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Properties

Label 6021.2.a.s
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $0$
Dimension $35$
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] = [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: \(35\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q + 4 q^{2} + 36 q^{4} + 14 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q + 4 q^{2} + 36 q^{4} + 14 q^{5} + 2 q^{7} + 15 q^{8} + 7 q^{10} + 30 q^{11} + 20 q^{14} + 34 q^{16} + 28 q^{17} + 2 q^{19} + 33 q^{20} + 8 q^{23} + 27 q^{25} + 20 q^{26} - 12 q^{28} + 39 q^{29} + 49 q^{32} - 5 q^{34} + 31 q^{35} + 26 q^{38} + 60 q^{41} + 2 q^{43} + 69 q^{44} - 15 q^{46} + 29 q^{47} + 35 q^{49} + 7 q^{50} + 2 q^{52} + 19 q^{53} + 20 q^{55} + 87 q^{56} + 8 q^{58} + 87 q^{59} - 8 q^{61} + 28 q^{62} + 67 q^{64} + 33 q^{65} + 26 q^{67} + 37 q^{68} - 56 q^{70} + 61 q^{71} - 16 q^{73} + 64 q^{74} + 5 q^{76} + 35 q^{77} - 18 q^{79} + 60 q^{80} + 26 q^{82} + 72 q^{83} - 23 q^{85} + 29 q^{86} - 46 q^{88} + 48 q^{89} + 6 q^{91} + 32 q^{92} - 21 q^{94} + 47 q^{95} - 49 q^{97} + 28 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.61106 0 4.81762 −0.184269 0 −4.16918 −7.35697 0 0.481138
1.2 −2.56113 0 4.55938 2.31479 0 0.335324 −6.55491 0 −5.92848
1.3 −2.52358 0 4.36847 4.38233 0 0.500266 −5.97704 0 −11.0592
1.4 −2.11116 0 2.45701 −1.14921 0 −0.801831 −0.964813 0 2.42617
1.5 −2.05952 0 2.24163 −0.0530226 0 0.321930 −0.497639 0 0.109201
1.6 −2.04642 0 2.18783 −4.05257 0 −4.59226 −0.384375 0 8.29325
1.7 −1.96187 0 1.84894 −0.732762 0 2.35429 0.296368 0 1.43758
1.8 −1.61982 0 0.623803 1.43034 0 3.99494 2.22919 0 −2.31689
1.9 −1.59701 0 0.550431 −1.58329 0 −2.55623 2.31497 0 2.52853
1.10 −1.41132 0 −0.00818088 −0.0525096 0 −2.04104 2.83418 0 0.0741077
1.11 −1.13326 0 −0.715718 4.26680 0 −0.430082 3.07762 0 −4.83540
1.12 −0.965400 0 −1.06800 2.76632 0 3.52339 2.96185 0 −2.67060
1.13 −0.941060 0 −1.11441 −2.88068 0 1.18539 2.93084 0 2.71090
1.14 −0.660797 0 −1.56335 0.251217 0 −4.35265 2.35465 0 −0.166003
1.15 −0.413605 0 −1.82893 0.215001 0 0.812311 1.58366 0 −0.0889253
1.16 −0.306610 0 −1.90599 2.77514 0 −1.39648 1.19761 0 −0.850885
1.17 −0.0361660 0 −1.99869 −2.14054 0 4.70710 0.144617 0 0.0774147
1.18 0.184073 0 −1.96612 3.73214 0 4.37279 −0.730057 0 0.686988
1.19 0.238471 0 −1.94313 −0.830062 0 1.83678 −0.940324 0 −0.197946
1.20 0.444304 0 −1.80259 −1.53364 0 −0.0368149 −1.68951 0 −0.681403
See all 35 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.35
Significant digits:
Format:

Atkin-Lehner signs

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

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 6021.2.a.s yes 35
3.b odd 2 1 6021.2.a.p ✓ 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.p ✓ 35 3.b odd 2 1
6021.2.a.s yes 35 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{35} - 4 T_{2}^{34} - 45 T_{2}^{33} + 191 T_{2}^{32} + 900 T_{2}^{31} - 4116 T_{2}^{30} + \cdots + 336 \) Copy content Toggle raw display
\( T_{5}^{35} - 14 T_{5}^{34} - 3 T_{5}^{33} + 887 T_{5}^{32} - 2828 T_{5}^{31} - 21605 T_{5}^{30} + \cdots - 81 \) Copy content Toggle raw display