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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 39.7
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.8

$q$-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} +(-3.12138 - 9.50037i) 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} +(1.43949 + 19.9481i) 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} +(-14.5114 - 44.1674i) 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} +(6.90005 - 39.4004i) 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} +(31.8464 - 38.5462i) 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} +(6.69224 + 92.7393i) 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} +(-30.9229 - 94.1184i) 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} +(-32.5313 + 73.0871i) 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} +(-39.3713 - 119.832i) 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 \(-0.217832\pi\)
0.774837 + 0.632161i \(0.217832\pi\)
\(4\) 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 \(-0.249765\pi\)
0.707630 + 0.706584i \(0.249765\pi\)
\(8\) −5.92811 − 5.37191i −0.741014 − 0.671489i
\(9\) 12.6134 1.40149
\(10\) −3.12138 − 9.50037i −0.312138 − 0.950037i
\(11\) 2.36298i 0.214816i 0.994215 + 0.107408i \(0.0342552\pi\)
−0.994215 + 0.107408i \(0.965745\pi\)
\(12\) 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
\(20\) 1.43949 + 19.9481i 0.0719747 + 0.997406i
\(21\) 46.0570 2.19319
\(22\) 1.14817 − 4.58437i 0.0521896 − 0.208380i
\(23\) 6.86595 0.298520 0.149260 − 0.988798i \(-0.452311\pi\)
0.149260 + 0.988798i \(0.452311\pi\)
\(24\) −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 \(0.714342\pi\)
\(30\) −14.5114 − 44.1674i −0.483712 − 1.47225i
\(31\) 46.4134i 1.49721i 0.663018 + 0.748603i \(0.269275\pi\)
−0.663018 + 0.748603i \(0.730725\pi\)
\(32\) −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
\(40\) 6.90005 − 39.4004i 0.172501 − 0.985009i
\(41\) −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 \(-0.484166\pi\)
0.0497234 + 0.998763i \(0.484166\pi\)
\(44\) −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 \(-0.643631\pi\)
−0.436073 + 0.899911i \(0.643631\pi\)
\(48\) 41.3335 + 61.8432i 0.861115 + 1.28840i
\(49\) 49.1449 1.00296
\(50\) 31.8464 − 38.5462i 0.636927 − 0.770924i
\(51\) − 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 \(0.203635\pi\)
−0.802252 + 0.596985i \(0.796365\pi\)
\(60\) 6.69224 + 92.7393i 0.111537 + 1.54566i
\(61\) 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 \(-0.482656\pi\)
0.0544597 + 0.998516i \(0.482656\pi\)
\(68\) 40.5346 − 75.8464i 0.596098 − 1.11539i
\(69\) 31.9200 0.462608
\(70\) −30.9229 − 94.1184i −0.441756 − 1.34455i
\(71\) − 31.8531i − 0.448636i −0.974516 − 0.224318i \(-0.927985\pi\)
0.974516 − 0.224318i \(-0.0720154\pi\)
\(72\) −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 \(-0.662274\pi\)
0.488002 − 0.872842i \(-0.337726\pi\)
\(80\) −32.5313 + 73.0871i −0.406641 + 0.913588i
\(81\) −35.4223 −0.437312
\(82\) 106.111 + 26.5760i 1.29404 + 0.324097i
\(83\) 99.8691 1.20324 0.601621 − 0.798782i \(-0.294522\pi\)
0.601621 + 0.798782i \(0.294522\pi\)
\(84\) 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 \(0.357747\pi\)
\(90\) −39.3713 − 119.832i −0.437459 − 1.33147i
\(91\) − 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.39.7 ✓ 108
4.3 odd 2 inner 380.3.h.a.39.101 yes 108
5.4 even 2 inner 380.3.h.a.39.102 yes 108
20.19 odd 2 inner 380.3.h.a.39.8 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.7 ✓ 108 1.1 even 1 trivial
380.3.h.a.39.8 yes 108 20.19 odd 2 inner
380.3.h.a.39.101 yes 108 4.3 odd 2 inner
380.3.h.a.39.102 yes 108 5.4 even 2 inner