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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(47,864)] 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("864.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.bh (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: \(6.89907473464\)
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 815.16
Character \(\chi\) \(=\) 864.815
Dual form 864.2.bh.b.335.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0447181 + 1.73147i) q^{3} +(4.00434 - 1.45746i) q^{5} +(1.46723 - 0.258712i) q^{7} +(-2.99600 - 0.154856i) q^{9} +(-1.57835 + 4.33649i) q^{11} +(-2.82693 + 3.36900i) q^{13} +(2.34449 + 6.99858i) q^{15} +(2.43381 + 1.40516i) q^{17} +(-0.516826 - 0.895168i) q^{19} +(0.382341 + 2.55204i) q^{21} +(-0.345539 + 1.95965i) q^{23} +(10.0803 - 8.45837i) q^{25} +(0.402105 - 5.18057i) q^{27} +(-0.783266 + 0.657238i) q^{29} +(6.04883 + 1.06657i) q^{31} +(-7.43793 - 2.92679i) q^{33} +(5.49822 - 3.17440i) q^{35} +(5.15589 + 2.97676i) q^{37} +(-5.70692 - 5.04540i) q^{39} +(2.77589 - 3.30817i) q^{41} +(2.28164 + 0.830449i) q^{43} +(-12.2227 + 3.74645i) q^{45} +(-0.806279 - 4.57264i) q^{47} +(-4.49202 + 1.63496i) q^{49} +(-2.54184 + 4.15125i) q^{51} +3.14544 q^{53} +19.6651i q^{55} +(1.57307 - 0.854840i) q^{57} +(-2.77458 - 7.62310i) q^{59} +(-2.25575 + 0.397750i) q^{61} +(-4.43588 + 0.547892i) q^{63} +(-6.40978 + 17.6107i) q^{65} +(-7.24112 - 6.07602i) q^{67} +(-3.37763 - 0.685923i) q^{69} +(2.21770 - 3.84117i) q^{71} +(6.12067 + 10.6013i) q^{73} +(14.1947 + 17.8320i) q^{75} +(-1.19390 + 6.77096i) q^{77} +(-2.19872 - 2.62033i) q^{79} +(8.95204 + 0.927899i) q^{81} +(-3.23224 - 3.85203i) q^{83} +(11.7938 + 2.07956i) q^{85} +(-1.10296 - 1.38559i) q^{87} +(-3.14150 + 1.81374i) q^{89} +(-3.27615 + 5.67445i) q^{91} +(-2.11723 + 10.4257i) q^{93} +(-3.37421 - 2.83130i) q^{95} +(-5.20281 - 1.89367i) q^{97} +(5.40028 - 12.7477i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 12 q^{3} - 12 q^{9} + 30 q^{11} - 18 q^{17} + 6 q^{19} - 12 q^{25} - 18 q^{27} - 30 q^{33} + 18 q^{35} + 18 q^{41} + 42 q^{43} - 12 q^{49} + 18 q^{51} - 36 q^{57} + 84 q^{59} - 12 q^{65} - 30 q^{67}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.0447181 + 1.73147i −0.0258180 + 0.999667i
\(4\) 0 0
\(5\) 4.00434 − 1.45746i 1.79079 − 0.651796i 0.791626 − 0.611006i \(-0.209235\pi\)
0.999168 − 0.0407891i \(-0.0129872\pi\)
\(6\) 0 0
\(7\) 1.46723 − 0.258712i 0.554561 − 0.0977840i 0.110657 − 0.993859i \(-0.464704\pi\)
0.443903 + 0.896075i \(0.353593\pi\)
\(8\) 0 0
\(9\) −2.99600 − 0.154856i −0.998667 − 0.0516188i
\(10\) 0 0
\(11\) −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 \(0.431005\pi\)
\(14\) 0 0
\(15\) 2.34449 + 6.99858i 0.605344 + 1.80702i
\(16\) 0 0
\(17\) 2.43381 + 1.40516i 0.590286 + 0.340802i 0.765211 − 0.643780i \(-0.222635\pi\)
−0.174924 + 0.984582i \(0.555968\pi\)
\(18\) 0 0
\(19\) −0.516826 − 0.895168i −0.118568 − 0.205366i 0.800632 − 0.599156i \(-0.204497\pi\)
−0.919200 + 0.393790i \(0.871164\pi\)
\(20\) 0 0
\(21\) 0.382341 + 2.55204i 0.0834337 + 0.556900i
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 0.402105 − 5.18057i 0.0773851 − 0.997001i
\(28\) 0 0
\(29\) −0.783266 + 0.657238i −0.145449 + 0.122046i −0.712609 − 0.701562i \(-0.752487\pi\)
0.567160 + 0.823608i \(0.308042\pi\)
\(30\) 0 0
\(31\) 6.04883 + 1.06657i 1.08640 + 0.191562i 0.688045 − 0.725668i \(-0.258469\pi\)
0.398358 + 0.917230i \(0.369580\pi\)
\(32\) 0 0
\(33\) −7.43793 − 2.92679i −1.29478 − 0.509490i
\(34\) 0 0
\(35\) 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 \(0.503891\pi\)
\(38\) 0 0
\(39\) −5.70692 − 5.04540i −0.913838 − 0.807911i
\(40\) 0 0
\(41\) 2.77589 − 3.30817i 0.433520 − 0.516650i −0.504414 − 0.863462i \(-0.668291\pi\)
0.937935 + 0.346812i \(0.112736\pi\)
\(42\) 0 0
\(43\) 2.28164 + 0.830449i 0.347947 + 0.126642i 0.510081 − 0.860126i \(-0.329615\pi\)
−0.162134 + 0.986769i \(0.551838\pi\)
\(44\) 0 0
\(45\) −12.2227 + 3.74645i −1.82205 + 0.558488i
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) −2.54184 + 4.15125i −0.355928 + 0.581291i
\(52\) 0 0
\(53\) 3.14544 0.432059 0.216030 − 0.976387i \(-0.430689\pi\)
0.216030 + 0.976387i \(0.430689\pi\)
\(54\) 0 0
\(55\) 19.6651i 2.65165i
\(56\) 0 0
\(57\) 1.57307 − 0.854840i 0.208358 − 0.113226i
\(58\) 0 0
\(59\) −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 \(0.491290\pi\)
\(62\) 0 0
\(63\) −4.43588 + 0.547892i −0.558869 + 0.0690279i
\(64\) 0 0
\(65\) −6.40978 + 17.6107i −0.795035 + 2.18434i
\(66\) 0 0
\(67\) −7.24112 − 6.07602i −0.884643 − 0.742304i 0.0824851 − 0.996592i \(-0.473714\pi\)
−0.967128 + 0.254288i \(0.918159\pi\)
\(68\) 0 0
\(69\) −3.37763 − 0.685923i −0.406618 − 0.0825754i
\(70\) 0 0
\(71\) 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 \(0.579136\pi\)
\(74\) 0 0
\(75\) 14.1947 + 17.8320i 1.63906 + 2.05906i
\(76\) 0 0
\(77\) −1.19390 + 6.77096i −0.136058 + 0.771622i
\(78\) 0 0
\(79\) −2.19872 − 2.62033i −0.247375 − 0.294811i 0.628041 − 0.778180i \(-0.283857\pi\)
−0.875416 + 0.483370i \(0.839413\pi\)
\(80\) 0 0
\(81\) 8.95204 + 0.927899i 0.994671 + 0.103100i
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) −1.10296 − 1.38559i −0.118250 − 0.148551i
\(88\) 0 0
\(89\) −3.14150 + 1.81374i −0.332998 + 0.192257i −0.657171 − 0.753741i \(-0.728247\pi\)
0.324173 + 0.945998i \(0.394914\pi\)
\(90\) 0 0
\(91\) −3.27615 + 5.67445i −0.343433 + 0.594844i
\(92\) 0 0
\(93\) −2.11723 + 10.4257i −0.219547 + 1.08109i
\(94\) 0 0
\(95\) −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 \(0.701805\pi\)
\(98\) 0 0
\(99\) 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 864.2.bh.b.815.16 192
4.3 odd 2 216.2.v.b.59.6 yes 192
8.3 odd 2 inner 864.2.bh.b.815.15 192
8.5 even 2 216.2.v.b.59.23 yes 192
12.11 even 2 648.2.v.b.611.27 192
24.5 odd 2 648.2.v.b.611.10 192
27.11 odd 18 inner 864.2.bh.b.335.15 192
108.11 even 18 216.2.v.b.11.23 yes 192
108.43 odd 18 648.2.v.b.35.10 192
216.11 even 18 inner 864.2.bh.b.335.16 192
216.173 odd 18 216.2.v.b.11.6 ✓ 192
216.205 even 18 648.2.v.b.35.27 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.v.b.11.6 ✓ 192 216.173 odd 18
216.2.v.b.11.23 yes 192 108.11 even 18
216.2.v.b.59.6 yes 192 4.3 odd 2
216.2.v.b.59.23 yes 192 8.5 even 2
648.2.v.b.35.10 192 108.43 odd 18
648.2.v.b.35.27 192 216.205 even 18
648.2.v.b.611.10 192 24.5 odd 2
648.2.v.b.611.27 192 12.11 even 2
864.2.bh.b.335.15 192 27.11 odd 18 inner
864.2.bh.b.335.16 192 216.11 even 18 inner
864.2.bh.b.815.15 192 8.3 odd 2 inner
864.2.bh.b.815.16 192 1.1 even 1 trivial