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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] 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("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), 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: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.101
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.102

$q$-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+(1.94008 - 0.485899i) q^{2} -4.64902 q^{3} +(3.52780 - 1.88537i) q^{4} +(2.66799 + 4.22869i) q^{5} +(-9.01947 + 2.25896i) q^{6} -9.90681 q^{7} +(5.92811 - 5.37191i) q^{8} +12.6134 q^{9} +(7.23082 + 6.90762i) q^{10} -2.36298i q^{11} +(-16.4008 + 8.76511i) q^{12} -16.5387i q^{13} +(-19.2200 + 4.81372i) q^{14} +(-12.4035 - 19.6593i) q^{15} +(8.89079 - 13.3024i) q^{16} -21.4996i q^{17} +(24.4710 - 6.12886i) q^{18} -4.35890i q^{19} +(17.3848 + 9.88787i) q^{20} +46.0570 q^{21} +(-1.14817 - 4.58437i) q^{22} -6.86595 q^{23} +(-27.5600 + 24.9742i) q^{24} +(-10.7637 + 22.5642i) q^{25} +(-8.03617 - 32.0865i) q^{26} -16.7989 q^{27} +(-34.9493 + 18.6780i) q^{28} -36.1705 q^{29} +(-33.6163 - 32.1137i) q^{30} -46.4134i q^{31} +(10.7852 - 30.1277i) q^{32} +10.9856i q^{33} +(-10.4467 - 41.7109i) q^{34} +(-26.4312 - 41.8929i) q^{35} +(44.4977 - 23.7809i) q^{36} -17.7479i q^{37} +(-2.11799 - 8.45660i) q^{38} +76.8890i q^{39} +(38.5323 + 10.7360i) q^{40} -54.6944 q^{41} +(89.3542 - 22.3791i) q^{42} -4.27621 q^{43} +(-4.45508 - 8.33613i) q^{44} +(33.6524 + 53.3383i) q^{45} +(-13.3205 + 3.33616i) q^{46} +40.9908 q^{47} +(-41.3335 + 61.8432i) q^{48} +49.1449 q^{49} +(-9.91850 + 49.0064i) q^{50} +99.9523i q^{51} +(-31.1816 - 58.3455i) q^{52} +10.7606i q^{53} +(-32.5912 + 8.16258i) q^{54} +(9.99232 - 6.30440i) q^{55} +(-58.7287 + 53.2185i) q^{56} +20.2646i q^{57} +(-70.1736 + 17.5752i) q^{58} -70.4443i q^{59} +(-80.8222 - 45.9690i) q^{60} +101.474 q^{61} +(-22.5522 - 90.0456i) q^{62} -124.959 q^{63} +(6.28509 - 63.6906i) q^{64} +(69.9373 - 44.1251i) q^{65} +(5.33787 + 21.3128i) q^{66} -7.29760 q^{67} +(-40.5346 - 75.8464i) q^{68} +31.9200 q^{69} +(-71.6344 - 68.4325i) q^{70} +31.8531i q^{71} +(74.7738 - 67.7582i) q^{72} +83.7571i q^{73} +(-8.62369 - 34.4323i) q^{74} +(50.0407 - 104.901i) q^{75} +(-8.21812 - 15.3773i) q^{76} +23.4096i q^{77} +(37.3603 + 149.171i) q^{78} +137.909i q^{79} +(79.9723 + 2.10584i) q^{80} -35.4223 q^{81} +(-106.111 + 26.5760i) q^{82} -99.8691 q^{83} +(162.480 - 86.8343i) q^{84} +(90.9153 - 57.3607i) q^{85} +(-8.29618 + 2.07781i) q^{86} +168.158 q^{87} +(-12.6937 - 14.0080i) q^{88} +76.9267 q^{89} +(91.2054 + 87.1288i) q^{90} +163.846i q^{91} +(-24.2217 + 12.9448i) q^{92} +215.777i q^{93} +(79.5254 - 19.9174i) q^{94} +(18.4325 - 11.6295i) q^{95} +(-50.1407 + 140.064i) q^{96} -155.574i q^{97} +(95.3450 - 23.8795i) q^{98} -29.8053i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.94008 − 0.485899i 0.970039 − 0.242950i
\(3\) −4.64902 −1.54967 −0.774837 − 0.632161i \(-0.782168\pi\)
−0.774837 + 0.632161i \(0.782168\pi\)
\(4\) 3.52780 − 1.88537i 0.881951 − 0.471341i
\(5\) 2.66799 + 4.22869i 0.533597 + 0.845739i
\(6\) −9.01947 + 2.25896i −1.50324 + 0.376493i
\(7\) −9.90681 −1.41526 −0.707630 − 0.706584i \(-0.750235\pi\)
−0.707630 + 0.706584i \(0.750235\pi\)
\(8\) 5.92811 − 5.37191i 0.741014 − 0.671489i
\(9\) 12.6134 1.40149
\(10\) 7.23082 + 6.90762i 0.723082 + 0.690762i
\(11\) − 2.36298i − 0.214816i −0.994215 − 0.107408i \(-0.965745\pi\)
0.994215 − 0.107408i \(-0.0342552\pi\)
\(12\) −16.4008 + 8.76511i −1.36674 + 0.730426i
\(13\) − 16.5387i − 1.27221i −0.771602 − 0.636106i \(-0.780544\pi\)
0.771602 − 0.636106i \(-0.219456\pi\)
\(14\) −19.2200 + 4.81372i −1.37286 + 0.343837i
\(15\) −12.4035 − 19.6593i −0.826902 − 1.31062i
\(16\) 8.89079 − 13.3024i 0.555675 − 0.831400i
\(17\) − 21.4996i − 1.26468i −0.774689 − 0.632342i \(-0.782094\pi\)
0.774689 − 0.632342i \(-0.217906\pi\)
\(18\) 24.4710 − 6.12886i 1.35950 − 0.340492i
\(19\) − 4.35890i − 0.229416i
\(20\) 17.3848 + 9.88787i 0.869238 + 0.494394i
\(21\) 46.0570 2.19319
\(22\) −1.14817 − 4.58437i −0.0521896 − 0.208380i
\(23\) −6.86595 −0.298520 −0.149260 − 0.988798i \(-0.547689\pi\)
−0.149260 + 0.988798i \(0.547689\pi\)
\(24\) −27.5600 + 24.9742i −1.14833 + 1.04059i
\(25\) −10.7637 + 22.5642i −0.430548 + 0.902568i
\(26\) −8.03617 − 32.0865i −0.309083 − 1.23409i
\(27\) −16.7989 −0.622182
\(28\) −34.9493 + 18.6780i −1.24819 + 0.667070i
\(29\) −36.1705 −1.24726 −0.623629 − 0.781720i \(-0.714342\pi\)
−0.623629 + 0.781720i \(0.714342\pi\)
\(30\) −33.6163 − 32.1137i −1.12054 − 1.07046i
\(31\) − 46.4134i − 1.49721i −0.663018 − 0.748603i \(-0.730725\pi\)
0.663018 − 0.748603i \(-0.269275\pi\)
\(32\) 10.7852 − 30.1277i 0.337038 − 0.941491i
\(33\) 10.9856i 0.332896i
\(34\) −10.4467 − 41.7109i −0.307255 − 1.22679i
\(35\) −26.4312 − 41.8929i −0.755178 − 1.19694i
\(36\) 44.4977 − 23.7809i 1.23605 − 0.660581i
\(37\) − 17.7479i − 0.479673i −0.970813 − 0.239836i \(-0.922906\pi\)
0.970813 − 0.239836i \(-0.0770938\pi\)
\(38\) −2.11799 − 8.45660i −0.0557365 − 0.222542i
\(39\) 76.8890i 1.97151i
\(40\) 38.5323 + 10.7360i 0.963308 + 0.268400i
\(41\) −54.6944 −1.33401 −0.667005 − 0.745053i \(-0.732424\pi\)
−0.667005 + 0.745053i \(0.732424\pi\)
\(42\) 89.3542 − 22.3791i 2.12748 − 0.532835i
\(43\) −4.27621 −0.0994467 −0.0497234 − 0.998763i \(-0.515834\pi\)
−0.0497234 + 0.998763i \(0.515834\pi\)
\(44\) −4.45508 − 8.33613i −0.101252 − 0.189457i
\(45\) 33.6524 + 53.3383i 0.747832 + 1.18530i
\(46\) −13.3205 + 3.33616i −0.289576 + 0.0725252i
\(47\) 40.9908 0.872145 0.436073 − 0.899911i \(-0.356369\pi\)
0.436073 + 0.899911i \(0.356369\pi\)
\(48\) −41.3335 + 61.8432i −0.861115 + 1.28840i
\(49\) 49.1449 1.00296
\(50\) −9.91850 + 49.0064i −0.198370 + 0.980127i
\(51\) 99.9523i 1.95985i
\(52\) −31.1816 − 58.3455i −0.599646 − 1.12203i
\(53\) 10.7606i 0.203031i 0.994834 + 0.101515i \(0.0323691\pi\)
−0.994834 + 0.101515i \(0.967631\pi\)
\(54\) −32.5912 + 8.16258i −0.603541 + 0.151159i
\(55\) 9.99232 − 6.30440i 0.181679 − 0.114625i
\(56\) −58.7287 + 53.2185i −1.04873 + 0.950331i
\(57\) 20.2646i 0.355520i
\(58\) −70.1736 + 17.5752i −1.20989 + 0.303021i
\(59\) − 70.4443i − 1.19397i −0.802252 − 0.596985i \(-0.796365\pi\)
0.802252 − 0.596985i \(-0.203635\pi\)
\(60\) −80.8222 − 45.9690i −1.34704 − 0.766149i
\(61\) 101.474 1.66351 0.831755 − 0.555142i \(-0.187336\pi\)
0.831755 + 0.555142i \(0.187336\pi\)
\(62\) −22.5522 − 90.0456i −0.363746 − 1.45235i
\(63\) −124.959 −1.98347
\(64\) 6.28509 − 63.6906i 0.0982045 − 0.995166i
\(65\) 69.9373 − 44.1251i 1.07596 − 0.678848i
\(66\) 5.33787 + 21.3128i 0.0808769 + 0.322922i
\(67\) −7.29760 −0.108919 −0.0544597 − 0.998516i \(-0.517344\pi\)
−0.0544597 + 0.998516i \(0.517344\pi\)
\(68\) −40.5346 − 75.8464i −0.596098 − 1.11539i
\(69\) 31.9200 0.462608
\(70\) −71.6344 − 68.4325i −1.02335 − 0.977608i
\(71\) 31.8531i 0.448636i 0.974516 + 0.224318i \(0.0720154\pi\)
−0.974516 + 0.224318i \(0.927985\pi\)
\(72\) 74.7738 − 67.7582i 1.03853 − 0.941087i
\(73\) 83.7571i 1.14736i 0.819080 + 0.573679i \(0.194484\pi\)
−0.819080 + 0.573679i \(0.805516\pi\)
\(74\) −8.62369 − 34.4323i −0.116536 − 0.465301i
\(75\) 50.0407 − 104.901i 0.667210 − 1.39869i
\(76\) −8.21812 − 15.3773i −0.108133 − 0.202333i
\(77\) 23.4096i 0.304021i
\(78\) 37.3603 + 149.171i 0.478979 + 1.91245i
\(79\) 137.909i 1.74568i 0.488002 + 0.872842i \(0.337726\pi\)
−0.488002 + 0.872842i \(0.662274\pi\)
\(80\) 79.9723 + 2.10584i 0.999653 + 0.0263230i
\(81\) −35.4223 −0.437312
\(82\) −106.111 + 26.5760i −1.29404 + 0.324097i
\(83\) −99.8691 −1.20324 −0.601621 − 0.798782i \(-0.705478\pi\)
−0.601621 + 0.798782i \(0.705478\pi\)
\(84\) 162.480 − 86.8343i 1.93429 − 1.03374i
\(85\) 90.9153 − 57.3607i 1.06959 − 0.674832i
\(86\) −8.29618 + 2.07781i −0.0964672 + 0.0241606i
\(87\) 168.158 1.93285
\(88\) −12.6937 − 14.0080i −0.144247 − 0.159182i
\(89\) 76.9267 0.864345 0.432173 − 0.901791i \(-0.357747\pi\)
0.432173 + 0.901791i \(0.357747\pi\)
\(90\) 91.2054 + 87.1288i 1.01339 + 0.968098i
\(91\) 163.846i 1.80051i
\(92\) −24.2217 + 12.9448i −0.263280 + 0.140705i
\(93\) 215.777i 2.32018i
\(94\) 79.5254 − 19.9174i 0.846015 − 0.211887i
\(95\) 18.4325 − 11.6295i 0.194026 − 0.122416i
\(96\) −50.1407 + 140.064i −0.522299 + 1.45901i
\(97\) − 155.574i − 1.60386i −0.597421 − 0.801928i \(-0.703808\pi\)
0.597421 − 0.801928i \(-0.296192\pi\)
\(98\) 95.3450 − 23.8795i 0.972908 − 0.243668i
\(99\) − 29.8053i − 0.301063i
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.h.a.39.101 yes 108
4.3 odd 2 inner 380.3.h.a.39.7 ✓ 108
5.4 even 2 inner 380.3.h.a.39.8 yes 108
20.19 odd 2 inner 380.3.h.a.39.102 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.7 ✓ 108 4.3 odd 2 inner
380.3.h.a.39.8 yes 108 5.4 even 2 inner
380.3.h.a.39.101 yes 108 1.1 even 1 trivial
380.3.h.a.39.102 yes 108 20.19 odd 2 inner