Properties

Label 512.2.k.a
Level $512$
Weight $2$
Character orbit 512.k
Analytic conductor $4.088$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.k (of order \(32\), degree \(16\), not minimal)

Newform invariants

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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240q + 16q^{3} - 16q^{5} + 16q^{7} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q + 16q^{3} - 16q^{5} + 16q^{7} - 16q^{9} + 16q^{11} - 16q^{13} + 16q^{15} - 16q^{17} + 16q^{19} - 16q^{21} + 16q^{23} - 16q^{25} + 16q^{27} - 16q^{29} + 16q^{31} - 16q^{33} + 16q^{35} - 16q^{37} + 16q^{39} - 16q^{41} + 16q^{43} - 16q^{45} + 16q^{47} - 16q^{49} + 16q^{51} - 16q^{53} + 16q^{55} - 16q^{57} + 16q^{59} - 16q^{61} + 16q^{67} - 16q^{69} + 16q^{71} - 16q^{73} + 16q^{75} - 16q^{77} + 16q^{79} - 16q^{81} + 16q^{83} - 16q^{85} + 16q^{87} - 16q^{89} + 16q^{91} - 16q^{93} + 16q^{95} - 16q^{97} + 16q^{99} + O(q^{100}) \)

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.

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 990-1000
Embeddings: e.g. 1-3 or 497.15
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 \(S_{2}^{\mathrm{new}}(512, [\chi])\).