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Properties

Label 512.2.k.a.497.11
Level $512$
Weight $2$
Character 512.497
Analytic conductor $4.088$
Analytic rank $0$
Dimension $240$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [512,2,Mod(17,512)] 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("512.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.k (of order \(32\), degree \(16\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(15\) over \(\Q(\zeta_{32})\)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 497.11
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.30167 - 0.695755i) q^{3} +(-2.50395 - 3.05107i) q^{5} +(-1.90175 + 1.27071i) q^{7} +(-0.456450 + 0.683126i) q^{9} +(-4.47003 + 1.35597i) q^{11} +(-2.41987 - 1.98594i) q^{13} +(-5.38211 - 2.22934i) q^{15} +(4.12129 - 1.70710i) q^{17} +(-0.371315 - 3.77002i) q^{19} +(-1.59134 + 2.97719i) q^{21} +(-1.98987 + 0.395809i) q^{23} +(-2.06382 + 10.3755i) q^{25} +(-0.552861 + 5.61329i) q^{27} +(0.870012 - 2.86805i) q^{29} +(-0.377059 - 0.377059i) q^{31} +(-4.87507 + 4.87507i) q^{33} +(8.63892 + 2.62059i) q^{35} +(1.87150 + 0.184327i) q^{37} +(-4.53160 - 0.901391i) q^{39} +(-2.35748 - 11.8518i) q^{41} +(-2.04620 - 1.09372i) q^{43} +(3.22720 - 0.317851i) q^{45} +(-0.807983 - 1.95064i) q^{47} +(-0.676829 + 1.63401i) q^{49} +(4.17683 - 5.08948i) q^{51} +(-2.23072 - 7.35371i) q^{53} +(15.3299 + 10.2431i) q^{55} +(-3.10634 - 4.64897i) q^{57} +(9.34237 - 7.66709i) q^{59} +(4.03888 + 7.55620i) q^{61} -1.87915i q^{63} +12.3559i q^{65} +(-3.52705 - 6.59865i) q^{67} +(-2.31476 + 1.89967i) q^{69} +(-5.81321 - 8.70008i) q^{71} +(-4.18253 - 2.79468i) q^{73} +(4.53243 + 14.9414i) q^{75} +(6.77784 - 8.25882i) q^{77} +(-0.189443 + 0.457356i) q^{79} +(2.24261 + 5.41414i) q^{81} +(-8.89052 + 0.875640i) q^{83} +(-15.5280 - 8.29988i) q^{85} +(-0.862992 - 4.33855i) q^{87} +(5.75001 + 1.14375i) q^{89} +(7.12555 + 0.701806i) q^{91} +(-0.753146 - 0.228464i) q^{93} +(-10.5729 + 10.5729i) q^{95} +(0.489078 + 0.489078i) q^{97} +(1.11405 - 3.67253i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{9}{32}\right)\) \(1\)

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
\(3\) 1.30167 − 0.695755i 0.751518 − 0.401695i −0.0507096 − 0.998713i \(-0.516148\pi\)
0.802227 + 0.597019i \(0.203648\pi\)
\(4\) 0 0
\(5\) −2.50395 − 3.05107i −1.11980 − 1.36448i −0.923346 − 0.383968i \(-0.874557\pi\)
−0.196454 − 0.980513i \(-0.562943\pi\)
\(6\) 0 0
\(7\) −1.90175 + 1.27071i −0.718794 + 0.480283i −0.860387 − 0.509642i \(-0.829778\pi\)
0.141592 + 0.989925i \(0.454778\pi\)
\(8\) 0 0
\(9\) −0.456450 + 0.683126i −0.152150 + 0.227709i
\(10\) 0 0
\(11\) −4.47003 + 1.35597i −1.34776 + 0.408840i −0.879983 − 0.475004i \(-0.842446\pi\)
−0.467781 + 0.883844i \(0.654946\pi\)
\(12\) 0 0
\(13\) −2.41987 − 1.98594i −0.671153 − 0.550801i 0.235921 − 0.971772i \(-0.424189\pi\)
−0.907074 + 0.420972i \(0.861689\pi\)
\(14\) 0 0
\(15\) −5.38211 − 2.22934i −1.38965 − 0.575614i
\(16\) 0 0
\(17\) 4.12129 − 1.70710i 0.999561 − 0.414032i 0.177925 − 0.984044i \(-0.443062\pi\)
0.821636 + 0.570013i \(0.193062\pi\)
\(18\) 0 0
\(19\) −0.371315 − 3.77002i −0.0851855 − 0.864903i −0.938907 − 0.344172i \(-0.888160\pi\)
0.853721 − 0.520731i \(-0.174340\pi\)
\(20\) 0 0
\(21\) −1.59134 + 2.97719i −0.347260 + 0.649677i
\(22\) 0 0
\(23\) −1.98987 + 0.395809i −0.414916 + 0.0825319i −0.398134 − 0.917327i \(-0.630342\pi\)
−0.0167818 + 0.999859i \(0.505342\pi\)
\(24\) 0 0
\(25\) −2.06382 + 10.3755i −0.412765 + 2.07511i
\(26\) 0 0
\(27\) −0.552861 + 5.61329i −0.106398 + 1.08028i
\(28\) 0 0
\(29\) 0.870012 − 2.86805i 0.161557 − 0.532583i −0.838364 − 0.545110i \(-0.816488\pi\)
0.999922 + 0.0125277i \(0.00398778\pi\)
\(30\) 0 0
\(31\) −0.377059 − 0.377059i −0.0677218 − 0.0677218i 0.672435 − 0.740156i \(-0.265249\pi\)
−0.740156 + 0.672435i \(0.765249\pi\)
\(32\) 0 0
\(33\) −4.87507 + 4.87507i −0.848640 + 0.848640i
\(34\) 0 0
\(35\) 8.63892 + 2.62059i 1.46024 + 0.442960i
\(36\) 0 0
\(37\) 1.87150 + 0.184327i 0.307673 + 0.0303032i 0.250676 − 0.968071i \(-0.419347\pi\)
0.0569976 + 0.998374i \(0.481847\pi\)
\(38\) 0 0
\(39\) −4.53160 − 0.901391i −0.725636 − 0.144338i
\(40\) 0 0
\(41\) −2.35748 − 11.8518i −0.368176 − 1.85094i −0.508804 − 0.860882i \(-0.669912\pi\)
0.140628 − 0.990062i \(-0.455088\pi\)
\(42\) 0 0
\(43\) −2.04620 − 1.09372i −0.312042 − 0.166790i 0.307940 − 0.951406i \(-0.400360\pi\)
−0.619983 + 0.784616i \(0.712860\pi\)
\(44\) 0 0
\(45\) 3.22720 − 0.317851i 0.481082 − 0.0473824i
\(46\) 0 0
\(47\) −0.807983 − 1.95064i −0.117856 − 0.284531i 0.853932 − 0.520384i \(-0.174211\pi\)
−0.971789 + 0.235854i \(0.924211\pi\)
\(48\) 0 0
\(49\) −0.676829 + 1.63401i −0.0966898 + 0.233430i
\(50\) 0 0
\(51\) 4.17683 − 5.08948i 0.584873 − 0.712670i
\(52\) 0 0
\(53\) −2.23072 − 7.35371i −0.306413 − 1.01011i −0.966013 − 0.258493i \(-0.916774\pi\)
0.659600 − 0.751617i \(-0.270726\pi\)
\(54\) 0 0
\(55\) 15.3299 + 10.2431i 2.06708 + 1.38118i
\(56\) 0 0
\(57\) −3.10634 − 4.64897i −0.411445 − 0.615771i
\(58\) 0 0
\(59\) 9.34237 − 7.66709i 1.21627 − 0.998169i 0.216513 − 0.976280i \(-0.430532\pi\)
0.999760 − 0.0218897i \(-0.00696826\pi\)
\(60\) 0 0
\(61\) 4.03888 + 7.55620i 0.517125 + 0.967473i 0.995980 + 0.0895786i \(0.0285520\pi\)
−0.478855 + 0.877894i \(0.658948\pi\)
\(62\) 0 0
\(63\) − 1.87915i − 0.236751i
\(64\) 0 0
\(65\) 12.3559i 1.53256i
\(66\) 0 0
\(67\) −3.52705 − 6.59865i −0.430898 − 0.806154i 0.568971 − 0.822358i \(-0.307342\pi\)
−0.999869 + 0.0162043i \(0.994842\pi\)
\(68\) 0 0
\(69\) −2.31476 + 1.89967i −0.278664 + 0.228694i
\(70\) 0 0
\(71\) −5.81321 − 8.70008i −0.689901 − 1.03251i −0.996735 − 0.0807464i \(-0.974270\pi\)
0.306834 − 0.951763i \(-0.400730\pi\)
\(72\) 0 0
\(73\) −4.18253 − 2.79468i −0.489528 − 0.327092i 0.286189 − 0.958173i \(-0.407612\pi\)
−0.775717 + 0.631081i \(0.782612\pi\)
\(74\) 0 0
\(75\) 4.53243 + 14.9414i 0.523359 + 1.72529i
\(76\) 0 0
\(77\) 6.77784 − 8.25882i 0.772407 − 0.941180i
\(78\) 0 0
\(79\) −0.189443 + 0.457356i −0.0213140 + 0.0514565i −0.934178 − 0.356807i \(-0.883865\pi\)
0.912864 + 0.408263i \(0.133865\pi\)
\(80\) 0 0
\(81\) 2.24261 + 5.41414i 0.249179 + 0.601571i
\(82\) 0 0
\(83\) −8.89052 + 0.875640i −0.975861 + 0.0961139i −0.573358 − 0.819305i \(-0.694360\pi\)
−0.402503 + 0.915419i \(0.631860\pi\)
\(84\) 0 0
\(85\) −15.5280 − 8.29988i −1.68425 − 0.900249i
\(86\) 0 0
\(87\) −0.862992 − 4.33855i −0.0925225 − 0.465142i
\(88\) 0 0
\(89\) 5.75001 + 1.14375i 0.609500 + 0.121237i 0.490183 − 0.871619i \(-0.336930\pi\)
0.119316 + 0.992856i \(0.461930\pi\)
\(90\) 0 0
\(91\) 7.12555 + 0.701806i 0.746961 + 0.0735692i
\(92\) 0 0
\(93\) −0.753146 − 0.228464i −0.0780976 − 0.0236907i
\(94\) 0 0
\(95\) −10.5729 + 10.5729i −1.08475 + 1.08475i
\(96\) 0 0
\(97\) 0.489078 + 0.489078i 0.0496583 + 0.0496583i 0.731500 − 0.681842i \(-0.238821\pi\)
−0.681842 + 0.731500i \(0.738821\pi\)
\(98\) 0 0
\(99\) 1.11405 − 3.67253i 0.111966 − 0.369103i
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 512.2.k.a.497.11 240
4.3 odd 2 128.2.k.a.101.11 ✓ 240
128.19 odd 32 128.2.k.a.109.11 yes 240
128.109 even 32 inner 512.2.k.a.273.11 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.11 ✓ 240 4.3 odd 2
128.2.k.a.109.11 yes 240 128.19 odd 32
512.2.k.a.273.11 240 128.109 even 32 inner
512.2.k.a.497.11 240 1.1 even 1 trivial