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Properties

Label 512.2.k.a.497.3
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 497.3
Character \(\chi\) \(=\) 512.497
Dual form 512.2.k.a.273.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.98529 + 1.06116i) q^{3} +(-0.212591 - 0.259043i) q^{5} +(0.792164 - 0.529307i) q^{7} +(1.14862 - 1.71903i) q^{9} +(-2.34047 + 0.709975i) q^{11} +(-2.81226 - 2.30796i) q^{13} +(0.696942 + 0.288683i) q^{15} +(-1.82571 + 0.756236i) q^{17} +(-0.157890 - 1.60309i) q^{19} +(-1.01100 + 1.89144i) q^{21} +(7.27219 - 1.44653i) q^{23} +(0.953543 - 4.79379i) q^{25} +(0.205763 - 2.08914i) q^{27} +(1.10598 - 3.64593i) q^{29} +(-7.16783 - 7.16783i) q^{31} +(3.89313 - 3.89313i) q^{33} +(-0.305520 - 0.0926785i) q^{35} +(0.968069 + 0.0953465i) q^{37} +(8.03228 + 1.59772i) q^{39} +(-2.34900 - 11.8092i) q^{41} +(-7.65676 - 4.09262i) q^{43} +(-0.689487 + 0.0679086i) q^{45} +(1.74223 + 4.20612i) q^{47} +(-2.33143 + 5.62856i) q^{49} +(2.82209 - 3.43873i) q^{51} +(2.50640 + 8.26251i) q^{53} +(0.681478 + 0.455349i) q^{55} +(2.01459 + 3.01505i) q^{57} +(2.07085 - 1.69950i) q^{59} +(3.63666 + 6.80371i) q^{61} -1.96972i q^{63} +1.21915i q^{65} +(-5.45136 - 10.1988i) q^{67} +(-12.9024 + 10.5887i) q^{69} +(-1.79285 - 2.68319i) q^{71} +(2.04993 + 1.36972i) q^{73} +(3.19392 + 10.5289i) q^{75} +(-1.47825 + 1.80125i) q^{77} +(-4.58780 + 11.0759i) q^{79} +(4.18196 + 10.0961i) q^{81} +(7.11006 - 0.700280i) q^{83} +(0.584028 + 0.312169i) q^{85} +(1.67322 + 8.41187i) q^{87} +(-13.6682 - 2.71877i) q^{89} +(-3.44939 - 0.339735i) q^{91} +(21.8365 + 6.62402i) q^{93} +(-0.381702 + 0.381702i) q^{95} +(-10.5511 - 10.5511i) q^{97} +(-1.46784 + 4.83883i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
\(3\) −1.98529 + 1.06116i −1.14621 + 0.612662i −0.931401 − 0.363994i \(-0.881413\pi\)
−0.214809 + 0.976656i \(0.568913\pi\)
\(4\) 0 0
\(5\) −0.212591 − 0.259043i −0.0950736 − 0.115847i 0.723300 − 0.690533i \(-0.242624\pi\)
−0.818374 + 0.574686i \(0.805124\pi\)
\(6\) 0 0
\(7\) 0.792164 − 0.529307i 0.299410 − 0.200059i −0.396786 − 0.917911i \(-0.629874\pi\)
0.696196 + 0.717852i \(0.254874\pi\)
\(8\) 0 0
\(9\) 1.14862 − 1.71903i 0.382872 − 0.573009i
\(10\) 0 0
\(11\) −2.34047 + 0.709975i −0.705680 + 0.214066i −0.622672 − 0.782483i \(-0.713953\pi\)
−0.0830081 + 0.996549i \(0.526453\pi\)
\(12\) 0 0
\(13\) −2.81226 − 2.30796i −0.779980 − 0.640113i 0.157750 − 0.987479i \(-0.449576\pi\)
−0.937730 + 0.347366i \(0.887076\pi\)
\(14\) 0 0
\(15\) 0.696942 + 0.288683i 0.179950 + 0.0745376i
\(16\) 0 0
\(17\) −1.82571 + 0.756236i −0.442801 + 0.183414i −0.592933 − 0.805252i \(-0.702030\pi\)
0.150132 + 0.988666i \(0.452030\pi\)
\(18\) 0 0
\(19\) −0.157890 − 1.60309i −0.0362225 − 0.367773i −0.996159 − 0.0875572i \(-0.972094\pi\)
0.959937 − 0.280216i \(-0.0904061\pi\)
\(20\) 0 0
\(21\) −1.01100 + 1.89144i −0.220618 + 0.412747i
\(22\) 0 0
\(23\) 7.27219 − 1.44653i 1.51636 − 0.301622i 0.634419 − 0.772990i \(-0.281240\pi\)
0.881937 + 0.471368i \(0.156240\pi\)
\(24\) 0 0
\(25\) 0.953543 − 4.79379i 0.190709 − 0.958757i
\(26\) 0 0
\(27\) 0.205763 − 2.08914i 0.0395990 − 0.402056i
\(28\) 0 0
\(29\) 1.10598 − 3.64593i 0.205376 − 0.677032i −0.792377 − 0.610031i \(-0.791157\pi\)
0.997753 − 0.0670011i \(-0.0213431\pi\)
\(30\) 0 0
\(31\) −7.16783 − 7.16783i −1.28738 − 1.28738i −0.936372 − 0.351008i \(-0.885839\pi\)
−0.351008 − 0.936372i \(-0.614161\pi\)
\(32\) 0 0
\(33\) 3.89313 − 3.89313i 0.677707 − 0.677707i
\(34\) 0 0
\(35\) −0.305520 − 0.0926785i −0.0516423 − 0.0156655i
\(36\) 0 0
\(37\) 0.968069 + 0.0953465i 0.159150 + 0.0156749i 0.177278 − 0.984161i \(-0.443271\pi\)
−0.0181285 + 0.999836i \(0.505771\pi\)
\(38\) 0 0
\(39\) 8.03228 + 1.59772i 1.28619 + 0.255840i
\(40\) 0 0
\(41\) −2.34900 − 11.8092i −0.366852 − 1.84429i −0.517473 − 0.855699i \(-0.673127\pi\)
0.150621 − 0.988592i \(-0.451873\pi\)
\(42\) 0 0
\(43\) −7.65676 − 4.09262i −1.16764 − 0.624119i −0.230504 − 0.973071i \(-0.574038\pi\)
−0.937141 + 0.348952i \(0.886538\pi\)
\(44\) 0 0
\(45\) −0.689487 + 0.0679086i −0.102783 + 0.0101232i
\(46\) 0 0
\(47\) 1.74223 + 4.20612i 0.254130 + 0.613525i 0.998530 − 0.0542099i \(-0.0172640\pi\)
−0.744399 + 0.667735i \(0.767264\pi\)
\(48\) 0 0
\(49\) −2.33143 + 5.62856i −0.333061 + 0.804080i
\(50\) 0 0
\(51\) 2.82209 − 3.43873i 0.395172 − 0.481518i
\(52\) 0 0
\(53\) 2.50640 + 8.26251i 0.344281 + 1.13494i 0.942667 + 0.333735i \(0.108309\pi\)
−0.598386 + 0.801208i \(0.704191\pi\)
\(54\) 0 0
\(55\) 0.681478 + 0.455349i 0.0918905 + 0.0613992i
\(56\) 0 0
\(57\) 2.01459 + 3.01505i 0.266839 + 0.399353i
\(58\) 0 0
\(59\) 2.07085 − 1.69950i 0.269601 − 0.221256i −0.489872 − 0.871794i \(-0.662957\pi\)
0.759474 + 0.650538i \(0.225457\pi\)
\(60\) 0 0
\(61\) 3.63666 + 6.80371i 0.465626 + 0.871125i 0.999670 + 0.0256689i \(0.00817157\pi\)
−0.534044 + 0.845457i \(0.679328\pi\)
\(62\) 0 0
\(63\) − 1.96972i − 0.248162i
\(64\) 0 0
\(65\) 1.21915i 0.151217i
\(66\) 0 0
\(67\) −5.45136 − 10.1988i −0.665990 − 1.24598i −0.957230 − 0.289329i \(-0.906568\pi\)
0.291240 − 0.956650i \(-0.405932\pi\)
\(68\) 0 0
\(69\) −12.9024 + 10.5887i −1.55327 + 1.27474i
\(70\) 0 0
\(71\) −1.79285 − 2.68319i −0.212772 − 0.318436i 0.709697 − 0.704507i \(-0.248832\pi\)
−0.922469 + 0.386071i \(0.873832\pi\)
\(72\) 0 0
\(73\) 2.04993 + 1.36972i 0.239926 + 0.160314i 0.669721 − 0.742613i \(-0.266414\pi\)
−0.429794 + 0.902927i \(0.641414\pi\)
\(74\) 0 0
\(75\) 3.19392 + 10.5289i 0.368802 + 1.21578i
\(76\) 0 0
\(77\) −1.47825 + 1.80125i −0.168462 + 0.205271i
\(78\) 0 0
\(79\) −4.58780 + 11.0759i −0.516168 + 1.24614i 0.424072 + 0.905628i \(0.360600\pi\)
−0.940240 + 0.340512i \(0.889400\pi\)
\(80\) 0 0
\(81\) 4.18196 + 10.0961i 0.464662 + 1.12179i
\(82\) 0 0
\(83\) 7.11006 − 0.700280i 0.780431 − 0.0768657i 0.300047 − 0.953924i \(-0.402997\pi\)
0.480384 + 0.877059i \(0.340497\pi\)
\(84\) 0 0
\(85\) 0.584028 + 0.312169i 0.0633467 + 0.0338595i
\(86\) 0 0
\(87\) 1.67322 + 8.41187i 0.179389 + 0.901847i
\(88\) 0 0
\(89\) −13.6682 − 2.71877i −1.44882 − 0.288189i −0.592893 − 0.805281i \(-0.702014\pi\)
−0.855930 + 0.517092i \(0.827014\pi\)
\(90\) 0 0
\(91\) −3.44939 − 0.339735i −0.361594 − 0.0356139i
\(92\) 0 0
\(93\) 21.8365 + 6.62402i 2.26434 + 0.686879i
\(94\) 0 0
\(95\) −0.381702 + 0.381702i −0.0391618 + 0.0391618i
\(96\) 0 0
\(97\) −10.5511 − 10.5511i −1.07131 − 1.07131i −0.997254 − 0.0740512i \(-0.976407\pi\)
−0.0740512 − 0.997254i \(-0.523593\pi\)
\(98\) 0 0
\(99\) −1.46784 + 4.83883i −0.147524 + 0.486321i
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.497.3 240
4.3 odd 2 128.2.k.a.101.10 ✓ 240
128.19 odd 32 128.2.k.a.109.10 yes 240
128.109 even 32 inner 512.2.k.a.273.3 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.10 ✓ 240 4.3 odd 2
128.2.k.a.109.10 yes 240 128.19 odd 32
512.2.k.a.273.3 240 128.109 even 32 inner
512.2.k.a.497.3 240 1.1 even 1 trivial