Properties

Label 6021.2.a.t
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $0$
Dimension $40$
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: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 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 −2.70299 0 5.30613 −3.42111 0 −1.82484 −8.93642 0 9.24720
1.2 −2.69102 0 5.24161 −2.40662 0 2.96509 −8.72327 0 6.47626
1.3 −2.65109 0 5.02828 1.58126 0 −3.54711 −8.02824 0 −4.19207
1.4 −2.63563 0 4.94652 3.15326 0 2.40096 −7.76593 0 −8.31081
1.5 −2.32144 0 3.38909 −1.08423 0 4.03069 −3.22470 0 2.51697
1.6 −2.16137 0 2.67154 −3.69668 0 0.733541 −1.45144 0 7.98990
1.7 −2.08210 0 2.33515 0.145427 0 4.41479 −0.697822 0 −0.302795
1.8 −2.00467 0 2.01871 4.10087 0 1.86508 −0.0375024 0 −8.22090
1.9 −1.74618 0 1.04913 −3.84450 0 0.0845192 1.66038 0 6.71318
1.10 −1.69867 0 0.885473 −0.550686 0 0.164494 1.89321 0 0.935432
1.11 −1.47084 0 0.163365 −1.43270 0 −5.26244 2.70139 0 2.10726
1.12 −1.37704 0 −0.103765 2.60531 0 −2.14080 2.89697 0 −3.58761
1.13 −1.35612 0 −0.160951 −1.12934 0 −1.90121 2.93050 0 1.53151
1.14 −1.23529 0 −0.474050 1.76412 0 −1.12531 3.05618 0 −2.17920
1.15 −1.01187 0 −0.976110 −1.54012 0 4.83269 3.01145 0 1.55841
1.16 −0.968595 0 −1.06182 2.59691 0 3.83305 2.96567 0 −2.51535
1.17 −0.538428 0 −1.71010 −1.52796 0 0.425996 1.99762 0 0.822698
1.18 −0.536475 0 −1.71219 −2.69435 0 −1.96571 1.99150 0 1.44545
1.19 −0.323417 0 −1.89540 3.36365 0 3.36190 1.25984 0 −1.08786
1.20 −0.243699 0 −1.94061 −0.214655 0 −3.34538 0.960324 0 0.0523112
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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.t 40
3.b odd 2 1 inner 6021.2.a.t 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.t 40 1.a even 1 1 trivial
6021.2.a.t 40 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{40} - 63 T_{2}^{38} + 1823 T_{2}^{36} - 32146 T_{2}^{34} + 386488 T_{2}^{32} - 3358868 T_{2}^{30} + 21829705 T_{2}^{28} - 108281137 T_{2}^{26} + 414740672 T_{2}^{24} - 1233202486 T_{2}^{22} + \cdots + 57132 \) Copy content Toggle raw display
\( T_{5}^{40} - 120 T_{5}^{38} + 6568 T_{5}^{36} - 217444 T_{5}^{34} + 4869438 T_{5}^{32} - 78157240 T_{5}^{30} + 929658705 T_{5}^{28} - 8359000882 T_{5}^{26} + 57453294656 T_{5}^{24} + \cdots + 337334448 \) Copy content Toggle raw display