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Properties

Label 512.2.k.a
Level $512$
Weight $2$
Character orbit 512.k
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: $\mathrm{SU}(2)[C_{32}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 16 q^{3} - 16 q^{5} + 16 q^{7} - 16 q^{9} + 16 q^{11} - 16 q^{13} + 16 q^{15} - 16 q^{17} + 16 q^{19} - 16 q^{21} + 16 q^{23} - 16 q^{25} + 16 q^{27} - 16 q^{29} + 16 q^{31} - 16 q^{33} + 16 q^{35}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −1.46662 + 2.74386i 0 −0.382125 − 0.313602i 0 −1.98950 − 1.32934i 0 −3.71107 − 5.55401i 0
17.2 0 −1.30307 + 2.43788i 0 0.180606 + 0.148219i 0 3.54546 + 2.36900i 0 −2.57853 − 3.85905i 0
17.3 0 −0.995218 + 1.86192i 0 −1.80486 − 1.48121i 0 −0.154904 − 0.103503i 0 −0.809586 − 1.21163i 0
17.4 0 −0.785927 + 1.47037i 0 1.70310 + 1.39770i 0 −0.0978572 − 0.0653861i 0 0.122416 + 0.183208i 0
17.5 0 −0.537163 + 1.00496i 0 3.03892 + 2.49397i 0 3.06141 + 2.04557i 0 0.945307 + 1.41475i 0
17.6 0 −0.486285 + 0.909775i 0 −2.96357 − 2.43214i 0 1.58228 + 1.05724i 0 1.07549 + 1.60959i 0
17.7 0 −0.247709 + 0.463431i 0 −2.05124 − 1.68341i 0 −1.35283 − 0.903935i 0 1.51330 + 2.26482i 0
17.8 0 −0.169198 + 0.316547i 0 −0.290696 − 0.238568i 0 −3.87217 − 2.58730i 0 1.59514 + 2.38729i 0
17.9 0 0.337004 − 0.630491i 0 1.52625 + 1.25256i 0 −0.541333 − 0.361707i 0 1.38276 + 2.06945i 0
17.10 0 0.544320 − 1.01835i 0 0.722871 + 0.593245i 0 0.369041 + 0.246585i 0 0.925956 + 1.38579i 0
17.11 0 0.596348 − 1.11569i 0 0.521459 + 0.427951i 0 3.16215 + 2.11288i 0 0.777579 + 1.16373i 0
17.12 0 1.05189 − 1.96795i 0 0.157199 + 0.129010i 0 −1.83383 − 1.22532i 0 −1.09966 − 1.64575i 0
17.13 0 1.06292 − 1.98859i 0 −2.42690 − 1.99171i 0 −2.87834 − 1.92325i 0 −1.15797 − 1.73303i 0
17.14 0 1.32215 − 2.47358i 0 −2.12126 − 1.74087i 0 3.19792 + 2.13678i 0 −2.70377 − 4.04648i 0
17.15 0 1.52098 − 2.84556i 0 2.99516 + 2.45806i 0 0.848139 + 0.566708i 0 −4.11709 − 6.16166i 0
49.1 0 −0.317663 − 3.22529i 0 1.50066 + 0.802122i 0 −0.303583 + 1.52621i 0 −7.35922 + 1.46384i 0
49.2 0 −0.247362 − 2.51150i 0 −3.44799 − 1.84299i 0 0.702840 − 3.53342i 0 −3.30411 + 0.657229i 0
49.3 0 −0.222853 − 2.26266i 0 1.42929 + 0.763973i 0 −0.778071 + 3.91163i 0 −2.12763 + 0.423212i 0
49.4 0 −0.166619 − 1.69171i 0 0.195184 + 0.104328i 0 0.644963 − 3.24245i 0 0.108228 − 0.0215280i 0
49.5 0 −0.166477 − 1.69027i 0 2.32067 + 1.24042i 0 0.239955 − 1.20633i 0 0.113053 − 0.0224877i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
128.k even 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.k.a 240
4.b odd 2 1 128.2.k.a ✓ 240
128.k even 32 1 inner 512.2.k.a 240
128.l odd 32 1 128.2.k.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.k.a ✓ 240 4.b odd 2 1
128.2.k.a ✓ 240 128.l odd 32 1
512.2.k.a 240 1.a even 1 1 trivial
512.2.k.a 240 128.k even 32 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(512, [\chi])\).