Properties

Label 6021.2.a.k
Level $6021$
Weight $2$
Character orbit 6021.a
Self dual yes
Analytic conductor $48.078$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} - \beta_1) q^{5} + q^{7} - 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} - \beta_1) q^{5} + q^{7} - 2 \beta_{2} q^{8} + ( - \beta_{3} + 1) q^{10} + (2 \beta_{2} - 2 \beta_1) q^{11} + q^{13} + \beta_{2} q^{14} - 4 q^{16} + ( - 3 \beta_{2} + 2 \beta_1) q^{17} + 3 \beta_{3} q^{19} + ( - 2 \beta_{3} + 2) q^{22} + ( - 4 \beta_{2} + 3 \beta_1) q^{23} + ( - \beta_{3} - 2) q^{25} + \beta_{2} q^{26} + ( - 3 \beta_{2} + 3 \beta_1) q^{29} + (\beta_{3} - 5) q^{31} + (2 \beta_{3} - 4) q^{34} + (\beta_{2} - \beta_1) q^{35} - 9 q^{37} + ( - 3 \beta_{2} + 6 \beta_1) q^{38} + (2 \beta_{3} - 2) q^{40} + (2 \beta_{2} + \beta_1) q^{41} + ( - 2 \beta_{3} + 4) q^{43} + (3 \beta_{3} - 5) q^{46} + (8 \beta_{2} - 2 \beta_1) q^{47} - 6 q^{49} + ( - \beta_{2} - 2 \beta_1) q^{50} + ( - 6 \beta_{2} - \beta_1) q^{53} + ( - 2 \beta_{3} + 6) q^{55} - 2 \beta_{2} q^{56} + (3 \beta_{3} - 3) q^{58} + ( - 6 \beta_{2} + 6 \beta_1) q^{59} + ( - 3 \beta_{3} - 4) q^{61} + ( - 6 \beta_{2} + 2 \beta_1) q^{62} + 8 q^{64} + (\beta_{2} - \beta_1) q^{65} - 9 q^{67} + ( - \beta_{3} + 1) q^{70} + ( - 5 \beta_{2} + 6 \beta_1) q^{71} + ( - 5 \beta_{3} + 2) q^{73} - 9 \beta_{2} q^{74} + (2 \beta_{2} - 2 \beta_1) q^{77} + (\beta_{3} + 8) q^{79} + ( - 4 \beta_{2} + 4 \beta_1) q^{80} + (\beta_{3} + 5) q^{82} + (5 \beta_{2} - 5 \beta_1) q^{83} + (3 \beta_{3} - 7) q^{85} + (6 \beta_{2} - 4 \beta_1) q^{86} + (4 \beta_{3} - 4) q^{88} + ( - 2 \beta_{2} + 2 \beta_1) q^{89} + q^{91} + ( - 2 \beta_{3} + 14) q^{94} + ( - 9 \beta_{2} + 3 \beta_1) q^{95} + ( - 3 \beta_{3} - 4) q^{97} - 6 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 4 q^{10} + 4 q^{13} - 16 q^{16} + 8 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 36 q^{37} - 8 q^{40} + 16 q^{43} - 20 q^{46} - 24 q^{49} + 24 q^{55} - 12 q^{58} - 16 q^{61} + 32 q^{64} - 36 q^{67} + 4 q^{70} + 8 q^{73} + 32 q^{79} + 20 q^{82} - 28 q^{85} - 16 q^{88} + 4 q^{91} + 56 q^{94} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 4\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.874032
−2.28825
2.28825
−0.874032
−1.41421 0 0 −2.28825 0 1.00000 2.82843 0 3.23607
1.2 −1.41421 0 0 0.874032 0 1.00000 2.82843 0 −1.23607
1.3 1.41421 0 0 −0.874032 0 1.00000 −2.82843 0 −1.23607
1.4 1.41421 0 0 2.28825 0 1.00000 −2.82843 0 3.23607
\(n\): e.g. 2-40 or 990-1000
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.k 4
3.b odd 2 1 inner 6021.2.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6021.2.a.k 4 1.a even 1 1 trivial
6021.2.a.k 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 24T^{2} + 64 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 70T^{2} + 100 \) Copy content Toggle raw display
$29$ \( T^{4} - 54T^{2} + 324 \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$37$ \( (T + 9)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 30T^{2} + 100 \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 216T^{2} + 7744 \) Copy content Toggle raw display
$53$ \( T^{4} - 174T^{2} + 6724 \) Copy content Toggle raw display
$59$ \( T^{4} - 216T^{2} + 5184 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 29)^{2} \) Copy content Toggle raw display
$67$ \( (T + 9)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 196T^{2} + 6724 \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 121)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 16 T + 59)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 150T^{2} + 2500 \) Copy content Toggle raw display
$89$ \( T^{4} - 24T^{2} + 64 \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 29)^{2} \) Copy content Toggle raw display
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