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Properties

Label 1050.3.e.e
Level $1050$
Weight $3$
Character orbit 1050.e
Analytic conductor $28.610$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(701,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{4} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{4} - 44 q^{9} + 96 q^{16} + 80 q^{19} - 28 q^{21} + 224 q^{31} - 128 q^{34} + 88 q^{36} + 92 q^{39} - 144 q^{46} + 168 q^{49} - 284 q^{51} + 144 q^{54} - 192 q^{64} + 224 q^{66} + 152 q^{69} - 160 q^{76} + 72 q^{79} - 212 q^{81} + 56 q^{84} + 168 q^{91} + 128 q^{94} + 876 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1 − 1.41421i −1.28756 − 2.70965i −2.00000 0 −3.83202 + 1.82088i −2.64575 2.82843i −5.68438 + 6.97766i 0
701.2 − 1.41421i 1.28756 − 2.70965i −2.00000 0 −3.83202 − 1.82088i 2.64575 2.82843i −5.68438 − 6.97766i 0
701.3 1.41421i −1.28756 + 2.70965i −2.00000 0 −3.83202 − 1.82088i −2.64575 − 2.82843i −5.68438 − 6.97766i 0
701.4 1.41421i 1.28756 + 2.70965i −2.00000 0 −3.83202 + 1.82088i 2.64575 − 2.82843i −5.68438 + 6.97766i 0
701.5 − 1.41421i −2.22285 + 2.01468i −2.00000 0 2.84919 + 3.14358i 2.64575 2.82843i 0.882103 − 8.95667i 0
701.6 − 1.41421i 2.22285 + 2.01468i −2.00000 0 2.84919 − 3.14358i −2.64575 2.82843i 0.882103 + 8.95667i 0
701.7 1.41421i −2.22285 − 2.01468i −2.00000 0 2.84919 − 3.14358i 2.64575 − 2.82843i 0.882103 + 8.95667i 0
701.8 1.41421i 2.22285 − 2.01468i −2.00000 0 2.84919 + 3.14358i −2.64575 − 2.82843i 0.882103 − 8.95667i 0
701.9 − 1.41421i −2.46556 + 1.70910i −2.00000 0 2.41703 + 3.48683i −2.64575 2.82843i 3.15796 − 8.42777i 0
701.10 − 1.41421i 2.46556 + 1.70910i −2.00000 0 2.41703 − 3.48683i 2.64575 2.82843i 3.15796 + 8.42777i 0
701.11 1.41421i −2.46556 − 1.70910i −2.00000 0 2.41703 − 3.48683i −2.64575 − 2.82843i 3.15796 + 8.42777i 0
701.12 1.41421i 2.46556 − 1.70910i −2.00000 0 2.41703 + 3.48683i 2.64575 − 2.82843i 3.15796 − 8.42777i 0
701.13 − 1.41421i −0.554262 + 2.94835i −2.00000 0 4.16960 + 0.783844i 2.64575 2.82843i −8.38559 − 3.26832i 0
701.14 − 1.41421i 0.554262 + 2.94835i −2.00000 0 4.16960 − 0.783844i −2.64575 2.82843i −8.38559 + 3.26832i 0
701.15 1.41421i −0.554262 − 2.94835i −2.00000 0 4.16960 − 0.783844i 2.64575 − 2.82843i −8.38559 + 3.26832i 0
701.16 1.41421i 0.554262 − 2.94835i −2.00000 0 4.16960 + 0.783844i −2.64575 − 2.82843i −8.38559 − 3.26832i 0
701.17 − 1.41421i −2.79991 − 1.07726i −2.00000 0 −1.52347 + 3.95968i 2.64575 2.82843i 6.67904 + 6.03245i 0
701.18 − 1.41421i 2.79991 − 1.07726i −2.00000 0 −1.52347 − 3.95968i −2.64575 2.82843i 6.67904 − 6.03245i 0
701.19 1.41421i −2.79991 + 1.07726i −2.00000 0 −1.52347 − 3.95968i 2.64575 − 2.82843i 6.67904 − 6.03245i 0
701.20 1.41421i 2.79991 + 1.07726i −2.00000 0 −1.52347 + 3.95968i −2.64575 − 2.82843i 6.67904 + 6.03245i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.e.e 24
3.b odd 2 1 inner 1050.3.e.e 24
5.b even 2 1 inner 1050.3.e.e 24
5.c odd 4 2 210.3.c.a ✓ 24
15.d odd 2 1 inner 1050.3.e.e 24
15.e even 4 2 210.3.c.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.c.a ✓ 24 5.c odd 4 2
210.3.c.a ✓ 24 15.e even 4 2
1050.3.e.e 24 1.a even 1 1 trivial
1050.3.e.e 24 3.b odd 2 1 inner
1050.3.e.e 24 5.b even 2 1 inner
1050.3.e.e 24 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{12} + 526T_{11}^{10} + 76657T_{11}^{8} + 4209048T_{11}^{6} + 95989320T_{11}^{4} + 817879680T_{11}^{2} + 2289048336 \) Copy content Toggle raw display
\( T_{13}^{12} - 1082 T_{13}^{10} + 379213 T_{13}^{8} - 48657988 T_{13}^{6} + 2156635604 T_{13}^{4} + \cdots + 989983296 \) Copy content Toggle raw display