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Properties

Label 507.2.f.g
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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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] = [507,2,Mod(239,507)] 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("507.239"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{9} - 8 q^{16} + 112 q^{22} - 84 q^{27} + 128 q^{40} - 56 q^{42} - 188 q^{48} + 8 q^{55} + 56 q^{61} - 92 q^{66} - 72 q^{81} - 112 q^{87} + 296 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 −1.82526 − 1.82526i 1.43862 − 0.964559i 4.66316i 0.624703 + 0.624703i −4.38643 − 0.865285i −1.18894 − 1.18894i 4.86096 − 4.86096i 1.13925 − 2.77527i − 2.28049i
239.2 −1.82526 − 1.82526i 1.43862 + 0.964559i 4.66316i 0.624703 + 0.624703i −0.865285 − 4.38643i 1.18894 + 1.18894i 4.86096 − 4.86096i 1.13925 + 2.77527i − 2.28049i
239.3 −1.42721 − 1.42721i −1.57050 − 0.730430i 2.07385i 1.72251 + 1.72251i 1.19895 + 3.28391i −2.20041 − 2.20041i 0.105393 − 0.105393i 1.93294 + 2.29428i − 4.91676i
239.4 −1.42721 − 1.42721i −1.57050 + 0.730430i 2.07385i 1.72251 + 1.72251i 3.28391 + 1.19895i 2.20041 + 2.20041i 0.105393 − 0.105393i 1.93294 − 2.29428i − 4.91676i
239.5 −1.38407 − 1.38407i 0.526444 − 1.65011i 1.83129i 1.04664 + 1.04664i −3.01250 + 1.55523i 3.17096 + 3.17096i −0.233508 + 0.233508i −2.44571 − 1.73738i − 2.89724i
239.6 −1.38407 − 1.38407i 0.526444 + 1.65011i 1.83129i 1.04664 + 1.04664i 1.55523 − 3.01250i −3.17096 − 3.17096i −0.233508 + 0.233508i −2.44571 + 1.73738i − 2.89724i
239.7 −0.928351 − 0.928351i −1.37245 − 1.05658i − 0.276330i −2.12536 − 2.12536i 0.293240 + 2.25500i 2.06528 + 2.06528i −2.11323 + 2.11323i 0.767265 + 2.90022i 3.94616i
239.8 −0.928351 − 0.928351i −1.37245 + 1.05658i − 0.276330i −2.12536 − 2.12536i 2.25500 + 0.293240i −2.06528 − 2.06528i −2.11323 + 2.11323i 0.767265 − 2.90022i 3.94616i
239.9 −0.540287 − 0.540287i 0.0858391 − 1.72992i − 1.41618i 0.996141 + 0.996141i −0.981032 + 0.888277i −1.80254 − 1.80254i −1.84572 + 1.84572i −2.98526 − 0.296990i − 1.07640i
239.10 −0.540287 − 0.540287i 0.0858391 + 1.72992i − 1.41618i 0.996141 + 0.996141i 0.888277 − 0.981032i 1.80254 + 1.80254i −1.84572 + 1.84572i −2.98526 + 0.296990i − 1.07640i
239.11 −0.249216 − 0.249216i 0.892053 − 1.48467i − 1.87578i −2.45719 − 2.45719i −0.592316 + 0.147689i 0.821655 + 0.821655i −0.965906 + 0.965906i −1.40848 − 2.64881i 1.22474i
239.12 −0.249216 − 0.249216i 0.892053 + 1.48467i − 1.87578i −2.45719 − 2.45719i 0.147689 − 0.592316i −0.821655 − 0.821655i −0.965906 + 0.965906i −1.40848 + 2.64881i 1.22474i
239.13 0.249216 + 0.249216i 0.892053 − 1.48467i − 1.87578i 2.45719 + 2.45719i 0.592316 − 0.147689i −0.821655 − 0.821655i 0.965906 − 0.965906i −1.40848 − 2.64881i 1.22474i
239.14 0.249216 + 0.249216i 0.892053 + 1.48467i − 1.87578i 2.45719 + 2.45719i −0.147689 + 0.592316i 0.821655 + 0.821655i 0.965906 − 0.965906i −1.40848 + 2.64881i 1.22474i
239.15 0.540287 + 0.540287i 0.0858391 − 1.72992i − 1.41618i −0.996141 − 0.996141i 0.981032 − 0.888277i 1.80254 + 1.80254i 1.84572 − 1.84572i −2.98526 − 0.296990i − 1.07640i
239.16 0.540287 + 0.540287i 0.0858391 + 1.72992i − 1.41618i −0.996141 − 0.996141i −0.888277 + 0.981032i −1.80254 − 1.80254i 1.84572 − 1.84572i −2.98526 + 0.296990i − 1.07640i
239.17 0.928351 + 0.928351i −1.37245 − 1.05658i − 0.276330i 2.12536 + 2.12536i −0.293240 − 2.25500i −2.06528 − 2.06528i 2.11323 − 2.11323i 0.767265 + 2.90022i 3.94616i
239.18 0.928351 + 0.928351i −1.37245 + 1.05658i − 0.276330i 2.12536 + 2.12536i −2.25500 − 0.293240i 2.06528 + 2.06528i 2.11323 − 2.11323i 0.767265 − 2.90022i 3.94616i
239.19 1.38407 + 1.38407i 0.526444 − 1.65011i 1.83129i −1.04664 − 1.04664i 3.01250 − 1.55523i −3.17096 − 3.17096i 0.233508 − 0.233508i −2.44571 − 1.73738i − 2.89724i
239.20 1.38407 + 1.38407i 0.526444 + 1.65011i 1.83129i −1.04664 − 1.04664i −1.55523 + 3.01250i 3.17096 + 3.17096i 0.233508 − 0.233508i −2.44571 + 1.73738i − 2.89724i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
13.d odd 4 2 inner
39.d odd 2 1 inner
39.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.g ✓ 48
3.b odd 2 1 inner 507.2.f.g ✓ 48
13.b even 2 1 inner 507.2.f.g ✓ 48
13.c even 3 2 507.2.k.k 96
13.d odd 4 2 inner 507.2.f.g ✓ 48
13.e even 6 2 507.2.k.k 96
13.f odd 12 4 507.2.k.k 96
39.d odd 2 1 inner 507.2.f.g ✓ 48
39.f even 4 2 inner 507.2.f.g ✓ 48
39.h odd 6 2 507.2.k.k 96
39.i odd 6 2 507.2.k.k 96
39.k even 12 4 507.2.k.k 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.f.g ✓ 48 1.a even 1 1 trivial
507.2.f.g ✓ 48 3.b odd 2 1 inner
507.2.f.g ✓ 48 13.b even 2 1 inner
507.2.f.g ✓ 48 13.d odd 4 2 inner
507.2.f.g ✓ 48 39.d odd 2 1 inner
507.2.f.g ✓ 48 39.f even 4 2 inner
507.2.k.k 96 13.c even 3 2
507.2.k.k 96 13.e even 6 2
507.2.k.k 96 13.f odd 12 4
507.2.k.k 96 39.h odd 6 2
507.2.k.k 96 39.i odd 6 2
507.2.k.k 96 39.k even 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{24} + 79T_{2}^{20} + 1885T_{2}^{16} + 16327T_{2}^{12} + 37725T_{2}^{8} + 11531T_{2}^{4} + 169 \) Copy content Toggle raw display
\( T_{5}^{24} + 272T_{5}^{20} + 22390T_{5}^{16} + 611594T_{5}^{12} + 4403113T_{5}^{8} + 10383698T_{5}^{4} + 4826809 \) Copy content Toggle raw display
\( T_{7}^{24} + 623T_{7}^{20} + 104321T_{7}^{16} + 6865831T_{7}^{12} + 175807737T_{7}^{8} + 1229789799T_{7}^{4} + 1698181681 \) Copy content Toggle raw display