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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,2,Mod(19,405)] 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("405.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.p (of order \(18\), degree \(6\), not 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 289.4
Character \(\chi\) \(=\) 405.289
Dual form 405.2.p.a.199.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.06000 + 1.26326i) q^{2} +(-0.124928 - 0.708504i) q^{4} +(0.908813 + 2.04305i) q^{5} +(-1.22191 - 0.215456i) q^{7} +(-1.82882 - 1.05587i) q^{8} +(-3.54425 - 1.01757i) q^{10} +(-3.30366 + 1.20243i) q^{11} +(0.380059 + 0.452936i) q^{13} +(1.56741 - 1.31521i) q^{14} +(4.62449 - 1.68318i) q^{16} +(-4.46967 + 2.58056i) q^{17} +(-1.41748 + 2.45515i) q^{19} +(1.33397 - 0.899132i) q^{20} +(1.98290 - 5.44797i) q^{22} +(6.59324 - 1.16257i) q^{23} +(-3.34812 + 3.71350i) q^{25} -0.975040 q^{26} +0.892646i q^{28} +(-7.21421 - 6.05345i) q^{29} +(0.196771 + 1.11595i) q^{31} +(-1.33116 + 3.65733i) q^{32} +(1.47793 - 8.38176i) q^{34} +(-0.670302 - 2.69224i) q^{35} +(-4.06570 + 2.34734i) q^{37} +(-1.59896 - 4.39311i) q^{38} +(0.495142 - 4.69597i) q^{40} +(-4.43129 + 3.71830i) q^{41} +(-0.739520 - 2.03181i) q^{43} +(1.26465 + 2.19044i) q^{44} +(-5.52022 + 9.56131i) q^{46} +(6.14268 + 1.08312i) q^{47} +(-5.13120 - 1.86760i) q^{49} +(-1.14211 - 8.16587i) q^{50} +(0.273427 - 0.325858i) q^{52} -8.21098i q^{53} +(-5.45905 - 5.65676i) q^{55} +(2.00717 + 1.68421i) q^{56} +(15.2942 - 2.69677i) q^{58} +(12.6981 + 4.62173i) q^{59} +(-1.38440 + 7.85133i) q^{61} +(-1.61831 - 0.934331i) q^{62} +(1.71215 + 2.96553i) q^{64} +(-0.579970 + 1.18811i) q^{65} +(2.83000 + 3.37266i) q^{67} +(2.38673 + 2.84439i) q^{68} +(4.11152 + 2.00701i) q^{70} +(3.43788 + 5.95458i) q^{71} +(1.23957 + 0.715668i) q^{73} +(1.34436 - 7.62423i) q^{74} +(1.91656 + 0.697573i) q^{76} +(4.29586 - 0.757476i) q^{77} +(-0.970806 - 0.814603i) q^{79} +(7.64162 + 7.91838i) q^{80} -9.53928i q^{82} +(0.982501 - 1.17090i) q^{83} +(-9.33432 - 6.78651i) q^{85} +(3.35061 + 1.21952i) q^{86} +(7.31144 + 1.28920i) q^{88} +(-6.52774 + 11.3064i) q^{89} +(-0.366811 - 0.635335i) q^{91} +(-1.64736 - 4.52610i) q^{92} +(-7.87952 + 6.61170i) q^{94} +(-6.30422 - 0.664715i) q^{95} +(4.74011 + 13.0234i) q^{97} +(7.79835 - 4.50238i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} + 9 q^{5} - 3 q^{10} + 6 q^{11} + 18 q^{14} - 24 q^{16} - 6 q^{19} + 57 q^{20} + 3 q^{25} - 48 q^{26} + 30 q^{29} - 30 q^{31} - 24 q^{34} + 12 q^{35} - 9 q^{40} + 12 q^{41} - 78 q^{44} - 6 q^{46}+ \cdots - 87 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06000 + 1.26326i −0.749535 + 0.893261i −0.997138 − 0.0756008i \(-0.975913\pi\)
0.247604 + 0.968861i \(0.420357\pi\)
\(3\) 0 0
\(4\) −0.124928 − 0.708504i −0.0624641 − 0.354252i
\(5\) 0.908813 + 2.04305i 0.406434 + 0.913680i
\(6\) 0 0
\(7\) −1.22191 − 0.215456i −0.461840 − 0.0814348i −0.0621128 − 0.998069i \(-0.519784\pi\)
−0.399727 + 0.916634i \(0.630895\pi\)
\(8\) −1.82882 − 1.05587i −0.646587 − 0.373307i
\(9\) 0 0
\(10\) −3.54425 − 1.01757i −1.12079 − 0.321784i
\(11\) −3.30366 + 1.20243i −0.996092 + 0.362548i −0.788076 − 0.615578i \(-0.788923\pi\)
−0.208016 + 0.978125i \(0.566701\pi\)
\(12\) 0 0
\(13\) 0.380059 + 0.452936i 0.105409 + 0.125622i 0.816170 − 0.577812i \(-0.196093\pi\)
−0.710760 + 0.703434i \(0.751649\pi\)
\(14\) 1.56741 − 1.31521i 0.418907 − 0.351505i
\(15\) 0 0
\(16\) 4.62449 − 1.68318i 1.15612 − 0.420794i
\(17\) −4.46967 + 2.58056i −1.08405 + 0.625879i −0.931987 − 0.362491i \(-0.881926\pi\)
−0.152067 + 0.988370i \(0.548593\pi\)
\(18\) 0 0
\(19\) −1.41748 + 2.45515i −0.325192 + 0.563250i −0.981551 − 0.191199i \(-0.938762\pi\)
0.656359 + 0.754449i \(0.272096\pi\)
\(20\) 1.33397 − 0.899132i 0.298285 − 0.201052i
\(21\) 0 0
\(22\) 1.98290 − 5.44797i 0.422756 − 1.16151i
\(23\) 6.59324 − 1.16257i 1.37479 − 0.242412i 0.563043 − 0.826427i \(-0.309630\pi\)
0.811742 + 0.584016i \(0.198519\pi\)
\(24\) 0 0
\(25\) −3.34812 + 3.71350i −0.669624 + 0.742701i
\(26\) −0.975040 −0.191221
\(27\) 0 0
\(28\) 0.892646i 0.168694i
\(29\) −7.21421 − 6.05345i −1.33965 − 1.12410i −0.981717 − 0.190348i \(-0.939038\pi\)
−0.357929 − 0.933749i \(-0.616517\pi\)
\(30\) 0 0
\(31\) 0.196771 + 1.11595i 0.0353412 + 0.200430i 0.997366 − 0.0725319i \(-0.0231079\pi\)
−0.962025 + 0.272962i \(0.911997\pi\)
\(32\) −1.33116 + 3.65733i −0.235318 + 0.646531i
\(33\) 0 0
\(34\) 1.47793 − 8.38176i 0.253463 − 1.43746i
\(35\) −0.670302 − 2.69224i −0.113302 − 0.455072i
\(36\) 0 0
\(37\) −4.06570 + 2.34734i −0.668398 + 0.385900i −0.795469 − 0.605994i \(-0.792776\pi\)
0.127072 + 0.991894i \(0.459442\pi\)
\(38\) −1.59896 − 4.39311i −0.259386 − 0.712657i
\(39\) 0 0
\(40\) 0.495142 − 4.69597i 0.0782888 − 0.742498i
\(41\) −4.43129 + 3.71830i −0.692052 + 0.580700i −0.919500 − 0.393090i \(-0.871406\pi\)
0.227448 + 0.973790i \(0.426962\pi\)
\(42\) 0 0
\(43\) −0.739520 − 2.03181i −0.112776 − 0.309849i 0.870446 − 0.492264i \(-0.163831\pi\)
−0.983222 + 0.182415i \(0.941608\pi\)
\(44\) 1.26465 + 2.19044i 0.190653 + 0.330221i
\(45\) 0 0
\(46\) −5.52022 + 9.56131i −0.813913 + 1.40974i
\(47\) 6.14268 + 1.08312i 0.896002 + 0.157989i 0.602641 − 0.798012i \(-0.294115\pi\)
0.293361 + 0.956002i \(0.405226\pi\)
\(48\) 0 0
\(49\) −5.13120 − 1.86760i −0.733029 − 0.266801i
\(50\) −1.14211 − 8.16587i −0.161519 − 1.15483i
\(51\) 0 0
\(52\) 0.273427 − 0.325858i 0.0379175 − 0.0451883i
\(53\) − 8.21098i − 1.12787i −0.825821 − 0.563933i \(-0.809288\pi\)
0.825821 − 0.563933i \(-0.190712\pi\)
\(54\) 0 0
\(55\) −5.45905 − 5.65676i −0.736098 − 0.762758i
\(56\) 2.00717 + 1.68421i 0.268219 + 0.225063i
\(57\) 0 0
\(58\) 15.2942 − 2.69677i 2.00822 − 0.354104i
\(59\) 12.6981 + 4.62173i 1.65315 + 0.601698i 0.989265 − 0.146136i \(-0.0466836\pi\)
0.663886 + 0.747833i \(0.268906\pi\)
\(60\) 0 0
\(61\) −1.38440 + 7.85133i −0.177254 + 1.00526i 0.758255 + 0.651958i \(0.226052\pi\)
−0.935509 + 0.353302i \(0.885059\pi\)
\(62\) −1.61831 − 0.934331i −0.205525 − 0.118660i
\(63\) 0 0
\(64\) 1.71215 + 2.96553i 0.214019 + 0.370691i
\(65\) −0.579970 + 1.18811i −0.0719364 + 0.147367i
\(66\) 0 0
\(67\) 2.83000 + 3.37266i 0.345739 + 0.412036i 0.910691 − 0.413088i \(-0.135550\pi\)
−0.564952 + 0.825124i \(0.691105\pi\)
\(68\) 2.38673 + 2.84439i 0.289433 + 0.344933i
\(69\) 0 0
\(70\) 4.11152 + 2.00701i 0.491421 + 0.239884i
\(71\) 3.43788 + 5.95458i 0.408001 + 0.706679i 0.994666 − 0.103151i \(-0.0328925\pi\)
−0.586664 + 0.809830i \(0.699559\pi\)
\(72\) 0 0
\(73\) 1.23957 + 0.715668i 0.145081 + 0.0837626i 0.570783 − 0.821101i \(-0.306640\pi\)
−0.425702 + 0.904863i \(0.639973\pi\)
\(74\) 1.34436 − 7.62423i 0.156278 − 0.886298i
\(75\) 0 0
\(76\) 1.91656 + 0.697573i 0.219845 + 0.0800171i
\(77\) 4.29586 − 0.757476i 0.489559 − 0.0863224i
\(78\) 0 0
\(79\) −0.970806 − 0.814603i −0.109224 − 0.0916500i 0.586540 − 0.809920i \(-0.300490\pi\)
−0.695764 + 0.718270i \(0.744934\pi\)
\(80\) 7.64162 + 7.91838i 0.854359 + 0.885302i
\(81\) 0 0
\(82\) − 9.53928i − 1.05344i
\(83\) 0.982501 − 1.17090i 0.107844 − 0.128523i −0.709423 − 0.704783i \(-0.751044\pi\)
0.817266 + 0.576260i \(0.195489\pi\)
\(84\) 0 0
\(85\) −9.33432 − 6.78651i −1.01245 − 0.736101i
\(86\) 3.35061 + 1.21952i 0.361305 + 0.131504i
\(87\) 0 0
\(88\) 7.31144 + 1.28920i 0.779402 + 0.137430i
\(89\) −6.52774 + 11.3064i −0.691939 + 1.19847i 0.279262 + 0.960215i \(0.409910\pi\)
−0.971202 + 0.238259i \(0.923423\pi\)
\(90\) 0 0
\(91\) −0.366811 − 0.635335i −0.0384522 − 0.0666012i
\(92\) −1.64736 − 4.52610i −0.171750 − 0.471878i
\(93\) 0 0
\(94\) −7.87952 + 6.61170i −0.812711 + 0.681945i
\(95\) −6.30422 − 0.664715i −0.646799 − 0.0681983i
\(96\) 0 0
\(97\) 4.74011 + 13.0234i 0.481286 + 1.32232i 0.908392 + 0.418120i \(0.137311\pi\)
−0.427106 + 0.904202i \(0.640467\pi\)
\(98\) 7.79835 − 4.50238i 0.787753 − 0.454809i
\(99\) 0 0
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 405.2.p.a.289.4 96
3.2 odd 2 135.2.p.a.124.13 yes 96
5.4 even 2 inner 405.2.p.a.289.13 96
15.2 even 4 675.2.l.h.151.13 96
15.8 even 4 675.2.l.h.151.4 96
15.14 odd 2 135.2.p.a.124.4 yes 96
27.5 odd 18 135.2.p.a.49.4 ✓ 96
27.22 even 9 inner 405.2.p.a.199.13 96
135.32 even 36 675.2.l.h.76.13 96
135.49 even 18 inner 405.2.p.a.199.4 96
135.59 odd 18 135.2.p.a.49.13 yes 96
135.113 even 36 675.2.l.h.76.4 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.p.a.49.4 ✓ 96 27.5 odd 18
135.2.p.a.49.13 yes 96 135.59 odd 18
135.2.p.a.124.4 yes 96 15.14 odd 2
135.2.p.a.124.13 yes 96 3.2 odd 2
405.2.p.a.199.4 96 135.49 even 18 inner
405.2.p.a.199.13 96 27.22 even 9 inner
405.2.p.a.289.4 96 1.1 even 1 trivial
405.2.p.a.289.13 96 5.4 even 2 inner
675.2.l.h.76.4 96 135.113 even 36
675.2.l.h.76.13 96 135.32 even 36
675.2.l.h.151.4 96 15.8 even 4
675.2.l.h.151.13 96 15.2 even 4