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Properties

Label 4004.2.m.b
Level $4004$
Weight $2$
Character orbit 4004.m
Analytic conductor $31.972$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4004,2,Mod(2157,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4004.2157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.m (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: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(30\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2157.1 0 −2.58184 0 1.48218i 0 1.00000i 0 3.66591 0
2157.2 0 −2.58184 0 − 1.48218i 0 − 1.00000i 0 3.66591 0
2157.3 0 −2.50189 0 − 2.36927i 0 − 1.00000i 0 3.25946 0
2157.4 0 −2.50189 0 2.36927i 0 1.00000i 0 3.25946 0
2157.5 0 −2.42937 0 1.20006i 0 1.00000i 0 2.90185 0
2157.6 0 −2.42937 0 − 1.20006i 0 − 1.00000i 0 2.90185 0
2157.7 0 −2.41023 0 − 3.42759i 0 1.00000i 0 2.80920 0
2157.8 0 −2.41023 0 3.42759i 0 − 1.00000i 0 2.80920 0
2157.9 0 −1.49114 0 − 1.37882i 0 1.00000i 0 −0.776492 0
2157.10 0 −1.49114 0 1.37882i 0 − 1.00000i 0 −0.776492 0
2157.11 0 −0.822838 0 − 0.417533i 0 − 1.00000i 0 −2.32294 0
2157.12 0 −0.822838 0 0.417533i 0 1.00000i 0 −2.32294 0
2157.13 0 −0.0797053 0 0.579061i 0 − 1.00000i 0 −2.99365 0
2157.14 0 −0.0797053 0 − 0.579061i 0 1.00000i 0 −2.99365 0
2157.15 0 0.0840880 0 − 3.49341i 0 − 1.00000i 0 −2.99293 0
2157.16 0 0.0840880 0 3.49341i 0 1.00000i 0 −2.99293 0
2157.17 0 0.705440 0 − 2.65594i 0 1.00000i 0 −2.50236 0
2157.18 0 0.705440 0 2.65594i 0 − 1.00000i 0 −2.50236 0
2157.19 0 0.726270 0 − 2.96350i 0 1.00000i 0 −2.47253 0
2157.20 0 0.726270 0 2.96350i 0 − 1.00000i 0 −2.47253 0
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2157.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.m.b ✓ 30
13.b even 2 1 inner 4004.2.m.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.b ✓ 30 1.a even 1 1 trivial
4004.2.m.b ✓ 30 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 27 T_{3}^{13} + 282 T_{3}^{11} - 18 T_{3}^{10} - 1437 T_{3}^{9} + 270 T_{3}^{8} + 3702 T_{3}^{7} + \cdots + 6 \) acting on \(S_{2}^{\mathrm{new}}(4004, [\chi])\). Copy content Toggle raw display