Properties

Label 1584.2.o.c
Level $1584$
Weight $2$
Character orbit 1584.o
Analytic conductor $12.648$
Analytic rank $0$
Dimension $4$
CM discriminant -132
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1584,2,Mod(703,1584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1584, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1584.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1584.o (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: \(12.6483036802\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 5x^{2} + 6x + 75 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{7} - \beta_{3} q^{11} + \beta_{2} q^{17} + 3 \beta_1 q^{19} - 2 \beta_{3} q^{23} - 5 q^{25} - \beta_{2} q^{29} + 4 q^{37} - \beta_{2} q^{41} - 5 \beta_1 q^{43} - 4 \beta_{3} q^{47} - q^{49} - 2 \beta_{3} q^{59} - 4 \beta_{3} q^{71} + \beta_{2} q^{77} + 7 \beta_1 q^{79} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{25} + 16 q^{37} - 4 q^{49} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 5x^{2} + 6x + 75 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 29\nu - 15 ) / 35 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} + 12\nu - 5 ) / 35 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{3} + 3\beta_{2} - 3\beta _1 + 10 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
2.94949 + 1.65831i
2.94949 1.65831i
−1.94949 + 1.65831i
−1.94949 1.65831i
0 0 0 0 0 −2.44949 0 0 0
703.2 0 0 0 0 0 −2.44949 0 0 0
703.3 0 0 0 0 0 2.44949 0 0 0
703.4 0 0 0 0 0 2.44949 0 0 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
132.d odd 2 1 CM by \(\Q(\sqrt{-33}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
11.b odd 2 1 inner
12.b even 2 1 inner
33.d even 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.2.o.c 4
3.b odd 2 1 inner 1584.2.o.c 4
4.b odd 2 1 inner 1584.2.o.c 4
11.b odd 2 1 inner 1584.2.o.c 4
12.b even 2 1 inner 1584.2.o.c 4
33.d even 2 1 inner 1584.2.o.c 4
44.c even 2 1 inner 1584.2.o.c 4
132.d odd 2 1 CM 1584.2.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1584.2.o.c 4 1.a even 1 1 trivial
1584.2.o.c 4 3.b odd 2 1 inner
1584.2.o.c 4 4.b odd 2 1 inner
1584.2.o.c 4 11.b odd 2 1 inner
1584.2.o.c 4 12.b even 2 1 inner
1584.2.o.c 4 33.d even 2 1 inner
1584.2.o.c 4 44.c even 2 1 inner
1584.2.o.c 4 132.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1584, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{83} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 66)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 66)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T - 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 66)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 294)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T - 16)^{4} \) Copy content Toggle raw display
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