Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(8,81)] 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("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (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: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 8.15
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(7.92414 + 1.39724i) q^{2} +(-179.722 - 65.4133i) q^{4} +(625.377 + 745.295i) q^{5} +(845.976 - 307.910i) q^{7} +(-3116.64 - 1799.40i) q^{8} +(3914.22 + 6779.62i) q^{10} +(15914.2 - 18965.9i) q^{11} +(4116.37 + 23345.1i) q^{13} +(7133.86 - 1257.89i) q^{14} +(15324.1 + 12858.5i) q^{16} +(-90629.4 + 52324.9i) q^{17} +(14265.1 - 24708.0i) q^{19} +(-63641.5 - 174854. i) q^{20} +(152607. - 128052. i) q^{22} +(-110317. + 303093. i) q^{23} +(-96537.3 + 547490. i) q^{25} +190741. i q^{26} -172182. q^{28} +(909304. + 160335. i) q^{29} +(1.44205e6 + 524864. i) q^{31} +(695658. + 829053. i) q^{32} +(-791271. + 287999. i) q^{34} +(758538. + 437942. i) q^{35} +(432930. + 749857. i) q^{37} +(147562. - 175857. i) q^{38} +(-607996. - 3.44812e6i) q^{40} +(-1.02949e6 + 181526. i) q^{41} +(2.74773e6 + 2.30562e6i) q^{43} +(-4.10075e6 + 2.36757e6i) q^{44} +(-1.29766e6 + 2.24761e6i) q^{46} +(42655.1 + 117194. i) q^{47} +(-3.79523e6 + 3.18457e6i) q^{49} +(-1.52995e6 + 4.20350e6i) q^{50} +(787279. - 4.46488e6i) q^{52} +6.71561e6i q^{53} +2.40876e7 q^{55} +(-3.19066e6 - 562599. i) q^{56} +(6.98142e6 + 2.54103e6i) q^{58} +(-5.75360e6 - 6.85688e6i) q^{59} +(-3.21847e6 + 1.17143e6i) q^{61} +(1.06937e7 + 6.17399e6i) q^{62} +(1.79356e6 + 3.10654e6i) q^{64} +(-1.48247e7 + 1.76674e7i) q^{65} +(4.19505e6 + 2.37913e7i) q^{67} +(1.97108e7 - 3.47555e6i) q^{68} +(5.39885e6 + 4.53017e6i) q^{70} +(-5.60407e6 + 3.23551e6i) q^{71} +(-2.52549e6 + 4.37427e6i) q^{73} +(2.38287e6 + 6.54688e6i) q^{74} +(-4.17999e6 + 3.50742e6i) q^{76} +(7.62329e6 - 2.09448e7i) q^{77} +(1.07849e7 - 6.11640e7i) q^{79} +1.94624e7i q^{80} -8.41144e6 q^{82} +(2.06052e7 + 3.63324e6i) q^{83} +(-9.56750e7 - 3.48229e7i) q^{85} +(1.85519e7 + 2.21092e7i) q^{86} +(-8.37261e7 + 3.04738e7i) q^{88} +(-5.38713e7 - 3.11026e7i) q^{89} +(1.06705e7 + 1.84819e7i) q^{91} +(3.96527e7 - 4.72562e7i) q^{92} +(174257. + 988260. i) q^{94} +(2.73358e7 - 4.82004e6i) q^{95} +(1.11846e8 + 9.38502e7i) q^{97} +(-3.45235e7 + 1.99322e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.92414 + 1.39724i 0.495259 + 0.0873275i 0.415698 − 0.909503i \(-0.363537\pi\)
0.0795604 + 0.996830i \(0.474648\pi\)
\(3\) 0 0
\(4\) −179.722 − 65.4133i −0.702038 − 0.255521i
\(5\) 625.377 + 745.295i 1.00060 + 1.19247i 0.981266 + 0.192659i \(0.0617112\pi\)
0.0193372 + 0.999813i \(0.493844\pi\)
\(6\) 0 0
\(7\) 845.976 − 307.910i 0.352343 − 0.128242i −0.159784 − 0.987152i \(-0.551080\pi\)
0.512127 + 0.858910i \(0.328858\pi\)
\(8\) −3116.64 − 1799.40i −0.760899 − 0.439305i
\(9\) 0 0
\(10\) 3914.22 + 6779.62i 0.391422 + 0.677962i
\(11\) 15914.2 − 18965.9i 1.08696 − 1.29539i 0.134443 − 0.990921i \(-0.457075\pi\)
0.952521 − 0.304472i \(-0.0984801\pi\)
\(12\) 0 0
\(13\) 4116.37 + 23345.1i 0.144125 + 0.817376i 0.968065 + 0.250699i \(0.0806603\pi\)
−0.823940 + 0.566677i \(0.808229\pi\)
\(14\) 7133.86 − 1257.89i 0.185700 − 0.0327440i
\(15\) 0 0
\(16\) 15324.1 + 12858.5i 0.233828 + 0.196205i
\(17\) −90629.4 + 52324.9i −1.08511 + 0.626488i −0.932270 − 0.361763i \(-0.882175\pi\)
−0.152839 + 0.988251i \(0.548842\pi\)
\(18\) 0 0
\(19\) 14265.1 − 24708.0i 0.109462 − 0.189593i −0.806091 − 0.591792i \(-0.798421\pi\)
0.915552 + 0.402199i \(0.131754\pi\)
\(20\) −63641.5 − 174854.i −0.397759 − 1.09284i
\(21\) 0 0
\(22\) 152607. − 128052.i 0.651452 − 0.546633i
\(23\) −110317. + 303093.i −0.394213 + 1.08309i 0.570846 + 0.821057i \(0.306615\pi\)
−0.965059 + 0.262034i \(0.915607\pi\)
\(24\) 0 0
\(25\) −96537.3 + 547490.i −0.247135 + 1.40157i
\(26\) 190741.i 0.417399i
\(27\) 0 0
\(28\) −172182. −0.280127
\(29\) 909304. + 160335.i 1.28563 + 0.226692i 0.774370 − 0.632733i \(-0.218067\pi\)
0.511263 + 0.859425i \(0.329178\pi\)
\(30\) 0 0
\(31\) 1.44205e6 + 524864.i 1.56147 + 0.568330i 0.971073 − 0.238782i \(-0.0767480\pi\)
0.590399 + 0.807111i \(0.298970\pi\)
\(32\) 695658. + 829053.i 0.663431 + 0.790647i
\(33\) 0 0
\(34\) −791271. + 287999.i −0.592119 + 0.215514i
\(35\) 758538. + 437942.i 0.505481 + 0.291840i
\(36\) 0 0
\(37\) 432930. + 749857.i 0.230999 + 0.400103i 0.958102 − 0.286426i \(-0.0924671\pi\)
−0.727103 + 0.686528i \(0.759134\pi\)
\(38\) 147562. − 175857.i 0.0707685 − 0.0843386i
\(39\) 0 0
\(40\) −607996. − 3.44812e6i −0.237499 − 1.34692i
\(41\) −1.02949e6 + 181526.i −0.364322 + 0.0642398i −0.352813 − 0.935694i \(-0.614775\pi\)
−0.0115094 + 0.999934i \(0.503664\pi\)
\(42\) 0 0
\(43\) 2.74773e6 + 2.30562e6i 0.803710 + 0.674393i 0.949098 − 0.314982i \(-0.101998\pi\)
−0.145387 + 0.989375i \(0.546443\pi\)
\(44\) −4.10075e6 + 2.36757e6i −1.09409 + 0.631673i
\(45\) 0 0
\(46\) −1.29766e6 + 2.24761e6i −0.289821 + 0.501985i
\(47\) 42655.1 + 117194.i 0.00874137 + 0.0240167i 0.943987 − 0.329984i \(-0.107043\pi\)
−0.935245 + 0.354001i \(0.884821\pi\)
\(48\) 0 0
\(49\) −3.79523e6 + 3.18457e6i −0.658345 + 0.552417i
\(50\) −1.52995e6 + 4.20350e6i −0.244792 + 0.672560i
\(51\) 0 0
\(52\) 787279. − 4.46488e6i 0.107675 − 0.610656i
\(53\) 6.71561e6i 0.851103i 0.904934 + 0.425551i \(0.139920\pi\)
−0.904934 + 0.425551i \(0.860080\pi\)
\(54\) 0 0
\(55\) 2.40876e7 2.63234
\(56\) −3.19066e6 − 562599.i −0.324435 − 0.0572067i
\(57\) 0 0
\(58\) 6.98142e6 + 2.54103e6i 0.616924 + 0.224542i
\(59\) −5.75360e6 − 6.85688e6i −0.474823 − 0.565872i 0.474467 − 0.880273i \(-0.342641\pi\)
−0.949290 + 0.314401i \(0.898196\pi\)
\(60\) 0 0
\(61\) −3.21847e6 + 1.17143e6i −0.232451 + 0.0846051i −0.455619 − 0.890175i \(-0.650582\pi\)
0.223169 + 0.974780i \(0.428360\pi\)
\(62\) 1.06937e7 + 6.17399e6i 0.723702 + 0.417830i
\(63\) 0 0
\(64\) 1.79356e6 + 3.10654e6i 0.106905 + 0.185164i
\(65\) −1.48247e7 + 1.76674e7i −0.830486 + 0.989734i
\(66\) 0 0
\(67\) 4.19505e6 + 2.37913e7i 0.208180 + 1.18065i 0.892357 + 0.451330i \(0.149050\pi\)
−0.684178 + 0.729315i \(0.739839\pi\)
\(68\) 1.97108e7 − 3.47555e6i 0.921868 − 0.162550i
\(69\) 0 0
\(70\) 5.39885e6 + 4.53017e6i 0.224858 + 0.188679i
\(71\) −5.60407e6 + 3.23551e6i −0.220531 + 0.127324i −0.606196 − 0.795315i \(-0.707305\pi\)
0.385665 + 0.922639i \(0.373972\pi\)
\(72\) 0 0
\(73\) −2.52549e6 + 4.37427e6i −0.0889311 + 0.154033i −0.907060 − 0.421002i \(-0.861678\pi\)
0.818128 + 0.575035i \(0.195012\pi\)
\(74\) 2.38287e6 + 6.54688e6i 0.0794645 + 0.218327i
\(75\) 0 0
\(76\) −4.17999e6 + 3.50742e6i −0.125291 + 0.105132i
\(77\) 7.62329e6 − 2.09448e7i 0.216860 − 0.595818i
\(78\) 0 0
\(79\) 1.07849e7 − 6.11640e7i 0.276889 − 1.57032i −0.456007 − 0.889976i \(-0.650721\pi\)
0.732896 − 0.680340i \(-0.238168\pi\)
\(80\) 1.94624e7i 0.475156i
\(81\) 0 0
\(82\) −8.41144e6 −0.186044
\(83\) 2.06052e7 + 3.63324e6i 0.434174 + 0.0765565i 0.386464 − 0.922305i \(-0.373697\pi\)
0.0477101 + 0.998861i \(0.484808\pi\)
\(84\) 0 0
\(85\) −9.56750e7 − 3.48229e7i −1.83283 − 0.667097i
\(86\) 1.85519e7 + 2.21092e7i 0.339151 + 0.404185i
\(87\) 0 0
\(88\) −8.37261e7 + 3.04738e7i −1.39614 + 0.508155i
\(89\) −5.38713e7 − 3.11026e7i −0.858613 − 0.495720i 0.00493471 − 0.999988i \(-0.498429\pi\)
−0.863548 + 0.504267i \(0.831763\pi\)
\(90\) 0 0
\(91\) 1.06705e7 + 1.84819e7i 0.155604 + 0.269514i
\(92\) 3.96527e7 − 4.72562e7i 0.553505 − 0.659641i
\(93\) 0 0
\(94\) 174257. + 988260.i 0.00223192 + 0.0126578i
\(95\) 2.73358e7 − 4.82004e6i 0.335612 − 0.0591775i
\(96\) 0 0
\(97\) 1.11846e8 + 9.38502e7i 1.26338 + 1.06010i 0.995313 + 0.0967035i \(0.0308299\pi\)
0.268069 + 0.963400i \(0.413615\pi\)
\(98\) −3.45235e7 + 1.99322e7i −0.374292 + 0.216098i
\(99\) 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 81.9.f.a.8.15 138
3.2 odd 2 27.9.f.a.2.9 ✓ 138
27.13 even 9 27.9.f.a.14.9 yes 138
27.14 odd 18 inner 81.9.f.a.71.15 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.9 ✓ 138 3.2 odd 2
27.9.f.a.14.9 yes 138 27.13 even 9
81.9.f.a.8.15 138 1.1 even 1 trivial
81.9.f.a.71.15 138 27.14 odd 18 inner