−0.918090 + 0.396372i \(0.870269\pi\)
\(12\) 9.00908 − 1.43465i 0.750757 − 0.119554i
\(13\) 12.4280 + 12.4280i 0.956003 + 0.956003i 0.999072 − 0.0430694i \(-0.0137137\pi\)
−0.0430694 + 0.999072i \(0.513714\pi\)
\(14\) 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 0.943761 + 0.330628i \(0.107261\pi\)
0.330628 + 0.943761i \(0.392739\pi\)
\(18\) 2.34466 − 7.22643i 0.130259 − 0.401468i
\(19\) −11.2885 − 15.2830i −0.594131 − 0.804368i
\(20\) 16.4436 − 11.3845i 0.822182 − 0.569224i
\(21\) 17.9129i 0.852993i
\(22\) 5.38241 − 16.5890i 0.244655 − 0.754047i
\(23\) 15.1312 + 15.1312i 0.657876 + 0.657876i 0.954877 − 0.297001i \(-0.0959865\pi\)
−0.297001 + 0.954877i \(0.595986\pi\)
\(24\) −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
\(28\) −31.0262 + 4.94076i −1.10808 + 0.176456i
\(29\) −6.81256 −0.234916 −0.117458 − 0.993078i \(-0.537475\pi\)
−0.117458 + 0.993078i \(0.537475\pi\)
\(30\) 3.04475 + 22.6023i 0.101492 + 0.753411i
\(31\) −30.7465 −0.991822 −0.495911 − 0.868373i \(-0.665166\pi\)
−0.495911 + 0.868373i \(0.665166\pi\)
\(32\) −0.0671278 − 31.9999i −0.00209774 − 0.999998i
\(33\) 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 \(-0.531568\pi\)
0.995086 + 0.0990107i \(0.0315678\pi\)
\(38\) 12.0417 + 36.0416i 0.316887 + 0.948463i
\(39\) 40.0844 1.02781
\(40\) −38.3088 + 11.5079i −0.957721 + 0.287698i
\(41\) 32.6770i 0.797000i 0.917168 + 0.398500i \(0.130469\pi\)
−0.917168 + 0.398500i \(0.869531\pi\)
\(42\) 11.0565 − 34.0769i 0.263249 − 0.811355i
\(43\) −30.2479 − 30.2479i −0.703439 − 0.703439i 0.261708 − 0.965147i \(-0.415714\pi\)
−0.965147 + 0.261708i \(0.915714\pi\)
\(44\) −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 \(-0.596014\pi\)
0.954852 + 0.297082i \(0.0960136\pi\)
\(48\) 32.5409 + 16.5117i 0.677936 + 0.343994i
\(49\) − 12.6897i − 0.258974i
\(50\) 27.4658 + 41.7807i 0.549317 + 0.835614i
\(51\) 69.8754 1.37011
\(52\) 11.0562 + 69.4288i 0.212619 + 1.33517i
\(53\) 21.5207 + 21.5207i 0.406050 + 0.406050i 0.880359 − 0.474309i \(-0.157302\pi\)
−0.474309 + 0.880359i \(0.657302\pi\)
\(54\) −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
\(57\) −42.8508 − 6.44178i −0.751769 − 0.113014i
\(58\) 12.9600 + 4.20496i 0.223449 + 0.0724993i
\(59\) − 26.9423i − 0.456649i −0.973585 − 0.228325i \(-0.926675\pi\)
0.973585 − 0.228325i \(-0.0733247\pi\)
\(60\) 8.15872 − 44.8774i 0.135979 − 0.747956i
\(61\) 72.3852 1.18664 0.593321 − 0.804966i \(-0.297816\pi\)
0.593321 + 0.804966i \(0.297816\pi\)
\(62\) 58.4912 + 18.9778i 0.943407 + 0.306094i
\(63\) −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 0.957601 − 0.288099i \(-0.0930232\pi\)
−0.288099 + 0.957601i \(0.593023\pi\)
\(68\) 19.2732 + 121.029i 0.283429 + 1.77983i
\(69\) 48.8029 0.707288
\(70\) −10.4858 − 77.8397i −0.149796 − 1.11200i
\(71\) −102.438 −1.44279 −0.721397 − 0.692522i \(-0.756500\pi\)
−0.721397 + 0.692522i \(0.756500\pi\)
\(72\) 24.5628 − 17.8932i 0.341150 − 0.248517i
\(73\) 31.3053 − 31.3053i 0.428840 − 0.428840i −0.459393 − 0.888233i \(-0.651933\pi\)
0.888233 + 0.459393i \(0.151933\pi\)
\(74\) −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
\(78\) −76.2555 − 24.7416i −0.977635 − 0.317199i
\(79\) 46.5275i 0.588955i 0.955658 + 0.294478i \(0.0951457\pi\)
−0.955658 + 0.294478i \(0.904854\pi\)
\(80\) 79.9808 + 1.75326i 0.999760 + 0.0219158i
\(81\) 32.3825 0.399784
\(82\) 20.1694 − 62.1638i 0.245969 − 0.758095i
\(83\) 56.4992 + 56.4992i 0.680714 + 0.680714i 0.960161 − 0.279447i \(-0.0901512\pi\)
−0.279447 + 0.960161i \(0.590151\pi\)
\(84\) −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
\(88\) 56.3865 − 41.0758i 0.640756 − 0.466770i
\(89\) −33.8164 −0.379959 −0.189980 − 0.981788i \(-0.560842\pi\)
−0.189980 + 0.981788i \(0.560842\pi\)
\(90\) −30.2060 − 23.0340i −0.335622 − 0.255934i
\(91\) −138.046 −1.51699
\(92\) 13.4609 + 84.5296i 0.146314 + 0.918800i
\(93\) −49.5837 + 49.5837i −0.533158 + 0.533158i
\(94\) −77.8941 + 39.7302i −0.828661 + 0.422662i
\(95\) −90.8009 + 27.9319i −0.955799 + 0.294020i
\(96\) −51.7133 − 51.4968i −0.538680 − 0.536425i
\(97\) −111.020 + 111.020i −1.14454 + 1.14454i −0.156927 + 0.987610i \(0.550159\pi\)
−0.987610 + 0.156927i \(0.949841\pi\)
\(98\) −7.83257 + 24.1406i −0.0799241 + 0.246333i
\(99\) −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.227.11 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.69 yes 232
5.3 odd 4 inner 380.3.j.a.303.48 yes 232
19.18 odd 2 inner 380.3.j.a.227.106 yes 232
20.3 even 4 inner 380.3.j.a.303.106 yes 232
76.75 even 2 inner 380.3.j.a.227.48 yes 232
95.18 even 4 inner 380.3.j.a.303.69 yes 232
380.303 odd 4 inner 380.3.j.a.303.11 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.11 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.48 yes 232 76.75 even 2 inner
380.3.j.a.227.69 yes 232 4.3 odd 2 inner
380.3.j.a.227.106 yes 232 19.18 odd 2 inner
380.3.j.a.303.11 yes 232 380.303 odd 4 inner
380.3.j.a.303.48 yes 232 5.3 odd 4 inner
380.3.j.a.303.69 yes 232 95.18 even 4 inner
380.3.j.a.303.106 yes 232 20.3 even 4 inner