Properties

Label 7569.2.a.y
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
Dimension $6$
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] = [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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.11973625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 15x^{3} + 13x^{2} - 27x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 841)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} + 2) q^{4} + (\beta_{5} - \beta_1 + 1) q^{5} + (\beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} + 2) q^{4} + (\beta_{5} - \beta_1 + 1) q^{5} + (\beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{8}+ \cdots + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 8 q^{4} + 2 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 8 q^{4} + 2 q^{5} - 4 q^{7} - 3 q^{8} + 15 q^{10} - 13 q^{11} - 6 q^{13} + 6 q^{14} + 4 q^{16} - 3 q^{17} - 6 q^{19} + 5 q^{20} - 4 q^{22} + 5 q^{23} + 6 q^{25} - 9 q^{26} - 5 q^{28} + 16 q^{31} - 33 q^{32} + 4 q^{35} - 10 q^{37} - 41 q^{38} + 7 q^{40} - 30 q^{41} - 2 q^{43} - 16 q^{44} - 20 q^{46} - 5 q^{47} - 12 q^{49} + 8 q^{50} + 13 q^{53} + 10 q^{55} - 16 q^{56} + 17 q^{59} - 4 q^{61} + q^{64} - 32 q^{65} - 9 q^{67} - 16 q^{68} - 28 q^{70} + 9 q^{73} + 36 q^{74} + 24 q^{76} - 13 q^{77} + 6 q^{79} - 5 q^{80} + 19 q^{82} + 24 q^{83} - 26 q^{85} - 17 q^{86} + 21 q^{88} - 9 q^{89} + 16 q^{91} + 23 q^{92} - 37 q^{94} - 31 q^{95} - 10 q^{97} + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 8x^{4} + 15x^{3} + 13x^{2} - 27x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 12\nu^{2} + 16\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 15\nu^{2} + 16\nu - 24 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 9\nu^{3} + 7\nu^{2} + 19\nu - 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 6\beta_{4} - 9\beta_{3} + \beta_{2} + 2\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 8\beta_{4} - 11\beta_{3} + 10\beta_{2} + 28\beta _1 + 16 \) 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
2.67904
1.94573
0.855835
0.511256
−1.80156
−2.19030
−2.67904 0 5.17728 −0.0973457 0 0.0377061 −8.51206 0 0.260793
1.2 −1.94573 0 1.78585 −3.21726 0 −2.53022 0.416673 0 6.25991
1.3 −0.855835 0 −1.26755 2.81313 0 −0.766737 2.79648 0 −2.40758
1.4 −0.511256 0 −1.73862 −2.20381 0 −1.30206 1.91139 0 1.12671
1.5 1.80156 0 1.24563 1.40413 0 3.53302 −1.35905 0 2.52963
1.6 2.19030 0 2.79741 3.30116 0 −2.97171 1.74657 0 7.23053
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

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 7569.2.a.y 6
3.b odd 2 1 841.2.a.h yes 6
29.b even 2 1 7569.2.a.bc 6
87.d odd 2 1 841.2.a.g 6
87.f even 4 2 841.2.b.d 12
87.h odd 14 6 841.2.d.o 36
87.j odd 14 6 841.2.d.n 36
87.k even 28 12 841.2.e.l 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.g 6 87.d odd 2 1
841.2.a.h yes 6 3.b odd 2 1
841.2.b.d 12 87.f even 4 2
841.2.d.n 36 87.j odd 14 6
841.2.d.o 36 87.h odd 14 6
841.2.e.l 72 87.k even 28 12
7569.2.a.y 6 1.a even 1 1 trivial
7569.2.a.bc 6 29.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{6} + 2T_{2}^{5} - 8T_{2}^{4} - 15T_{2}^{3} + 13T_{2}^{2} + 27T_{2} + 9 \) Copy content Toggle raw display
\( T_{5}^{6} - 2T_{5}^{5} - 16T_{5}^{4} + 29T_{5}^{3} + 59T_{5}^{2} - 87T_{5} - 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 4T_{7}^{5} - 7T_{7}^{4} - 49T_{7}^{3} - 65T_{7}^{2} - 24T_{7} + 1 \) Copy content Toggle raw display
\( T_{19}^{6} + 6T_{19}^{5} - 54T_{19}^{4} - 299T_{19}^{3} + 711T_{19}^{2} + 3555T_{19} + 1055 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + \cdots - 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 13 T^{5} + \cdots - 81 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + \cdots + 29 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots - 36 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 1055 \) Copy content Toggle raw display
$23$ \( T^{6} - 5 T^{5} + \cdots - 81 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 16 T^{5} + \cdots + 199 \) Copy content Toggle raw display
$37$ \( T^{6} + 10 T^{5} + \cdots - 1616 \) Copy content Toggle raw display
$41$ \( T^{6} + 30 T^{5} + \cdots - 12816 \) Copy content Toggle raw display
$43$ \( T^{6} + 2 T^{5} + \cdots - 4751 \) Copy content Toggle raw display
$47$ \( T^{6} + 5 T^{5} + \cdots + 1341 \) Copy content Toggle raw display
$53$ \( T^{6} - 13 T^{5} + \cdots - 29124 \) Copy content Toggle raw display
$59$ \( T^{6} - 17 T^{5} + \cdots - 1305 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots - 76421 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( T^{6} - 162 T^{4} + \cdots + 1611 \) Copy content Toggle raw display
$73$ \( T^{6} - 9 T^{5} + \cdots + 33251 \) Copy content Toggle raw display
$79$ \( T^{6} - 6 T^{5} + \cdots + 95125 \) Copy content Toggle raw display
$83$ \( T^{6} - 24 T^{5} + \cdots - 4311 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + \cdots - 585405 \) Copy content Toggle raw display
$97$ \( T^{6} + 10 T^{5} + \cdots + 3824 \) Copy content Toggle raw display
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