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Properties

Label 6021.2.a.o
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $1$
Dimension $30$
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: \(1\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 32 q^{4} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + 32 q^{4} - 20 q^{7} - 30 q^{10} - 22 q^{13} + 40 q^{16} - 56 q^{19} - 32 q^{22} + 38 q^{25} - 56 q^{28} - 30 q^{31} - 20 q^{34} + 8 q^{37} - 56 q^{40} - 56 q^{43} - 56 q^{46} + 34 q^{49} - 42 q^{52} - 80 q^{55} - 12 q^{58} - 46 q^{61} - 2 q^{64} - 40 q^{67} - 38 q^{70} - 42 q^{73} - 172 q^{76} - 124 q^{79} - 14 q^{82} - 34 q^{85} - 68 q^{88} - 90 q^{91} - 80 q^{94} - 38 q^{97}+O(q^{100}) \) 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 −2.81836 0 5.94314 3.63819 0 −3.78079 −11.1132 0 −10.2537
1.2 −2.46595 0 4.08089 −2.98652 0 −3.37186 −5.13136 0 7.36460
1.3 −2.45457 0 4.02493 −0.00507495 0 −0.674826 −4.97035 0 0.0124568
1.4 −2.30967 0 3.33455 1.91273 0 −0.513141 −3.08237 0 −4.41775
1.5 −2.21680 0 2.91420 2.76232 0 −0.455549 −2.02660 0 −6.12351
1.6 −2.00204 0 2.00815 3.21168 0 4.46796 −0.0163190 0 −6.42990
1.7 −1.85933 0 1.45712 −1.44601 0 1.81391 1.00940 0 2.68862
1.8 −1.73290 0 1.00293 −2.43764 0 −3.28701 1.72782 0 4.22417
1.9 −1.72554 0 0.977485 −1.44752 0 0.0997377 1.76439 0 2.49776
1.10 −0.841433 0 −1.29199 −1.54140 0 −2.11294 2.76999 0 1.29698
1.11 −0.705496 0 −1.50228 2.77052 0 2.04618 2.47084 0 −1.95459
1.12 −0.623430 0 −1.61133 4.19098 0 0.517171 2.25142 0 −2.61278
1.13 −0.611838 0 −1.62565 −0.865277 0 −3.42018 2.21831 0 0.529409
1.14 −0.536057 0 −1.71264 3.41198 0 −5.12771 1.99019 0 −1.82902
1.15 −0.0224203 0 −1.99950 −0.325123 0 3.79904 0.0896701 0 0.00728936
1.16 0.0224203 0 −1.99950 0.325123 0 3.79904 −0.0896701 0 0.00728936
1.17 0.536057 0 −1.71264 −3.41198 0 −5.12771 −1.99019 0 −1.82902
1.18 0.611838 0 −1.62565 0.865277 0 −3.42018 −2.21831 0 0.529409
1.19 0.623430 0 −1.61133 −4.19098 0 0.517171 −2.25142 0 −2.61278
1.20 0.705496 0 −1.50228 −2.77052 0 2.04618 −2.47084 0 −1.95459
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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.o ✓ 30
3.b odd 2 1 inner 6021.2.a.o ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.o ✓ 30 1.a even 1 1 trivial
6021.2.a.o ✓ 30 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{30} - 46 T_{2}^{28} + 940 T_{2}^{26} - 11253 T_{2}^{24} + 87607 T_{2}^{22} - 465452 T_{2}^{20} + \cdots - 7 \) Copy content Toggle raw display
\( T_{5}^{30} - 94 T_{5}^{28} + 3925 T_{5}^{26} - 96192 T_{5}^{24} + 1539723 T_{5}^{22} - 16945610 T_{5}^{20} + \cdots - 6727 \) Copy content Toggle raw display