Properties

Label 6021.2.a.p
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $1$
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: \(1\)
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

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 −2.76954 0 5.67037 0.397024 0 1.86856 −10.1653 0 −1.09958
1.2 −2.71512 0 5.37185 2.73826 0 0.227565 −9.15497 0 −7.43468
1.3 −2.57522 0 4.63175 −3.44083 0 −2.38378 −6.77735 0 8.86089
1.4 −2.51287 0 4.31451 −2.91287 0 3.68266 −5.81606 0 7.31966
1.5 −2.50224 0 4.26123 −0.804329 0 3.78595 −5.65814 0 2.01263
1.6 −2.29192 0 3.25289 −2.24227 0 −1.95732 −2.87153 0 5.13910
1.7 −1.77853 0 1.16318 0.209108 0 1.24629 1.48830 0 −0.371906
1.8 −1.74295 0 1.03787 3.36855 0 −3.59458 1.67694 0 −5.87121
1.9 −1.72263 0 0.967467 −3.66365 0 0.816918 1.77868 0 6.31112
1.10 −1.58898 0 0.524854 −1.96674 0 3.69318 2.34398 0 3.12510
1.11 −1.32840 0 −0.235355 0.0596677 0 −3.41560 2.96944 0 −0.0792625
1.12 −1.29059 0 −0.334371 −0.472881 0 −4.11430 3.01272 0 0.610296
1.13 −1.20962 0 −0.536819 −3.36438 0 −4.87159 3.06859 0 4.06962
1.14 −1.17334 0 −0.623271 4.01472 0 1.71701 3.07799 0 −4.71064
1.15 −0.890971 0 −1.20617 1.02214 0 1.73110 2.85661 0 −0.910699
1.16 −0.444304 0 −1.80259 1.53364 0 −0.0368149 1.68951 0 −0.681403
1.17 −0.238471 0 −1.94313 0.830062 0 1.83678 0.940324 0 −0.197946
1.18 −0.184073 0 −1.96612 −3.73214 0 4.37279 0.730057 0 0.686988
1.19 0.0361660 0 −1.99869 2.14054 0 4.70710 −0.144617 0 0.0774147
1.20 0.306610 0 −1.90599 −2.77514 0 −1.39648 −1.19761 0 −0.850885
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.p 35
3.b odd 2 1 6021.2.a.s yes 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.p 35 1.a even 1 1 trivial
6021.2.a.s yes 35 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{35} + 4 T_{2}^{34} - 45 T_{2}^{33} - 191 T_{2}^{32} + 900 T_{2}^{31} + 4116 T_{2}^{30} - 10519 T_{2}^{29} - 52982 T_{2}^{28} + 79265 T_{2}^{27} + 454465 T_{2}^{26} - 398432 T_{2}^{25} - 2744239 T_{2}^{24} + 1314611 T_{2}^{23} + \cdots - 336 \) Copy content Toggle raw display
\( T_{5}^{35} + 14 T_{5}^{34} - 3 T_{5}^{33} - 887 T_{5}^{32} - 2828 T_{5}^{31} + 21605 T_{5}^{30} + 117281 T_{5}^{29} - 229098 T_{5}^{28} - 2245013 T_{5}^{27} + 238516 T_{5}^{26} + 24887450 T_{5}^{25} + 20420308 T_{5}^{24} + \cdots + 81 \) Copy content Toggle raw display