Properties

Label 7569.2.a.bf
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.8.14884000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{6} - \beta_1) q^{5} + ( - \beta_{4} - \beta_{3} - 1) q^{7} + (\beta_{7} + \beta_{6}) q^{8} + ( - 2 \beta_{4} - \beta_{3} - 3) q^{10} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5}) q^{11}+ \cdots + (3 \beta_{7} + 4 \beta_{6} + \cdots + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} - 12 q^{10} + 2 q^{13} - 2 q^{16} + 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} + 10 q^{31} - 36 q^{34} + 18 q^{37} - 10 q^{40} + 28 q^{43} + 26 q^{46} + 4 q^{49} + 44 q^{52} + 14 q^{55}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 5\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 7 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 8\nu^{5} + 16\nu^{3} - 7\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 12\nu^{5} - 40\nu^{3} + 27\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 12\nu^{5} + 44\nu^{3} - 43\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{7} + 7\beta_{6} + \beta_{5} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{4} + 8\beta_{3} + 24\beta_{2} + 71 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32\beta_{7} + 40\beta_{6} + 12\beta_{5} + 95\beta_1 \) 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
−2.30927
−2.08529
−1.13370
−0.183172
0.183172
1.13370
2.08529
2.30927
−2.30927 0 3.33275 3.03582 0 −4.39250 −3.07768 0 −7.01054
1.2 −2.08529 0 2.34841 −0.992398 0 2.45140 −0.726543 0 2.06943
1.3 −1.13370 0 −0.714715 0.407162 0 2.15643 3.07768 0 −0.461601
1.4 −0.183172 0 −1.96645 3.26086 0 −0.215332 0.726543 0 −0.597298
1.5 0.183172 0 −1.96645 −3.26086 0 −0.215332 −0.726543 0 −0.597298
1.6 1.13370 0 −0.714715 −0.407162 0 2.15643 −3.07768 0 −0.461601
1.7 2.08529 0 2.34841 0.992398 0 2.45140 0.726543 0 2.06943
1.8 2.30927 0 3.33275 −3.03582 0 −4.39250 3.07768 0 −7.01054
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7569.2.a.bf 8
3.b odd 2 1 inner 7569.2.a.bf 8
29.b even 2 1 7569.2.a.bg yes 8
87.d odd 2 1 7569.2.a.bg yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7569.2.a.bf 8 1.a even 1 1 trivial
7569.2.a.bf 8 3.b odd 2 1 inner
7569.2.a.bg yes 8 29.b even 2 1
7569.2.a.bg yes 8 87.d odd 2 1

Hecke kernels

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

\( T_{2}^{8} - 11T_{2}^{6} + 36T_{2}^{4} - 31T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 21T_{5}^{6} + 121T_{5}^{4} - 116T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} - 15T_{7}^{2} + 20T_{7} + 5 \) Copy content Toggle raw display
\( T_{19}^{2} - 5T_{19} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} - 15 T^{2} + 20 T + 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 54 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} - 14 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 84 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 5)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 149 T^{6} + \cdots + 633616 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{3} - 10 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 9 T^{3} - 39 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 166 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + \cdots + 3041)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 221 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$53$ \( (T^{4} - 68 T^{2} + 976)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 261 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$61$ \( (T^{4} - 27 T^{3} + \cdots + 941)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 19 T^{3} + \cdots - 4519)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 465 T^{6} + \cdots + 93702400 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8 T^{3} + \cdots + 6151)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 7 T^{3} + \cdots + 1991)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 526 T^{6} + \cdots + 91317136 \) Copy content Toggle raw display
$89$ \( T^{8} - 274 T^{6} + \cdots + 190096 \) Copy content Toggle raw display
$97$ \( (T^{4} + 3 T^{3} + \cdots + 4516)^{2} \) Copy content Toggle raw display
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