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Properties

Label 6030.2.d.k
Level $6030$
Weight $2$
Character orbit 6030.d
Analytic conductor $48.150$
Analytic rank $0$
Dimension $24$
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] = [6030,2,Mod(2411,6030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("6030.2411");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.d (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: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8} + 24 q^{10} - 12 q^{11} + 24 q^{16} + 4 q^{19} - 24 q^{20} + 12 q^{22} + 24 q^{25} - 24 q^{32} - 16 q^{37} - 4 q^{38} + 24 q^{40} - 8 q^{41} - 12 q^{44} - 20 q^{49} - 24 q^{50} - 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} + 16 q^{74} + 4 q^{76} - 24 q^{80} + 8 q^{82} + 12 q^{88} - 4 q^{95} + 20 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2411.1 −1.00000 0 1.00000 −1.00000 0 − 4.73932i −1.00000 0 1.00000
2411.2 −1.00000 0 1.00000 −1.00000 0 − 3.69493i −1.00000 0 1.00000
2411.3 −1.00000 0 1.00000 −1.00000 0 − 3.48375i −1.00000 0 1.00000
2411.4 −1.00000 0 1.00000 −1.00000 0 − 3.30563i −1.00000 0 1.00000
2411.5 −1.00000 0 1.00000 −1.00000 0 − 2.95623i −1.00000 0 1.00000
2411.6 −1.00000 0 1.00000 −1.00000 0 − 2.79591i −1.00000 0 1.00000
2411.7 −1.00000 0 1.00000 −1.00000 0 − 2.65115i −1.00000 0 1.00000
2411.8 −1.00000 0 1.00000 −1.00000 0 − 2.64016i −1.00000 0 1.00000
2411.9 −1.00000 0 1.00000 −1.00000 0 − 1.58772i −1.00000 0 1.00000
2411.10 −1.00000 0 1.00000 −1.00000 0 − 1.26508i −1.00000 0 1.00000
2411.11 −1.00000 0 1.00000 −1.00000 0 − 0.376360i −1.00000 0 1.00000
2411.12 −1.00000 0 1.00000 −1.00000 0 − 0.0653098i −1.00000 0 1.00000
2411.13 −1.00000 0 1.00000 −1.00000 0 0.0653098i −1.00000 0 1.00000
2411.14 −1.00000 0 1.00000 −1.00000 0 0.376360i −1.00000 0 1.00000
2411.15 −1.00000 0 1.00000 −1.00000 0 1.26508i −1.00000 0 1.00000
2411.16 −1.00000 0 1.00000 −1.00000 0 1.58772i −1.00000 0 1.00000
2411.17 −1.00000 0 1.00000 −1.00000 0 2.64016i −1.00000 0 1.00000
2411.18 −1.00000 0 1.00000 −1.00000 0 2.65115i −1.00000 0 1.00000
2411.19 −1.00000 0 1.00000 −1.00000 0 2.79591i −1.00000 0 1.00000
2411.20 −1.00000 0 1.00000 −1.00000 0 2.95623i −1.00000 0 1.00000
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2411.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6030.2.d.k ✓ 24
3.b odd 2 1 6030.2.d.l yes 24
67.b odd 2 1 6030.2.d.l yes 24
201.d even 2 1 inner 6030.2.d.k ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6030.2.d.k ✓ 24 1.a even 1 1 trivial
6030.2.d.k ✓ 24 201.d even 2 1 inner
6030.2.d.l yes 24 3.b odd 2 1
6030.2.d.l yes 24 67.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6030, [\chi])\):

\( T_{7}^{24} + 94 T_{7}^{22} + 3817 T_{7}^{20} + 88210 T_{7}^{18} + 1283288 T_{7}^{16} + 12247040 T_{7}^{14} + \cdots + 331776 \) Copy content Toggle raw display
\( T_{11}^{12} + 6 T_{11}^{11} - 59 T_{11}^{10} - 352 T_{11}^{9} + 1154 T_{11}^{8} + 7160 T_{11}^{7} + \cdots - 145152 \) Copy content Toggle raw display
\( T_{41}^{12} + 4 T_{41}^{11} - 176 T_{41}^{10} - 764 T_{41}^{9} + 10058 T_{41}^{8} + 40928 T_{41}^{7} + \cdots + 15316992 \) Copy content Toggle raw display