Properties

Label 8568.2.a.bj
Level $8568$
Weight $2$
Character orbit 8568.a
Self dual yes
Analytic conductor $68.416$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{5} - q^{7} + (\beta_{2} - 1) q^{11} + ( - 2 \beta_{2} - 2) q^{13} - q^{17} + ( - 2 \beta_1 + 4) q^{19} + (2 \beta_{3} - 2) q^{23} + (3 \beta_{2} - \beta_1 + 3) q^{25} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{29} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{31} + ( - \beta_{3} - \beta_1) q^{35} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{37} + (3 \beta_{2} - \beta_1 + 2) q^{41} + ( - \beta_{2} + 3 \beta_1 + 4) q^{43} + ( - 4 \beta_{3} - 2 \beta_{2} - 2) q^{47} + q^{49} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 1) q^{53}+ \cdots + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61} + 8 q^{65} + 15 q^{67} - 2 q^{71} - 5 q^{73} + 4 q^{77} - 24 q^{79} - 10 q^{83} - q^{85} + 16 q^{89} + 8 q^{91} - 22 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} + x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 3\nu^{2} + 3\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 3\beta_{2} + 6\beta _1 + 4 \) 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
−1.48718
1.26498
−1.88301
3.10522
0 0 0 −4.02435 0 −1.00000 0 0 0
1.2 0 0 0 −0.163765 0 −1.00000 0 0 0
1.3 0 0 0 1.78180 0 −1.00000 0 0 0
1.4 0 0 0 3.40632 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.bj 4
3.b odd 2 1 952.2.a.g 4
12.b even 2 1 1904.2.a.q 4
21.c even 2 1 6664.2.a.o 4
24.f even 2 1 7616.2.a.bp 4
24.h odd 2 1 7616.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.g 4 3.b odd 2 1
1904.2.a.q 4 12.b even 2 1
6664.2.a.o 4 21.c even 2 1
7616.2.a.bj 4 24.h odd 2 1
7616.2.a.bp 4 24.f even 2 1
8568.2.a.bj 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8568))\):

\( T_{5}^{4} - T_{5}^{3} - 15T_{5}^{2} + 22T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} - 15 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 14 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$31$ \( T^{4} - 5 T^{3} + \cdots - 400 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots - 304 \) Copy content Toggle raw display
$41$ \( T^{4} - 7 T^{3} + \cdots + 964 \) Copy content Toggle raw display
$43$ \( T^{4} - 19 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 2816 \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$59$ \( T^{4} - 44 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 23 T^{3} + \cdots - 1964 \) Copy content Toggle raw display
$67$ \( T^{4} - 15 T^{3} + \cdots - 176 \) Copy content Toggle raw display
$71$ \( T^{4} + 2 T^{3} + \cdots - 704 \) Copy content Toggle raw display
$73$ \( T^{4} + 5 T^{3} + \cdots + 244 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + \cdots - 256 \) Copy content Toggle raw display
$83$ \( T^{4} + 10 T^{3} + \cdots + 4400 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + \cdots + 7744 \) Copy content Toggle raw display
$97$ \( T^{4} + 15 T^{3} + \cdots - 5900 \) Copy content Toggle raw display
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