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Properties

Label 512.2.k.a.273.2
Level $512$
Weight $2$
Character 512.273
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 273.2
Character \(\chi\) \(=\) 512.273
Dual form 512.2.k.a.497.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.34878 - 1.25545i) q^{3} +(-2.35055 + 2.86416i) q^{5} +(1.77512 + 1.18610i) q^{7} +(2.27391 + 3.40315i) q^{9} +(-3.00424 - 0.911326i) q^{11} +(4.40033 - 3.61126i) q^{13} +(9.11675 - 3.77628i) q^{15} +(-2.12200 - 0.878960i) q^{17} +(0.222958 - 2.26373i) q^{19} +(-2.68028 - 5.01446i) q^{21} +(-4.14830 - 0.825148i) q^{23} +(-1.70285 - 8.56080i) q^{25} +(-0.285305 - 2.89675i) q^{27} +(-1.05879 - 3.49035i) q^{29} +(0.733820 - 0.733820i) q^{31} +(5.91218 + 5.91218i) q^{33} +(-7.56968 + 2.29624i) q^{35} +(1.40268 - 0.138152i) q^{37} +(-14.8692 + 2.95766i) q^{39} +(1.67780 - 8.43488i) q^{41} +(2.27833 - 1.21779i) q^{43} +(-15.0921 - 1.48644i) q^{45} +(4.22422 - 10.1982i) q^{47} +(-0.934561 - 2.25623i) q^{49} +(3.88062 + 4.72855i) q^{51} +(-0.810406 + 2.67155i) q^{53} +(9.67181 - 6.46250i) q^{55} +(-3.36567 + 5.03709i) q^{57} +(6.48788 + 5.32446i) q^{59} +(4.18727 - 7.83384i) q^{61} +8.73808i q^{63} +21.0917i q^{65} +(-1.02445 + 1.91661i) q^{67} +(8.70752 + 7.14608i) q^{69} +(4.85914 - 7.27222i) q^{71} +(-9.35312 + 6.24955i) q^{73} +(-6.74804 + 22.2453i) q^{75} +(-4.25196 - 5.18103i) q^{77} +(1.50619 + 3.63626i) q^{79} +(1.73229 - 4.18212i) q^{81} +(0.375929 + 0.0370258i) q^{83} +(7.50535 - 4.01169i) q^{85} +(-1.89510 + 9.52732i) q^{87} +(5.47983 - 1.09001i) q^{89} +(12.0944 - 1.19120i) q^{91} +(-2.64486 + 0.802308i) q^{93} +(5.95960 + 5.95960i) q^{95} +(-4.20900 + 4.20900i) q^{97} +(-3.73000 - 12.2962i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{23}{32}\right)\) \(1\)

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
\(3\) −2.34878 − 1.25545i −1.35607 − 0.724835i −0.376591 − 0.926380i \(-0.622904\pi\)
−0.979479 + 0.201545i \(0.935404\pi\)
\(4\) 0 0
\(5\) −2.35055 + 2.86416i −1.05120 + 1.28089i −0.0927320 + 0.995691i \(0.529560\pi\)
−0.958468 + 0.285200i \(0.907940\pi\)
\(6\) 0 0
\(7\) 1.77512 + 1.18610i 0.670932 + 0.448302i 0.843811 − 0.536641i \(-0.180307\pi\)
−0.172879 + 0.984943i \(0.555307\pi\)
\(8\) 0 0
\(9\) 2.27391 + 3.40315i 0.757971 + 1.13438i
\(10\) 0 0
\(11\) −3.00424 − 0.911326i −0.905812 − 0.274775i −0.197192 − 0.980365i \(-0.563182\pi\)
−0.708621 + 0.705590i \(0.750682\pi\)
\(12\) 0 0
\(13\) 4.40033 − 3.61126i 1.22043 − 1.00158i 0.220777 − 0.975324i \(-0.429141\pi\)
0.999655 − 0.0262590i \(-0.00835946\pi\)
\(14\) 0 0
\(15\) 9.11675 − 3.77628i 2.35394 − 0.975032i
\(16\) 0 0
\(17\) −2.12200 − 0.878960i −0.514660 − 0.213179i 0.110210 − 0.993908i \(-0.464848\pi\)
−0.624869 + 0.780729i \(0.714848\pi\)
\(18\) 0 0
\(19\) 0.222958 − 2.26373i 0.0511500 − 0.519334i −0.935136 − 0.354289i \(-0.884723\pi\)
0.986286 − 0.165045i \(-0.0527771\pi\)
\(20\) 0 0
\(21\) −2.68028 − 5.01446i −0.584886 − 1.09424i
\(22\) 0 0
\(23\) −4.14830 − 0.825148i −0.864981 − 0.172055i −0.257392 − 0.966307i \(-0.582863\pi\)
−0.607589 + 0.794252i \(0.707863\pi\)
\(24\) 0 0
\(25\) −1.70285 − 8.56080i −0.340570 − 1.71216i
\(26\) 0 0
\(27\) −0.285305 − 2.89675i −0.0549069 − 0.557480i
\(28\) 0 0
\(29\) −1.05879 − 3.49035i −0.196612 − 0.648142i −0.998702 − 0.0509386i \(-0.983779\pi\)
0.802090 − 0.597203i \(-0.203721\pi\)
\(30\) 0 0
\(31\) 0.733820 − 0.733820i 0.131798 − 0.131798i −0.638130 − 0.769928i \(-0.720292\pi\)
0.769928 + 0.638130i \(0.220292\pi\)
\(32\) 0 0
\(33\) 5.91218 + 5.91218i 1.02918 + 1.02918i
\(34\) 0 0
\(35\) −7.56968 + 2.29624i −1.27951 + 0.388135i
\(36\) 0 0
\(37\) 1.40268 − 0.138152i 0.230600 − 0.0227121i 0.0179429 − 0.999839i \(-0.494288\pi\)
0.212657 + 0.977127i \(0.431788\pi\)
\(38\) 0 0
\(39\) −14.8692 + 2.95766i −2.38097 + 0.473605i
\(40\) 0 0
\(41\) 1.67780 − 8.43488i 0.262028 − 1.31731i −0.595702 − 0.803206i \(-0.703126\pi\)
0.857730 − 0.514100i \(-0.171874\pi\)
\(42\) 0 0
\(43\) 2.27833 − 1.21779i 0.347442 − 0.185711i −0.288440 − 0.957498i \(-0.593137\pi\)
0.635882 + 0.771787i \(0.280637\pi\)
\(44\) 0 0
\(45\) −15.0921 − 1.48644i −2.24980 − 0.221586i
\(46\) 0 0
\(47\) 4.22422 − 10.1982i 0.616166 − 1.48756i −0.239957 − 0.970783i \(-0.577133\pi\)
0.856123 − 0.516772i \(-0.172867\pi\)
\(48\) 0 0
\(49\) −0.934561 − 2.25623i −0.133509 − 0.322319i
\(50\) 0 0
\(51\) 3.88062 + 4.72855i 0.543395 + 0.662129i
\(52\) 0 0
\(53\) −0.810406 + 2.67155i −0.111318 + 0.366966i −0.994844 − 0.101416i \(-0.967663\pi\)
0.883526 + 0.468381i \(0.155163\pi\)
\(54\) 0 0
\(55\) 9.67181 − 6.46250i 1.30415 − 0.871403i
\(56\) 0 0
\(57\) −3.36567 + 5.03709i −0.445794 + 0.667179i
\(58\) 0 0
\(59\) 6.48788 + 5.32446i 0.844650 + 0.693186i 0.953670 − 0.300853i \(-0.0972715\pi\)
−0.109021 + 0.994039i \(0.534771\pi\)
\(60\) 0 0
\(61\) 4.18727 − 7.83384i 0.536125 − 1.00302i −0.457384 − 0.889269i \(-0.651213\pi\)
0.993509 − 0.113751i \(-0.0362866\pi\)
\(62\) 0 0
\(63\) 8.73808i 1.10089i
\(64\) 0 0
\(65\) 21.0917i 2.61611i
\(66\) 0 0
\(67\) −1.02445 + 1.91661i −0.125157 + 0.234152i −0.936694 − 0.350149i \(-0.886131\pi\)
0.811537 + 0.584300i \(0.198631\pi\)
\(68\) 0 0
\(69\) 8.70752 + 7.14608i 1.04826 + 0.860287i
\(70\) 0 0
\(71\) 4.85914 − 7.27222i 0.576674 − 0.863053i −0.422386 − 0.906416i \(-0.638807\pi\)
0.999059 + 0.0433628i \(0.0138071\pi\)
\(72\) 0 0
\(73\) −9.35312 + 6.24955i −1.09470 + 0.731455i −0.965562 − 0.260172i \(-0.916221\pi\)
−0.129137 + 0.991627i \(0.541221\pi\)
\(74\) 0 0
\(75\) −6.74804 + 22.2453i −0.779196 + 2.56867i
\(76\) 0 0
\(77\) −4.25196 − 5.18103i −0.484556 − 0.590433i
\(78\) 0 0
\(79\) 1.50619 + 3.63626i 0.169459 + 0.409111i 0.985679 − 0.168630i \(-0.0539343\pi\)
−0.816220 + 0.577741i \(0.803934\pi\)
\(80\) 0 0
\(81\) 1.73229 − 4.18212i 0.192477 − 0.464680i
\(82\) 0 0
\(83\) 0.375929 + 0.0370258i 0.0412636 + 0.00406411i 0.118628 − 0.992939i \(-0.462150\pi\)
−0.0773643 + 0.997003i \(0.524650\pi\)
\(84\) 0 0
\(85\) 7.50535 − 4.01169i 0.814070 − 0.435129i
\(86\) 0 0
\(87\) −1.89510 + 9.52732i −0.203176 + 1.02144i
\(88\) 0 0
\(89\) 5.47983 − 1.09001i 0.580861 − 0.115540i 0.104090 − 0.994568i \(-0.466807\pi\)
0.476771 + 0.879028i \(0.341807\pi\)
\(90\) 0 0
\(91\) 12.0944 − 1.19120i 1.26784 − 0.124871i
\(92\) 0 0
\(93\) −2.64486 + 0.802308i −0.274259 + 0.0831955i
\(94\) 0 0
\(95\) 5.95960 + 5.95960i 0.611442 + 0.611442i
\(96\) 0 0
\(97\) −4.20900 + 4.20900i −0.427359 + 0.427359i −0.887728 − 0.460369i \(-0.847717\pi\)
0.460369 + 0.887728i \(0.347717\pi\)
\(98\) 0 0
\(99\) −3.73000 − 12.2962i −0.374879 − 1.23581i
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.273.2 240
4.3 odd 2 128.2.k.a.109.7 yes 240
128.27 odd 32 128.2.k.a.101.7 ✓ 240
128.101 even 32 inner 512.2.k.a.497.2 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.7 ✓ 240 128.27 odd 32
128.2.k.a.109.7 yes 240 4.3 odd 2
512.2.k.a.273.2 240 1.1 even 1 trivial
512.2.k.a.497.2 240 128.101 even 32 inner