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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 109.9
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.957543 - 1.04073i) q^{2} +(-0.166221 + 1.99308i) q^{4} +(3.57149 - 1.47936i) q^{5} +(1.49427 - 1.49427i) q^{7} +(2.23341 - 1.73547i) q^{8} +(-4.95946 - 2.30039i) q^{10} +(-2.05347 - 4.95752i) q^{11} +(-5.69413 - 2.35859i) q^{13} +(-2.98596 - 0.124297i) q^{14} +(-3.94474 - 0.662583i) q^{16} +3.31060i q^{17} +(-2.88336 - 1.19433i) q^{19} +(2.35482 + 7.36416i) q^{20} +(-3.19313 + 6.88415i) q^{22} +(0.270391 + 0.270391i) q^{23} +(7.03148 - 7.03148i) q^{25} +(2.99774 + 8.18448i) q^{26} +(2.72983 + 3.22659i) q^{28} +(1.92998 - 4.65940i) q^{29} -2.53582 q^{31} +(3.08769 + 4.73985i) q^{32} +(3.44543 - 3.17005i) q^{34} +(3.12621 - 7.54735i) q^{35} +(-2.34297 + 0.970490i) q^{37} +(1.51797 + 4.14440i) q^{38} +(5.40923 - 9.50223i) q^{40} +(7.10581 + 7.10581i) q^{41} +(0.386274 + 0.932547i) q^{43} +(10.2221 - 3.26869i) q^{44} +(0.0224918 - 0.540314i) q^{46} -6.02870i q^{47} +2.53429i q^{49} +(-14.0508 - 0.584896i) q^{50} +(5.64734 - 10.9568i) q^{52} +(0.373721 + 0.902243i) q^{53} +(-14.6679 - 14.6679i) q^{55} +(0.744064 - 5.93061i) q^{56} +(-6.69720 + 2.45299i) q^{58} +(-7.07472 + 2.93044i) q^{59} +(-3.06103 + 7.38997i) q^{61} +(2.42816 + 2.63910i) q^{62} +(1.97628 - 7.75205i) q^{64} -23.8257 q^{65} +(1.29458 - 3.12539i) q^{67} +(-6.59830 - 0.550292i) q^{68} +(-10.8482 + 3.97338i) q^{70} +(6.96387 - 6.96387i) q^{71} +(11.2517 + 11.2517i) q^{73} +(3.25351 + 1.50910i) q^{74} +(2.85966 - 5.54824i) q^{76} +(-10.4764 - 4.33945i) q^{77} +0.302934i q^{79} +(-15.0688 + 3.46928i) q^{80} +(0.591078 - 14.1993i) q^{82} +(10.2268 + 4.23610i) q^{83} +(4.89757 + 11.8238i) q^{85} +(0.600652 - 1.29496i) q^{86} +(-13.1899 - 7.50846i) q^{88} +(9.29031 - 9.29031i) q^{89} +(-12.0330 + 4.98422i) q^{91} +(-0.583856 + 0.493967i) q^{92} +(-6.27422 + 5.77274i) q^{94} -12.0647 q^{95} +1.22161 q^{97} +(2.63750 - 2.42669i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.957543 − 1.04073i −0.677085 − 0.735904i
\(3\) 0 0
\(4\) −0.166221 + 1.99308i −0.0831105 + 0.996540i
\(5\) 3.57149 − 1.47936i 1.59722 − 0.661589i 0.606199 − 0.795313i \(-0.292694\pi\)
0.991019 + 0.133724i \(0.0426937\pi\)
\(6\) 0 0
\(7\) 1.49427 − 1.49427i 0.564783 − 0.564783i −0.365879 − 0.930662i \(-0.619232\pi\)
0.930662 + 0.365879i \(0.119232\pi\)
\(8\) 2.23341 − 1.73547i 0.789631 − 0.613582i
\(9\) 0 0
\(10\) −4.95946 − 2.30039i −1.56832 − 0.727447i
\(11\) −2.05347 − 4.95752i −0.619145 − 1.49475i −0.852698 − 0.522405i \(-0.825035\pi\)
0.233552 − 0.972344i \(-0.424965\pi\)
\(12\) 0 0
\(13\) −5.69413 − 2.35859i −1.57927 − 0.654154i −0.590970 − 0.806693i \(-0.701255\pi\)
−0.988297 + 0.152539i \(0.951255\pi\)
\(14\) −2.98596 − 0.124297i −0.798032 − 0.0332199i
\(15\) 0 0
\(16\) −3.94474 − 0.662583i −0.986185 − 0.165646i
\(17\) 3.31060i 0.802939i 0.915872 + 0.401470i \(0.131500\pi\)
−0.915872 + 0.401470i \(0.868500\pi\)
\(18\) 0 0
\(19\) −2.88336 − 1.19433i −0.661488 − 0.273997i 0.0265768 − 0.999647i \(-0.491539\pi\)
−0.688064 + 0.725650i \(0.741539\pi\)
\(20\) 2.35482 + 7.36416i 0.526555 + 1.64668i
\(21\) 0 0
\(22\) −3.19313 + 6.88415i −0.680778 + 1.46770i
\(23\) 0.270391 + 0.270391i 0.0563805 + 0.0563805i 0.734735 − 0.678354i \(-0.237307\pi\)
−0.678354 + 0.734735i \(0.737307\pi\)
\(24\) 0 0
\(25\) 7.03148 − 7.03148i 1.40630 − 1.40630i
\(26\) 2.99774 + 8.18448i 0.587904 + 1.60511i
\(27\) 0 0
\(28\) 2.72983 + 3.22659i 0.515890 + 0.609768i
\(29\) 1.92998 − 4.65940i 0.358389 − 0.865228i −0.637138 − 0.770750i \(-0.719882\pi\)
0.995527 − 0.0944781i \(-0.0301182\pi\)
\(30\) 0 0
\(31\) −2.53582 −0.455448 −0.227724 − 0.973726i \(-0.573128\pi\)
−0.227724 + 0.973726i \(0.573128\pi\)
\(32\) 3.08769 + 4.73985i 0.545832 + 0.837894i
\(33\) 0 0
\(34\) 3.44543 − 3.17005i 0.590887 − 0.543659i
\(35\) 3.12621 − 7.54735i 0.528427 − 1.27573i
\(36\) 0 0
\(37\) −2.34297 + 0.970490i −0.385182 + 0.159548i −0.566867 − 0.823809i \(-0.691845\pi\)
0.181685 + 0.983357i \(0.441845\pi\)
\(38\) 1.51797 + 4.14440i 0.246248 + 0.672311i
\(39\) 0 0
\(40\) 5.40923 − 9.50223i 0.855274 − 1.50243i
\(41\) 7.10581 + 7.10581i 1.10974 + 1.10974i 0.993184 + 0.116557i \(0.0371857\pi\)
0.116557 + 0.993184i \(0.462814\pi\)
\(42\) 0 0
\(43\) 0.386274 + 0.932547i 0.0589062 + 0.142212i 0.950592 − 0.310442i \(-0.100477\pi\)
−0.891686 + 0.452654i \(0.850477\pi\)
\(44\) 10.2221 − 3.26869i 1.54104 − 0.492774i
\(45\) 0 0
\(46\) 0.0224918 − 0.540314i 0.00331623 − 0.0796650i
\(47\) − 6.02870i − 0.879376i −0.898151 − 0.439688i \(-0.855089\pi\)
0.898151 − 0.439688i \(-0.144911\pi\)
\(48\) 0 0
\(49\) 2.53429i 0.362041i
\(50\) −14.0508 − 0.584896i −1.98708 − 0.0827167i
\(51\) 0 0
\(52\) 5.64734 − 10.9568i 0.783145 − 1.51944i
\(53\) 0.373721 + 0.902243i 0.0513346 + 0.123933i 0.947466 − 0.319855i \(-0.103634\pi\)
−0.896132 + 0.443788i \(0.853634\pi\)
\(54\) 0 0
\(55\) −14.6679 − 14.6679i −1.97782 − 1.97782i
\(56\) 0.744064 − 5.93061i 0.0994298 − 0.792511i
\(57\) 0 0
\(58\) −6.69720 + 2.45299i −0.879385 + 0.322093i
\(59\) −7.07472 + 2.93044i −0.921050 + 0.381511i −0.792276 − 0.610163i \(-0.791104\pi\)
−0.128774 + 0.991674i \(0.541104\pi\)
\(60\) 0 0
\(61\) −3.06103 + 7.38997i −0.391924 + 0.946189i 0.597596 + 0.801797i \(0.296123\pi\)
−0.989521 + 0.144392i \(0.953877\pi\)
\(62\) 2.42816 + 2.63910i 0.308377 + 0.335166i
\(63\) 0 0
\(64\) 1.97628 − 7.75205i 0.247035 − 0.969007i
\(65\) −23.8257 −2.95521
\(66\) 0 0
\(67\) 1.29458 − 3.12539i 0.158158 − 0.381827i −0.824860 − 0.565337i \(-0.808746\pi\)
0.983018 + 0.183510i \(0.0587461\pi\)
\(68\) −6.59830 − 0.550292i −0.800162 − 0.0667327i
\(69\) 0 0
\(70\) −10.8482 + 3.97338i −1.29661 + 0.474910i
\(71\) 6.96387 − 6.96387i 0.826459 − 0.826459i −0.160566 − 0.987025i \(-0.551332\pi\)
0.987025 + 0.160566i \(0.0513320\pi\)
\(72\) 0 0
\(73\) 11.2517 + 11.2517i 1.31691 + 1.31691i 0.916206 + 0.400708i \(0.131236\pi\)
0.400708 + 0.916206i \(0.368764\pi\)
\(74\) 3.25351 + 1.50910i 0.378213 + 0.175430i
\(75\) 0 0
\(76\) 2.85966 − 5.54824i 0.328026 − 0.636427i
\(77\) −10.4764 − 4.33945i −1.19389 − 0.494526i
\(78\) 0 0
\(79\) 0.302934i 0.0340828i 0.999855 + 0.0170414i \(0.00542471\pi\)
−0.999855 + 0.0170414i \(0.994575\pi\)
\(80\) −15.0688 + 3.46928i −1.68474 + 0.387877i
\(81\) 0 0
\(82\) 0.591078 − 14.1993i 0.0652737 − 1.56805i
\(83\) 10.2268 + 4.23610i 1.12254 + 0.464972i 0.865239 − 0.501359i \(-0.167166\pi\)
0.257302 + 0.966331i \(0.417166\pi\)
\(84\) 0 0
\(85\) 4.89757 + 11.8238i 0.531216 + 1.28247i
\(86\) 0.600652 − 1.29496i 0.0647700 − 0.139639i
\(87\) 0 0
\(88\) −13.1899 − 7.50846i −1.40605 − 0.800404i
\(89\) 9.29031 − 9.29031i 0.984771 − 0.984771i −0.0151150 − 0.999886i \(-0.504811\pi\)
0.999886 + 0.0151150i \(0.00481142\pi\)
\(90\) 0 0
\(91\) −12.0330 + 4.98422i −1.26140 + 0.522488i
\(92\) −0.583856 + 0.493967i −0.0608712 + 0.0514996i
\(93\) 0 0
\(94\) −6.27422 + 5.77274i −0.647136 + 0.595412i
\(95\) −12.0647 −1.23781
\(96\) 0 0
\(97\) 1.22161 0.124036 0.0620178 − 0.998075i \(-0.480246\pi\)
0.0620178 + 0.998075i \(0.480246\pi\)
\(98\) 2.63750 − 2.42669i 0.266427 − 0.245133i
\(99\) 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.109.9 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.24 yes 128
32.5 even 8 inner 864.2.v.a.325.9 yes 128
96.5 odd 8 inner 864.2.v.a.325.24 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.9 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.24 yes 128 3.2 odd 2 inner
864.2.v.a.325.9 yes 128 32.5 even 8 inner
864.2.v.a.325.24 yes 128 96.5 odd 8 inner