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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [135,2,Mod(4,135)] 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("135.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.p (of order \(18\), degree \(6\), 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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 124.4
Character \(\chi\) \(=\) 135.124
Dual form 135.2.p.a.49.4

$q$-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} +(0.958575 + 1.44261i) q^{3} +(-0.124928 - 0.708504i) q^{4} +(-2.00944 + 0.980894i) q^{5} +(-2.83849 - 0.318243i) q^{6} +(1.22191 + 0.215456i) q^{7} +(-1.82882 - 1.05587i) q^{8} +(-1.16227 + 2.76571i) q^{9} +(0.890884 - 3.57820i) q^{10} +(3.30366 - 1.20243i) q^{11} +(0.902344 - 0.859377i) q^{12} +(-0.380059 - 0.452936i) q^{13} +(-1.56741 + 1.31521i) q^{14} +(-3.34125 - 1.95858i) q^{15} +(4.62449 - 1.68318i) q^{16} +(-4.46967 + 2.58056i) q^{17} +(-2.26180 - 4.39990i) q^{18} +(-1.41748 + 2.45515i) q^{19} +(0.946003 + 1.30115i) q^{20} +(0.860474 + 1.96928i) q^{21} +(-1.98290 + 5.44797i) q^{22} +(6.59324 - 1.16257i) q^{23} +(-0.229849 - 3.65042i) q^{24} +(3.07569 - 3.94210i) q^{25} +0.975040 q^{26} +(-5.10397 + 0.974431i) q^{27} -0.892646i q^{28} +(7.21421 + 6.05345i) q^{29} +(6.01593 - 2.14477i) q^{30} +(0.196771 + 1.11595i) q^{31} +(-1.33116 + 3.65733i) q^{32} +(4.90146 + 3.61329i) q^{33} +(1.47793 - 8.38176i) q^{34} +(-2.66670 + 0.765621i) q^{35} +(2.10471 + 0.477957i) q^{36} +(4.06570 - 2.34734i) q^{37} +(-1.59896 - 4.39311i) q^{38} +(0.289098 - 0.982452i) q^{39} +(4.71061 + 0.327827i) q^{40} +(4.43129 - 3.71830i) q^{41} +(-3.39982 - 1.00044i) q^{42} +(0.739520 + 2.03181i) q^{43} +(-1.26465 - 2.19044i) q^{44} +(-0.377357 - 6.69758i) q^{45} +(-5.52022 + 9.56131i) q^{46} +(6.14268 + 1.08312i) q^{47} +(6.86110 + 5.05791i) q^{48} +(-5.13120 - 1.86760i) q^{49} +(1.71966 + 8.06403i) q^{50} +(-8.00727 - 3.97434i) q^{51} +(-0.273427 + 0.325858i) q^{52} -8.21098i q^{53} +(4.17926 - 7.48054i) q^{54} +(-5.45905 + 5.65676i) q^{55} +(-2.00717 - 1.68421i) q^{56} +(-4.90059 + 0.308566i) q^{57} +(-15.2942 + 2.69677i) q^{58} +(-12.6981 - 4.62173i) q^{59} +(-0.970247 + 2.61197i) q^{60} +(-1.38440 + 7.85133i) q^{61} +(-1.61831 - 0.934331i) q^{62} +(-2.01608 + 3.12903i) q^{63} +(1.71215 + 2.96553i) q^{64} +(1.20799 + 0.537351i) q^{65} +(-9.76008 + 2.36173i) q^{66} +(-2.83000 - 3.37266i) q^{67} +(2.38673 + 2.84439i) q^{68} +(7.99725 + 8.39709i) q^{69} +(1.85953 - 4.18030i) q^{70} +(-3.43788 - 5.95458i) q^{71} +(5.04582 - 3.83078i) q^{72} +(-1.23957 - 0.715668i) q^{73} +(-1.34436 + 7.62423i) q^{74} +(8.63520 + 0.658243i) q^{75} +(1.91656 + 0.697573i) q^{76} +(4.29586 - 0.757476i) q^{77} +(0.934649 + 1.40661i) q^{78} +(-0.970806 - 0.814603i) q^{79} +(-7.64162 + 7.91838i) q^{80} +(-6.29826 - 6.42899i) q^{81} +9.53928i q^{82} +(0.982501 - 1.17090i) q^{83} +(1.28774 - 0.855668i) q^{84} +(6.45027 - 9.56976i) q^{85} +(-3.35061 - 1.21952i) q^{86} +(-1.81742 + 16.2100i) q^{87} +(-7.31144 - 1.28920i) q^{88} +(6.52774 - 11.3064i) q^{89} +(8.86079 + 6.62275i) q^{90} +(-0.366811 - 0.635335i) q^{91} +(-1.64736 - 4.52610i) q^{92} +(-1.42126 + 1.35358i) q^{93} +(-7.87952 + 6.61170i) q^{94} +(0.440100 - 6.32387i) q^{95} +(-6.55214 + 1.58548i) q^{96} +(-4.74011 - 13.0234i) q^{97} +(7.79835 - 4.50238i) q^{98} +(-0.514163 + 10.5345i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} - 9 q^{5} - 6 q^{6} - 18 q^{9} - 3 q^{10} - 6 q^{11} - 18 q^{14} - 21 q^{15} - 24 q^{16} - 6 q^{19} - 57 q^{20} + 24 q^{21} - 30 q^{24} + 3 q^{25} + 48 q^{26} - 30 q^{29} - 51 q^{30} - 30 q^{31}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.06000 + 1.26326i −0.749535 + 0.893261i −0.997138 − 0.0756008i \(-0.975913\pi\)
0.247604 + 0.968861i \(0.420357\pi\)
\(3\) 0.958575 + 1.44261i 0.553433 + 0.832893i
\(4\) −0.124928 − 0.708504i −0.0624641 − 0.354252i
\(5\) −2.00944 + 0.980894i −0.898649 + 0.438669i
\(6\) −2.83849 − 0.318243i −1.15881 − 0.129922i
\(7\) 1.22191 + 0.215456i 0.461840 + 0.0814348i 0.399727 − 0.916634i \(-0.369105\pi\)
0.0621128 + 0.998069i \(0.480216\pi\)
\(8\) −1.82882 − 1.05587i −0.646587 − 0.373307i
\(9\) −1.16227 + 2.76571i −0.387423 + 0.921902i
\(10\) 0.890884 − 3.57820i 0.281722 − 1.13153i
\(11\) 3.30366 − 1.20243i 0.996092 − 0.362548i 0.208016 − 0.978125i \(-0.433299\pi\)
0.788076 + 0.615578i \(0.211077\pi\)
\(12\) 0.902344 − 0.859377i 0.260484 − 0.248081i
\(13\) −0.380059 − 0.452936i −0.105409 − 0.125622i 0.710760 − 0.703434i \(-0.248351\pi\)
−0.816170 + 0.577812i \(0.803907\pi\)
\(14\) −1.56741 + 1.31521i −0.418907 + 0.351505i
\(15\) −3.34125 − 1.95858i −0.862707 − 0.505704i
\(16\) 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 \(0.548593\pi\)
\(18\) −2.26180 − 4.39990i −0.533112 − 1.03707i
\(19\) −1.41748 + 2.45515i −0.325192 + 0.563250i −0.981551 − 0.191199i \(-0.938762\pi\)
0.656359 + 0.754449i \(0.272096\pi\)
\(20\) 0.946003 + 1.30115i 0.211533 + 0.290947i
\(21\) 0.860474 + 1.96928i 0.187771 + 0.429732i
\(22\) −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 \(0.198519\pi\)
\(24\) −0.229849 − 3.65042i −0.0469177 − 0.745139i
\(25\) 3.07569 − 3.94210i 0.615138 − 0.788419i
\(26\) 0.975040 0.191221
\(27\) −5.10397 + 0.974431i −0.982259 + 0.187529i
\(28\) − 0.892646i − 0.168694i
\(29\) 7.21421 + 6.05345i 1.33965 + 1.12410i 0.981717 + 0.190348i \(0.0609616\pi\)
0.357929 + 0.933749i \(0.383483\pi\)
\(30\) 6.01593 − 2.14477i 1.09835 − 0.391579i
\(31\) 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
\(33\) 4.90146 + 3.61329i 0.853234 + 0.628992i
\(34\) 1.47793 − 8.38176i 0.253463 − 1.43746i
\(35\) −2.66670 + 0.765621i −0.450754 + 0.129414i
\(36\) 2.10471 + 0.477957i 0.350785 + 0.0796595i
\(37\) 4.06570 − 2.34734i 0.668398 − 0.385900i −0.127072 − 0.991894i \(-0.540558\pi\)
0.795469 + 0.605994i \(0.207224\pi\)
\(38\) −1.59896 − 4.39311i −0.259386 − 0.712657i
\(39\) 0.289098 − 0.982452i 0.0462927 − 0.157318i
\(40\) 4.71061 + 0.327827i 0.744813 + 0.0518341i
\(41\) 4.43129 − 3.71830i 0.692052 − 0.580700i −0.227448 − 0.973790i \(-0.573038\pi\)
0.919500 + 0.393090i \(0.128594\pi\)
\(42\) −3.39982 − 1.00044i −0.524603 − 0.154371i
\(43\) 0.739520 + 2.03181i 0.112776 + 0.309849i 0.983222 − 0.182415i \(-0.0583916\pi\)
−0.870446 + 0.492264i \(0.836169\pi\)
\(44\) −1.26465 − 2.19044i −0.190653 − 0.330221i
\(45\) −0.377357 − 6.69758i −0.0562530 − 0.998417i
\(46\) −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 \(0.405226\pi\)
\(48\) 6.86110 + 5.05791i 0.990314 + 0.730046i
\(49\) −5.13120 − 1.86760i −0.733029 − 0.266801i
\(50\) 1.71966 + 8.06403i 0.243196 + 1.14043i
\(51\) −8.00727 − 3.97434i −1.12124 − 0.556519i
\(52\) −0.273427 + 0.325858i −0.0379175 + 0.0451883i
\(53\) − 8.21098i − 1.12787i −0.825821 − 0.563933i \(-0.809288\pi\)
0.825821 − 0.563933i \(-0.190712\pi\)
\(54\) 4.17926 − 7.48054i 0.568725 − 1.01797i
\(55\) −5.45905 + 5.65676i −0.736098 + 0.762758i
\(56\) −2.00717 − 1.68421i −0.268219 − 0.225063i
\(57\) −4.90059 + 0.308566i −0.649099 + 0.0408706i
\(58\) −15.2942 + 2.69677i −2.00822 + 0.354104i
\(59\) −12.6981 − 4.62173i −1.65315 − 0.601698i −0.663886 − 0.747833i \(-0.731094\pi\)
−0.989265 + 0.146136i \(0.953316\pi\)
\(60\) −0.970247 + 2.61197i −0.125258 + 0.337204i
\(61\) −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
\(63\) −2.01608 + 3.12903i −0.254002 + 0.394221i
\(64\) 1.71215 + 2.96553i 0.214019 + 0.370691i
\(65\) 1.20799 + 0.537351i 0.149832 + 0.0666502i
\(66\) −9.76008 + 2.36173i −1.20138 + 0.290709i
\(67\) −2.83000 − 3.37266i −0.345739 − 0.412036i 0.564952 − 0.825124i \(-0.308895\pi\)
−0.910691 + 0.413088i \(0.864450\pi\)
\(68\) 2.38673 + 2.84439i 0.289433 + 0.344933i
\(69\) 7.99725 + 8.39709i 0.962755 + 1.01089i
\(70\) 1.85953 − 4.18030i 0.222256 − 0.499641i
\(71\) −3.43788 − 5.95458i −0.408001 − 0.706679i 0.586664 − 0.809830i \(-0.300441\pi\)
−0.994666 + 0.103151i \(0.967107\pi\)
\(72\) 5.04582 − 3.83078i 0.594655 − 0.451462i
\(73\) −1.23957 − 0.715668i −0.145081 − 0.0837626i 0.425702 − 0.904863i \(-0.360027\pi\)
−0.570783 + 0.821101i \(0.693360\pi\)
\(74\) −1.34436 + 7.62423i −0.156278 + 0.886298i
\(75\) 8.63520 + 0.658243i 0.997107 + 0.0760073i
\(76\) 1.91656 + 0.697573i 0.219845 + 0.0800171i
\(77\) 4.29586 − 0.757476i 0.489559 − 0.0863224i
\(78\) 0.934649 + 1.40661i 0.105828 + 0.159267i
\(79\) −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
\(81\) −6.29826 − 6.42899i −0.699807 − 0.714332i
\(82\) 9.53928i 1.05344i
\(83\) 0.982501 − 1.17090i 0.107844 − 0.128523i −0.709423 − 0.704783i \(-0.751044\pi\)
0.817266 + 0.576260i \(0.195489\pi\)
\(84\) 1.28774 − 0.855668i 0.140504 − 0.0933610i
\(85\) 6.45027 − 9.56976i 0.699630 − 1.03799i
\(86\) −3.35061 − 1.21952i −0.361305 − 0.131504i
\(87\) −1.81742 + 16.2100i −0.194848 + 1.73789i
\(88\) −7.31144 − 1.28920i −0.779402 − 0.137430i
\(89\) 6.52774 − 11.3064i 0.691939 − 1.19847i −0.279262 − 0.960215i \(-0.590090\pi\)
0.971202 − 0.238259i \(-0.0765767\pi\)
\(90\) 8.86079 + 6.62275i 0.934010 + 0.698099i
\(91\) −0.366811 − 0.635335i −0.0384522 − 0.0666012i
\(92\) −1.64736 − 4.52610i −0.171750 − 0.471878i
\(93\) −1.42126 + 1.35358i −0.147378 + 0.140360i
\(94\) −7.87952 + 6.61170i −0.812711 + 0.681945i
\(95\) 0.440100 − 6.32387i 0.0451533 − 0.648815i
\(96\) −6.55214 + 1.58548i −0.668724 + 0.161817i
\(97\) −4.74011 − 13.0234i −0.481286 − 1.32232i −0.908392 − 0.418120i \(-0.862689\pi\)
0.427106 − 0.904202i \(-0.359533\pi\)
\(98\) 7.79835 − 4.50238i 0.787753 − 0.454809i
\(99\) −0.514163 + 10.5345i −0.0516753 + 1.05876i
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 135.2.p.a.124.4 yes 96
3.2 odd 2 405.2.p.a.289.13 96
5.2 odd 4 675.2.l.h.151.4 96
5.3 odd 4 675.2.l.h.151.13 96
5.4 even 2 inner 135.2.p.a.124.13 yes 96
15.14 odd 2 405.2.p.a.289.4 96
27.5 odd 18 405.2.p.a.199.4 96
27.22 even 9 inner 135.2.p.a.49.13 yes 96
135.22 odd 36 675.2.l.h.76.4 96
135.49 even 18 inner 135.2.p.a.49.4 ✓ 96
135.59 odd 18 405.2.p.a.199.13 96
135.103 odd 36 675.2.l.h.76.13 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.p.a.49.4 ✓ 96 135.49 even 18 inner
135.2.p.a.49.13 yes 96 27.22 even 9 inner
135.2.p.a.124.4 yes 96 1.1 even 1 trivial
135.2.p.a.124.13 yes 96 5.4 even 2 inner
405.2.p.a.199.4 96 27.5 odd 18
405.2.p.a.199.13 96 135.59 odd 18
405.2.p.a.289.4 96 15.14 odd 2
405.2.p.a.289.13 96 3.2 odd 2
675.2.l.h.76.4 96 135.22 odd 36
675.2.l.h.76.13 96 135.103 odd 36
675.2.l.h.151.4 96 5.2 odd 4
675.2.l.h.151.13 96 5.3 odd 4