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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] 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("864.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 109.20
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.567462 + 1.29537i) q^{2} +(-1.35597 + 1.47015i) q^{4} +(-2.16011 + 0.894745i) q^{5} +(1.46314 - 1.46314i) q^{7} +(-2.67385 - 0.922239i) q^{8} +(-2.38480 - 2.29041i) q^{10} +(-2.39513 - 5.78235i) q^{11} +(0.0552727 + 0.0228947i) q^{13} +(2.72558 + 1.06503i) q^{14} +(-0.322664 - 3.98696i) q^{16} +3.26457i q^{17} +(-0.456261 - 0.188990i) q^{19} +(1.61364 - 4.38892i) q^{20} +(6.13115 - 6.38384i) q^{22} +(-6.25659 - 6.25659i) q^{23} +(0.329954 - 0.329954i) q^{25} +(0.00170799 + 0.0845906i) q^{26} +(0.167050 + 4.13501i) q^{28} +(3.28735 - 7.93636i) q^{29} +6.60653 q^{31} +(4.98150 - 2.68042i) q^{32} +(-4.22883 + 1.85252i) q^{34} +(-1.85140 + 4.46967i) q^{35} +(-0.590451 + 0.244573i) q^{37} +(-0.0140990 - 0.698273i) q^{38} +(6.60097 - 0.400280i) q^{40} +(-1.11963 - 1.11963i) q^{41} +(-3.75382 - 9.06253i) q^{43} +(11.7486 + 4.31953i) q^{44} +(4.55424 - 11.6550i) q^{46} +12.0850i q^{47} +2.71846i q^{49} +(0.614649 + 0.240177i) q^{50} +(-0.108607 + 0.0502144i) q^{52} +(0.152093 + 0.367185i) q^{53} +(10.3475 + 10.3475i) q^{55} +(-5.26157 + 2.56285i) q^{56} +(12.1460 - 0.245242i) q^{58} +(-4.24644 + 1.75893i) q^{59} +(0.578617 - 1.39691i) q^{61} +(3.74895 + 8.55791i) q^{62} +(6.29895 + 4.93186i) q^{64} -0.139880 q^{65} +(3.44098 - 8.30727i) q^{67} +(-4.79940 - 4.42667i) q^{68} +(-6.84048 + 0.138117i) q^{70} +(2.39497 - 2.39497i) q^{71} +(-9.83836 - 9.83836i) q^{73} +(-0.651871 - 0.626068i) q^{74} +(0.896522 - 0.414506i) q^{76} +(-11.9648 - 4.95597i) q^{77} -3.62239i q^{79} +(4.26431 + 8.32356i) q^{80} +(0.814992 - 2.08569i) q^{82} +(-9.86593 - 4.08660i) q^{83} +(-2.92096 - 7.05182i) q^{85} +(9.60920 - 10.0052i) q^{86} +(1.07150 + 17.6700i) q^{88} +(-6.31778 + 6.31778i) q^{89} +(0.114370 - 0.0473735i) q^{91} +(17.6819 - 0.714329i) q^{92} +(-15.6546 + 6.85778i) q^{94} +1.15467 q^{95} -4.47320 q^{97} +(-3.52141 + 1.54262i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.567462 + 1.29537i 0.401256 + 0.915966i
\(3\) 0 0
\(4\) −1.35597 + 1.47015i −0.677987 + 0.735073i
\(5\) −2.16011 + 0.894745i −0.966029 + 0.400142i −0.809232 − 0.587489i \(-0.800117\pi\)
−0.156796 + 0.987631i \(0.550117\pi\)
\(6\) 0 0
\(7\) 1.46314 − 1.46314i 0.553014 − 0.553014i −0.374295 − 0.927309i \(-0.622115\pi\)
0.927309 + 0.374295i \(0.122115\pi\)
\(8\) −2.67385 − 0.922239i −0.945349 − 0.326061i
\(9\) 0 0
\(10\) −2.38480 − 2.29041i −0.754141 − 0.724290i
\(11\) −2.39513 − 5.78235i −0.722158 − 1.74344i −0.667108 − 0.744961i \(-0.732468\pi\)
−0.0550506 − 0.998484i \(-0.517532\pi\)
\(12\) 0 0
\(13\) 0.0552727 + 0.0228947i 0.0153299 + 0.00634985i 0.390335 − 0.920673i \(-0.372359\pi\)
−0.375005 + 0.927023i \(0.622359\pi\)
\(14\) 2.72558 + 1.06503i 0.728442 + 0.284642i
\(15\) 0 0
\(16\) −0.322664 − 3.98696i −0.0806661 − 0.996741i
\(17\) 3.26457i 0.791775i 0.918299 + 0.395887i \(0.129563\pi\)
−0.918299 + 0.395887i \(0.870437\pi\)
\(18\) 0 0
\(19\) −0.456261 − 0.188990i −0.104674 − 0.0433572i 0.329732 − 0.944075i \(-0.393042\pi\)
−0.434406 + 0.900717i \(0.643042\pi\)
\(20\) 1.61364 − 4.38892i 0.360821 − 0.981393i
\(21\) 0 0
\(22\) 6.13115 − 6.38384i 1.30717 − 1.36104i
\(23\) −6.25659 − 6.25659i −1.30459 − 1.30459i −0.925262 − 0.379328i \(-0.876155\pi\)
−0.379328 − 0.925262i \(-0.623845\pi\)
\(24\) 0 0
\(25\) 0.329954 − 0.329954i 0.0659908 − 0.0659908i
\(26\) 0.00170799 + 0.0845906i 0.000334964 + 0.0165896i
\(27\) 0 0
\(28\) 0.167050 + 4.13501i 0.0315694 + 0.781443i
\(29\) 3.28735 − 7.93636i 0.610445 − 1.47374i −0.252068 − 0.967710i \(-0.581111\pi\)
0.862513 − 0.506035i \(-0.168889\pi\)
\(30\) 0 0
\(31\) 6.60653 1.18657 0.593284 − 0.804994i \(-0.297831\pi\)
0.593284 + 0.804994i \(0.297831\pi\)
\(32\) 4.98150 − 2.68042i 0.880613 − 0.473836i
\(33\) 0 0
\(34\) −4.22883 + 1.85252i −0.725239 + 0.317704i
\(35\) −1.85140 + 4.46967i −0.312943 + 0.755512i
\(36\) 0 0
\(37\) −0.590451 + 0.244573i −0.0970696 + 0.0402076i −0.430690 − 0.902500i \(-0.641730\pi\)
0.333620 + 0.942708i \(0.391730\pi\)
\(38\) −0.0140990 − 0.698273i −0.00228715 − 0.113275i
\(39\) 0 0
\(40\) 6.60097 − 0.400280i 1.04370 − 0.0632898i
\(41\) −1.11963 − 1.11963i −0.174857 − 0.174857i 0.614252 − 0.789110i \(-0.289458\pi\)
−0.789110 + 0.614252i \(0.789458\pi\)
\(42\) 0 0
\(43\) −3.75382 − 9.06253i −0.572453 − 1.38202i −0.899461 − 0.437002i \(-0.856040\pi\)
0.327008 − 0.945022i \(-0.393960\pi\)
\(44\) 11.7486 + 4.31953i 1.77117 + 0.651194i
\(45\) 0 0
\(46\) 4.55424 − 11.6550i 0.671486 − 1.71843i
\(47\) 12.0850i 1.76278i 0.472389 + 0.881390i \(0.343392\pi\)
−0.472389 + 0.881390i \(0.656608\pi\)
\(48\) 0 0
\(49\) 2.71846i 0.388351i
\(50\) 0.614649 + 0.240177i 0.0869245 + 0.0339661i
\(51\) 0 0
\(52\) −0.108607 + 0.0502144i −0.0150611 + 0.00696348i
\(53\) 0.152093 + 0.367185i 0.0208916 + 0.0504367i 0.933981 − 0.357322i \(-0.116310\pi\)
−0.913090 + 0.407759i \(0.866310\pi\)
\(54\) 0 0
\(55\) 10.3475 + 10.3475i 1.39525 + 1.39525i
\(56\) −5.26157 + 2.56285i −0.703107 + 0.342475i
\(57\) 0 0
\(58\) 12.1460 − 0.245242i 1.59484 − 0.0322018i
\(59\) −4.24644 + 1.75893i −0.552840 + 0.228994i −0.641573 − 0.767062i \(-0.721718\pi\)
0.0887336 + 0.996055i \(0.471718\pi\)
\(60\) 0 0
\(61\) 0.578617 − 1.39691i 0.0740844 − 0.178855i −0.882499 − 0.470314i \(-0.844141\pi\)
0.956583 + 0.291459i \(0.0941406\pi\)
\(62\) 3.74895 + 8.55791i 0.476117 + 1.08686i
\(63\) 0 0
\(64\) 6.29895 + 4.93186i 0.787369 + 0.616482i
\(65\) −0.139880 −0.0173500
\(66\) 0 0
\(67\) 3.44098 − 8.30727i 0.420383 − 1.01489i −0.561852 − 0.827238i \(-0.689911\pi\)
0.982235 − 0.187657i \(-0.0600892\pi\)
\(68\) −4.79940 − 4.42667i −0.582012 − 0.536813i
\(69\) 0 0
\(70\) −6.84048 + 0.138117i −0.817593 + 0.0165082i
\(71\) 2.39497 − 2.39497i 0.284231 − 0.284231i −0.550563 − 0.834794i \(-0.685587\pi\)
0.834794 + 0.550563i \(0.185587\pi\)
\(72\) 0 0
\(73\) −9.83836 − 9.83836i −1.15149 − 1.15149i −0.986253 − 0.165240i \(-0.947160\pi\)
−0.165240 − 0.986253i \(-0.552840\pi\)
\(74\) −0.651871 − 0.626068i −0.0757785 − 0.0727790i
\(75\) 0 0
\(76\) 0.896522 − 0.414506i 0.102838 − 0.0475471i
\(77\) −11.9648 − 4.95597i −1.36351 − 0.564786i
\(78\) 0 0
\(79\) − 3.62239i − 0.407550i −0.979018 − 0.203775i \(-0.934679\pi\)
0.979018 − 0.203775i \(-0.0653212\pi\)
\(80\) 4.26431 + 8.32356i 0.476764 + 0.930603i
\(81\) 0 0
\(82\) 0.814992 − 2.08569i 0.0900008 − 0.230326i
\(83\) −9.86593 − 4.08660i −1.08293 − 0.448563i −0.231392 − 0.972861i \(-0.574328\pi\)
−0.851535 + 0.524298i \(0.824328\pi\)
\(84\) 0 0
\(85\) −2.92096 − 7.05182i −0.316822 − 0.764877i
\(86\) 9.60920 − 10.0052i 1.03619 − 1.07889i
\(87\) 0 0
\(88\) 1.07150 + 17.6700i 0.114222 + 1.88363i
\(89\) −6.31778 + 6.31778i −0.669684 + 0.669684i −0.957643 − 0.287959i \(-0.907023\pi\)
0.287959 + 0.957643i \(0.407023\pi\)
\(90\) 0 0
\(91\) 0.114370 − 0.0473735i 0.0119892 − 0.00496609i
\(92\) 17.6819 − 0.714329i 1.84347 − 0.0744739i
\(93\) 0 0
\(94\) −15.6546 + 6.85778i −1.61465 + 0.707326i
\(95\) 1.15467 0.118467
\(96\) 0 0
\(97\) −4.47320 −0.454185 −0.227093 − 0.973873i \(-0.572922\pi\)
−0.227093 + 0.973873i \(0.572922\pi\)
\(98\) −3.52141 + 1.54262i −0.355716 + 0.155828i
\(99\) 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 864.2.v.a.109.20 yes 128
3.2 odd 2 inner 864.2.v.a.109.13 ✓ 128
32.5 even 8 inner 864.2.v.a.325.20 yes 128
96.5 odd 8 inner 864.2.v.a.325.13 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.13 ✓ 128 3.2 odd 2 inner
864.2.v.a.109.20 yes 128 1.1 even 1 trivial
864.2.v.a.325.13 yes 128 96.5 odd 8 inner
864.2.v.a.325.20 yes 128 32.5 even 8 inner