Properties

Label 6040.2.a.q
Level $6040$
Weight $2$
Character orbit 6040.a
Self dual yes
Analytic conductor $48.230$
Analytic rank $1$
Dimension $23$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.a (trivial)

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: yes
Analytic conductor: \(48.2296428209\)
Analytic rank: \(1\)
Dimension: \(23\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 23 q - 4 q^{3} - 23 q^{5} - 9 q^{7} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q - 4 q^{3} - 23 q^{5} - 9 q^{7} + 37 q^{9} - 17 q^{11} - 10 q^{13} + 4 q^{15} - 5 q^{21} - 17 q^{23} + 23 q^{25} - 16 q^{27} - 5 q^{29} - 2 q^{31} - 3 q^{33} + 9 q^{35} - 10 q^{37} - 19 q^{39} - 6 q^{41} - 23 q^{43} - 37 q^{45} - 20 q^{47} + 68 q^{49} - 32 q^{51} - 23 q^{53} + 17 q^{55} + 8 q^{57} - 13 q^{59} + 4 q^{61} - 33 q^{63} + 10 q^{65} - 23 q^{67} + 27 q^{69} - 64 q^{71} - 5 q^{73} - 4 q^{75} - 50 q^{77} - 2 q^{79} + 47 q^{81} - 22 q^{83} - 38 q^{87} + q^{89} - 6 q^{91} - 43 q^{93} + 31 q^{97} - 68 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.35813 0 −1.00000 0 −3.21673 0 8.27706 0
1.2 0 −3.21791 0 −1.00000 0 3.89140 0 7.35496 0
1.3 0 −3.04643 0 −1.00000 0 −4.80583 0 6.28073 0
1.4 0 −2.90197 0 −1.00000 0 0.345703 0 5.42142 0
1.5 0 −2.61857 0 −1.00000 0 4.97833 0 3.85689 0
1.6 0 −2.12620 0 −1.00000 0 −4.25186 0 1.52074 0
1.7 0 −1.37961 0 −1.00000 0 −1.90795 0 −1.09667 0
1.8 0 −1.37145 0 −1.00000 0 3.46098 0 −1.11913 0
1.9 0 −1.32609 0 −1.00000 0 2.45028 0 −1.24148 0
1.10 0 −0.877804 0 −1.00000 0 −0.669807 0 −2.22946 0
1.11 0 −0.739679 0 −1.00000 0 1.72019 0 −2.45288 0
1.12 0 −0.717713 0 −1.00000 0 −4.74749 0 −2.48489 0
1.13 0 −0.618582 0 −1.00000 0 −3.29286 0 −2.61736 0
1.14 0 0.512071 0 −1.00000 0 −0.0450824 0 −2.73778 0
1.15 0 1.23778 0 −1.00000 0 2.85194 0 −1.46790 0
1.16 0 1.34680 0 −1.00000 0 −1.73300 0 −1.18612 0
1.17 0 1.71820 0 −1.00000 0 3.71115 0 −0.0477919 0
1.18 0 2.01176 0 −1.00000 0 −2.39634 0 1.04718 0
1.19 0 2.34338 0 −1.00000 0 0.957310 0 2.49143 0
1.20 0 2.46975 0 −1.00000 0 −2.30951 0 3.09967 0
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(151\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6040.2.a.q 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.q 23 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):

\( T_{3}^{23} + 4 T_{3}^{22} - 45 T_{3}^{21} - 188 T_{3}^{20} + 846 T_{3}^{19} + 3771 T_{3}^{18} - 8547 T_{3}^{17} - 42303 T_{3}^{16} + 48752 T_{3}^{15} + 291821 T_{3}^{14} - 139453 T_{3}^{13} - 1280053 T_{3}^{12} + \cdots + 155000 \) Copy content Toggle raw display
\( T_{7}^{23} + 9 T_{7}^{22} - 74 T_{7}^{21} - 857 T_{7}^{20} + 1757 T_{7}^{19} + 34264 T_{7}^{18} + 336 T_{7}^{17} - 750187 T_{7}^{16} - 801360 T_{7}^{15} + 9793378 T_{7}^{14} + 17263757 T_{7}^{13} - 77324914 T_{7}^{12} + \cdots + 27713536 \) Copy content Toggle raw display
\( T_{11}^{23} + 17 T_{11}^{22} - 20 T_{11}^{21} - 1838 T_{11}^{20} - 6534 T_{11}^{19} + 70413 T_{11}^{18} + 451128 T_{11}^{17} - 958945 T_{11}^{16} - 12358265 T_{11}^{15} - 5968462 T_{11}^{14} + 164667377 T_{11}^{13} + \cdots + 2081775616 \) Copy content Toggle raw display