Properties

Label 7569.2.a.v
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, 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("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q - 2 q^{4} + ( - \beta_{3} + 3 \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{4} + ( - \beta_{3} + 3 \beta_1 - 1) q^{7} + (3 \beta_{3} + 4 \beta_{2}) q^{13} + 4 q^{16} + (2 \beta_{3} + 5 \beta_{2}) q^{19} - 5 q^{25} + (2 \beta_{3} - 6 \beta_1 + 2) q^{28} + ( - \beta_{3} + 6 \beta_1 - 1) q^{31} + (7 \beta_{3} + 3 \beta_{2}) q^{37} + ( - 6 \beta_{3} + 7 \beta_1 - 6) q^{43} + ( - 5 \beta_{3} + 3 \beta_{2} + 7) q^{49} + ( - 6 \beta_{3} - 8 \beta_{2}) q^{52} + (4 \beta_{3} + 5 \beta_1 + 4) q^{61} - 8 q^{64} + (2 \beta_{3} - 7 \beta_{2}) q^{67} + ( - 9 \beta_{3} - 8 \beta_{2}) q^{73} + ( - 4 \beta_{3} - 10 \beta_{2}) q^{76} + ( - 3 \beta_{3} + 7 \beta_{2}) q^{79} + (21 \beta_{3} + 5 \beta_{2} + \cdots + 10) q^{91}+ \cdots + ( - 11 \beta_{3} + 8 \beta_1 - 11) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + q^{7} - 2 q^{13} + 16 q^{16} + q^{19} - 20 q^{25} - 2 q^{28} + 4 q^{31} - 11 q^{37} - 5 q^{43} + 41 q^{49} + 4 q^{52} + 13 q^{61} - 32 q^{64} - 11 q^{67} + 10 q^{73} - 2 q^{76} + 13 q^{79} + 2 q^{91} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{15} + \zeta_{15}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
−1.95630
−0.209057
1.82709
1.33826
0 0 −2.00000 0 0 −5.25085 0 0 0
1.2 0 0 −2.00000 0 0 −2.24520 0 0 0
1.3 0 0 −2.00000 0 0 3.86324 0 0 0
1.4 0 0 −2.00000 0 0 4.63282 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(29\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7569.2.a.v yes 4
3.b odd 2 1 CM 7569.2.a.v yes 4
29.b even 2 1 7569.2.a.u 4
87.d odd 2 1 7569.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7569.2.a.u 4 29.b even 2 1
7569.2.a.u 4 87.d odd 2 1
7569.2.a.v yes 4 1.a even 1 1 trivial
7569.2.a.v yes 4 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{3} - 34T_{7}^{2} + 34T_{7} + 211 \) Copy content Toggle raw display
\( T_{19}^{4} - T_{19}^{3} - 94T_{19}^{2} + 94T_{19} + 1711 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + \cdots + 211 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 601 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} + \cdots + 1711 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 2581 \) Copy content Toggle raw display
$37$ \( T^{4} + 11 T^{3} + \cdots - 899 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots + 4495 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 13 T^{3} + \cdots - 4379 \) Copy content Toggle raw display
$67$ \( T^{4} + 11 T^{3} + \cdots - 3449 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 10 T^{3} + \cdots + 145 \) Copy content Toggle raw display
$79$ \( T^{4} - 13 T^{3} + \cdots - 6989 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + \cdots - 9599 \) Copy content Toggle raw display
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