Properties

Label 7569.2.a.w
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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: \(4\)
Coefficient field: 4.4.7600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + \beta_{3} q^{5} + ( - 4 \beta_{2} - 2) q^{7} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{4} + \beta_{3} q^{5} + ( - 4 \beta_{2} - 2) q^{7} + (\beta_{3} + \beta_1) q^{8} + (4 \beta_{2} + 1) q^{10} + ( - \beta_{3} + \beta_1) q^{11} + ( - \beta_{2} - 3) q^{13} + ( - 4 \beta_{3} - 2 \beta_1) q^{14} + 3 \beta_{2} q^{16} + (2 \beta_{3} + \beta_1) q^{17} + (3 \beta_{2} - 3) q^{19} + (2 \beta_{3} + \beta_1) q^{20} + ( - 3 \beta_{2} + 4) q^{22} + ( - 3 \beta_{3} - 2 \beta_1) q^{23} + ( - 3 \beta_{2} - 1) q^{25} + ( - \beta_{3} - 3 \beta_1) q^{26} + ( - 10 \beta_{2} - 10) q^{28} + (2 \beta_{2} - 5) q^{31} + (\beta_{3} - 2 \beta_1) q^{32} + (9 \beta_{2} + 7) q^{34} + (2 \beta_{3} - 4 \beta_1) q^{35} + (3 \beta_{2} + 4) q^{37} + (3 \beta_{3} - 3 \beta_1) q^{38} + (\beta_{2} + 5) q^{40} + (\beta_{3} + 3 \beta_1) q^{41} + (2 \beta_{2} + 5) q^{43} + ( - \beta_{3} + 2 \beta_1) q^{44} + ( - 14 \beta_{2} - 13) q^{46} + (\beta_{3} - 3 \beta_1) q^{47} + 13 q^{49} + ( - 3 \beta_{3} - \beta_1) q^{50} + ( - 5 \beta_{2} - 10) q^{52} + ( - \beta_{3} - 3 \beta_1) q^{53} + (7 \beta_{2} - 3) q^{55} + ( - 2 \beta_{3} - 6 \beta_1) q^{56} + ( - 3 \beta_{3} - 4 \beta_1) q^{59} + ( - 8 \beta_{2} - 8) q^{61} + (2 \beta_{3} - 5 \beta_1) q^{62} + ( - 4 \beta_{2} - 9) q^{64} + ( - 2 \beta_{3} - \beta_1) q^{65} + (5 \beta_{2} - 2) q^{67} + (5 \beta_{3} + 5 \beta_1) q^{68} + (4 \beta_{2} - 18) q^{70} + ( - 2 \beta_{3} - 5 \beta_1) q^{71} + (6 \beta_{2} + 2) q^{73} + (3 \beta_{3} + 4 \beta_1) q^{74} + (3 \beta_{2} - 6) q^{76} + ( - 6 \beta_{3} + 2 \beta_1) q^{77} + ( - 4 \beta_{2} - 7) q^{79} + ( - 3 \beta_{3} + 3 \beta_1) q^{80} + (7 \beta_{2} + 16) q^{82} + (2 \beta_{3} + \beta_1) q^{83} + ( - 2 \beta_{2} + 9) q^{85} + (2 \beta_{3} + 5 \beta_1) q^{86} + (4 \beta_{2} + 1) q^{88} - 4 \beta_{3} q^{89} + (10 \beta_{2} + 10) q^{91} + ( - 8 \beta_{3} - 9 \beta_1) q^{92} + (\beta_{2} - 14) q^{94} + ( - 6 \beta_{3} + 3 \beta_1) q^{95} + (3 \beta_{2} + 1) q^{97} + 13 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} + 22 q^{22} + 2 q^{25} - 20 q^{28} - 24 q^{31} + 10 q^{34} + 10 q^{37} + 18 q^{40} + 16 q^{43} - 24 q^{46} + 52 q^{49} - 30 q^{52} - 26 q^{55} - 16 q^{61} - 28 q^{64} - 18 q^{67} - 80 q^{70} - 4 q^{73} - 30 q^{76} - 20 q^{79} + 50 q^{82} + 40 q^{85} - 4 q^{88} + 20 q^{91} - 58 q^{94} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} + 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) 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
−2.37024
−1.83901
1.83901
2.37024
−2.37024 0 3.61803 −1.46489 0 −4.47214 −3.83513 0 3.47214
1.2 −1.83901 0 1.38197 2.97558 0 4.47214 1.13657 0 −5.47214
1.3 1.83901 0 1.38197 −2.97558 0 4.47214 −1.13657 0 −5.47214
1.4 2.37024 0 3.61803 1.46489 0 −4.47214 3.83513 0 3.47214
\(n\): 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7569.2.a.w 4
3.b odd 2 1 inner 7569.2.a.w 4
29.b even 2 1 7569.2.a.x yes 4
87.d odd 2 1 7569.2.a.x yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7569.2.a.w 4 1.a even 1 1 trivial
7569.2.a.w 4 3.b odd 2 1 inner
7569.2.a.x yes 4 29.b even 2 1
7569.2.a.x yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{4} - 9T_{2}^{2} + 19 \) Copy content Toggle raw display
\( T_{5}^{4} - 11T_{5}^{2} + 19 \) Copy content Toggle raw display
\( T_{7}^{2} - 20 \) Copy content Toggle raw display
\( T_{19}^{2} + 9T_{19} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 9T^{2} + 19 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 11T^{2} + 19 \) Copy content Toggle raw display
$7$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 24T^{2} + 19 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 45T^{2} + 475 \) Copy content Toggle raw display
$19$ \( (T^{2} + 9 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 111T^{2} + 2299 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12 T + 31)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 5 T - 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 80T^{2} + 475 \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 104T^{2} + 2299 \) Copy content Toggle raw display
$53$ \( T^{4} - 80T^{2} + 475 \) Copy content Toggle raw display
$59$ \( T^{4} - 195T^{2} + 475 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9 T - 11)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 229T^{2} + 2299 \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 45T^{2} + 475 \) Copy content Toggle raw display
$89$ \( T^{4} - 176T^{2} + 4864 \) Copy content Toggle raw display
$97$ \( (T^{2} + T - 11)^{2} \) Copy content Toggle raw display
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