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Properties

Label 512.2.k.a.113.15
Level $512$
Weight $2$
Character 512.113
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 113.15
Character \(\chi\) \(=\) 512.113
Dual form 512.2.k.a.145.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.11090 + 0.943682i) q^{3} +(-2.52957 - 0.249141i) q^{5} +(1.75170 + 2.62161i) q^{7} +(6.29277 + 4.20469i) q^{9} +(1.57480 - 2.94625i) q^{11} +(0.0312066 + 0.316846i) q^{13} +(-7.63415 - 3.16217i) q^{15} +(-3.42808 + 1.41996i) q^{17} +(-0.234333 + 0.285535i) q^{19} +(2.97541 + 9.80861i) q^{21} +(-0.0973467 - 0.489395i) q^{23} +(1.43275 + 0.284991i) q^{25} +(9.42128 + 11.4799i) q^{27} +(4.96138 - 2.65191i) q^{29} +(1.02952 + 1.02952i) q^{31} +(7.67938 - 7.67938i) q^{33} +(-3.77791 - 7.06797i) q^{35} +(-1.22495 + 1.00529i) q^{37} +(-0.201921 + 1.01513i) q^{39} +(-8.45379 + 1.68156i) q^{41} +(4.50048 - 1.36521i) q^{43} +(-14.8705 - 12.2039i) q^{45} +(-1.50649 - 3.63698i) q^{47} +(-1.12558 + 2.71739i) q^{49} +(-12.0044 + 1.18233i) q^{51} +(-5.27790 - 2.82110i) q^{53} +(-4.71761 + 7.06041i) q^{55} +(-0.998441 + 0.667137i) q^{57} +(1.29265 - 13.1245i) q^{59} +(4.30925 - 14.2057i) q^{61} +23.8625i q^{63} -0.809261i q^{65} +(-1.60436 + 5.28888i) q^{67} +(0.158997 - 1.61432i) q^{69} +(7.40781 - 4.94974i) q^{71} +(-4.11433 + 6.15753i) q^{73} +(4.18820 + 2.23864i) q^{75} +(10.4825 - 1.03244i) q^{77} +(1.35491 - 3.27103i) q^{79} +(9.78660 + 23.6270i) q^{81} +(-10.5477 - 8.65630i) q^{83} +(9.02534 - 2.73781i) q^{85} +(17.9369 - 3.56787i) q^{87} +(-1.18408 + 5.95276i) q^{89} +(-0.775981 + 0.636831i) q^{91} +(2.23119 + 4.17427i) q^{93} +(0.663901 - 0.663901i) q^{95} +(-5.22348 - 5.22348i) q^{97} +(22.2979 - 11.9185i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 3.11090 + 0.943682i 1.79608 + 0.544835i 0.998309 − 0.0581257i \(-0.0185124\pi\)
0.797771 + 0.602961i \(0.206012\pi\)
\(4\) 0 0
\(5\) −2.52957 − 0.249141i −1.13126 − 0.111419i −0.484984 − 0.874523i \(-0.661175\pi\)
−0.646276 + 0.763104i \(0.723675\pi\)
\(6\) 0 0
\(7\) 1.75170 + 2.62161i 0.662081 + 0.990874i 0.998786 + 0.0492639i \(0.0156875\pi\)
−0.336705 + 0.941610i \(0.609312\pi\)
\(8\) 0 0
\(9\) 6.29277 + 4.20469i 2.09759 + 1.40156i
\(10\) 0 0
\(11\) 1.57480 − 2.94625i 0.474821 − 0.888328i −0.524497 − 0.851412i \(-0.675747\pi\)
0.999318 − 0.0369155i \(-0.0117532\pi\)
\(12\) 0 0
\(13\) 0.0312066 + 0.316846i 0.00865516 + 0.0878773i 0.998542 − 0.0539829i \(-0.0171916\pi\)
−0.989887 + 0.141860i \(0.954692\pi\)
\(14\) 0 0
\(15\) −7.63415 − 3.16217i −1.97113 − 0.816468i
\(16\) 0 0
\(17\) −3.42808 + 1.41996i −0.831431 + 0.344390i −0.757469 − 0.652871i \(-0.773564\pi\)
−0.0739617 + 0.997261i \(0.523564\pi\)
\(18\) 0 0
\(19\) −0.234333 + 0.285535i −0.0537596 + 0.0655063i −0.799196 − 0.601070i \(-0.794741\pi\)
0.745436 + 0.666577i \(0.232241\pi\)
\(20\) 0 0
\(21\) 2.97541 + 9.80861i 0.649288 + 2.14041i
\(22\) 0 0
\(23\) −0.0973467 − 0.489395i −0.0202982 − 0.102046i 0.969308 − 0.245849i \(-0.0790668\pi\)
−0.989606 + 0.143803i \(0.954067\pi\)
\(24\) 0 0
\(25\) 1.43275 + 0.284991i 0.286549 + 0.0569982i
\(26\) 0 0
\(27\) 9.42128 + 11.4799i 1.81313 + 2.20930i
\(28\) 0 0
\(29\) 4.96138 − 2.65191i 0.921304 − 0.492447i 0.0586712 − 0.998277i \(-0.481314\pi\)
0.862633 + 0.505830i \(0.168814\pi\)
\(30\) 0 0
\(31\) 1.02952 + 1.02952i 0.184907 + 0.184907i 0.793490 − 0.608583i \(-0.208262\pi\)
−0.608583 + 0.793490i \(0.708262\pi\)
\(32\) 0 0
\(33\) 7.67938 − 7.67938i 1.33681 − 1.33681i
\(34\) 0 0
\(35\) −3.77791 − 7.06797i −0.638583 − 1.19470i
\(36\) 0 0
\(37\) −1.22495 + 1.00529i −0.201381 + 0.165269i −0.729679 − 0.683790i \(-0.760330\pi\)
0.528298 + 0.849059i \(0.322830\pi\)
\(38\) 0 0
\(39\) −0.201921 + 1.01513i −0.0323333 + 0.162550i
\(40\) 0 0
\(41\) −8.45379 + 1.68156i −1.32026 + 0.262616i −0.804416 − 0.594066i \(-0.797522\pi\)
−0.515844 + 0.856683i \(0.672522\pi\)
\(42\) 0 0
\(43\) 4.50048 − 1.36521i 0.686317 − 0.208192i 0.0721800 − 0.997392i \(-0.477004\pi\)
0.614137 + 0.789200i \(0.289504\pi\)
\(44\) 0 0
\(45\) −14.8705 − 12.2039i −2.21676 − 1.81925i
\(46\) 0 0
\(47\) −1.50649 − 3.63698i −0.219744 − 0.530508i 0.775110 − 0.631826i \(-0.217694\pi\)
−0.994854 + 0.101318i \(0.967694\pi\)
\(48\) 0 0
\(49\) −1.12558 + 2.71739i −0.160797 + 0.388198i
\(50\) 0 0
\(51\) −12.0044 + 1.18233i −1.68095 + 0.165559i
\(52\) 0 0
\(53\) −5.27790 − 2.82110i −0.724975 − 0.387507i 0.0672499 − 0.997736i \(-0.478578\pi\)
−0.792225 + 0.610229i \(0.791078\pi\)
\(54\) 0 0
\(55\) −4.71761 + 7.06041i −0.636123 + 0.952025i
\(56\) 0 0
\(57\) −0.998441 + 0.667137i −0.132247 + 0.0883645i
\(58\) 0 0
\(59\) 1.29265 − 13.1245i 0.168289 − 1.70867i −0.426252 − 0.904605i \(-0.640166\pi\)
0.594541 − 0.804065i \(-0.297334\pi\)
\(60\) 0 0
\(61\) 4.30925 − 14.2057i 0.551743 − 1.81885i −0.0219176 − 0.999760i \(-0.506977\pi\)
0.573661 − 0.819093i \(-0.305523\pi\)
\(62\) 0 0
\(63\) 23.8625i 3.00640i
\(64\) 0 0
\(65\) − 0.809261i − 0.100376i
\(66\) 0 0
\(67\) −1.60436 + 5.28888i −0.196004 + 0.646140i 0.802753 + 0.596311i \(0.203368\pi\)
−0.998758 + 0.0498287i \(0.984132\pi\)
\(68\) 0 0
\(69\) 0.158997 − 1.61432i 0.0191410 − 0.194342i
\(70\) 0 0
\(71\) 7.40781 − 4.94974i 0.879146 − 0.587426i −0.0320106 − 0.999488i \(-0.510191\pi\)
0.911156 + 0.412061i \(0.135191\pi\)
\(72\) 0 0
\(73\) −4.11433 + 6.15753i −0.481546 + 0.720685i −0.990103 − 0.140344i \(-0.955179\pi\)
0.508557 + 0.861028i \(0.330179\pi\)
\(74\) 0 0
\(75\) 4.18820 + 2.23864i 0.483611 + 0.258496i
\(76\) 0 0
\(77\) 10.4825 − 1.03244i 1.19459 − 0.117657i
\(78\) 0 0
\(79\) 1.35491 − 3.27103i 0.152439 − 0.368020i −0.829150 − 0.559026i \(-0.811175\pi\)
0.981589 + 0.191006i \(0.0611751\pi\)
\(80\) 0 0
\(81\) 9.78660 + 23.6270i 1.08740 + 2.62522i
\(82\) 0 0
\(83\) −10.5477 − 8.65630i −1.15776 − 0.950152i −0.158456 − 0.987366i \(-0.550652\pi\)
−0.999307 + 0.0372140i \(0.988152\pi\)
\(84\) 0 0
\(85\) 9.02534 − 2.73781i 0.978936 − 0.296957i
\(86\) 0 0
\(87\) 17.9369 − 3.56787i 1.92304 − 0.382516i
\(88\) 0 0
\(89\) −1.18408 + 5.95276i −0.125512 + 0.630992i 0.865900 + 0.500217i \(0.166746\pi\)
−0.991412 + 0.130775i \(0.958254\pi\)
\(90\) 0 0
\(91\) −0.775981 + 0.636831i −0.0813449 + 0.0667581i
\(92\) 0 0
\(93\) 2.23119 + 4.17427i 0.231364 + 0.432851i
\(94\) 0 0
\(95\) 0.663901 − 0.663901i 0.0681148 − 0.0681148i
\(96\) 0 0
\(97\) −5.22348 − 5.22348i −0.530364 − 0.530364i 0.390316 − 0.920681i \(-0.372366\pi\)
−0.920681 + 0.390316i \(0.872366\pi\)
\(98\) 0 0
\(99\) 22.2979 − 11.9185i 2.24103 − 1.19785i
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.113.15 240
4.3 odd 2 128.2.k.a.5.12 ✓ 240
128.51 odd 32 128.2.k.a.77.12 yes 240
128.77 even 32 inner 512.2.k.a.145.15 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.5.12 ✓ 240 4.3 odd 2
128.2.k.a.77.12 yes 240 128.51 odd 32
512.2.k.a.113.15 240 1.1 even 1 trivial
512.2.k.a.145.15 240 128.77 even 32 inner