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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [648,2,Mod(35,648)] 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("648.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.v (of order \(18\), degree \(6\), not 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 611.27
Character \(\chi\) \(=\) 648.611
Dual form 648.2.v.b.35.27

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.26354 - 0.635192i) q^{2} +(1.19306 - 1.60518i) q^{4} +(-4.00434 + 1.45746i) q^{5} +(-1.46723 + 0.258712i) q^{7} +(0.487886 - 2.78603i) q^{8} +(-4.13387 + 4.38508i) q^{10} +(-1.57835 + 4.33649i) q^{11} +(-2.82693 + 3.36900i) q^{13} +(-1.68957 + 1.25886i) q^{14} +(-1.15320 - 3.83016i) q^{16} +(-2.43381 - 1.40516i) q^{17} +(0.516826 + 0.895168i) q^{19} +(-2.43794 + 8.16652i) q^{20} +(0.760191 + 6.48188i) q^{22} +(-0.345539 + 1.95965i) q^{23} +(10.0803 - 8.45837i) q^{25} +(-1.43197 + 6.05250i) q^{26} +(-1.33522 + 2.66383i) q^{28} +(0.783266 - 0.657238i) q^{29} +(-6.04883 - 1.06657i) q^{31} +(-3.89000 - 4.10705i) q^{32} +(-3.96777 - 0.229540i) q^{34} +(5.49822 - 3.17440i) q^{35} +(5.15589 + 2.97676i) q^{37} +(1.22163 + 0.802797i) q^{38} +(2.10687 + 11.8673i) q^{40} +(-2.77589 + 3.30817i) q^{41} +(-2.28164 - 0.830449i) q^{43} +(5.07777 + 7.70724i) q^{44} +(0.808150 + 2.69557i) q^{46} +(-0.806279 - 4.57264i) q^{47} +(-4.49202 + 1.63496i) q^{49} +(7.36416 - 17.0904i) q^{50} +(2.03515 + 8.55715i) q^{52} -3.14544 q^{53} -19.6651i q^{55} +(0.00493996 + 4.21397i) q^{56} +(0.572215 - 1.32797i) q^{58} +(-2.77458 - 7.62310i) q^{59} +(-2.25575 + 0.397750i) q^{61} +(-8.32042 + 2.49451i) q^{62} +(-7.52394 - 2.71853i) q^{64} +(6.40978 - 17.6107i) q^{65} +(7.24112 + 6.07602i) q^{67} +(-5.15923 + 2.23026i) q^{68} +(4.93086 - 7.50340i) q^{70} +(2.21770 - 3.84117i) q^{71} +(6.12067 + 10.6013i) q^{73} +(8.40549 + 0.486268i) q^{74} +(2.05351 + 0.238394i) q^{76} +(1.19390 - 6.77096i) q^{77} +(2.19872 + 2.62033i) q^{79} +(10.2001 + 13.6565i) q^{80} +(-1.40612 + 5.94322i) q^{82} +(-3.23224 - 3.85203i) q^{83} +(11.7938 + 2.07956i) q^{85} +(-3.41044 + 0.399974i) q^{86} +(11.3115 + 6.51305i) q^{88} +(3.14150 - 1.81374i) q^{89} +(3.27615 - 5.67445i) q^{91} +(2.73334 + 2.89263i) q^{92} +(-3.92327 - 5.26556i) q^{94} +(-3.37421 - 2.83130i) q^{95} +(-5.20281 - 1.89367i) q^{97} +(-4.63733 + 4.91913i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 6 q^{2} - 6 q^{4} + 9 q^{8} - 3 q^{10} + 30 q^{11} - 9 q^{14} - 6 q^{16} + 18 q^{17} - 6 q^{19} + 27 q^{20} - 18 q^{22} - 12 q^{25} - 12 q^{28} + 36 q^{32} + 12 q^{34} + 18 q^{35} + 102 q^{38} + 9 q^{40}+ \cdots - 162 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{18}\right)\)

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 0
\(4\) 1.19306 − 1.60518i 0.596532 − 0.802590i
\(5\) −4.00434 + 1.45746i −1.79079 + 0.651796i −0.791626 + 0.611006i \(0.790765\pi\)
−0.999168 + 0.0407891i \(0.987013\pi\)
\(6\) 0 0
\(7\) −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
\(9\) 0 0
\(10\) −4.13387 + 4.38508i −1.30724 + 1.38668i
\(11\) −1.57835 + 4.33649i −0.475891 + 1.30750i 0.437060 + 0.899432i \(0.356020\pi\)
−0.912952 + 0.408068i \(0.866203\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) −1.15320 − 3.83016i −0.288300 − 0.957540i
\(17\) −2.43381 − 1.40516i −0.590286 − 0.340802i 0.174924 − 0.984582i \(-0.444032\pi\)
−0.765211 + 0.643780i \(0.777365\pi\)
\(18\) 0 0
\(19\) 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 0
\(22\) 0.760191 + 6.48188i 0.162073 + 1.38194i
\(23\) −0.345539 + 1.95965i −0.0720498 + 0.408615i 0.927357 + 0.374178i \(0.122075\pi\)
−0.999407 + 0.0344371i \(0.989036\pi\)
\(24\) 0 0
\(25\) 10.0803 − 8.45837i 2.01606 − 1.69167i
\(26\) −1.43197 + 6.05250i −0.280833 + 1.18699i
\(27\) 0 0
\(28\) −1.33522 + 2.66383i −0.252332 + 0.503416i
\(29\) 0.783266 − 0.657238i 0.145449 − 0.122046i −0.567160 − 0.823608i \(-0.691958\pi\)
0.712609 + 0.701562i \(0.247513\pi\)
\(30\) 0 0
\(31\) −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
\(33\) 0 0
\(34\) −3.96777 − 0.229540i −0.680466 − 0.0393659i
\(35\) 5.49822 − 3.17440i 0.929368 − 0.536571i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) 2.10687 + 11.8673i 0.333125 + 1.87638i
\(41\) −2.77589 + 3.30817i −0.433520 + 0.516650i −0.937935 − 0.346812i \(-0.887264\pi\)
0.504414 + 0.863462i \(0.331709\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) 0.808150 + 2.69557i 0.119155 + 0.397441i
\(47\) −0.806279 − 4.57264i −0.117608 − 0.666987i −0.985426 − 0.170105i \(-0.945589\pi\)
0.867818 − 0.496882i \(-0.165522\pi\)
\(48\) 0 0
\(49\) −4.49202 + 1.63496i −0.641717 + 0.233566i
\(50\) 7.36416 − 17.0904i 1.04145 − 2.41695i
\(51\) 0 0
\(52\) 2.03515 + 8.55715i 0.282224 + 1.18666i
\(53\) −3.14544 −0.432059 −0.216030 − 0.976387i \(-0.569311\pi\)
−0.216030 + 0.976387i \(0.569311\pi\)
\(54\) 0 0
\(55\) − 19.6651i − 2.65165i
\(56\) 0.00493996 + 4.21397i 0.000660129 + 0.563115i
\(57\) 0 0
\(58\) 0.572215 − 1.32797i 0.0751355 − 0.174371i
\(59\) −2.77458 − 7.62310i −0.361220 − 0.992443i −0.978599 − 0.205777i \(-0.934028\pi\)
0.617379 − 0.786666i \(-0.288195\pi\)
\(60\) 0 0
\(61\) −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
\(63\) 0 0
\(64\) −7.52394 − 2.71853i −0.940492 − 0.339816i
\(65\) 6.40978 − 17.6107i 0.795035 − 2.18434i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) 4.93086 − 7.50340i 0.589351 − 0.896827i
\(71\) 2.21770 − 3.84117i 0.263193 − 0.455863i −0.703896 − 0.710303i \(-0.748558\pi\)
0.967089 + 0.254440i \(0.0818912\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) 2.05351 + 0.238394i 0.235554 + 0.0273457i
\(77\) 1.19390 − 6.77096i 0.136058 − 0.771622i
\(78\) 0 0
\(79\) 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
\(81\) 0 0
\(82\) −1.40612 + 5.94322i −0.155280 + 0.656319i
\(83\) −3.23224 − 3.85203i −0.354784 − 0.422816i 0.558903 − 0.829233i \(-0.311222\pi\)
−0.913688 + 0.406417i \(0.866778\pi\)
\(84\) 0 0
\(85\) 11.7938 + 2.07956i 1.27921 + 0.225560i
\(86\) −3.41044 + 0.399974i −0.367757 + 0.0431303i
\(87\) 0 0
\(88\) 11.3115 + 6.51305i 1.20581 + 0.694293i
\(89\) 3.14150 − 1.81374i 0.332998 − 0.192257i −0.324173 − 0.945998i \(-0.605086\pi\)
0.657171 + 0.753741i \(0.271753\pi\)
\(90\) 0 0
\(91\) 3.27615 − 5.67445i 0.343433 − 0.594844i
\(92\) 2.73334 + 2.89263i 0.284970 + 0.301578i
\(93\) 0 0
\(94\) −3.92327 − 5.26556i −0.404654 − 0.543101i
\(95\) −3.37421 − 2.83130i −0.346187 − 0.290485i
\(96\) 0 0
\(97\) −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
\(99\) 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 648.2.v.b.611.27 192
3.2 odd 2 216.2.v.b.59.6 yes 192
8.3 odd 2 inner 648.2.v.b.611.10 192
12.11 even 2 864.2.bh.b.815.16 192
24.5 odd 2 864.2.bh.b.815.15 192
24.11 even 2 216.2.v.b.59.23 yes 192
27.11 odd 18 inner 648.2.v.b.35.10 192
27.16 even 9 216.2.v.b.11.23 yes 192
108.43 odd 18 864.2.bh.b.335.15 192
216.11 even 18 inner 648.2.v.b.35.27 192
216.43 odd 18 216.2.v.b.11.6 ✓ 192
216.205 even 18 864.2.bh.b.335.16 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.6 ✓ 192 216.43 odd 18
216.2.v.b.11.23 yes 192 27.16 even 9
216.2.v.b.59.6 yes 192 3.2 odd 2
216.2.v.b.59.23 yes 192 24.11 even 2
648.2.v.b.35.10 192 27.11 odd 18 inner
648.2.v.b.35.27 192 216.11 even 18 inner
648.2.v.b.611.10 192 8.3 odd 2 inner
648.2.v.b.611.27 192 1.1 even 1 trivial
864.2.bh.b.335.15 192 108.43 odd 18
864.2.bh.b.335.16 192 216.205 even 18
864.2.bh.b.815.15 192 24.5 odd 2
864.2.bh.b.815.16 192 12.11 even 2