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Properties

Label 380.3.j.a.227.12
Level $380$
Weight $3$
Character 380.227
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(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 227.12
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.12

$q$-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+(-1.89506 + 0.639336i) q^{2} +(-0.113263 + 0.113263i) q^{3} +(3.18250 - 2.42316i) q^{4} +(-1.31466 + 4.82407i) q^{5} +(0.142227 - 0.287054i) q^{6} +(-6.53716 + 6.53716i) q^{7} +(-4.48181 + 6.62672i) q^{8} +8.97434i q^{9} +(-0.592839 - 9.98241i) q^{10} -4.16055i q^{11} +(-0.0860052 + 0.634916i) q^{12} +(-7.13071 - 7.13071i) q^{13} +(8.20887 - 16.5678i) q^{14} +(-0.397487 - 0.695294i) q^{15} +(4.25659 - 15.4234i) q^{16} +(15.0048 + 15.0048i) q^{17} +(-5.73762 - 17.0069i) q^{18} +(1.04794 - 18.9711i) q^{19} +(7.50558 + 18.5382i) q^{20} -1.48084i q^{21} +(2.65999 + 7.88448i) q^{22} +(-8.80745 - 8.80745i) q^{23} +(-0.242940 - 1.25819i) q^{24} +(-21.5433 - 12.6841i) q^{25} +(18.0721 + 8.95420i) q^{26} +(-2.03584 - 2.03584i) q^{27} +(-4.96391 + 36.6451i) q^{28} -18.3645 q^{29} +(1.19779 + 1.06350i) q^{30} +23.5672 q^{31} +(1.79426 + 31.9497i) q^{32} +(0.471238 + 0.471238i) q^{33} +(-38.0282 - 18.8419i) q^{34} +(-22.9416 - 40.1299i) q^{35} +(21.7463 + 28.5608i) q^{36} +(-41.5872 + 41.5872i) q^{37} +(10.1430 + 36.6213i) q^{38} +1.61530 q^{39} +(-26.0757 - 30.3325i) q^{40} -35.5687i q^{41} +(0.946757 + 2.80629i) q^{42} +(-27.2761 - 27.2761i) q^{43} +(-10.0817 - 13.2409i) q^{44} +(-43.2929 - 11.7982i) q^{45} +(22.3216 + 11.0597i) q^{46} +(37.0991 - 37.0991i) q^{47} +(1.26479 + 2.22902i) q^{48} -36.4690i q^{49} +(48.9352 + 10.2636i) q^{50} -3.39900 q^{51} +(-39.9723 - 5.41462i) q^{52} +(-5.21359 - 5.21359i) q^{53} +(5.15961 + 2.55644i) q^{54} +(20.0708 + 5.46972i) q^{55} +(-14.0216 - 72.6183i) q^{56} +(2.03004 + 2.26742i) q^{57} +(34.8018 - 11.7411i) q^{58} -34.5881i q^{59} +(-2.94981 - 1.24960i) q^{60} -13.4088 q^{61} +(-44.6612 + 15.0674i) q^{62} +(-58.6668 - 58.6668i) q^{63} +(-23.8268 - 59.3994i) q^{64} +(43.7736 - 25.0246i) q^{65} +(-1.19430 - 0.591744i) q^{66} +(-31.1774 - 31.1774i) q^{67} +(84.1121 + 11.3937i) q^{68} +1.99512 q^{69} +(69.1322 + 61.3812i) q^{70} -70.1730 q^{71} +(-59.4704 - 40.2213i) q^{72} +(-41.2283 + 41.2283i) q^{73} +(52.2220 - 105.398i) q^{74} +(3.87671 - 1.00343i) q^{75} +(-42.6349 - 62.9148i) q^{76} +(27.1982 + 27.1982i) q^{77} +(-3.06109 + 1.03272i) q^{78} +49.6890i q^{79} +(68.8076 + 40.8107i) q^{80} -80.3079 q^{81} +(22.7404 + 67.4049i) q^{82} +(-44.2541 - 44.2541i) q^{83} +(-3.58832 - 4.71278i) q^{84} +(-92.1108 + 52.6581i) q^{85} +(69.1285 + 34.2512i) q^{86} +(2.08002 - 2.08002i) q^{87} +(27.5708 + 18.6468i) q^{88} +36.8085 q^{89} +(89.5856 - 5.32034i) q^{90} +93.2293 q^{91} +(-49.3716 - 6.68783i) q^{92} +(-2.66930 + 2.66930i) q^{93} +(-46.5861 + 94.0237i) q^{94} +(90.1401 + 29.9959i) q^{95} +(-3.82195 - 3.41550i) q^{96} +(3.61538 - 3.61538i) q^{97} +(23.3160 + 69.1110i) q^{98} +37.3382 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89506 + 0.639336i −0.947530 + 0.319668i
\(3\) −0.113263 + 0.113263i −0.0377545 + 0.0377545i −0.725732 − 0.687978i \(-0.758499\pi\)
0.687978 + 0.725732i \(0.258499\pi\)
\(4\) 3.18250 − 2.42316i 0.795624 − 0.605790i
\(5\) −1.31466 + 4.82407i −0.262933 + 0.964814i
\(6\) 0.142227 − 0.287054i 0.0237046 − 0.0478424i
\(7\) −6.53716 + 6.53716i −0.933881 + 0.933881i −0.997946 − 0.0640650i \(-0.979593\pi\)
0.0640650 + 0.997946i \(0.479593\pi\)
\(8\) −4.48181 + 6.62672i −0.560226 + 0.828340i
\(9\) 8.97434i 0.997149i
\(10\) −0.592839 − 9.98241i −0.0592839 − 0.998241i
\(11\) − 4.16055i − 0.378231i −0.981955 − 0.189116i \(-0.939438\pi\)
0.981955 − 0.189116i \(-0.0605621\pi\)
\(12\) −0.0860052 + 0.634916i −0.00716710 + 0.0529097i
\(13\) −7.13071 − 7.13071i −0.548516 − 0.548516i 0.377495 − 0.926012i \(-0.376786\pi\)
−0.926012 + 0.377495i \(0.876786\pi\)
\(14\) 8.20887 − 16.5678i 0.586348 − 1.18341i
\(15\) −0.397487 − 0.695294i −0.0264992 − 0.0463529i
\(16\) 4.25659 − 15.4234i 0.266037 − 0.963963i
\(17\) 15.0048 + 15.0048i 0.882638 + 0.882638i 0.993802 − 0.111164i \(-0.0354578\pi\)
−0.111164 + 0.993802i \(0.535458\pi\)
\(18\) −5.73762 − 17.0069i −0.318757 − 0.944828i
\(19\) 1.04794 − 18.9711i 0.0551549 − 0.998478i
\(20\) 7.50558 + 18.5382i 0.375279 + 0.926912i
\(21\) − 1.48084i − 0.0705163i
\(22\) 2.65999 + 7.88448i 0.120909 + 0.358386i
\(23\) −8.80745 − 8.80745i −0.382933 − 0.382933i 0.489225 − 0.872158i \(-0.337280\pi\)
−0.872158 + 0.489225i \(0.837280\pi\)
\(24\) −0.242940 − 1.25819i −0.0101225 − 0.0524246i
\(25\) −21.5433 − 12.6841i −0.861733 − 0.507362i
\(26\) 18.0721 + 8.95420i 0.695079 + 0.344392i
\(27\) −2.03584 − 2.03584i −0.0754013 − 0.0754013i
\(28\) −4.96391 + 36.6451i −0.177283 + 1.30875i
\(29\) −18.3645 −0.633258 −0.316629 − 0.948549i \(-0.602551\pi\)
−0.316629 + 0.948549i \(0.602551\pi\)
\(30\) 1.19779 + 1.06350i 0.0399263 + 0.0354498i
\(31\) 23.5672 0.760232 0.380116 − 0.924939i \(-0.375884\pi\)
0.380116 + 0.924939i \(0.375884\pi\)
\(32\) 1.79426 + 31.9497i 0.0560707 + 0.998427i
\(33\) 0.471238 + 0.471238i 0.0142799 + 0.0142799i
\(34\) −38.0282 − 18.8419i −1.11848 − 0.554174i
\(35\) −22.9416 − 40.1299i −0.655474 − 1.14657i
\(36\) 21.7463 + 28.5608i 0.604063 + 0.793356i
\(37\) −41.5872 + 41.5872i −1.12398 + 1.12398i −0.132841 + 0.991137i \(0.542410\pi\)
−0.991137 + 0.132841i \(0.957590\pi\)
\(38\) 10.1430 + 36.6213i 0.266921 + 0.963718i
\(39\) 1.61530 0.0414179
\(40\) −26.0757 − 30.3325i −0.651892 − 0.758312i
\(41\) − 35.5687i − 0.867530i −0.901026 − 0.433765i \(-0.857185\pi\)
0.901026 − 0.433765i \(-0.142815\pi\)
\(42\) 0.946757 + 2.80629i 0.0225418 + 0.0668163i
\(43\) −27.2761 − 27.2761i −0.634328 − 0.634328i 0.314822 − 0.949151i \(-0.398055\pi\)
−0.949151 + 0.314822i \(0.898055\pi\)
\(44\) −10.0817 − 13.2409i −0.229129 − 0.300930i
\(45\) −43.2929 − 11.7982i −0.962064 − 0.262183i
\(46\) 22.3216 + 11.0597i 0.485251 + 0.240429i
\(47\) 37.0991 − 37.0991i 0.789342 − 0.789342i −0.192044 − 0.981386i \(-0.561512\pi\)
0.981386 + 0.192044i \(0.0615118\pi\)
\(48\) 1.26479 + 2.22902i 0.0263498 + 0.0464380i
\(49\) − 36.4690i − 0.744266i
\(50\) 48.9352 + 10.2636i 0.978705 + 0.205272i
\(51\) −3.39900 −0.0666471
\(52\) −39.9723 − 5.41462i −0.768699 − 0.104127i
\(53\) −5.21359 − 5.21359i −0.0983696 − 0.0983696i 0.656209 − 0.754579i \(-0.272159\pi\)
−0.754579 + 0.656209i \(0.772159\pi\)
\(54\) 5.15961 + 2.55644i 0.0955484 + 0.0473416i
\(55\) 20.0708 + 5.46972i 0.364923 + 0.0994494i
\(56\) −14.0216 − 72.6183i −0.250386 − 1.29675i
\(57\) 2.03004 + 2.26742i 0.0356147 + 0.0397793i
\(58\) 34.8018 − 11.7411i 0.600031 − 0.202432i
\(59\) − 34.5881i − 0.586239i −0.956076 − 0.293119i \(-0.905307\pi\)
0.956076 − 0.293119i \(-0.0946933\pi\)
\(60\) −2.94981 − 1.24960i −0.0491635 − 0.0208266i
\(61\) −13.4088 −0.219817 −0.109908 − 0.993942i \(-0.535056\pi\)
−0.109908 + 0.993942i \(0.535056\pi\)
\(62\) −44.6612 + 15.0674i −0.720342 + 0.243022i
\(63\) −58.6668 − 58.6668i −0.931218 − 0.931218i
\(64\) −23.8268 − 59.3994i −0.372294 − 0.928115i
\(65\) 43.7736 − 25.0246i 0.673439 − 0.384994i
\(66\) −1.19430 − 0.591744i −0.0180955 − 0.00896582i
\(67\) −31.1774 − 31.1774i −0.465335 − 0.465335i 0.435065 − 0.900399i \(-0.356726\pi\)
−0.900399 + 0.435065i \(0.856726\pi\)
\(68\) 84.1121 + 11.3937i 1.23694 + 0.167555i
\(69\) 1.99512 0.0289148
\(70\) 69.1322 + 61.3812i 0.987602 + 0.876874i
\(71\) −70.1730 −0.988352 −0.494176 − 0.869362i \(-0.664530\pi\)
−0.494176 + 0.869362i \(0.664530\pi\)
\(72\) −59.4704 − 40.2213i −0.825978 − 0.558629i
\(73\) −41.2283 + 41.2283i −0.564771 + 0.564771i −0.930659 − 0.365888i \(-0.880765\pi\)
0.365888 + 0.930659i \(0.380765\pi\)
\(74\) 52.2220 − 105.398i 0.705703 − 1.42430i
\(75\) 3.87671 − 1.00343i 0.0516895 − 0.0133791i
\(76\) −42.6349 − 62.9148i −0.560985 − 0.827826i
\(77\) 27.1982 + 27.1982i 0.353223 + 0.353223i
\(78\) −3.06109 + 1.03272i −0.0392447 + 0.0132400i
\(79\) 49.6890i 0.628975i 0.949262 + 0.314487i \(0.101833\pi\)
−0.949262 + 0.314487i \(0.898167\pi\)
\(80\) 68.8076 + 40.8107i 0.860095 + 0.510133i
\(81\) −80.3079 −0.991456
\(82\) 22.7404 + 67.4049i 0.277322 + 0.822011i
\(83\) −44.2541 − 44.2541i −0.533182 − 0.533182i 0.388336 − 0.921518i \(-0.373050\pi\)
−0.921518 + 0.388336i \(0.873050\pi\)
\(84\) −3.58832 − 4.71278i −0.0427181 − 0.0561045i
\(85\) −92.1108 + 52.6581i −1.08366 + 0.619507i
\(86\) 69.1285 + 34.2512i 0.803819 + 0.398270i
\(87\) 2.08002 − 2.08002i 0.0239083 − 0.0239083i
\(88\) 27.5708 + 18.6468i 0.313304 + 0.211895i
\(89\) 36.8085 0.413578 0.206789 − 0.978386i \(-0.433699\pi\)
0.206789 + 0.978386i \(0.433699\pi\)
\(90\) 89.5856 − 5.32034i 0.995395 − 0.0591149i
\(91\) 93.2293 1.02450
\(92\) −49.3716 − 6.68783i −0.536647 − 0.0726938i
\(93\) −2.66930 + 2.66930i −0.0287021 + 0.0287021i
\(94\) −46.5861 + 94.0237i −0.495597 + 1.00025i
\(95\) 90.1401 + 29.9959i 0.948844 + 0.315747i
\(96\) −3.82195 − 3.41550i −0.0398120 − 0.0355782i
\(97\) 3.61538 − 3.61538i 0.0372719 − 0.0372719i −0.688225 − 0.725497i \(-0.741610\pi\)
0.725497 + 0.688225i \(0.241610\pi\)
\(98\) 23.3160 + 69.1110i 0.237918 + 0.705214i
\(99\) 37.3382 0.377153
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.227.12 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.47 yes 232
5.3 odd 4 inner 380.3.j.a.303.70 yes 232
19.18 odd 2 inner 380.3.j.a.227.105 yes 232
20.3 even 4 inner 380.3.j.a.303.105 yes 232
76.75 even 2 inner 380.3.j.a.227.70 yes 232
95.18 even 4 inner 380.3.j.a.303.47 yes 232
380.303 odd 4 inner 380.3.j.a.303.12 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.12 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.47 yes 232 4.3 odd 2 inner
380.3.j.a.227.70 yes 232 76.75 even 2 inner
380.3.j.a.227.105 yes 232 19.18 odd 2 inner
380.3.j.a.303.12 yes 232 380.303 odd 4 inner
380.3.j.a.303.47 yes 232 95.18 even 4 inner
380.3.j.a.303.70 yes 232 5.3 odd 4 inner
380.3.j.a.303.105 yes 232 20.3 even 4 inner