Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [216,2,Mod(11,216)] 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("216.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(216, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.72476868366\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 59.6
Character \(\chi\) \(=\) 216.59
Dual form 216.2.v.b.11.6

$q$-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+(-1.26354 + 0.635192i) q^{2} +(0.0447181 - 1.73147i) q^{3} +(1.19306 - 1.60518i) q^{4} +(4.00434 - 1.45746i) q^{5} +(1.04331 + 2.21619i) q^{6} +(-1.46723 + 0.258712i) q^{7} +(-0.487886 + 2.78603i) q^{8} +(-2.99600 - 0.154856i) q^{9} +(-4.13387 + 4.38508i) q^{10} +(1.57835 - 4.33649i) q^{11} +(-2.72597 - 2.13754i) q^{12} +(-2.82693 + 3.36900i) q^{13} +(1.68957 - 1.25886i) q^{14} +(-2.34449 - 6.99858i) q^{15} +(-1.15320 - 3.83016i) q^{16} +(2.43381 + 1.40516i) q^{17} +(3.88393 - 1.70737i) q^{18} +(0.516826 + 0.895168i) q^{19} +(2.43794 - 8.16652i) q^{20} +(0.382341 + 2.55204i) q^{21} +(0.760191 + 6.48188i) q^{22} +(0.345539 - 1.95965i) q^{23} +(4.80212 + 0.969347i) q^{24} +(10.0803 - 8.45837i) q^{25} +(1.43197 - 6.05250i) q^{26} +(-0.402105 + 5.18057i) q^{27} +(-1.33522 + 2.66383i) q^{28} +(-0.783266 + 0.657238i) q^{29} +(7.40779 + 7.35378i) q^{30} +(-6.04883 - 1.06657i) q^{31} +(3.89000 + 4.10705i) q^{32} +(-7.43793 - 2.92679i) q^{33} +(-3.96777 - 0.229540i) q^{34} +(-5.49822 + 3.17440i) q^{35} +(-3.82299 + 4.62436i) q^{36} +(5.15589 + 2.97676i) q^{37} +(-1.22163 - 0.802797i) q^{38} +(5.70692 + 5.04540i) q^{39} +(2.10687 + 11.8673i) q^{40} +(2.77589 - 3.30817i) q^{41} +(-2.10414 - 2.98174i) q^{42} +(-2.28164 - 0.830449i) q^{43} +(-5.07777 - 7.70724i) q^{44} +(-12.2227 + 3.74645i) q^{45} +(0.808150 + 2.69557i) q^{46} +(0.806279 + 4.57264i) q^{47} +(-6.68339 + 1.82546i) q^{48} +(-4.49202 + 1.63496i) q^{49} +(-7.36416 + 17.0904i) q^{50} +(2.54184 - 4.15125i) q^{51} +(2.03515 + 8.55715i) q^{52} +3.14544 q^{53} +(-2.78258 - 6.80127i) q^{54} -19.6651i q^{55} +(-0.00493996 - 4.21397i) q^{56} +(1.57307 - 0.854840i) q^{57} +(0.572215 - 1.32797i) q^{58} +(2.77458 + 7.62310i) q^{59} +(-14.0311 - 4.58642i) q^{60} +(-2.25575 + 0.397750i) q^{61} +(8.32042 - 2.49451i) q^{62} +(4.43588 - 0.547892i) q^{63} +(-7.52394 - 2.71853i) q^{64} +(-6.40978 + 17.6107i) q^{65} +(11.2572 - 1.02639i) q^{66} +(7.24112 + 6.07602i) q^{67} +(5.15923 - 2.23026i) q^{68} +(-3.37763 - 0.685923i) q^{69} +(4.93086 - 7.50340i) q^{70} +(-2.21770 + 3.84117i) q^{71} +(1.89314 - 8.27140i) q^{72} +(6.12067 + 10.6013i) q^{73} +(-8.40549 - 0.486268i) q^{74} +(-14.1947 - 17.8320i) q^{75} +(2.05351 + 0.238394i) q^{76} +(-1.19390 + 6.77096i) q^{77} +(-10.4157 - 2.75008i) q^{78} +(2.19872 + 2.62033i) q^{79} +(-10.2001 - 13.6565i) q^{80} +(8.95204 + 0.927899i) q^{81} +(-1.40612 + 5.94322i) q^{82} +(3.23224 + 3.85203i) q^{83} +(4.55264 + 2.43102i) q^{84} +(11.7938 + 2.07956i) q^{85} +(3.41044 - 0.399974i) q^{86} +(1.10296 + 1.38559i) q^{87} +(11.3115 + 6.51305i) q^{88} +(-3.14150 + 1.81374i) q^{89} +(13.0641 - 12.4975i) q^{90} +(3.27615 - 5.67445i) q^{91} +(-2.73334 - 2.89263i) q^{92} +(-2.11723 + 10.4257i) q^{93} +(-3.92327 - 5.26556i) q^{94} +(3.37421 + 2.83130i) q^{95} +(7.28521 - 6.55177i) q^{96} +(-5.20281 - 1.89367i) q^{97} +(4.63733 - 4.91913i) q^{98} +(-5.40028 + 12.7477i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 6 q^{6} - 9 q^{8} - 12 q^{9} - 3 q^{10} - 30 q^{11} - 3 q^{12} + 9 q^{14} - 6 q^{16} - 18 q^{17} + 63 q^{18} - 6 q^{19} - 27 q^{20} - 18 q^{22} - 30 q^{24} - 12 q^{25}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26354 + 0.635192i −0.893457 + 0.449148i
\(3\) 0.0447181 − 1.73147i 0.0258180 − 0.999667i
\(4\) 1.19306 − 1.60518i 0.596532 − 0.802590i
\(5\) 4.00434 − 1.45746i 1.79079 − 0.651796i 0.791626 − 0.611006i \(-0.209235\pi\)
0.999168 − 0.0407891i \(-0.0129872\pi\)
\(6\) 1.04331 + 2.21619i 0.425931 + 0.904755i
\(7\) −1.46723 + 0.258712i −0.554561 + 0.0977840i −0.443903 − 0.896075i \(-0.646407\pi\)
−0.110657 + 0.993859i \(0.535296\pi\)
\(8\) −0.487886 + 2.78603i −0.172494 + 0.985011i
\(9\) −2.99600 − 0.154856i −0.998667 − 0.0516188i
\(10\) −4.13387 + 4.38508i −1.30724 + 1.38668i
\(11\) 1.57835 − 4.33649i 0.475891 − 1.30750i −0.437060 − 0.899432i \(-0.643980\pi\)
0.912952 − 0.408068i \(-0.133797\pi\)
\(12\) −2.72597 − 2.13754i −0.786921 − 0.617054i
\(13\) −2.82693 + 3.36900i −0.784048 + 0.934392i −0.999109 − 0.0422087i \(-0.986561\pi\)
0.215061 + 0.976601i \(0.431005\pi\)
\(14\) 1.68957 − 1.25886i 0.451557 − 0.336446i
\(15\) −2.34449 − 6.99858i −0.605344 − 1.80702i
\(16\) −1.15320 − 3.83016i −0.288300 − 0.957540i
\(17\) 2.43381 + 1.40516i 0.590286 + 0.340802i 0.765211 − 0.643780i \(-0.222635\pi\)
−0.174924 + 0.984582i \(0.555968\pi\)
\(18\) 3.88393 − 1.70737i 0.915451 − 0.402430i
\(19\) 0.516826 + 0.895168i 0.118568 + 0.205366i 0.919200 − 0.393790i \(-0.128836\pi\)
−0.800632 + 0.599156i \(0.795503\pi\)
\(20\) 2.43794 − 8.16652i 0.545140 − 1.82609i
\(21\) 0.382341 + 2.55204i 0.0834337 + 0.556900i
\(22\) 0.760191 + 6.48188i 0.162073 + 1.38194i
\(23\) 0.345539 − 1.95965i 0.0720498 − 0.408615i −0.927357 − 0.374178i \(-0.877925\pi\)
0.999407 − 0.0344371i \(-0.0109638\pi\)
\(24\) 4.80212 + 0.969347i 0.980229 + 0.197867i
\(25\) 10.0803 − 8.45837i 2.01606 − 1.69167i
\(26\) 1.43197 − 6.05250i 0.280833 − 1.18699i
\(27\) −0.402105 + 5.18057i −0.0773851 + 0.997001i
\(28\) −1.33522 + 2.66383i −0.252332 + 0.503416i
\(29\) −0.783266 + 0.657238i −0.145449 + 0.122046i −0.712609 − 0.701562i \(-0.752487\pi\)
0.567160 + 0.823608i \(0.308042\pi\)
\(30\) 7.40779 + 7.35378i 1.35247 + 1.34261i
\(31\) −6.04883 − 1.06657i −1.08640 − 0.191562i −0.398358 − 0.917230i \(-0.630420\pi\)
−0.688045 + 0.725668i \(0.741531\pi\)
\(32\) 3.89000 + 4.10705i 0.687661 + 0.726031i
\(33\) −7.43793 − 2.92679i −1.29478 − 0.509490i
\(34\) −3.96777 − 0.229540i −0.680466 − 0.0393659i
\(35\) −5.49822 + 3.17440i −0.929368 + 0.536571i
\(36\) −3.82299 + 4.62436i −0.637165 + 0.770727i
\(37\) 5.15589 + 2.97676i 0.847624 + 0.489376i 0.859848 − 0.510549i \(-0.170558\pi\)
−0.0122246 + 0.999925i \(0.503891\pi\)
\(38\) −1.22163 − 0.802797i −0.198175 − 0.130231i
\(39\) 5.70692 + 5.04540i 0.913838 + 0.807911i
\(40\) 2.10687 + 11.8673i 0.333125 + 1.87638i
\(41\) 2.77589 − 3.30817i 0.433520 − 0.516650i −0.504414 − 0.863462i \(-0.668291\pi\)
0.937935 + 0.346812i \(0.112736\pi\)
\(42\) −2.10414 − 2.98174i −0.324675 − 0.460092i
\(43\) −2.28164 − 0.830449i −0.347947 − 0.126642i 0.162134 − 0.986769i \(-0.448162\pi\)
−0.510081 + 0.860126i \(0.670385\pi\)
\(44\) −5.07777 − 7.70724i −0.765502 − 1.16191i
\(45\) −12.2227 + 3.74645i −1.82205 + 0.558488i
\(46\) 0.808150 + 2.69557i 0.119155 + 0.397441i
\(47\) 0.806279 + 4.57264i 0.117608 + 0.666987i 0.985426 + 0.170105i \(0.0544108\pi\)
−0.867818 + 0.496882i \(0.834478\pi\)
\(48\) −6.68339 + 1.82546i −0.964664 + 0.263482i
\(49\) −4.49202 + 1.63496i −0.641717 + 0.233566i
\(50\) −7.36416 + 17.0904i −1.04145 + 2.41695i
\(51\) 2.54184 − 4.15125i 0.355928 − 0.581291i
\(52\) 2.03515 + 8.55715i 0.282224 + 1.18666i
\(53\) 3.14544 0.432059 0.216030 − 0.976387i \(-0.430689\pi\)
0.216030 + 0.976387i \(0.430689\pi\)
\(54\) −2.78258 − 6.80127i −0.378661 − 0.925535i
\(55\) − 19.6651i − 2.65165i
\(56\) −0.00493996 − 4.21397i −0.000660129 − 0.563115i
\(57\) 1.57307 − 0.854840i 0.208358 − 0.113226i
\(58\) 0.572215 − 1.32797i 0.0751355 − 0.174371i
\(59\) 2.77458 + 7.62310i 0.361220 + 0.992443i 0.978599 + 0.205777i \(0.0659723\pi\)
−0.617379 + 0.786666i \(0.711805\pi\)
\(60\) −14.0311 − 4.58642i −1.81141 − 0.592105i
\(61\) −2.25575 + 0.397750i −0.288820 + 0.0509267i −0.316181 − 0.948699i \(-0.602401\pi\)
0.0273613 + 0.999626i \(0.491290\pi\)
\(62\) 8.32042 − 2.49451i 1.05669 − 0.316803i
\(63\) 4.43588 − 0.547892i 0.558869 − 0.0690279i
\(64\) −7.52394 − 2.71853i −0.940492 − 0.339816i
\(65\) −6.40978 + 17.6107i −0.795035 + 2.18434i
\(66\) 11.2572 − 1.02639i 1.38566 − 0.126340i
\(67\) 7.24112 + 6.07602i 0.884643 + 0.742304i 0.967128 − 0.254288i \(-0.0818413\pi\)
−0.0824851 + 0.996592i \(0.526286\pi\)
\(68\) 5.15923 − 2.23026i 0.625649 − 0.270459i
\(69\) −3.37763 − 0.685923i −0.406618 − 0.0825754i
\(70\) 4.93086 − 7.50340i 0.589351 − 0.896827i
\(71\) −2.21770 + 3.84117i −0.263193 + 0.455863i −0.967089 − 0.254440i \(-0.918109\pi\)
0.703896 + 0.710303i \(0.251442\pi\)
\(72\) 1.89314 − 8.27140i 0.223109 − 0.974794i
\(73\) 6.12067 + 10.6013i 0.716370 + 1.24079i 0.962429 + 0.271534i \(0.0875309\pi\)
−0.246059 + 0.969255i \(0.579136\pi\)
\(74\) −8.40549 − 0.486268i −0.977118 − 0.0565275i
\(75\) −14.1947 − 17.8320i −1.63906 − 2.05906i
\(76\) 2.05351 + 0.238394i 0.235554 + 0.0273457i
\(77\) −1.19390 + 6.77096i −0.136058 + 0.771622i
\(78\) −10.4157 − 2.75008i −1.17935 − 0.311385i
\(79\) 2.19872 + 2.62033i 0.247375 + 0.294811i 0.875416 − 0.483370i \(-0.160587\pi\)
−0.628041 + 0.778180i \(0.716143\pi\)
\(80\) −10.2001 − 13.6565i −1.14041 − 1.52684i
\(81\) 8.95204 + 0.927899i 0.994671 + 0.103100i
\(82\) −1.40612 + 5.94322i −0.155280 + 0.656319i
\(83\) 3.23224 + 3.85203i 0.354784 + 0.422816i 0.913688 − 0.406417i \(-0.133222\pi\)
−0.558903 + 0.829233i \(0.688778\pi\)
\(84\) 4.55264 + 2.43102i 0.496733 + 0.265245i
\(85\) 11.7938 + 2.07956i 1.27921 + 0.225560i
\(86\) 3.41044 − 0.399974i 0.367757 − 0.0431303i
\(87\) 1.10296 + 1.38559i 0.118250 + 0.148551i
\(88\) 11.3115 + 6.51305i 1.20581 + 0.694293i
\(89\) −3.14150 + 1.81374i −0.332998 + 0.192257i −0.657171 − 0.753741i \(-0.728247\pi\)
0.324173 + 0.945998i \(0.394914\pi\)
\(90\) 13.0641 − 12.4975i 1.37708 − 1.31736i
\(91\) 3.27615 − 5.67445i 0.343433 − 0.594844i
\(92\) −2.73334 − 2.89263i −0.284970 − 0.301578i
\(93\) −2.11723 + 10.4257i −0.219547 + 1.08109i
\(94\) −3.92327 − 5.26556i −0.404654 − 0.543101i
\(95\) 3.37421 + 2.83130i 0.346187 + 0.290485i
\(96\) 7.28521 − 6.55177i 0.743543 − 0.668688i
\(97\) −5.20281 − 1.89367i −0.528265 − 0.192273i 0.0640983 − 0.997944i \(-0.479583\pi\)
−0.592364 + 0.805671i \(0.701805\pi\)
\(98\) 4.63733 − 4.91913i 0.468441 − 0.496907i
\(99\) −5.40028 + 12.7477i −0.542748 + 1.28119i
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 216.2.v.b.59.6 yes 192
3.2 odd 2 648.2.v.b.611.27 192
4.3 odd 2 864.2.bh.b.815.16 192
8.3 odd 2 inner 216.2.v.b.59.23 yes 192
8.5 even 2 864.2.bh.b.815.15 192
24.11 even 2 648.2.v.b.611.10 192
27.11 odd 18 inner 216.2.v.b.11.23 yes 192
27.16 even 9 648.2.v.b.35.10 192
108.11 even 18 864.2.bh.b.335.15 192
216.11 even 18 inner 216.2.v.b.11.6 ✓ 192
216.43 odd 18 648.2.v.b.35.27 192
216.173 odd 18 864.2.bh.b.335.16 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.6 ✓ 192 216.11 even 18 inner
216.2.v.b.11.23 yes 192 27.11 odd 18 inner
216.2.v.b.59.6 yes 192 1.1 even 1 trivial
216.2.v.b.59.23 yes 192 8.3 odd 2 inner
648.2.v.b.35.10 192 27.16 even 9
648.2.v.b.35.27 192 216.43 odd 18
648.2.v.b.611.10 192 24.11 even 2
648.2.v.b.611.27 192 3.2 odd 2
864.2.bh.b.335.15 192 108.11 even 18
864.2.bh.b.335.16 192 216.173 odd 18
864.2.bh.b.815.15 192 8.5 even 2
864.2.bh.b.815.16 192 4.3 odd 2