Properties

Label 567.2.a.i
Level $567$
Weight $2$
Character orbit 567.a
Self dual yes
Analytic conductor $4.528$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(1,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, 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("567.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{2} q^{4} - 2 \beta_1 q^{5} - q^{7} + (\beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{2} q^{4} - 2 \beta_1 q^{5} - q^{7} + (\beta_{3} + \beta_1) q^{8} + 2 q^{10} + (\beta_{3} - \beta_1) q^{11} + 4 q^{13} - \beta_{3} q^{14} + (\beta_{2} + 1) q^{16} + (2 \beta_{3} + 2 \beta_1) q^{17} - 2 \beta_{2} q^{19} + (2 \beta_{3} + 4 \beta_1) q^{20} + ( - \beta_{2} + 3) q^{22} + ( - 2 \beta_{3} - 2 \beta_1) q^{23} + (4 \beta_{2} + 7) q^{25} + 4 \beta_{3} q^{26} + \beta_{2} q^{28} - 4 \beta_{3} q^{29} + (4 \beta_{2} + 2) q^{31} + ( - 4 \beta_{3} - 3 \beta_1) q^{32} + ( - 2 \beta_{2} + 2) q^{34} + 2 \beta_1 q^{35} + 3 q^{37} + (6 \beta_{3} + 2 \beta_1) q^{38} + ( - 2 \beta_{2} - 4) q^{40} + 2 \beta_1 q^{41} + ( - 2 \beta_{2} - 5) q^{43} + (4 \beta_{3} + 3 \beta_1) q^{44} + (2 \beta_{2} - 2) q^{46} - 6 \beta_{3} q^{47} + q^{49} + ( - 5 \beta_{3} - 4 \beta_1) q^{50} - 4 \beta_{2} q^{52} + ( - 5 \beta_{3} - 5 \beta_1) q^{53} + (2 \beta_{2} + 8) q^{55} + ( - \beta_{3} - \beta_1) q^{56} + (4 \beta_{2} - 8) q^{58} + (2 \beta_{3} + 2 \beta_1) q^{59} + ( - 2 \beta_{2} + 6) q^{61} + ( - 10 \beta_{3} - 4 \beta_1) q^{62} + (2 \beta_{2} - 7) q^{64} - 8 \beta_1 q^{65} + ( - 2 \beta_{2} + 3) q^{67} + (4 \beta_{3} - 2 \beta_1) q^{68} - 2 q^{70} + ( - \beta_{3} + 5 \beta_1) q^{71} + ( - 4 \beta_{2} + 4) q^{73} + 3 \beta_{3} q^{74} + ( - 2 \beta_{2} + 10) q^{76} + ( - \beta_{3} + \beta_1) q^{77} + ( - 2 \beta_{2} - 5) q^{79} + ( - 2 \beta_{3} - 6 \beta_1) q^{80} - 2 q^{82} + ( - 4 \beta_{3} + 2 \beta_1) q^{83} + ( - 4 \beta_{2} - 8) q^{85} + (\beta_{3} + 2 \beta_1) q^{86} + ( - 2 \beta_{2} - 1) q^{88} + 4 \beta_1 q^{89} - 4 q^{91} + ( - 4 \beta_{3} + 2 \beta_1) q^{92} + (6 \beta_{2} - 12) q^{94} + (4 \beta_{3} + 8 \beta_1) q^{95} + (2 \beta_{2} + 4) q^{97} + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{7} + 8 q^{10} + 16 q^{13} + 2 q^{16} + 4 q^{19} + 14 q^{22} + 20 q^{25} - 2 q^{28} + 12 q^{34} + 12 q^{37} - 12 q^{40} - 16 q^{43} - 12 q^{46} + 4 q^{49} + 8 q^{52} + 28 q^{55} - 40 q^{58} + 28 q^{61} - 32 q^{64} + 16 q^{67} - 8 q^{70} + 24 q^{73} + 44 q^{76} - 16 q^{79} - 8 q^{82} - 24 q^{85} - 16 q^{91} - 60 q^{94} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) 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
0.456850
2.18890
−2.18890
−0.456850
−2.18890 0 2.79129 −0.913701 0 −1.00000 −1.73205 0 2.00000
1.2 −0.456850 0 −1.79129 −4.37780 0 −1.00000 1.73205 0 2.00000
1.3 0.456850 0 −1.79129 4.37780 0 −1.00000 −1.73205 0 2.00000
1.4 2.18890 0 2.79129 0.913701 0 −1.00000 1.73205 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

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 567.2.a.i 4
3.b odd 2 1 inner 567.2.a.i 4
4.b odd 2 1 9072.2.a.ci 4
7.b odd 2 1 3969.2.a.u 4
9.c even 3 2 567.2.f.n 8
9.d odd 6 2 567.2.f.n 8
12.b even 2 1 9072.2.a.ci 4
21.c even 2 1 3969.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.i 4 1.a even 1 1 trivial
567.2.a.i 4 3.b odd 2 1 inner
567.2.f.n 8 9.c even 3 2
567.2.f.n 8 9.d odd 6 2
3969.2.a.u 4 7.b odd 2 1
3969.2.a.u 4 21.c even 2 1
9072.2.a.ci 4 4.b odd 2 1
9072.2.a.ci 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(567))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 20T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 20)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 80T^{2} + 256 \) Copy content Toggle raw display
$31$ \( (T^{2} - 84)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 20T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 180T^{2} + 1296 \) Copy content Toggle raw display
$53$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 14 T + 28)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 8 T - 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 150T^{2} + 2601 \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T - 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 132T^{2} + 3600 \) Copy content Toggle raw display
$89$ \( T^{4} - 80T^{2} + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T - 12)^{2} \) Copy content Toggle raw display
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