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Properties

Label 512.2.k.a.17.5
Level $512$
Weight $2$
Character 512.17
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 17.5
Character \(\chi\) \(=\) 512.17
Dual form 512.2.k.a.241.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.537163 + 1.00496i) q^{3} +(3.03892 + 2.49397i) q^{5} +(3.06141 + 2.04557i) q^{7} +(0.945307 + 1.41475i) q^{9} +(1.17836 - 3.88454i) q^{11} +(-2.86546 - 3.49157i) q^{13} +(-4.13874 + 1.71432i) q^{15} +(-0.993546 - 0.411540i) q^{17} +(-2.22717 - 0.219357i) q^{19} +(-3.70019 + 1.97779i) q^{21} +(-1.21113 - 0.240910i) q^{23} +(2.03965 + 10.2540i) q^{25} +(-5.33164 + 0.525120i) q^{27} +(3.38471 - 1.02674i) q^{29} +(2.90844 - 2.90844i) q^{31} +(3.27084 + 3.27084i) q^{33} +(4.20177 + 13.8514i) q^{35} +(-0.669587 - 6.79843i) q^{37} +(5.04811 - 1.00413i) q^{39} +(0.00412960 - 0.0207609i) q^{41} +(2.22242 + 4.15785i) q^{43} +(-0.655646 + 6.65689i) q^{45} +(1.51766 - 3.66395i) q^{47} +(2.50909 + 6.05748i) q^{49} +(0.947278 - 0.777411i) q^{51} +(-13.5390 - 4.10702i) q^{53} +(13.2689 - 8.86600i) q^{55} +(1.41680 - 2.12039i) q^{57} +(-7.41498 + 9.03518i) q^{59} +(-2.15405 - 1.15136i) q^{61} +6.26483i q^{63} -17.7570i q^{65} +(-0.577237 - 0.308539i) q^{67} +(0.892681 - 1.08774i) q^{69} +(-3.31428 + 4.96017i) q^{71} +(-5.15965 + 3.44757i) q^{73} +(-11.4005 - 3.45832i) q^{75} +(11.5535 - 9.48175i) q^{77} +(-0.00465201 - 0.0112309i) q^{79} +(0.382815 - 0.924197i) q^{81} +(0.504388 - 5.12114i) q^{83} +(-1.99293 - 3.72851i) q^{85} +(-0.786305 + 3.95302i) q^{87} +(-0.875154 + 0.174079i) q^{89} +(-1.63009 - 16.5506i) q^{91} +(1.36056 + 4.48518i) q^{93} +(-6.22112 - 6.22112i) q^{95} +(7.49995 - 7.49995i) q^{97} +(6.60958 - 2.00500i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{7}{32}\right)\) \(1\)

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
\(3\) −0.537163 + 1.00496i −0.310131 + 0.580215i −0.987906 − 0.155055i \(-0.950445\pi\)
0.677775 + 0.735270i \(0.262945\pi\)
\(4\) 0 0
\(5\) 3.03892 + 2.49397i 1.35904 + 1.11534i 0.981945 + 0.189167i \(0.0605787\pi\)
0.377100 + 0.926173i \(0.376921\pi\)
\(6\) 0 0
\(7\) 3.06141 + 2.04557i 1.15710 + 0.773152i 0.977572 − 0.210600i \(-0.0675419\pi\)
0.179531 + 0.983752i \(0.442542\pi\)
\(8\) 0 0
\(9\) 0.945307 + 1.41475i 0.315102 + 0.471584i
\(10\) 0 0
\(11\) 1.17836 − 3.88454i 0.355290 − 1.17123i −0.579182 − 0.815198i \(-0.696628\pi\)
0.934472 − 0.356036i \(-0.115872\pi\)
\(12\) 0 0
\(13\) −2.86546 − 3.49157i −0.794735 − 0.968387i 0.205212 − 0.978718i \(-0.434212\pi\)
−0.999947 + 0.0103304i \(0.996712\pi\)
\(14\) 0 0
\(15\) −4.13874 + 1.71432i −1.06862 + 0.442636i
\(16\) 0 0
\(17\) −0.993546 − 0.411540i −0.240970 − 0.0998131i 0.258930 − 0.965896i \(-0.416630\pi\)
−0.499900 + 0.866083i \(0.666630\pi\)
\(18\) 0 0
\(19\) −2.22717 − 0.219357i −0.510948 − 0.0503240i −0.160742 − 0.986997i \(-0.551389\pi\)
−0.350206 + 0.936673i \(0.613889\pi\)
\(20\) 0 0
\(21\) −3.70019 + 1.97779i −0.807448 + 0.431590i
\(22\) 0 0
\(23\) −1.21113 − 0.240910i −0.252539 − 0.0502331i 0.0671973 − 0.997740i \(-0.478594\pi\)
−0.319736 + 0.947507i \(0.603594\pi\)
\(24\) 0 0
\(25\) 2.03965 + 10.2540i 0.407931 + 2.05081i
\(26\) 0 0
\(27\) −5.33164 + 0.525120i −1.02607 + 0.101059i
\(28\) 0 0
\(29\) 3.38471 − 1.02674i 0.628524 − 0.190661i 0.0400772 − 0.999197i \(-0.487240\pi\)
0.588447 + 0.808536i \(0.299740\pi\)
\(30\) 0 0
\(31\) 2.90844 − 2.90844i 0.522372 − 0.522372i −0.395915 − 0.918287i \(-0.629573\pi\)
0.918287 + 0.395915i \(0.129573\pi\)
\(32\) 0 0
\(33\) 3.27084 + 3.27084i 0.569381 + 0.569381i
\(34\) 0 0
\(35\) 4.20177 + 13.8514i 0.710229 + 2.34131i
\(36\) 0 0
\(37\) −0.669587 − 6.79843i −0.110079 − 1.11765i −0.878511 − 0.477723i \(-0.841462\pi\)
0.768431 − 0.639932i \(-0.221038\pi\)
\(38\) 0 0
\(39\) 5.04811 − 1.00413i 0.808345 − 0.160790i
\(40\) 0 0
\(41\) 0.00412960 − 0.0207609i 0.000644935 − 0.00324231i −0.980461 − 0.196711i \(-0.936974\pi\)
0.981106 + 0.193469i \(0.0619739\pi\)
\(42\) 0 0
\(43\) 2.22242 + 4.15785i 0.338915 + 0.634066i 0.992425 − 0.122852i \(-0.0392041\pi\)
−0.653510 + 0.756918i \(0.726704\pi\)
\(44\) 0 0
\(45\) −0.655646 + 6.65689i −0.0977380 + 0.992350i
\(46\) 0 0
\(47\) 1.51766 − 3.66395i 0.221373 − 0.534442i −0.773704 − 0.633548i \(-0.781598\pi\)
0.995077 + 0.0991054i \(0.0315981\pi\)
\(48\) 0 0
\(49\) 2.50909 + 6.05748i 0.358441 + 0.865354i
\(50\) 0 0
\(51\) 0.947278 − 0.777411i 0.132645 − 0.108859i
\(52\) 0 0
\(53\) −13.5390 − 4.10702i −1.85973 − 0.564143i −0.998428 − 0.0560518i \(-0.982149\pi\)
−0.861303 − 0.508091i \(-0.830351\pi\)
\(54\) 0 0
\(55\) 13.2689 − 8.86600i 1.78918 − 1.19549i
\(56\) 0 0
\(57\) 1.41680 − 2.12039i 0.187660 − 0.280853i
\(58\) 0 0
\(59\) −7.41498 + 9.03518i −0.965349 + 1.17628i 0.0189591 + 0.999820i \(0.493965\pi\)
−0.984308 + 0.176460i \(0.943535\pi\)
\(60\) 0 0
\(61\) −2.15405 − 1.15136i −0.275798 − 0.147417i 0.327700 − 0.944782i \(-0.393727\pi\)
−0.603497 + 0.797365i \(0.706227\pi\)
\(62\) 0 0
\(63\) 6.26483i 0.789294i
\(64\) 0 0
\(65\) − 17.7570i − 2.20248i
\(66\) 0 0
\(67\) −0.577237 − 0.308539i −0.0705207 − 0.0376941i 0.435759 − 0.900063i \(-0.356480\pi\)
−0.506280 + 0.862369i \(0.668980\pi\)
\(68\) 0 0
\(69\) 0.892681 − 1.08774i 0.107466 − 0.130948i
\(70\) 0 0
\(71\) −3.31428 + 4.96017i −0.393333 + 0.588664i −0.974298 − 0.225265i \(-0.927675\pi\)
0.580965 + 0.813929i \(0.302675\pi\)
\(72\) 0 0
\(73\) −5.15965 + 3.44757i −0.603891 + 0.403507i −0.819588 − 0.572953i \(-0.805798\pi\)
0.215697 + 0.976460i \(0.430798\pi\)
\(74\) 0 0
\(75\) −11.4005 − 3.45832i −1.31642 − 0.399332i
\(76\) 0 0
\(77\) 11.5535 − 9.48175i 1.31665 − 1.08055i
\(78\) 0 0
\(79\) −0.00465201 − 0.0112309i −0.000523392 − 0.00126358i 0.923618 − 0.383315i \(-0.125218\pi\)
−0.924141 + 0.382052i \(0.875218\pi\)
\(80\) 0 0
\(81\) 0.382815 − 0.924197i 0.0425350 − 0.102689i
\(82\) 0 0
\(83\) 0.504388 − 5.12114i 0.0553638 − 0.562118i −0.926990 − 0.375086i \(-0.877613\pi\)
0.982354 − 0.187032i \(-0.0598868\pi\)
\(84\) 0 0
\(85\) −1.99293 − 3.72851i −0.216164 − 0.404414i
\(86\) 0 0
\(87\) −0.786305 + 3.95302i −0.0843008 + 0.423809i
\(88\) 0 0
\(89\) −0.875154 + 0.174079i −0.0927662 + 0.0184523i −0.241255 − 0.970462i \(-0.577559\pi\)
0.148489 + 0.988914i \(0.452559\pi\)
\(90\) 0 0
\(91\) −1.63009 − 16.5506i −0.170880 − 1.73497i
\(92\) 0 0
\(93\) 1.36056 + 4.48518i 0.141084 + 0.465092i
\(94\) 0 0
\(95\) −6.22112 − 6.22112i −0.638273 − 0.638273i
\(96\) 0 0
\(97\) 7.49995 − 7.49995i 0.761504 − 0.761504i −0.215090 − 0.976594i \(-0.569005\pi\)
0.976594 + 0.215090i \(0.0690045\pi\)
\(98\) 0 0
\(99\) 6.60958 − 2.00500i 0.664288 − 0.201510i
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.17.5 240
4.3 odd 2 128.2.k.a.45.11 yes 240
128.37 even 32 inner 512.2.k.a.241.5 240
128.91 odd 32 128.2.k.a.37.11 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.37.11 ✓ 240 128.91 odd 32
128.2.k.a.45.11 yes 240 4.3 odd 2
512.2.k.a.17.5 240 1.1 even 1 trivial
512.2.k.a.241.5 240 128.37 even 32 inner