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Properties

Label 507.2.m.b
Level $507$
Weight $2$
Character orbit 507.m
Analytic conductor $4.048$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,2,Mod(40,507)] 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("507.40"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [204] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9} - 6 q^{10} - 8 q^{11} - 21 q^{12} + 54 q^{13} - 30 q^{14} - 6 q^{15} - 45 q^{16} - 18 q^{17} - q^{18} - 20 q^{19}+ \cdots + 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −2.61147 − 0.643669i 0.120537 − 0.992709i 4.63454 + 2.43240i 2.12219 − 3.07452i −0.953754 + 2.51484i 1.40227 + 1.24231i −6.51088 − 5.76814i −0.970942 − 0.239316i −7.52100 + 6.66303i
40.2 −2.55822 − 0.630545i 0.120537 − 0.992709i 4.37599 + 2.29670i −1.41367 + 2.04805i −0.934307 + 2.46356i −3.49265 − 3.09422i −5.80225 − 5.14035i −0.970942 − 0.239316i 4.90787 − 4.34799i
40.3 −2.29214 − 0.564962i 0.120537 − 0.992709i 3.16382 + 1.66050i −1.62559 + 2.35507i −0.837130 + 2.20733i 3.18940 + 2.82556i −2.77972 − 2.46262i −0.970942 − 0.239316i 5.05660 − 4.47975i
40.4 −1.94540 − 0.479497i 0.120537 − 0.992709i 1.78373 + 0.936175i 1.40868 − 2.04083i −0.710492 + 1.87341i −0.476086 − 0.421776i −0.0217150 − 0.0192378i −0.970942 − 0.239316i −3.71901 + 3.29475i
40.5 −1.68275 − 0.414759i 0.120537 − 0.992709i 0.888694 + 0.466423i 0.797976 − 1.15607i −0.614568 + 1.62048i −1.62398 − 1.43872i 1.29250 + 1.14506i −0.970942 − 0.239316i −1.82228 + 1.61440i
40.6 −1.18717 − 0.292610i 0.120537 − 0.992709i −0.447170 − 0.234693i −0.808035 + 1.17064i −0.433574 + 1.14324i −0.0218939 − 0.0193963i 2.29259 + 2.03106i −0.970942 − 0.239316i 1.30181 − 1.15330i
40.7 −1.09223 − 0.269211i 0.120537 − 0.992709i −0.650419 − 0.341366i −2.21140 + 3.20377i −0.398902 + 1.05182i −0.833262 − 0.738206i 2.30254 + 2.03987i −0.970942 − 0.239316i 3.27785 − 2.90392i
40.8 −0.374061 − 0.0921977i 0.120537 − 0.992709i −1.63949 − 0.860471i 0.205107 − 0.297149i −0.136614 + 0.360220i −0.762777 − 0.675762i 1.11067 + 0.983969i −0.970942 − 0.239316i −0.104119 + 0.0922415i
40.9 0.203567 + 0.0501748i 0.120537 − 0.992709i −1.73199 − 0.909018i −0.892774 + 1.29341i 0.0743463 − 0.196035i 2.37726 + 2.10607i −0.620832 − 0.550009i −0.970942 − 0.239316i −0.246636 + 0.218500i
40.10 0.479187 + 0.118109i 0.120537 − 0.992709i −1.55524 − 0.816254i 2.38960 − 3.46193i 0.175008 − 0.461457i −0.109460 − 0.0969731i −1.38767 − 1.22937i −0.970942 − 0.239316i 1.55395 − 1.37668i
40.11 0.781588 + 0.192644i 0.120537 − 0.992709i −1.19714 − 0.628310i −1.69825 + 2.46034i 0.285450 − 0.752669i 1.79485 + 1.59010i −2.01970 − 1.78930i −0.970942 − 0.239316i −1.80130 + 1.59582i
40.12 1.31905 + 0.325116i 0.120537 − 0.992709i −0.136731 − 0.0717618i 0.705120 − 1.02154i 0.481738 − 1.27024i −2.12000 − 1.87816i −2.19076 − 1.94084i −0.970942 − 0.239316i 1.26221 − 1.11822i
40.13 1.37667 + 0.339319i 0.120537 − 0.992709i 0.00917504 + 0.00481543i −2.23921 + 3.24406i 0.502784 − 1.32573i −3.33005 − 2.95017i −2.11159 − 1.87070i −0.970942 − 0.239316i −4.18343 + 3.70620i
40.14 1.45113 + 0.357672i 0.120537 − 0.992709i 0.206949 + 0.108615i 0.974467 − 1.41176i 0.529980 − 1.39744i 2.67610 + 2.37082i −1.97593 − 1.75052i −0.970942 − 0.239316i 1.91903 − 1.70011i
40.15 2.11067 + 0.520234i 0.120537 − 0.992709i 2.41339 + 1.26664i 1.14528 − 1.65922i 0.770855 − 2.03258i −0.139189 − 0.123310i 1.18063 + 1.04595i −0.970942 − 0.239316i 3.28049 − 2.90626i
40.16 2.38165 + 0.587024i 0.120537 − 0.992709i 3.55674 + 1.86672i −1.58170 + 2.29149i 0.869820 − 2.29353i 2.70346 + 2.39506i 3.70302 + 3.28059i −0.970942 − 0.239316i −5.11221 + 4.52903i
40.17 2.66897 + 0.657842i 0.120537 − 0.992709i 4.91973 + 2.58207i 0.113107 − 0.163863i 0.974754 − 2.57022i −1.95577 − 1.73266i 7.31692 + 6.48223i −0.970942 − 0.239316i 0.409674 − 0.362939i
79.1 −2.42650 − 1.27353i −0.970942 − 0.239316i 3.12991 + 4.53445i −0.0390694 − 0.103018i 2.05122 + 1.81722i −0.184062 − 1.51589i −1.15934 − 9.54803i 0.885456 + 0.464723i −0.0363936 + 0.299728i
79.2 −1.97300 − 1.03551i −0.970942 − 0.239316i 1.68431 + 2.44014i 1.49468 + 3.94114i 1.66785 + 1.47759i −0.165780 − 1.36532i −0.259180 − 2.13454i 0.885456 + 0.464723i 1.13209 − 9.32359i
79.3 −1.80313 − 0.946355i −0.970942 − 0.239316i 1.21956 + 1.76683i −0.526754 − 1.38893i 1.52426 + 1.35037i −0.593546 − 4.88829i −0.0360490 − 0.296891i 0.885456 + 0.464723i −0.364621 + 3.00293i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.g even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.m.b ✓ 204
169.g even 13 1 inner 507.2.m.b ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.m.b ✓ 204 1.a even 1 1 trivial
507.2.m.b ✓ 204 169.g even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{204} + T_{2}^{203} + 28 T_{2}^{202} + 32 T_{2}^{201} + 473 T_{2}^{200} + 587 T_{2}^{199} + \cdots + 43\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\). Copy content Toggle raw display