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Properties

Label 512.2.k.a.497.5
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 497.5
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.731810 + 0.391160i) q^{3} +(-0.0771150 - 0.0939649i) q^{5} +(3.30773 - 2.21015i) q^{7} +(-1.28417 + 1.92190i) q^{9} +(-0.0222437 + 0.00674755i) q^{11} +(0.466044 + 0.382472i) q^{13} +(0.0931888 + 0.0386001i) q^{15} +(4.59132 - 1.90179i) q^{17} +(-0.166080 - 1.68624i) q^{19} +(-1.55610 + 2.91126i) q^{21} +(3.40974 - 0.678239i) q^{23} +(0.972569 - 4.88943i) q^{25} +(0.432000 - 4.38616i) q^{27} +(-1.44800 + 4.77342i) q^{29} +(6.39673 + 6.39673i) q^{31} +(0.0136388 - 0.0136388i) q^{33} +(-0.462752 - 0.140374i) q^{35} +(8.75672 + 0.862462i) q^{37} +(-0.490664 - 0.0975990i) q^{39} +(0.0606195 + 0.304755i) q^{41} +(4.91504 + 2.62714i) q^{43} +(0.279620 - 0.0275402i) q^{45} +(-0.167395 - 0.404127i) q^{47} +(3.37751 - 8.15402i) q^{49} +(-2.61607 + 3.18769i) q^{51} +(1.85360 + 6.11049i) q^{53} +(0.00234935 + 0.00156979i) q^{55} +(0.781127 + 1.16904i) q^{57} +(-7.95889 + 6.53169i) q^{59} +(-6.59803 - 12.3440i) q^{61} +9.19534i q^{63} -0.0732861i q^{65} +(-5.13746 - 9.61151i) q^{67} +(-2.22998 + 1.83010i) q^{69} +(-3.99036 - 5.97200i) q^{71} +(-8.64165 - 5.77416i) q^{73} +(1.20082 + 3.95857i) q^{75} +(-0.0586630 + 0.0714811i) q^{77} +(0.504221 - 1.21730i) q^{79} +(-1.25411 - 3.02768i) q^{81} +(-16.2560 + 1.60107i) q^{83} +(-0.532760 - 0.284766i) q^{85} +(-0.807512 - 4.05964i) q^{87} +(2.00759 + 0.399335i) q^{89} +(2.38687 + 0.235086i) q^{91} +(-7.18334 - 2.17904i) q^{93} +(-0.145640 + 0.145640i) q^{95} +(5.29993 + 5.29993i) q^{97} +(0.0155966 - 0.0514152i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
\(3\) −0.731810 + 0.391160i −0.422510 + 0.225837i −0.668928 − 0.743327i \(-0.733247\pi\)
0.246418 + 0.969164i \(0.420747\pi\)
\(4\) 0 0
\(5\) −0.0771150 − 0.0939649i −0.0344869 − 0.0420224i 0.755481 − 0.655170i \(-0.227403\pi\)
−0.789968 + 0.613148i \(0.789903\pi\)
\(6\) 0 0
\(7\) 3.30773 − 2.21015i 1.25020 − 0.835360i 0.258766 − 0.965940i \(-0.416684\pi\)
0.991438 + 0.130580i \(0.0416840\pi\)
\(8\) 0 0
\(9\) −1.28417 + 1.92190i −0.428057 + 0.640633i
\(10\) 0 0
\(11\) −0.0222437 + 0.00674755i −0.00670673 + 0.00203446i −0.293636 − 0.955917i \(-0.594865\pi\)
0.286930 + 0.957952i \(0.407365\pi\)
\(12\) 0 0
\(13\) 0.466044 + 0.382472i 0.129257 + 0.106079i 0.696801 − 0.717265i \(-0.254606\pi\)
−0.567544 + 0.823343i \(0.692106\pi\)
\(14\) 0 0
\(15\) 0.0931888 + 0.0386001i 0.0240612 + 0.00996650i
\(16\) 0 0
\(17\) 4.59132 − 1.90179i 1.11356 − 0.461251i 0.251396 − 0.967884i \(-0.419110\pi\)
0.862162 + 0.506633i \(0.169110\pi\)
\(18\) 0 0
\(19\) −0.166080 − 1.68624i −0.0381013 − 0.386849i −0.995257 − 0.0972765i \(-0.968987\pi\)
0.957156 − 0.289572i \(-0.0935131\pi\)
\(20\) 0 0
\(21\) −1.55610 + 2.91126i −0.339570 + 0.635290i
\(22\) 0 0
\(23\) 3.40974 − 0.678239i 0.710979 − 0.141423i 0.173665 − 0.984805i \(-0.444439\pi\)
0.537314 + 0.843382i \(0.319439\pi\)
\(24\) 0 0
\(25\) 0.972569 − 4.88943i 0.194514 − 0.977887i
\(26\) 0 0
\(27\) 0.432000 − 4.38616i 0.0831383 − 0.844118i
\(28\) 0 0
\(29\) −1.44800 + 4.77342i −0.268887 + 0.886403i 0.713676 + 0.700476i \(0.247029\pi\)
−0.982564 + 0.185927i \(0.940471\pi\)
\(30\) 0 0
\(31\) 6.39673 + 6.39673i 1.14889 + 1.14889i 0.986772 + 0.162115i \(0.0518316\pi\)
0.162115 + 0.986772i \(0.448168\pi\)
\(32\) 0 0
\(33\) 0.0136388 − 0.0136388i 0.00237421 − 0.00237421i
\(34\) 0 0
\(35\) −0.462752 − 0.140374i −0.0782194 − 0.0237276i
\(36\) 0 0
\(37\) 8.75672 + 0.862462i 1.43960 + 0.141788i 0.787484 − 0.616335i \(-0.211383\pi\)
0.652112 + 0.758123i \(0.273883\pi\)
\(38\) 0 0
\(39\) −0.490664 − 0.0975990i −0.0785690 − 0.0156284i
\(40\) 0 0
\(41\) 0.0606195 + 0.304755i 0.00946718 + 0.0475947i 0.985230 − 0.171236i \(-0.0547759\pi\)
−0.975763 + 0.218830i \(0.929776\pi\)
\(42\) 0 0
\(43\) 4.91504 + 2.62714i 0.749536 + 0.400635i 0.801485 − 0.598014i \(-0.204043\pi\)
−0.0519494 + 0.998650i \(0.516543\pi\)
\(44\) 0 0
\(45\) 0.279620 − 0.0275402i 0.0416833 − 0.00410544i
\(46\) 0 0
\(47\) −0.167395 − 0.404127i −0.0244171 − 0.0589480i 0.911201 − 0.411963i \(-0.135157\pi\)
−0.935618 + 0.353015i \(0.885157\pi\)
\(48\) 0 0
\(49\) 3.37751 − 8.15402i 0.482501 − 1.16486i
\(50\) 0 0
\(51\) −2.61607 + 3.18769i −0.366323 + 0.446365i
\(52\) 0 0
\(53\) 1.85360 + 6.11049i 0.254611 + 0.839341i 0.987312 + 0.158790i \(0.0507594\pi\)
−0.732701 + 0.680551i \(0.761741\pi\)
\(54\) 0 0
\(55\) 0.00234935 + 0.00156979i 0.000316787 + 0.000211670i
\(56\) 0 0
\(57\) 0.781127 + 1.16904i 0.103463 + 0.154843i
\(58\) 0 0
\(59\) −7.95889 + 6.53169i −1.03616 + 0.850354i −0.988908 − 0.148529i \(-0.952546\pi\)
−0.0472513 + 0.998883i \(0.515046\pi\)
\(60\) 0 0
\(61\) −6.59803 − 12.3440i −0.844791 − 1.58049i −0.813925 − 0.580969i \(-0.802674\pi\)
−0.0308657 − 0.999524i \(-0.509826\pi\)
\(62\) 0 0
\(63\) 9.19534i 1.15850i
\(64\) 0 0
\(65\) − 0.0732861i − 0.00909002i
\(66\) 0 0
\(67\) −5.13746 − 9.61151i −0.627641 − 1.17423i −0.971908 − 0.235361i \(-0.924373\pi\)
0.344267 − 0.938872i \(-0.388127\pi\)
\(68\) 0 0
\(69\) −2.22998 + 1.83010i −0.268458 + 0.220318i
\(70\) 0 0
\(71\) −3.99036 − 5.97200i −0.473569 − 0.708746i 0.515387 − 0.856958i \(-0.327648\pi\)
−0.988956 + 0.148212i \(0.952648\pi\)
\(72\) 0 0
\(73\) −8.64165 − 5.77416i −1.01143 − 0.675815i −0.0647187 − 0.997904i \(-0.520615\pi\)
−0.946709 + 0.322089i \(0.895615\pi\)
\(74\) 0 0
\(75\) 1.20082 + 3.95857i 0.138658 + 0.457096i
\(76\) 0 0
\(77\) −0.0586630 + 0.0714811i −0.00668527 + 0.00814602i
\(78\) 0 0
\(79\) 0.504221 − 1.21730i 0.0567293 − 0.136957i −0.892974 − 0.450109i \(-0.851385\pi\)
0.949703 + 0.313153i \(0.101385\pi\)
\(80\) 0 0
\(81\) −1.25411 − 3.02768i −0.139345 − 0.336409i
\(82\) 0 0
\(83\) −16.2560 + 1.60107i −1.78433 + 0.175741i −0.935593 − 0.353081i \(-0.885134\pi\)
−0.848733 + 0.528822i \(0.822634\pi\)
\(84\) 0 0
\(85\) −0.532760 − 0.284766i −0.0577860 − 0.0308872i
\(86\) 0 0
\(87\) −0.807512 − 4.05964i −0.0865744 − 0.435239i
\(88\) 0 0
\(89\) 2.00759 + 0.399335i 0.212805 + 0.0423294i 0.300341 − 0.953832i \(-0.402899\pi\)
−0.0875366 + 0.996161i \(0.527899\pi\)
\(90\) 0 0
\(91\) 2.38687 + 0.235086i 0.250212 + 0.0246437i
\(92\) 0 0
\(93\) −7.18334 − 2.17904i −0.744878 − 0.225956i
\(94\) 0 0
\(95\) −0.145640 + 0.145640i −0.0149423 + 0.0149423i
\(96\) 0 0
\(97\) 5.29993 + 5.29993i 0.538126 + 0.538126i 0.922978 − 0.384852i \(-0.125748\pi\)
−0.384852 + 0.922978i \(0.625748\pi\)
\(98\) 0 0
\(99\) 0.0155966 − 0.0514152i 0.00156752 − 0.00516742i
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.497.5 240
4.3 odd 2 128.2.k.a.101.14 ✓ 240
128.19 odd 32 128.2.k.a.109.14 yes 240
128.109 even 32 inner 512.2.k.a.273.5 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.14 ✓ 240 4.3 odd 2
128.2.k.a.109.14 yes 240 128.19 odd 32
512.2.k.a.273.5 240 128.109 even 32 inner
512.2.k.a.497.5 240 1.1 even 1 trivial