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Properties

Label 6021.2.a.q
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.

\(\operatorname{Tr}(f)(q) = \) \( 35 q - 4 q^{2} + 36 q^{4} - 10 q^{5} - 2 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 35 q - 4 q^{2} + 36 q^{4} - 10 q^{5} - 2 q^{7} - 15 q^{8} - 7 q^{10} - 34 q^{11} + 2 q^{13} - 18 q^{14} + 42 q^{16} - 20 q^{17} - 14 q^{19} - 27 q^{20} + 8 q^{22} - 8 q^{23} + 27 q^{25} - 28 q^{26} - 4 q^{28} - 23 q^{29} + 12 q^{31} - 29 q^{32} - 21 q^{34} - 41 q^{35} - 8 q^{37} - 18 q^{38} + 16 q^{40} - 50 q^{41} + 2 q^{43} - 83 q^{44} - 5 q^{46} - 21 q^{47} + 43 q^{49} - 39 q^{50} + 6 q^{52} - 37 q^{53} + 20 q^{55} - 33 q^{56} - 32 q^{58} - 81 q^{59} - 6 q^{61} - 26 q^{62} - q^{64} - 29 q^{65} + 12 q^{67} - 55 q^{68} + 50 q^{70} - 43 q^{71} - 20 q^{73} - 48 q^{74} - 15 q^{76} - 29 q^{77} + 28 q^{79} - 88 q^{80} - 6 q^{82} - 64 q^{83} - 67 q^{85} - 41 q^{86} + 10 q^{88} - 50 q^{89} + 2 q^{91} - 32 q^{92} + 15 q^{94} - 25 q^{95} + 9 q^{97} - 74 q^{98}+O(q^{100}) \) 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 −2.77094 0 5.67810 −3.43273 0 −0.892996 −10.1918 0 9.51187
1.2 −2.66906 0 5.12389 0.717238 0 0.344401 −8.33785 0 −1.91435
1.3 −2.51467 0 4.32355 2.08612 0 −5.01087 −5.84296 0 −5.24589
1.4 −2.47604 0 4.13075 −1.44591 0 2.49797 −5.27582 0 3.58014
1.5 −2.44295 0 3.96802 −3.21125 0 3.58144 −4.80778 0 7.84492
1.6 −2.20217 0 2.84954 2.41764 0 4.65530 −1.87082 0 −5.32405
1.7 −2.19310 0 2.80969 −1.63738 0 −1.86919 −1.77573 0 3.59093
1.8 −1.97681 0 1.90779 2.49801 0 2.31744 0.182287 0 −4.93810
1.9 −1.88652 0 1.55895 2.71795 0 −4.19954 0.832040 0 −5.12747
1.10 −1.57738 0 0.488121 −3.82175 0 4.29453 2.38480 0 6.02835
1.11 −1.45613 0 0.120315 3.33141 0 −2.57328 2.73707 0 −4.85097
1.12 −1.00750 0 −0.984952 −1.59854 0 3.40695 3.00733 0 1.61053
1.13 −1.00287 0 −0.994248 −2.43870 0 −1.20780 3.00285 0 2.44570
1.14 −0.835810 0 −1.30142 1.10458 0 1.89974 2.75936 0 −0.923222
1.15 −0.802939 0 −1.35529 1.60074 0 0.576275 2.69409 0 −1.28529
1.16 −0.424438 0 −1.81985 1.02884 0 −4.02930 1.62129 0 −0.436677
1.17 −0.378423 0 −1.85680 −0.673616 0 3.71590 1.45950 0 0.254912
1.18 −0.145112 0 −1.97894 −4.00700 0 −4.35864 0.577391 0 0.581462
1.19 0.0961107 0 −1.99076 −2.13805 0 −1.72518 −0.383555 0 −0.205489
1.20 0.384611 0 −1.85207 1.68485 0 −1.18684 −1.48155 0 0.648011
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.q ✓ 35
3.b odd 2 1 6021.2.a.r yes 35
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.q ✓ 35 1.a even 1 1 trivial
6021.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{35} + 4 T_{2}^{34} - 45 T_{2}^{33} - 191 T_{2}^{32} + 898 T_{2}^{31} + 4104 T_{2}^{30} + \cdots - 534 \) Copy content Toggle raw display
\( T_{5}^{35} + 10 T_{5}^{34} - 51 T_{5}^{33} - 805 T_{5}^{32} + 348 T_{5}^{31} + 28343 T_{5}^{30} + \cdots - 4984161 \) Copy content Toggle raw display