Properties

Label 2808.2.a.y
Level $2808$
Weight $2$
Character orbit 2808.a
Self dual yes
Analytic conductor $22.422$
Analytic rank $1$
Dimension $3$
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] = [2808,2,Mod(1,2808)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2808, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2808.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2808.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: \(22.4219928876\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{11} + q^{13} - 2 q^{17} - 2 \beta_1 q^{19} + (\beta_{2} - \beta_1 - 2) q^{23} + (\beta_{2} - \beta_1) q^{25} + (\beta_{2} - \beta_1 - 2) q^{29} + (\beta_{2} - \beta_1 - 2) q^{31} + ( - 2 \beta_1 - 2) q^{35} + (\beta_{2} - \beta_1 + 2) q^{37} + (2 \beta_1 - 4) q^{41} + ( - 2 \beta_{2} - 1) q^{43} + (3 \beta_1 + 1) q^{47} + ( - \beta_{2} + 3 \beta_1 + 3) q^{49} + ( - 3 \beta_{2} + \beta_1 - 2) q^{53} + (2 \beta_1 + 2) q^{55} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{59} + (\beta_{2} + 3 \beta_1 + 3) q^{61} + (\beta_1 - 1) q^{65} + (2 \beta_1 - 6) q^{67} + ( - 3 \beta_1 - 3) q^{71} + (2 \beta_{2} - 2 \beta_1) q^{73} + (\beta_{2} - 3 \beta_1 - 10) q^{77} + (2 \beta_{2} - 2 \beta_1 - 4) q^{79} + (\beta_{2} - 2 \beta_1 - 5) q^{83} + ( - 2 \beta_1 + 2) q^{85} + (\beta_{2} + 2 \beta_1 - 5) q^{89} + ( - \beta_{2} - \beta_1) q^{91} + ( - 2 \beta_{2} - 8) q^{95} + ( - 2 \beta_{2} + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{5} - q^{7} + q^{11} + 3 q^{13} - 6 q^{17} - 2 q^{19} - 7 q^{23} - q^{25} - 7 q^{29} - 7 q^{31} - 8 q^{35} + 5 q^{37} - 10 q^{41} - 3 q^{43} + 6 q^{47} + 12 q^{49} - 5 q^{53} + 8 q^{55} - 11 q^{59} + 12 q^{61} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 2 q^{73} - 33 q^{77} - 14 q^{79} - 17 q^{83} + 4 q^{85} - 13 q^{89} - q^{91} - 24 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) 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
−2.17741
0.321637
2.85577
0 0 0 −3.17741 0 −0.741113 0 0 0
1.2 0 0 0 −0.678363 0 3.89655 0 0 0
1.3 0 0 0 1.85577 0 −4.15544 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(-1\)

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 2808.2.a.y 3
3.b odd 2 1 2808.2.a.ba yes 3
4.b odd 2 1 5616.2.a.cb 3
12.b even 2 1 5616.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2808.2.a.y 3 1.a even 1 1 trivial
2808.2.a.ba yes 3 3.b odd 2 1
5616.2.a.cb 3 4.b odd 2 1
5616.2.a.cd 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2808))\):

\( T_{5}^{3} + 2T_{5}^{2} - 5T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 16T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{3} - T_{11}^{2} - 16T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} - 5 T - 4 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 16 T - 12 \) Copy content Toggle raw display
$11$ \( T^{3} - T^{2} - 16 T + 12 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( (T + 2)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 24 T - 16 \) Copy content Toggle raw display
$23$ \( T^{3} + 7 T^{2} - 8 T - 72 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} - 8 T - 72 \) Copy content Toggle raw display
$31$ \( T^{3} + 7 T^{2} - 8 T - 72 \) Copy content Toggle raw display
$37$ \( T^{3} - 5 T^{2} - 16 T + 8 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} + 8 T - 48 \) Copy content Toggle raw display
$43$ \( T^{3} + 3 T^{2} - 53 T - 183 \) Copy content Toggle raw display
$47$ \( T^{3} - 6 T^{2} - 45 T + 104 \) Copy content Toggle raw display
$53$ \( T^{3} + 5 T^{2} - 136 T - 432 \) Copy content Toggle raw display
$59$ \( T^{3} + 11 T^{2} - 87 T - 841 \) Copy content Toggle raw display
$61$ \( T^{3} - 12 T^{2} - 11 T - 2 \) Copy content Toggle raw display
$67$ \( T^{3} + 16 T^{2} + 60 T + 16 \) Copy content Toggle raw display
$71$ \( T^{3} + 12 T^{2} - 9 T - 162 \) Copy content Toggle raw display
$73$ \( T^{3} + 2 T^{2} - 96 T - 288 \) Copy content Toggle raw display
$79$ \( T^{3} + 14 T^{2} - 32 T - 576 \) Copy content Toggle raw display
$83$ \( T^{3} + 17 T^{2} + 49 T - 211 \) Copy content Toggle raw display
$89$ \( T^{3} + 13 T^{2} + 25 T - 111 \) Copy content Toggle raw display
$97$ \( T^{3} - 6 T^{2} - 44 T - 24 \) Copy content Toggle raw display
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