Properties

Label 1008.2.b.i
Level $1008$
Weight $2$
Character orbit 1008.b
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(559,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, 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("1008.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.b (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 24x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} - \beta_{3} q^{7} - \beta_{2} q^{11} + 3 \beta_1 q^{17} - 2 \beta_{3} q^{19} + \beta_{2} q^{23} - q^{25} - 4 \beta_{3} q^{31} + \beta_{2} q^{35} - 8 q^{37} - 5 \beta_1 q^{41} + 7 q^{49} - 6 \beta_{3} q^{55} - \beta_{2} q^{71} + 7 \beta_1 q^{77} + 18 q^{85} + \beta_1 q^{89} + 2 \beta_{2} q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{25} - 32 q^{37} + 28 q^{49} + 72 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 24x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 15\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 33\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + 12 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -15\beta_{2} + 33\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
2.01563i
4.46512i
2.01563i
4.46512i
0 0 0 2.44949i 0 −2.64575 0 0 0
559.2 0 0 0 2.44949i 0 2.64575 0 0 0
559.3 0 0 0 2.44949i 0 −2.64575 0 0 0
559.4 0 0 0 2.44949i 0 2.64575 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
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.b.i 4
3.b odd 2 1 inner 1008.2.b.i 4
4.b odd 2 1 inner 1008.2.b.i 4
7.b odd 2 1 inner 1008.2.b.i 4
8.b even 2 1 4032.2.b.l 4
8.d odd 2 1 4032.2.b.l 4
12.b even 2 1 inner 1008.2.b.i 4
21.c even 2 1 inner 1008.2.b.i 4
24.f even 2 1 4032.2.b.l 4
24.h odd 2 1 4032.2.b.l 4
28.d even 2 1 inner 1008.2.b.i 4
56.e even 2 1 4032.2.b.l 4
56.h odd 2 1 4032.2.b.l 4
84.h odd 2 1 CM 1008.2.b.i 4
168.e odd 2 1 4032.2.b.l 4
168.i even 2 1 4032.2.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.b.i 4 1.a even 1 1 trivial
1008.2.b.i 4 3.b odd 2 1 inner
1008.2.b.i 4 4.b odd 2 1 inner
1008.2.b.i 4 7.b odd 2 1 inner
1008.2.b.i 4 12.b even 2 1 inner
1008.2.b.i 4 21.c even 2 1 inner
1008.2.b.i 4 28.d even 2 1 inner
1008.2.b.i 4 84.h odd 2 1 CM
4032.2.b.l 4 8.b even 2 1
4032.2.b.l 4 8.d odd 2 1
4032.2.b.l 4 24.f even 2 1
4032.2.b.l 4 24.h odd 2 1
4032.2.b.l 4 56.e even 2 1
4032.2.b.l 4 56.h odd 2 1
4032.2.b.l 4 168.e odd 2 1
4032.2.b.l 4 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{2} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 42 \) Copy content Toggle raw display
\( T_{19}^{2} - 28 \) 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
$5$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$37$ \( (T + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 150)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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