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Properties

Label 6022.2.a.e
Level $6022$
Weight $2$
Character orbit 6022.a
Self dual yes
Analytic conductor $48.086$
Analytic rank $0$
Dimension $68$
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] = [6022,2,Mod(1,6022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6022 = 2 \cdot 3011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6022.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: \(48.0859120972\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 68 q^{2} + 25 q^{3} + 68 q^{4} + 20 q^{5} + 25 q^{6} + 29 q^{7} + 68 q^{8} + 87 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 68 q^{2} + 25 q^{3} + 68 q^{4} + 20 q^{5} + 25 q^{6} + 29 q^{7} + 68 q^{8} + 87 q^{9} + 20 q^{10} + 46 q^{11} + 25 q^{12} + 30 q^{13} + 29 q^{14} + 13 q^{15} + 68 q^{16} + 73 q^{17} + 87 q^{18} + 56 q^{19} + 20 q^{20} - 5 q^{21} + 46 q^{22} + 63 q^{23} + 25 q^{24} + 88 q^{25} + 30 q^{26} + 67 q^{27} + 29 q^{28} + 43 q^{29} + 13 q^{30} + 68 q^{31} + 68 q^{32} + 26 q^{33} + 73 q^{34} + 50 q^{35} + 87 q^{36} + 8 q^{37} + 56 q^{38} + 6 q^{39} + 20 q^{40} + 64 q^{41} - 5 q^{42} + 52 q^{43} + 46 q^{44} + 7 q^{45} + 63 q^{46} + 94 q^{47} + 25 q^{48} + 91 q^{49} + 88 q^{50} + 20 q^{51} + 30 q^{52} + 38 q^{53} + 67 q^{54} + 37 q^{55} + 29 q^{56} + 4 q^{57} + 43 q^{58} + 84 q^{59} + 13 q^{60} + 26 q^{61} + 68 q^{62} + 22 q^{63} + 68 q^{64} - 20 q^{65} + 26 q^{66} + 54 q^{67} + 73 q^{68} - 11 q^{69} + 50 q^{70} + 46 q^{71} + 87 q^{72} + 62 q^{73} + 8 q^{74} + 54 q^{75} + 56 q^{76} + 67 q^{77} + 6 q^{78} + 67 q^{79} + 20 q^{80} + 120 q^{81} + 64 q^{82} + 130 q^{83} - 5 q^{84} - 24 q^{85} + 52 q^{86} + 72 q^{87} + 46 q^{88} + 61 q^{89} + 7 q^{90} + 43 q^{91} + 63 q^{92} + 40 q^{93} + 94 q^{94} + 55 q^{95} + 25 q^{96} + 41 q^{97} + 91 q^{98} + 106 q^{99}+O(q^{100}) \) 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 1.00000 −3.35476 1.00000 −3.57894 −3.35476 1.79257 1.00000 8.25439 −3.57894
1.2 1.00000 −3.25636 1.00000 2.14369 −3.25636 −2.21695 1.00000 7.60391 2.14369
1.3 1.00000 −3.01193 1.00000 3.00375 −3.01193 5.03538 1.00000 6.07172 3.00375
1.4 1.00000 −2.72056 1.00000 −0.689023 −2.72056 0.199511 1.00000 4.40143 −0.689023
1.5 1.00000 −2.64273 1.00000 −1.57154 −2.64273 −2.07612 1.00000 3.98402 −1.57154
1.6 1.00000 −2.61602 1.00000 −0.861265 −2.61602 0.881327 1.00000 3.84357 −0.861265
1.7 1.00000 −2.49368 1.00000 −0.272983 −2.49368 −4.45268 1.00000 3.21842 −0.272983
1.8 1.00000 −2.49125 1.00000 3.68938 −2.49125 1.92510 1.00000 3.20631 3.68938
1.9 1.00000 −2.43949 1.00000 −1.96660 −2.43949 4.60253 1.00000 2.95113 −1.96660
1.10 1.00000 −2.32468 1.00000 2.75364 −2.32468 1.28652 1.00000 2.40413 2.75364
1.11 1.00000 −2.31895 1.00000 2.61531 −2.31895 1.26807 1.00000 2.37754 2.61531
1.12 1.00000 −2.31495 1.00000 −4.12889 −2.31495 −1.35948 1.00000 2.35897 −4.12889
1.13 1.00000 −2.26646 1.00000 2.42379 −2.26646 −1.89494 1.00000 2.13682 2.42379
1.14 1.00000 −2.24505 1.00000 −1.17279 −2.24505 1.15296 1.00000 2.04026 −1.17279
1.15 1.00000 −1.47177 1.00000 0.895866 −1.47177 0.963361 1.00000 −0.833884 0.895866
1.16 1.00000 −1.21963 1.00000 −1.41374 −1.21963 −2.00566 1.00000 −1.51249 −1.41374
1.17 1.00000 −1.15681 1.00000 1.22743 −1.15681 3.85871 1.00000 −1.66179 1.22743
1.18 1.00000 −1.15100 1.00000 1.71186 −1.15100 3.72563 1.00000 −1.67520 1.71186
1.19 1.00000 −1.02270 1.00000 3.27073 −1.02270 4.67919 1.00000 −1.95408 3.27073
1.20 1.00000 −0.927049 1.00000 −3.29749 −0.927049 4.05370 1.00000 −2.14058 −3.29749
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.68
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3011\) \(1\)

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 6022.2.a.e ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6022.2.a.e ✓ 68 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{68} - 25 T_{3}^{67} + 167 T_{3}^{66} + 961 T_{3}^{65} - 17331 T_{3}^{64} + 33713 T_{3}^{63} + \cdots - 87273344 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6022))\). Copy content Toggle raw display