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Properties

Label 512.2.k.a.497.2
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 497.2
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.2

$q$-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 \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

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

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
\(3\) −2.34878 + 1.25545i −1.35607 + 0.724835i −0.979479 − 0.201545i \(-0.935404\pi\)
−0.376591 + 0.926380i \(0.622904\pi\)
\(4\) 0 0
\(5\) −2.35055 − 2.86416i −1.05120 − 1.28089i −0.958468 − 0.285200i \(-0.907940\pi\)
−0.0927320 − 0.995691i \(-0.529560\pi\)
\(6\) 0 0
\(7\) 1.77512 − 1.18610i 0.670932 − 0.448302i −0.172879 − 0.984943i \(-0.555307\pi\)
0.843811 + 0.536641i \(0.180307\pi\)
\(8\) 0 0
\(9\) 2.27391 − 3.40315i 0.757971 − 1.13438i
\(10\) 0 0
\(11\) −3.00424 + 0.911326i −0.905812 + 0.274775i −0.708621 − 0.705590i \(-0.750682\pi\)
−0.197192 + 0.980365i \(0.563182\pi\)
\(12\) 0 0
\(13\) 4.40033 + 3.61126i 1.22043 + 1.00158i 0.999655 + 0.0262590i \(0.00835946\pi\)
0.220777 + 0.975324i \(0.429141\pi\)
\(14\) 0 0
\(15\) 9.11675 + 3.77628i 2.35394 + 0.975032i
\(16\) 0 0
\(17\) −2.12200 + 0.878960i −0.514660 + 0.213179i −0.624869 − 0.780729i \(-0.714848\pi\)
0.110210 + 0.993908i \(0.464848\pi\)
\(18\) 0 0
\(19\) 0.222958 + 2.26373i 0.0511500 + 0.519334i 0.986286 + 0.165045i \(0.0527771\pi\)
−0.935136 + 0.354289i \(0.884723\pi\)
\(20\) 0 0
\(21\) −2.68028 + 5.01446i −0.584886 + 1.09424i
\(22\) 0 0
\(23\) −4.14830 + 0.825148i −0.864981 + 0.172055i −0.607589 − 0.794252i \(-0.707863\pi\)
−0.257392 + 0.966307i \(0.582863\pi\)
\(24\) 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 0.802090 + 0.597203i \(0.203721\pi\)
−0.998702 + 0.0509386i \(0.983779\pi\)
\(30\) 0 0
\(31\) 0.733820 + 0.733820i 0.131798 + 0.131798i 0.769928 − 0.638130i \(-0.220292\pi\)
−0.638130 + 0.769928i \(0.720292\pi\)
\(32\) 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 0.212657 − 0.977127i \(-0.431788\pi\)
0.0179429 + 0.999839i \(0.494288\pi\)
\(38\) 0 0
\(39\) −14.8692 − 2.95766i −2.38097 − 0.473605i
\(40\) 0 0
\(41\) 1.67780 + 8.43488i 0.262028 + 1.31731i 0.857730 + 0.514100i \(0.171874\pi\)
−0.595702 + 0.803206i \(0.703126\pi\)
\(42\) 0 0
\(43\) 2.27833 + 1.21779i 0.347442 + 0.185711i 0.635882 − 0.771787i \(-0.280637\pi\)
−0.288440 + 0.957498i \(0.593137\pi\)
\(44\) 0 0
\(45\) −15.0921 + 1.48644i −2.24980 + 0.221586i
\(46\) 0 0
\(47\) 4.22422 + 10.1982i 0.616166 + 1.48756i 0.856123 + 0.516772i \(0.172867\pi\)
−0.239957 + 0.970783i \(0.577133\pi\)
\(48\) 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 0.883526 − 0.468381i \(-0.155163\pi\)
−0.994844 + 0.101416i \(0.967663\pi\)
\(54\) 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 −0.109021 − 0.994039i \(-0.534771\pi\)
0.953670 + 0.300853i \(0.0972715\pi\)
\(60\) 0 0
\(61\) 4.18727 + 7.83384i 0.536125 + 1.00302i 0.993509 + 0.113751i \(0.0362866\pi\)
−0.457384 + 0.889269i \(0.651213\pi\)
\(62\) 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 0.811537 − 0.584300i \(-0.198631\pi\)
−0.936694 + 0.350149i \(0.886131\pi\)
\(68\) 0 0
\(69\) 8.70752 − 7.14608i 1.04826 − 0.860287i
\(70\) 0 0
\(71\) 4.85914 + 7.27222i 0.576674 + 0.863053i 0.999059 − 0.0433628i \(-0.0138071\pi\)
−0.422386 + 0.906416i \(0.638807\pi\)
\(72\) 0 0
\(73\) −9.35312 − 6.24955i −1.09470 − 0.731455i −0.129137 − 0.991627i \(-0.541221\pi\)
−0.965562 + 0.260172i \(0.916221\pi\)
\(74\) 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 −0.816220 − 0.577741i \(-0.803934\pi\)
0.985679 + 0.168630i \(0.0539343\pi\)
\(80\) 0 0
\(81\) 1.73229 + 4.18212i 0.192477 + 0.464680i
\(82\) 0 0
\(83\) 0.375929 − 0.0370258i 0.0412636 − 0.00406411i −0.0773643 − 0.997003i \(-0.524650\pi\)
0.118628 + 0.992939i \(0.462150\pi\)
\(84\) 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 0.476771 − 0.879028i \(-0.341807\pi\)
0.104090 + 0.994568i \(0.466807\pi\)
\(90\) 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 0.460369 − 0.887728i \(-0.347717\pi\)
−0.887728 + 0.460369i \(0.847717\pi\)
\(98\) 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.497.2 240
4.3 odd 2 128.2.k.a.101.7 ✓ 240
128.19 odd 32 128.2.k.a.109.7 yes 240
128.109 even 32 inner 512.2.k.a.273.2 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.7 ✓ 240 4.3 odd 2
128.2.k.a.109.7 yes 240 128.19 odd 32
512.2.k.a.273.2 240 128.109 even 32 inner
512.2.k.a.497.2 240 1.1 even 1 trivial