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Properties

Label 1008.2.k.b
Level $1008$
Weight $2$
Character orbit 1008.k
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(881,1008)]
 
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([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.k (of order \(2\), degree \(1\), not 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: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + (\beta_1 + 1) q^{7} - \beta_{2} q^{11} - 2 \beta_1 q^{13} - \beta_{3} q^{17} - 2 \beta_1 q^{19} + \beta_{2} q^{23} + 7 q^{25} + \beta_{2} q^{29} + ( - \beta_{3} + 2 \beta_{2}) q^{35}+ \cdots - 2 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 28 q^{25} - 32 q^{37} + 8 q^{43} - 20 q^{49} - 32 q^{67} + 16 q^{79} + 48 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{3} - 9\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{2} - 9\beta_1 ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
− 0.517638i
0.517638i
− 1.93185i
1.93185i
0 0 0 −3.46410 0 1.00000 − 2.44949i 0 0 0
881.2 0 0 0 −3.46410 0 1.00000 + 2.44949i 0 0 0
881.3 0 0 0 3.46410 0 1.00000 − 2.44949i 0 0 0
881.4 0 0 0 3.46410 0 1.00000 + 2.44949i 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.k.b 4
3.b odd 2 1 inner 1008.2.k.b 4
4.b odd 2 1 252.2.f.a ✓ 4
7.b odd 2 1 inner 1008.2.k.b 4
8.b even 2 1 4032.2.k.d 4
8.d odd 2 1 4032.2.k.a 4
12.b even 2 1 252.2.f.a ✓ 4
20.d odd 2 1 6300.2.d.c 4
20.e even 4 2 6300.2.f.b 8
21.c even 2 1 inner 1008.2.k.b 4
24.f even 2 1 4032.2.k.a 4
24.h odd 2 1 4032.2.k.d 4
28.d even 2 1 252.2.f.a ✓ 4
28.f even 6 2 1764.2.t.b 8
28.g odd 6 2 1764.2.t.b 8
36.f odd 6 2 2268.2.x.i 8
36.h even 6 2 2268.2.x.i 8
56.e even 2 1 4032.2.k.a 4
56.h odd 2 1 4032.2.k.d 4
60.h even 2 1 6300.2.d.c 4
60.l odd 4 2 6300.2.f.b 8
84.h odd 2 1 252.2.f.a ✓ 4
84.j odd 6 2 1764.2.t.b 8
84.n even 6 2 1764.2.t.b 8
140.c even 2 1 6300.2.d.c 4
140.j odd 4 2 6300.2.f.b 8
168.e odd 2 1 4032.2.k.a 4
168.i even 2 1 4032.2.k.d 4
252.s odd 6 2 2268.2.x.i 8
252.bi even 6 2 2268.2.x.i 8
420.o odd 2 1 6300.2.d.c 4
420.w even 4 2 6300.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.f.a ✓ 4 4.b odd 2 1
252.2.f.a ✓ 4 12.b even 2 1
252.2.f.a ✓ 4 28.d even 2 1
252.2.f.a ✓ 4 84.h odd 2 1
1008.2.k.b 4 1.a even 1 1 trivial
1008.2.k.b 4 3.b odd 2 1 inner
1008.2.k.b 4 7.b odd 2 1 inner
1008.2.k.b 4 21.c even 2 1 inner
1764.2.t.b 8 28.f even 6 2
1764.2.t.b 8 28.g odd 6 2
1764.2.t.b 8 84.j odd 6 2
1764.2.t.b 8 84.n even 6 2
2268.2.x.i 8 36.f odd 6 2
2268.2.x.i 8 36.h even 6 2
2268.2.x.i 8 252.s odd 6 2
2268.2.x.i 8 252.bi even 6 2
4032.2.k.a 4 8.d odd 2 1
4032.2.k.a 4 24.f even 2 1
4032.2.k.a 4 56.e even 2 1
4032.2.k.a 4 168.e odd 2 1
4032.2.k.d 4 8.b even 2 1
4032.2.k.d 4 24.h odd 2 1
4032.2.k.d 4 56.h odd 2 1
4032.2.k.d 4 168.i even 2 1
6300.2.d.c 4 20.d odd 2 1
6300.2.d.c 4 60.h even 2 1
6300.2.d.c 4 140.c even 2 1
6300.2.d.c 4 420.o odd 2 1
6300.2.f.b 8 20.e even 4 2
6300.2.f.b 8 60.l odd 4 2
6300.2.f.b 8 140.j odd 4 2
6300.2.f.b 8 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\). 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
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$67$ \( (T + 8)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
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