Properties

Label 572.2.f.b
Level $572$
Weight $2$
Character orbit 572.f
Analytic conductor $4.567$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [572,2,Mod(441,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.441");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{3} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{3} + 3) q^{9} - \beta_{2} q^{11} + ( - 3 \beta_{2} + 2) q^{13} + (2 \beta_{3} - 4) q^{17} + (4 \beta_{2} - \beta_1) q^{19} + ( - 5 \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{3} + 2) q^{23} + 5 q^{25} + 5 q^{27} + (2 \beta_{3} + 2) q^{29} + ( - 4 \beta_{2} - 2 \beta_1) q^{31} + \beta_1 q^{33} + ( - 2 \beta_{2} - 4 \beta_1) q^{37} + ( - 2 \beta_{3} + 3 \beta_1 + 2) q^{39} + ( - 3 \beta_{2} - 3 \beta_1) q^{41} + 2 q^{43} + 6 \beta_{2} q^{47} + (3 \beta_{3} - 2) q^{49} + (4 \beta_{3} - 14) q^{51} + (\beta_{3} - 5) q^{53} + (5 \beta_{2} - 5 \beta_1) q^{57} - 6 \beta_{2} q^{59} + 10 q^{61} + ( - 7 \beta_{2} + 4 \beta_1) q^{63} + (4 \beta_{2} - 4 \beta_1) q^{67} + ( - 2 \beta_{3} + 7) q^{69} + (6 \beta_{2} + 6 \beta_1) q^{71} + (4 \beta_{2} + 5 \beta_1) q^{73} + ( - 5 \beta_{3} + 5) q^{75} + (\beta_{3} - 2) q^{77} - 8 q^{79} + ( - 2 \beta_{3} - 4) q^{81} + (3 \beta_{2} - 3 \beta_1) q^{83} + ( - 2 \beta_{3} - 8) q^{87} - 6 \beta_{2} q^{89} + (3 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{91} + (10 \beta_{2} + 2 \beta_1) q^{93} + ( - 2 \beta_{2} + 2 \beta_1) q^{97} + ( - 2 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 10 q^{9} + 8 q^{13} - 12 q^{17} + 6 q^{23} + 20 q^{25} + 20 q^{27} + 12 q^{29} + 4 q^{39} + 8 q^{43} - 2 q^{49} - 48 q^{51} - 18 q^{53} + 40 q^{61} + 24 q^{69} + 10 q^{75} - 6 q^{77} - 32 q^{79} - 20 q^{81} - 36 q^{87} - 18 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{2} - 6\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
441.1
1.79129i
1.79129i
2.79129i
2.79129i
0 −1.79129 0 0 0 0.791288i 0 0.208712 0
441.2 0 −1.79129 0 0 0 0.791288i 0 0.208712 0
441.3 0 2.79129 0 0 0 3.79129i 0 4.79129 0
441.4 0 2.79129 0 0 0 3.79129i 0 4.79129 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.f.b 4
3.b odd 2 1 5148.2.e.b 4
4.b odd 2 1 2288.2.j.f 4
13.b even 2 1 inner 572.2.f.b 4
13.d odd 4 1 7436.2.a.i 2
13.d odd 4 1 7436.2.a.j 2
39.d odd 2 1 5148.2.e.b 4
52.b odd 2 1 2288.2.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.b 4 1.a even 1 1 trivial
572.2.f.b 4 13.b even 2 1 inner
2288.2.j.f 4 4.b odd 2 1
2288.2.j.f 4 52.b odd 2 1
5148.2.e.b 4 3.b odd 2 1
5148.2.e.b 4 39.d odd 2 1
7436.2.a.i 2 13.d odd 4 1
7436.2.a.j 2 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 5 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 51T^{2} + 225 \) Copy content Toggle raw display
$23$ \( (T^{2} - 3 T - 3)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$37$ \( (T^{2} + 84)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 99T^{2} + 2025 \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9 T + 15)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 240T^{2} + 2304 \) Copy content Toggle raw display
$71$ \( T^{4} + 396 T^{2} + 32400 \) Copy content Toggle raw display
$73$ \( T^{4} + 267 T^{2} + 16641 \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 135T^{2} + 729 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
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