Properties

Label 8568.2.a.x
Level $8568$
Weight $2$
Character orbit 8568.a
Self dual yes
Analytic conductor $68.416$
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] = [8568,2,Mod(1,8568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8568.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8568.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: \(68.4158244518\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{5} + q^{7} + (2 \beta_1 - 2) q^{11} + q^{17} + (2 \beta_{2} + 2) q^{19} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + (\beta_{2} - 2 \beta_1 - 1) q^{25} + ( - 2 \beta_{2} + 2) q^{29} + ( - \beta_1 + 1) q^{31} + (\beta_1 - 1) q^{35} + ( - 2 \beta_{2} - 2) q^{37} + (3 \beta_{2} - 6 \beta_1 - 2) q^{41} + (\beta_{2} - 6 \beta_1) q^{43} + ( - 4 \beta_{2} - 4) q^{47} + q^{49} + ( - 2 \beta_{2} + \beta_1 - 7) q^{53} + (2 \beta_{2} - 4 \beta_1 + 8) q^{55} + (2 \beta_{2} - 6 \beta_1 - 4) q^{59} + (5 \beta_{2} - 4 \beta_1 + 2) q^{61} + ( - 4 \beta_{2} + 7 \beta_1 + 1) q^{67} + ( - 6 \beta_1 - 2) q^{71} + ( - \beta_{2} + 2 \beta_1 - 2) q^{73} + (2 \beta_1 - 2) q^{77} + 8 q^{79} + ( - 2 \beta_{2} + 8 \beta_1 - 2) q^{83} + (\beta_1 - 1) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{89} + ( - 2 \beta_{2} + 4 \beta_1) q^{95} + (2 \beta_{2} + \beta_1 + 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{17} + 4 q^{19} - 4 q^{23} - 4 q^{25} + 8 q^{29} + 3 q^{31} - 3 q^{35} - 4 q^{37} - 9 q^{41} - q^{43} - 8 q^{47} + 3 q^{49} - 19 q^{53} + 22 q^{55} - 14 q^{59} + q^{61} + 7 q^{67} - 6 q^{71} - 5 q^{73} - 6 q^{77} + 24 q^{79} - 4 q^{83} - 3 q^{85} - 10 q^{89} + 2 q^{95} + 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) 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
−1.86081
−0.254102
2.11491
0 0 0 −2.86081 0 1.00000 0 0 0
1.2 0 0 0 −1.25410 0 1.00000 0 0 0
1.3 0 0 0 1.11491 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(17\) \(-1\)

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 8568.2.a.x 3
3.b odd 2 1 952.2.a.f 3
12.b even 2 1 1904.2.a.m 3
21.c even 2 1 6664.2.a.j 3
24.f even 2 1 7616.2.a.be 3
24.h odd 2 1 7616.2.a.bc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.f 3 3.b odd 2 1
1904.2.a.m 3 12.b even 2 1
6664.2.a.j 3 21.c even 2 1
7616.2.a.bc 3 24.h odd 2 1
7616.2.a.be 3 24.f even 2 1
8568.2.a.x 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8568))\):

\( T_{5}^{3} + 3T_{5}^{2} - T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 6T_{11}^{2} - 4T_{11} - 32 \) Copy content Toggle raw display
\( T_{13} \) 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
$5$ \( T^{3} + 3T^{2} - T - 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 6 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( (T - 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$29$ \( T^{3} - 8T^{2} + 8 \) Copy content Toggle raw display
$31$ \( T^{3} - 3T^{2} - T + 4 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} + \cdots - 56 \) Copy content Toggle raw display
$41$ \( T^{3} + 9 T^{2} + \cdots - 1006 \) Copy content Toggle raw display
$43$ \( T^{3} + T^{2} + \cdots - 184 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} + \cdots - 448 \) Copy content Toggle raw display
$53$ \( T^{3} + 19 T^{2} + \cdots + 106 \) Copy content Toggle raw display
$59$ \( T^{3} + 14 T^{2} + \cdots - 928 \) Copy content Toggle raw display
$61$ \( T^{3} - T^{2} + \cdots + 124 \) Copy content Toggle raw display
$67$ \( T^{3} - 7 T^{2} + \cdots + 1508 \) Copy content Toggle raw display
$71$ \( T^{3} + 6 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$73$ \( T^{3} + 5 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$79$ \( (T - 8)^{3} \) Copy content Toggle raw display
$83$ \( T^{3} + 4 T^{2} + \cdots + 392 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$97$ \( T^{3} - 13 T^{2} + \cdots + 46 \) Copy content Toggle raw display
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