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Properties

Label 380.3.j.a.227.9
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 227.9
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.9

$q$-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+(-1.92854 - 0.529843i) q^{2} +(-1.41485 + 1.41485i) q^{3} +(3.43853 + 2.04365i) q^{4} +(-4.97296 + 0.519307i) q^{5} +(3.47824 - 1.97894i) q^{6} +(-9.66238 + 9.66238i) q^{7} +(-5.54854 - 5.76314i) q^{8} +4.99641i q^{9} +(9.86570 + 1.63338i) q^{10} +4.65546i q^{11} +(-7.75645 + 1.97355i) q^{12} +(-5.48728 - 5.48728i) q^{13} +(23.7538 - 13.5147i) q^{14} +(6.30124 - 7.77072i) q^{15} +(7.64701 + 14.0543i) q^{16} +(-13.1030 - 13.1030i) q^{17} +(2.64731 - 9.63578i) q^{18} +(-3.28612 + 18.7137i) q^{19} +(-18.1610 - 8.37732i) q^{20} -27.3416i q^{21} +(2.46666 - 8.97823i) q^{22} +(18.0148 + 18.0148i) q^{23} +(16.0043 + 0.303629i) q^{24} +(24.4606 - 5.16499i) q^{25} +(7.67505 + 13.4898i) q^{26} +(-19.8028 - 19.8028i) q^{27} +(-52.9709 + 13.4779i) q^{28} +45.8365 q^{29} +(-16.2695 + 11.6475i) q^{30} -47.5334 q^{31} +(-7.30100 - 31.1560i) q^{32} +(-6.58676 - 6.58676i) q^{33} +(18.3271 + 32.2121i) q^{34} +(43.0329 - 53.0683i) q^{35} +(-10.2109 + 17.1803i) q^{36} +(1.52848 - 1.52848i) q^{37} +(16.2527 - 34.3489i) q^{38} +15.5273 q^{39} +(30.5855 + 25.7784i) q^{40} +24.4625i q^{41} +(-14.4867 + 52.7293i) q^{42} +(23.4069 + 23.4069i) q^{43} +(-9.51411 + 16.0079i) q^{44} +(-2.59467 - 24.8470i) q^{45} +(-25.1972 - 44.2872i) q^{46} +(19.4081 - 19.4081i) q^{47} +(-30.7040 - 9.06532i) q^{48} -137.723i q^{49} +(-49.9100 - 2.99941i) q^{50} +37.0774 q^{51} +(-7.65414 - 30.0823i) q^{52} +(-53.2243 - 53.2243i) q^{53} +(27.6981 + 48.6828i) q^{54} +(-2.41761 - 23.1514i) q^{55} +(109.298 + 2.07356i) q^{56} +(-21.8276 - 31.1264i) q^{57} +(-88.3975 - 24.2861i) q^{58} -65.8450i q^{59} +(37.5476 - 13.8424i) q^{60} -39.0543 q^{61} +(91.6701 + 25.1853i) q^{62} +(-48.2772 - 48.2772i) q^{63} +(-2.42750 + 63.9539i) q^{64} +(30.1376 + 24.4385i) q^{65} +(9.21288 + 16.1928i) q^{66} +(34.3948 + 34.3948i) q^{67} +(-18.2771 - 71.8327i) q^{68} -50.9763 q^{69} +(-111.108 + 79.5438i) q^{70} -0.159283 q^{71} +(28.7950 - 27.7228i) q^{72} +(-22.7826 + 22.7826i) q^{73} +(-3.75758 + 2.13788i) q^{74} +(-27.3004 + 41.9158i) q^{75} +(-49.5436 + 57.6319i) q^{76} +(-44.9828 - 44.9828i) q^{77} +(-29.9451 - 8.22705i) q^{78} -83.5604i q^{79} +(-45.3268 - 65.9203i) q^{80} +11.0682 q^{81} +(12.9613 - 47.1769i) q^{82} +(59.6104 + 59.6104i) q^{83} +(55.8765 - 94.0149i) q^{84} +(71.9649 + 58.3560i) q^{85} +(-32.7391 - 57.5431i) q^{86} +(-64.8517 + 64.8517i) q^{87} +(26.8300 - 25.8310i) q^{88} +22.7873 q^{89} +(-8.16105 + 49.2931i) q^{90} +106.040 q^{91} +(25.1285 + 98.7601i) q^{92} +(67.2526 - 67.2526i) q^{93} +(-47.7125 + 27.1460i) q^{94} +(6.62360 - 94.7688i) q^{95} +(54.4108 + 33.7512i) q^{96} +(-50.2694 + 50.2694i) q^{97} +(-72.9716 + 265.604i) q^{98} -23.2606 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92854 − 0.529843i −0.964270 − 0.264922i
\(3\) −1.41485 + 1.41485i −0.471616 + 0.471616i −0.902437 − 0.430821i \(-0.858224\pi\)
0.430821 + 0.902437i \(0.358224\pi\)
\(4\) 3.43853 + 2.04365i 0.859633 + 0.510912i
\(5\) −4.97296 + 0.519307i −0.994592 + 0.103861i
\(6\) 3.47824 − 1.97894i 0.579706 − 0.329824i
\(7\) −9.66238 + 9.66238i −1.38034 + 1.38034i −0.536333 + 0.844006i \(0.680191\pi\)
−0.844006 + 0.536333i \(0.819809\pi\)
\(8\) −5.54854 − 5.76314i −0.693567 − 0.720392i
\(9\) 4.99641i 0.555157i
\(10\) 9.86570 + 1.63338i 0.986570 + 0.163338i
\(11\) 4.65546i 0.423223i 0.977354 + 0.211612i \(0.0678712\pi\)
−0.977354 + 0.211612i \(0.932129\pi\)
\(12\) −7.75645 + 1.97355i −0.646371 + 0.164463i
\(13\) −5.48728 − 5.48728i −0.422099 − 0.422099i 0.463827 − 0.885926i \(-0.346476\pi\)
−0.885926 + 0.463827i \(0.846476\pi\)
\(14\) 23.7538 − 13.5147i 1.69670 − 0.965338i
\(15\) 6.30124 − 7.77072i 0.420083 − 0.518048i
\(16\) 7.64701 + 14.0543i 0.477938 + 0.878393i
\(17\) −13.1030 − 13.1030i −0.770762 − 0.770762i 0.207478 − 0.978240i \(-0.433475\pi\)
−0.978240 + 0.207478i \(0.933475\pi\)
\(18\) 2.64731 − 9.63578i 0.147073 − 0.535321i
\(19\) −3.28612 + 18.7137i −0.172954 + 0.984930i
\(20\) −18.1610 − 8.37732i −0.908048 − 0.418866i
\(21\) − 27.3416i − 1.30198i
\(22\) 2.46666 − 8.97823i 0.112121 − 0.408101i
\(23\) 18.0148 + 18.0148i 0.783250 + 0.783250i 0.980378 − 0.197128i \(-0.0631614\pi\)
−0.197128 + 0.980378i \(0.563161\pi\)
\(24\) 16.0043 + 0.303629i 0.666846 + 0.0126512i
\(25\) 24.4606 − 5.16499i 0.978426 − 0.206600i
\(26\) 7.67505 + 13.4898i 0.295194 + 0.518840i
\(27\) −19.8028 − 19.8028i −0.733437 − 0.733437i
\(28\) −52.9709 + 13.4779i −1.89182 + 0.481354i
\(29\) 45.8365 1.58057 0.790284 − 0.612740i \(-0.209933\pi\)
0.790284 + 0.612740i \(0.209933\pi\)
\(30\) −16.2695 + 11.6475i −0.542315 + 0.388249i
\(31\) −47.5334 −1.53334 −0.766668 − 0.642043i \(-0.778087\pi\)
−0.766668 + 0.642043i \(0.778087\pi\)
\(32\) −7.30100 − 31.1560i −0.228156 − 0.973625i
\(33\) −6.58676 − 6.58676i −0.199599 − 0.199599i
\(34\) 18.3271 + 32.2121i 0.539031 + 0.947414i
\(35\) 43.0329 − 53.0683i 1.22951 − 1.51624i
\(36\) −10.2109 + 17.1803i −0.283636 + 0.477231i
\(37\) 1.52848 − 1.52848i 0.0413102 − 0.0413102i −0.686150 − 0.727460i \(-0.740701\pi\)
0.727460 + 0.686150i \(0.240701\pi\)
\(38\) 16.2527 − 34.3489i 0.427703 − 0.903919i
\(39\) 15.5273 0.398137
\(40\) 30.5855 + 25.7784i 0.764637 + 0.644461i
\(41\) 24.4625i 0.596646i 0.954465 + 0.298323i \(0.0964273\pi\)
−0.954465 + 0.298323i \(0.903573\pi\)
\(42\) −14.4867 + 52.7293i −0.344923 + 1.25546i
\(43\) 23.4069 + 23.4069i 0.544346 + 0.544346i 0.924800 − 0.380454i \(-0.124232\pi\)
−0.380454 + 0.924800i \(0.624232\pi\)
\(44\) −9.51411 + 16.0079i −0.216230 + 0.363817i
\(45\) −2.59467 − 24.8470i −0.0576594 − 0.552154i
\(46\) −25.1972 − 44.2872i −0.547765 − 0.962764i
\(47\) 19.4081 − 19.4081i 0.412938 − 0.412938i −0.469823 − 0.882761i \(-0.655682\pi\)
0.882761 + 0.469823i \(0.155682\pi\)
\(48\) −30.7040 − 9.06532i −0.639668 − 0.188861i
\(49\) − 137.723i − 2.81067i
\(50\) −49.9100 − 2.99941i −0.998199 − 0.0599883i
\(51\) 37.0774 0.727007
\(52\) −7.65414 − 30.0823i −0.147195 − 0.578505i
\(53\) −53.2243 − 53.2243i −1.00423 − 1.00423i −0.999991 − 0.00424187i \(-0.998650\pi\)
−0.00424187 − 0.999991i \(-0.501350\pi\)
\(54\) 27.6981 + 48.6828i 0.512928 + 0.901534i
\(55\) −2.41761 − 23.1514i −0.0439566 − 0.420934i
\(56\) 109.298 + 2.07356i 1.95174 + 0.0370279i
\(57\) −21.8276 − 31.1264i −0.382941 − 0.546076i
\(58\) −88.3975 − 24.2861i −1.52410 − 0.418727i
\(59\) − 65.8450i − 1.11602i −0.829835 − 0.558008i \(-0.811566\pi\)
0.829835 − 0.558008i \(-0.188434\pi\)
\(60\) 37.5476 − 13.8424i 0.625794 − 0.230706i
\(61\) −39.0543 −0.640234 −0.320117 − 0.947378i \(-0.603722\pi\)
−0.320117 + 0.947378i \(0.603722\pi\)
\(62\) 91.6701 + 25.1853i 1.47855 + 0.406214i
\(63\) −48.2772 − 48.2772i −0.766305 − 0.766305i
\(64\) −2.42750 + 63.9539i −0.0379297 + 0.999280i
\(65\) 30.1376 + 24.4385i 0.463656 + 0.375976i
\(66\) 9.21288 + 16.1928i 0.139589 + 0.245345i
\(67\) 34.3948 + 34.3948i 0.513355 + 0.513355i 0.915553 − 0.402198i \(-0.131754\pi\)
−0.402198 + 0.915553i \(0.631754\pi\)
\(68\) −18.2771 − 71.8327i −0.268781 − 1.05636i
\(69\) −50.9763 −0.738786
\(70\) −111.108 + 79.5438i −1.58726 + 1.13634i
\(71\) −0.159283 −0.00224342 −0.00112171 − 0.999999i \(-0.500357\pi\)
−0.00112171 + 0.999999i \(0.500357\pi\)
\(72\) 28.7950 − 27.7228i 0.399931 − 0.385038i
\(73\) −22.7826 + 22.7826i −0.312091 + 0.312091i −0.845719 − 0.533628i \(-0.820828\pi\)
0.533628 + 0.845719i \(0.320828\pi\)
\(74\) −3.75758 + 2.13788i −0.0507782 + 0.0288902i
\(75\) −27.3004 + 41.9158i −0.364005 + 0.558877i
\(76\) −49.5436 + 57.6319i −0.651889 + 0.758314i
\(77\) −44.9828 − 44.9828i −0.584192 − 0.584192i
\(78\) −29.9451 − 8.22705i −0.383912 − 0.105475i
\(79\) − 83.5604i − 1.05773i −0.848707 − 0.528863i \(-0.822618\pi\)
0.848707 − 0.528863i \(-0.177382\pi\)
\(80\) −45.3268 − 65.9203i −0.566585 − 0.824003i
\(81\) 11.0682 0.136644
\(82\) 12.9613 − 47.1769i 0.158064 − 0.575328i
\(83\) 59.6104 + 59.6104i 0.718197 + 0.718197i 0.968236 − 0.250039i \(-0.0804433\pi\)
−0.250039 + 0.968236i \(0.580443\pi\)
\(84\) 55.8765 − 94.0149i 0.665197 − 1.11923i
\(85\) 71.9649 + 58.3560i 0.846646 + 0.686541i
\(86\) −32.7391 − 57.5431i −0.380687 − 0.669105i
\(87\) −64.8517 + 64.8517i −0.745421 + 0.745421i
\(88\) 26.8300 − 25.8310i 0.304887 − 0.293534i
\(89\) 22.7873 0.256037 0.128018 − 0.991772i \(-0.459138\pi\)
0.128018 + 0.991772i \(0.459138\pi\)
\(90\) −8.16105 + 49.2931i −0.0906784 + 0.547701i
\(91\) 106.040 1.16528
\(92\) 25.1285 + 98.7601i 0.273136 + 1.07348i
\(93\) 67.2526 − 67.2526i 0.723146 − 0.723146i
\(94\) −47.7125 + 27.1460i −0.507579 + 0.288787i
\(95\) 6.62360 − 94.7688i 0.0697221 − 0.997566i
\(96\) 54.4108 + 33.7512i 0.566779 + 0.351575i
\(97\) −50.2694 + 50.2694i −0.518242 + 0.518242i −0.917039 − 0.398797i \(-0.869428\pi\)
0.398797 + 0.917039i \(0.369428\pi\)
\(98\) −72.9716 + 265.604i −0.744608 + 2.71025i
\(99\) −23.2606 −0.234955
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.227.9 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.67 yes 232
5.3 odd 4 inner 380.3.j.a.303.50 yes 232
19.18 odd 2 inner 380.3.j.a.227.108 yes 232
20.3 even 4 inner 380.3.j.a.303.108 yes 232
76.75 even 2 inner 380.3.j.a.227.50 yes 232
95.18 even 4 inner 380.3.j.a.303.67 yes 232
380.303 odd 4 inner 380.3.j.a.303.9 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.9 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.50 yes 232 76.75 even 2 inner
380.3.j.a.227.67 yes 232 4.3 odd 2 inner
380.3.j.a.227.108 yes 232 19.18 odd 2 inner
380.3.j.a.303.9 yes 232 380.303 odd 4 inner
380.3.j.a.303.50 yes 232 5.3 odd 4 inner
380.3.j.a.303.67 yes 232 95.18 even 4 inner
380.3.j.a.303.108 yes 232 20.3 even 4 inner