Properties

Label 6021.2.a.n
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
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] = [6021,2,Mod(1,6021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6021.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: \(48.0779270570\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 30 q^{4} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 30 q^{4} + 14 q^{7} + 20 q^{10} + 8 q^{13} + 22 q^{16} + 36 q^{19} + 50 q^{22} + 38 q^{25} - 16 q^{28} + 64 q^{31} + 12 q^{34} + 48 q^{40} + 30 q^{43} + 42 q^{46} + 96 q^{49} + 44 q^{52} - 12 q^{55} - 36 q^{58} + 22 q^{61} - 16 q^{64} + 10 q^{67} + 72 q^{70} + 6 q^{73} - 8 q^{76} + 76 q^{79} + 80 q^{82} + 98 q^{85} + 80 q^{88} + 80 q^{91} + 140 q^{94} + 80 q^{97}+O(q^{100}) \) 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 −2.62208 0 4.87530 −2.85914 0 −3.15839 −7.53926 0 7.49690
1.2 −2.61208 0 4.82295 −0.781189 0 −0.254267 −7.37378 0 2.04053
1.3 −2.48811 0 4.19069 2.67155 0 3.81403 −5.45068 0 −6.64711
1.4 −2.32514 0 3.40627 −3.10806 0 4.20004 −3.26977 0 7.22668
1.5 −2.02272 0 2.09142 0.405475 0 −4.34576 −0.184909 0 −0.820164
1.6 −2.01253 0 2.05027 −3.97367 0 1.72063 −0.101176 0 7.99712
1.7 −1.75221 0 1.07024 1.33036 0 −3.64016 1.62913 0 −2.33106
1.8 −1.71011 0 0.924479 3.76518 0 −2.83710 1.83926 0 −6.43887
1.9 −1.44814 0 0.0971206 1.39837 0 2.53252 2.75564 0 −2.02504
1.10 −1.31983 0 −0.258046 0.212807 0 4.34948 2.98024 0 −0.280869
1.11 −0.869700 0 −1.24362 −3.21655 0 1.50813 2.82098 0 2.79743
1.12 −0.795724 0 −1.36682 0.986468 0 −3.41517 2.67906 0 −0.784957
1.13 −0.556736 0 −1.69004 −4.00675 0 4.91682 2.05438 0 2.23070
1.14 −0.125963 0 −1.98413 1.92011 0 2.00612 0.501853 0 −0.241862
1.15 −0.118028 0 −1.98607 1.85904 0 −0.396926 0.470468 0 −0.219419
1.16 0.118028 0 −1.98607 −1.85904 0 −0.396926 −0.470468 0 −0.219419
1.17 0.125963 0 −1.98413 −1.92011 0 2.00612 −0.501853 0 −0.241862
1.18 0.556736 0 −1.69004 4.00675 0 4.91682 −2.05438 0 2.23070
1.19 0.795724 0 −1.36682 −0.986468 0 −3.41517 −2.67906 0 −0.784957
1.20 0.869700 0 −1.24362 3.21655 0 1.50813 −2.82098 0 2.79743
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(223\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6021.2.a.n 30
3.b odd 2 1 inner 6021.2.a.n 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.n 30 1.a even 1 1 trivial
6021.2.a.n 30 3.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{30} - 45 T_{2}^{28} + 902 T_{2}^{26} - 10629 T_{2}^{24} + 81847 T_{2}^{22} - 433021 T_{2}^{20} + 1610983 T_{2}^{18} - 4240555 T_{2}^{16} + 7837897 T_{2}^{14} - 9955497 T_{2}^{12} + 8359905 T_{2}^{10} + \cdots - 28 \) Copy content Toggle raw display
\( T_{5}^{30} - 94 T_{5}^{28} + 3888 T_{5}^{26} - 93260 T_{5}^{24} + 1439928 T_{5}^{22} - 15026061 T_{5}^{20} + 108358761 T_{5}^{18} - 543324836 T_{5}^{16} + 1883425344 T_{5}^{14} - 4440409747 T_{5}^{12} + \cdots - 4085872 \) Copy content Toggle raw display