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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 159.6
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.6

$q$-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+(-1.97327 + 0.325875i) q^{2} +(-0.501671 - 0.868919i) q^{3} +(3.78761 - 1.28608i) q^{4} +(-2.04342 - 4.56338i) q^{5} +(1.27309 + 1.55113i) q^{6} +6.90502 q^{7} +(-7.05489 + 3.77207i) q^{8} +(3.99665 - 6.92241i) q^{9} +(5.51932 + 8.33889i) q^{10} -6.00165i q^{11} +(-3.01763 - 2.64594i) q^{12} +(11.7221 + 6.76774i) q^{13} +(-13.6255 + 2.25017i) q^{14} +(-2.94008 + 4.06488i) q^{15} +(12.6920 - 9.74234i) q^{16} +(-13.1722 + 7.60495i) q^{17} +(-5.63065 + 14.9622i) q^{18} +(17.7989 - 6.64830i) q^{19} +(-13.6085 - 14.6563i) q^{20} +(-3.46405 - 5.99990i) q^{21} +(1.95578 + 11.8429i) q^{22} +(6.96578 - 12.0651i) q^{23} +(6.81686 + 4.23779i) q^{24} +(-16.6489 + 18.6498i) q^{25} +(-25.3363 - 9.53467i) q^{26} -17.0501 q^{27} +(26.1535 - 8.88040i) q^{28} +(-5.24051 + 9.07684i) q^{29} +(4.47695 - 8.97922i) q^{30} -21.2501i q^{31} +(-21.8700 + 23.3603i) q^{32} +(-5.21495 + 3.01085i) q^{33} +(23.5140 - 19.2991i) q^{34} +(-14.1099 - 31.5102i) q^{35} +(6.23500 - 31.3594i) q^{36} +0.879830i q^{37} +(-32.9555 + 18.9191i) q^{38} -13.5807i q^{39} +(31.6295 + 24.4862i) q^{40} +(2.72756 + 4.72427i) q^{41} +(8.79072 + 10.7106i) q^{42} +(-17.2054 - 29.8007i) q^{43} +(-7.71859 - 22.7319i) q^{44} +(-39.7564 - 4.09286i) q^{45} +(-9.81368 + 26.0777i) q^{46} +(31.5408 - 54.6303i) q^{47} +(-14.8325 - 6.14088i) q^{48} -1.32072 q^{49} +(26.7753 - 42.2266i) q^{50} +(13.2162 + 7.63037i) q^{51} +(53.1025 + 10.5581i) q^{52} +(-43.4426 - 25.0816i) q^{53} +(33.6445 - 5.55619i) q^{54} +(-27.3878 + 12.2639i) q^{55} +(-48.7141 + 26.0462i) q^{56} +(-14.7060 - 12.1305i) q^{57} +(7.38305 - 19.6188i) q^{58} +(25.3190 - 14.6179i) q^{59} +(-5.90814 + 19.1774i) q^{60} +(57.0667 - 98.8424i) q^{61} +(6.92488 + 41.9323i) q^{62} +(27.5970 - 47.7993i) q^{63} +(35.5429 - 53.2231i) q^{64} +(6.93066 - 67.3216i) q^{65} +(9.30935 - 7.64065i) q^{66} +(39.6785 - 68.7251i) q^{67} +(-40.1105 + 45.7451i) q^{68} -13.9781 q^{69} +(38.1110 + 57.5802i) q^{70} +(24.9299 - 14.3933i) q^{71} +(-2.08413 + 63.9125i) q^{72} +(-106.137 + 61.2784i) q^{73} +(-0.286714 - 1.73614i) q^{74} +(24.5574 + 5.11046i) q^{75} +(58.8650 - 48.0719i) q^{76} -41.4415i q^{77} +(4.42561 + 26.7984i) q^{78} +(-108.023 + 62.3672i) q^{79} +(-70.3931 - 38.0107i) q^{80} +(-27.4163 - 47.4865i) q^{81} +(-6.92174 - 8.43343i) q^{82} -24.6660 q^{83} +(-20.8368 - 18.2703i) q^{84} +(61.6206 + 44.5695i) q^{85} +(43.6623 + 53.1981i) q^{86} +10.5161 q^{87} +(22.6386 + 42.3409i) q^{88} +(-21.5215 + 37.2764i) q^{89} +(79.7840 - 4.87927i) q^{90} +(80.9411 + 46.7314i) q^{91} +(10.8670 - 54.6564i) q^{92} +(-18.4646 + 10.6606i) q^{93} +(-44.4360 + 118.079i) q^{94} +(-66.7093 - 67.6378i) q^{95} +(31.2697 + 7.28408i) q^{96} +(16.2959 - 9.40847i) q^{97} +(2.60615 - 0.430391i) q^{98} +(-41.5458 - 23.9865i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.97327 + 0.325875i −0.986636 + 0.162937i
\(3\) −0.501671 − 0.868919i −0.167224 − 0.289640i 0.770219 − 0.637779i \(-0.220147\pi\)
−0.937443 + 0.348140i \(0.886814\pi\)
\(4\) 3.78761 − 1.28608i 0.946903 − 0.321520i
\(5\) −2.04342 − 4.56338i −0.408684 − 0.912676i
\(6\) 1.27309 + 1.55113i 0.212182 + 0.258522i
\(7\) 6.90502 0.986431 0.493216 − 0.869907i \(-0.335821\pi\)
0.493216 + 0.869907i \(0.335821\pi\)
\(8\) −7.05489 + 3.77207i −0.881861 + 0.471509i
\(9\) 3.99665 − 6.92241i 0.444073 − 0.769156i
\(10\) 5.51932 + 8.33889i 0.551932 + 0.833889i
\(11\) − 6.00165i − 0.545604i −0.962070 − 0.272802i \(-0.912050\pi\)
0.962070 − 0.272802i \(-0.0879504\pi\)
\(12\) −3.01763 − 2.64594i −0.251469 − 0.220495i
\(13\) 11.7221 + 6.76774i 0.901698 + 0.520595i 0.877751 − 0.479118i \(-0.159043\pi\)
0.0239471 + 0.999713i \(0.492377\pi\)
\(14\) −13.6255 + 2.25017i −0.973249 + 0.160726i
\(15\) −2.94008 + 4.06488i −0.196006 + 0.270992i
\(16\) 12.6920 − 9.74234i 0.793250 − 0.608896i
\(17\) −13.1722 + 7.60495i −0.774833 + 0.447350i −0.834596 − 0.550862i \(-0.814299\pi\)
0.0597627 + 0.998213i \(0.480966\pi\)
\(18\) −5.63065 + 14.9622i −0.312814 + 0.831234i
\(19\) 17.7989 − 6.64830i 0.936783 − 0.349910i
\(20\) −13.6085 − 14.6563i −0.680427 − 0.732815i
\(21\) −3.46405 − 5.99990i −0.164955 − 0.285710i
\(22\) 1.95578 + 11.8429i 0.0888993 + 0.538313i
\(23\) 6.96578 − 12.0651i 0.302860 − 0.524569i −0.673923 − 0.738802i \(-0.735392\pi\)
0.976783 + 0.214233i \(0.0687252\pi\)
\(24\) 6.81686 + 4.23779i 0.284036 + 0.176575i
\(25\) −16.6489 + 18.6498i −0.665955 + 0.745992i
\(26\) −25.3363 − 9.53467i −0.974472 − 0.366718i
\(27\) −17.0501 −0.631485
\(28\) 26.1535 − 8.88040i 0.934054 − 0.317157i
\(29\) −5.24051 + 9.07684i −0.180707 + 0.312994i −0.942122 − 0.335271i \(-0.891172\pi\)
0.761414 + 0.648266i \(0.224505\pi\)
\(30\) 4.47695 − 8.97922i 0.149232 − 0.299307i
\(31\) − 21.2501i − 0.685488i −0.939429 − 0.342744i \(-0.888644\pi\)
0.939429 − 0.342744i \(-0.111356\pi\)
\(32\) −21.8700 + 23.3603i −0.683437 + 0.730009i
\(33\) −5.21495 + 3.01085i −0.158029 + 0.0912379i
\(34\) 23.5140 − 19.2991i 0.691589 − 0.567621i
\(35\) −14.1099 − 31.5102i −0.403139 − 0.900292i
\(36\) 6.23500 − 31.3594i 0.173195 − 0.871094i
\(37\) 0.879830i 0.0237792i 0.999929 + 0.0118896i \(0.00378466\pi\)
−0.999929 + 0.0118896i \(0.996215\pi\)
\(38\) −32.9555 + 18.9191i −0.867251 + 0.497871i
\(39\) − 13.5807i − 0.348223i
\(40\) 31.6295 + 24.4862i 0.790738 + 0.612155i
\(41\) 2.72756 + 4.72427i 0.0665258 + 0.115226i 0.897370 − 0.441279i \(-0.145475\pi\)
−0.830844 + 0.556505i \(0.812142\pi\)
\(42\) 8.79072 + 10.7106i 0.209303 + 0.255014i
\(43\) −17.2054 − 29.8007i −0.400126 − 0.693039i 0.593614 − 0.804750i \(-0.297700\pi\)
−0.993741 + 0.111710i \(0.964367\pi\)
\(44\) −7.71859 − 22.7319i −0.175423 − 0.516634i
\(45\) −39.7564 − 4.09286i −0.883476 − 0.0909525i
\(46\) −9.81368 + 26.0777i −0.213341 + 0.566906i
\(47\) 31.5408 − 54.6303i 0.671081 − 1.16235i −0.306517 − 0.951865i \(-0.599164\pi\)
0.977598 − 0.210481i \(-0.0675031\pi\)
\(48\) −14.8325 − 6.14088i −0.309011 − 0.127935i
\(49\) −1.32072 −0.0269536
\(50\) 26.7753 − 42.2266i 0.535505 − 0.844532i
\(51\) 13.2162 + 7.63037i 0.259141 + 0.149615i
\(52\) 53.1025 + 10.5581i 1.02120 + 0.203040i
\(53\) −43.4426 − 25.0816i −0.819672 − 0.473238i 0.0306313 − 0.999531i \(-0.490248\pi\)
−0.850303 + 0.526293i \(0.823582\pi\)
\(54\) 33.6445 − 5.55619i 0.623046 − 0.102892i
\(55\) −27.3878 + 12.2639i −0.497960 + 0.222980i
\(56\) −48.7141 + 26.0462i −0.869895 + 0.465111i
\(57\) −14.7060 − 12.1305i −0.258000 − 0.212816i
\(58\) 7.38305 − 19.6188i 0.127294 − 0.338256i
\(59\) 25.3190 − 14.6179i 0.429136 − 0.247762i −0.269842 − 0.962904i \(-0.586972\pi\)
0.698979 + 0.715143i \(0.253638\pi\)
\(60\) −5.90814 + 19.1774i −0.0984689 + 0.319623i
\(61\) 57.0667 − 98.8424i 0.935519 − 1.62037i 0.161814 − 0.986821i \(-0.448266\pi\)
0.773705 − 0.633546i \(-0.218401\pi\)
\(62\) 6.92488 + 41.9323i 0.111692 + 0.676328i
\(63\) 27.5970 − 47.7993i 0.438047 − 0.758720i
\(64\) 35.5429 − 53.2231i 0.555358 − 0.831611i
\(65\) 6.93066 − 67.3216i 0.106625 − 1.03572i
\(66\) 9.30935 − 7.64065i 0.141051 − 0.115767i
\(67\) 39.6785 − 68.7251i 0.592216 − 1.02575i −0.401718 − 0.915764i \(-0.631587\pi\)
0.993933 − 0.109984i \(-0.0350800\pi\)
\(68\) −40.1105 + 45.7451i −0.589860 + 0.672722i
\(69\) −13.9781 −0.202581
\(70\) 38.1110 + 57.5802i 0.544442 + 0.822575i
\(71\) 24.9299 − 14.3933i 0.351126 − 0.202723i −0.314055 − 0.949405i \(-0.601688\pi\)
0.665181 + 0.746682i \(0.268354\pi\)
\(72\) −2.08413 + 63.9125i −0.0289463 + 0.887673i
\(73\) −106.137 + 61.2784i −1.45394 + 0.839431i −0.998702 − 0.0509409i \(-0.983778\pi\)
−0.455235 + 0.890371i \(0.650445\pi\)
\(74\) −0.286714 − 1.73614i −0.00387452 − 0.0234614i
\(75\) 24.5574 + 5.11046i 0.327432 + 0.0681395i
\(76\) 58.8650 − 48.0719i 0.774540 − 0.632525i
\(77\) − 41.4415i − 0.538201i
\(78\) 4.42561 + 26.7984i 0.0567386 + 0.343570i
\(79\) −108.023 + 62.3672i −1.36738 + 0.789458i −0.990593 − 0.136841i \(-0.956305\pi\)
−0.376789 + 0.926299i \(0.622972\pi\)
\(80\) −70.3931 − 38.0107i −0.879913 − 0.475134i
\(81\) −27.4163 − 47.4865i −0.338473 − 0.586253i
\(82\) −6.92174 − 8.43343i −0.0844114 − 0.102847i
\(83\) −24.6660 −0.297181 −0.148591 − 0.988899i \(-0.547474\pi\)
−0.148591 + 0.988899i \(0.547474\pi\)
\(84\) −20.8368 − 18.2703i −0.248057 − 0.217503i
\(85\) 61.6206 + 44.5695i 0.724948 + 0.524347i
\(86\) 43.6623 + 53.1981i 0.507701 + 0.618582i
\(87\) 10.5161 0.120874
\(88\) 22.6386 + 42.3409i 0.257257 + 0.481147i
\(89\) −21.5215 + 37.2764i −0.241815 + 0.418836i −0.961231 − 0.275743i \(-0.911076\pi\)
0.719416 + 0.694579i \(0.244409\pi\)
\(90\) 79.7840 − 4.87927i 0.886489 − 0.0542142i
\(91\) 80.9411 + 46.7314i 0.889463 + 0.513532i
\(92\) 10.8670 − 54.6564i 0.118120 − 0.594091i
\(93\) −18.4646 + 10.6606i −0.198545 + 0.114630i
\(94\) −44.4360 + 118.079i −0.472723 + 1.25616i
\(95\) −66.7093 − 67.6378i −0.702203 − 0.711977i
\(96\) 31.2697 + 7.28408i 0.325727 + 0.0758759i
\(97\) 16.2959 − 9.40847i 0.167999 − 0.0969945i −0.413643 − 0.910439i \(-0.635744\pi\)
0.581642 + 0.813445i \(0.302410\pi\)
\(98\) 2.60615 − 0.430391i 0.0265934 − 0.00439174i
\(99\) −41.5458 − 23.9865i −0.419655 − 0.242288i
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.159.6 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.34 yes 232
5.4 even 2 inner 380.3.p.a.159.111 yes 232
19.11 even 3 inner 380.3.p.a.239.83 yes 232
20.19 odd 2 inner 380.3.p.a.159.83 yes 232
76.11 odd 6 inner 380.3.p.a.239.111 yes 232
95.49 even 6 inner 380.3.p.a.239.34 yes 232
380.239 odd 6 inner 380.3.p.a.239.6 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.6 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.34 yes 232 4.3 odd 2 inner
380.3.p.a.159.83 yes 232 20.19 odd 2 inner
380.3.p.a.159.111 yes 232 5.4 even 2 inner
380.3.p.a.239.6 yes 232 380.239 odd 6 inner
380.3.p.a.239.34 yes 232 95.49 even 6 inner
380.3.p.a.239.83 yes 232 19.11 even 3 inner
380.3.p.a.239.111 yes 232 76.11 odd 6 inner