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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 33.4
Character \(\chi\) \(=\) 380.33
Dual form 380.2.bh.a.357.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.886233 - 1.26567i) q^{3} +(-2.22818 - 0.187668i) q^{5} +(0.477272 + 0.127885i) q^{7} +(0.209544 - 0.575718i) q^{9} +(-1.30958 + 2.26826i) q^{11} +(-3.24748 - 2.27391i) q^{13} +(1.73716 + 2.98646i) q^{15} +(-3.14733 + 6.74948i) q^{17} +(-3.99559 - 1.74219i) q^{19} +(-0.261114 - 0.717405i) q^{21} +(-6.95648 - 0.608613i) q^{23} +(4.92956 + 0.836317i) q^{25} +(-5.39173 + 1.44471i) q^{27} +(7.14293 + 2.59981i) q^{29} +(-4.18291 + 2.41501i) q^{31} +(4.03147 - 0.352708i) q^{33} +(-1.03945 - 0.374519i) q^{35} +(-2.46586 - 2.46586i) q^{37} +6.12545i q^{39} +(7.64737 + 1.34844i) q^{41} +(-0.596842 - 6.82194i) q^{43} +(-0.574946 + 1.24348i) q^{45} +(3.78481 - 1.76489i) q^{47} +(-5.85074 - 3.37793i) q^{49} +(11.3319 - 1.99812i) q^{51} +(1.10972 - 12.6842i) q^{53} +(3.34367 - 4.80833i) q^{55} +(1.33598 + 6.60110i) q^{57} +(-9.64613 + 3.51090i) q^{59} +(4.23307 - 3.55197i) q^{61} +(0.173635 - 0.247977i) q^{63} +(6.80922 + 5.67612i) q^{65} +(-2.96957 - 6.36826i) q^{67} +(5.39476 + 9.34399i) q^{69} +(2.25576 - 2.68831i) q^{71} +(-7.90574 + 5.53566i) q^{73} +(-3.31024 - 6.98038i) q^{75} +(-0.915103 + 0.915103i) q^{77} +(2.00528 - 11.3725i) q^{79} +(5.19887 + 4.36237i) q^{81} +(-0.901564 + 3.36468i) q^{83} +(8.27949 - 14.4484i) q^{85} +(-3.03979 - 11.3446i) q^{87} +(1.27414 + 7.22603i) q^{89} +(-1.25913 - 1.50057i) q^{91} +(6.76364 + 3.15394i) q^{93} +(8.57594 + 4.63176i) q^{95} +(1.27464 + 0.594373i) q^{97} +(1.03146 + 1.22925i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.886233 − 1.26567i −0.511667 − 0.730736i 0.477322 − 0.878729i \(-0.341608\pi\)
−0.988988 + 0.147993i \(0.952719\pi\)
\(4\) 0 0
\(5\) −2.22818 − 0.187668i −0.996472 − 0.0839278i
\(6\) 0 0
\(7\) 0.477272 + 0.127885i 0.180392 + 0.0483359i 0.347884 − 0.937538i \(-0.386900\pi\)
−0.167492 + 0.985873i \(0.553567\pi\)
\(8\) 0 0
\(9\) 0.209544 − 0.575718i 0.0698481 − 0.191906i
\(10\) 0 0
\(11\) −1.30958 + 2.26826i −0.394854 + 0.683907i −0.993083 − 0.117418i \(-0.962538\pi\)
0.598229 + 0.801326i \(0.295871\pi\)
\(12\) 0 0
\(13\) −3.24748 − 2.27391i −0.900688 − 0.630668i 0.0287743 − 0.999586i \(-0.490840\pi\)
−0.929462 + 0.368918i \(0.879728\pi\)
\(14\) 0 0
\(15\) 1.73716 + 2.98646i 0.448532 + 0.771101i
\(16\) 0 0
\(17\) −3.14733 + 6.74948i −0.763341 + 1.63699i 0.00750861 + 0.999972i \(0.497610\pi\)
−0.770849 + 0.637018i \(0.780168\pi\)
\(18\) 0 0
\(19\) −3.99559 − 1.74219i −0.916652 − 0.399686i
\(20\) 0 0
\(21\) −0.261114 − 0.717405i −0.0569798 − 0.156551i
\(22\) 0 0
\(23\) −6.95648 − 0.608613i −1.45053 − 0.126905i −0.665514 − 0.746385i \(-0.731788\pi\)
−0.785012 + 0.619480i \(0.787343\pi\)
\(24\) 0 0
\(25\) 4.92956 + 0.836317i 0.985912 + 0.167263i
\(26\) 0 0
\(27\) −5.39173 + 1.44471i −1.03764 + 0.278035i
\(28\) 0 0
\(29\) 7.14293 + 2.59981i 1.32641 + 0.482773i 0.905506 − 0.424333i \(-0.139491\pi\)
0.420902 + 0.907106i \(0.361714\pi\)
\(30\) 0 0
\(31\) −4.18291 + 2.41501i −0.751274 + 0.433748i −0.826154 − 0.563445i \(-0.809476\pi\)
0.0748803 + 0.997193i \(0.476143\pi\)
\(32\) 0 0
\(33\) 4.03147 − 0.352708i 0.701789 − 0.0613986i
\(34\) 0 0
\(35\) −1.03945 − 0.374519i −0.175699 − 0.0633052i
\(36\) 0 0
\(37\) −2.46586 − 2.46586i −0.405384 − 0.405384i 0.474741 − 0.880125i \(-0.342542\pi\)
−0.880125 + 0.474741i \(0.842542\pi\)
\(38\) 0 0
\(39\) 6.12545i 0.980857i
\(40\) 0 0
\(41\) 7.64737 + 1.34844i 1.19432 + 0.210591i 0.735241 − 0.677805i \(-0.237069\pi\)
0.459078 + 0.888396i \(0.348180\pi\)
\(42\) 0 0
\(43\) −0.596842 − 6.82194i −0.0910176 − 1.04034i −0.895114 − 0.445837i \(-0.852906\pi\)
0.804097 − 0.594499i \(-0.202649\pi\)
\(44\) 0 0
\(45\) −0.574946 + 1.24348i −0.0857079 + 0.185367i
\(46\) 0 0
\(47\) 3.78481 − 1.76489i 0.552071 − 0.257435i −0.126504 − 0.991966i \(-0.540376\pi\)
0.678575 + 0.734531i \(0.262598\pi\)
\(48\) 0 0
\(49\) −5.85074 − 3.37793i −0.835821 − 0.482561i
\(50\) 0 0
\(51\) 11.3319 − 1.99812i 1.58678 − 0.279793i
\(52\) 0 0
\(53\) 1.10972 − 12.6842i 0.152432 − 1.74230i −0.407000 − 0.913428i \(-0.633425\pi\)
0.559432 − 0.828877i \(-0.311019\pi\)
\(54\) 0 0
\(55\) 3.34367 − 4.80833i 0.450860 − 0.648355i
\(56\) 0 0
\(57\) 1.33598 + 6.60110i 0.176955 + 0.874337i
\(58\) 0 0
\(59\) −9.64613 + 3.51090i −1.25582 + 0.457081i −0.882364 − 0.470568i \(-0.844049\pi\)
−0.373455 + 0.927648i \(0.621827\pi\)
\(60\) 0 0
\(61\) 4.23307 − 3.55197i 0.541989 − 0.454783i −0.330228 − 0.943901i \(-0.607126\pi\)
0.872218 + 0.489118i \(0.162681\pi\)
\(62\) 0 0
\(63\) 0.173635 − 0.247977i 0.0218760 − 0.0312421i
\(64\) 0 0
\(65\) 6.80922 + 5.67612i 0.844579 + 0.704036i
\(66\) 0 0
\(67\) −2.96957 − 6.36826i −0.362790 − 0.778006i −0.999973 − 0.00729393i \(-0.997678\pi\)
0.637183 − 0.770713i \(-0.280100\pi\)
\(68\) 0 0
\(69\) 5.39476 + 9.34399i 0.649453 + 1.12488i
\(70\) 0 0
\(71\) 2.25576 − 2.68831i 0.267709 − 0.319043i −0.615396 − 0.788218i \(-0.711004\pi\)
0.883105 + 0.469175i \(0.155449\pi\)
\(72\) 0 0
\(73\) −7.90574 + 5.53566i −0.925296 + 0.647900i −0.935984 − 0.352043i \(-0.885487\pi\)
0.0106872 + 0.999943i \(0.496598\pi\)
\(74\) 0 0
\(75\) −3.31024 − 6.98038i −0.382233 − 0.806025i
\(76\) 0 0
\(77\) −0.915103 + 0.915103i −0.104286 + 0.104286i
\(78\) 0 0
\(79\) 2.00528 − 11.3725i 0.225612 − 1.27951i −0.635899 − 0.771772i \(-0.719371\pi\)
0.861512 − 0.507738i \(-0.169518\pi\)
\(80\) 0 0
\(81\) 5.19887 + 4.36237i 0.577652 + 0.484708i
\(82\) 0 0
\(83\) −0.901564 + 3.36468i −0.0989596 + 0.369322i −0.997590 − 0.0693843i \(-0.977897\pi\)
0.898630 + 0.438706i \(0.144563\pi\)
\(84\) 0 0
\(85\) 8.27949 − 14.4484i 0.898036 − 1.56715i
\(86\) 0 0
\(87\) −3.03979 − 11.3446i −0.325899 − 1.21627i
\(88\) 0 0
\(89\) 1.27414 + 7.22603i 0.135059 + 0.765958i 0.974819 + 0.222999i \(0.0715848\pi\)
−0.839760 + 0.542958i \(0.817304\pi\)
\(90\) 0 0
\(91\) −1.25913 − 1.50057i −0.131993 − 0.157303i
\(92\) 0 0
\(93\) 6.76364 + 3.15394i 0.701357 + 0.327048i
\(94\) 0 0
\(95\) 8.57594 + 4.63176i 0.879873 + 0.475209i
\(96\) 0 0
\(97\) 1.27464 + 0.594373i 0.129420 + 0.0603494i 0.486250 − 0.873820i \(-0.338365\pi\)
−0.356830 + 0.934169i \(0.616142\pi\)
\(98\) 0 0
\(99\) 1.03146 + 1.22925i 0.103666 + 0.123544i
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 380.2.bh.a.33.4 ✓ 120
5.2 odd 4 inner 380.2.bh.a.337.4 yes 120
19.15 odd 18 inner 380.2.bh.a.53.4 yes 120
95.72 even 36 inner 380.2.bh.a.357.4 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.4 ✓ 120 1.1 even 1 trivial
380.2.bh.a.53.4 yes 120 19.15 odd 18 inner
380.2.bh.a.337.4 yes 120 5.2 odd 4 inner
380.2.bh.a.357.4 yes 120 95.72 even 36 inner