Properties

Label 1008.2.k.a
Level $1008$
Weight $2$
Character orbit 1008.k
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $4$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{7} + ( \beta_{1} - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{23} -5 q^{25} + ( 2 \beta_{1} - \beta_{3} ) q^{29} -4 \beta_{2} q^{37} -2 \beta_{2} q^{43} + 7 q^{49} + ( 2 \beta_{1} + \beta_{3} ) q^{53} + 4 q^{67} + ( \beta_{1} - 3 \beta_{3} ) q^{71} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{77} -8 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 20q^{25} + 28q^{49} + 16q^{67} - 32q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{3} - 16 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{3} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-16 \beta_{3} + 15 \beta_{1}\)\()/6\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.16372i
1.16372i
2.57794i
2.57794i
0 0 0 0 0 −2.64575 0 0 0
881.2 0 0 0 0 0 −2.64575 0 0 0
881.3 0 0 0 0 0 2.64575 0 0 0
881.4 0 0 0 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.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.a 4
3.b odd 2 1 inner 1008.2.k.a 4
4.b odd 2 1 63.2.c.a 4
7.b odd 2 1 CM 1008.2.k.a 4
8.b even 2 1 4032.2.k.b 4
8.d odd 2 1 4032.2.k.c 4
12.b even 2 1 63.2.c.a 4
20.d odd 2 1 1575.2.b.a 4
20.e even 4 2 1575.2.g.d 8
21.c even 2 1 inner 1008.2.k.a 4
24.f even 2 1 4032.2.k.c 4
24.h odd 2 1 4032.2.k.b 4
28.d even 2 1 63.2.c.a 4
28.f even 6 2 441.2.p.b 8
28.g odd 6 2 441.2.p.b 8
36.f odd 6 2 567.2.o.f 8
36.h even 6 2 567.2.o.f 8
56.e even 2 1 4032.2.k.c 4
56.h odd 2 1 4032.2.k.b 4
60.h even 2 1 1575.2.b.a 4
60.l odd 4 2 1575.2.g.d 8
84.h odd 2 1 63.2.c.a 4
84.j odd 6 2 441.2.p.b 8
84.n even 6 2 441.2.p.b 8
140.c even 2 1 1575.2.b.a 4
140.j odd 4 2 1575.2.g.d 8
168.e odd 2 1 4032.2.k.c 4
168.i even 2 1 4032.2.k.b 4
252.s odd 6 2 567.2.o.f 8
252.bi even 6 2 567.2.o.f 8
420.o odd 2 1 1575.2.b.a 4
420.w even 4 2 1575.2.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 4.b odd 2 1
63.2.c.a 4 12.b even 2 1
63.2.c.a 4 28.d even 2 1
63.2.c.a 4 84.h odd 2 1
441.2.p.b 8 28.f even 6 2
441.2.p.b 8 28.g odd 6 2
441.2.p.b 8 84.j odd 6 2
441.2.p.b 8 84.n even 6 2
567.2.o.f 8 36.f odd 6 2
567.2.o.f 8 36.h even 6 2
567.2.o.f 8 252.s odd 6 2
567.2.o.f 8 252.bi even 6 2
1008.2.k.a 4 1.a even 1 1 trivial
1008.2.k.a 4 3.b odd 2 1 inner
1008.2.k.a 4 7.b odd 2 1 CM
1008.2.k.a 4 21.c even 2 1 inner
1575.2.b.a 4 20.d odd 2 1
1575.2.b.a 4 60.h even 2 1
1575.2.b.a 4 140.c even 2 1
1575.2.b.a 4 420.o odd 2 1
1575.2.g.d 8 20.e even 4 2
1575.2.g.d 8 60.l odd 4 2
1575.2.g.d 8 140.j odd 4 2
1575.2.g.d 8 420.w even 4 2
4032.2.k.b 4 8.b even 2 1
4032.2.k.b 4 24.h odd 2 1
4032.2.k.b 4 56.h odd 2 1
4032.2.k.b 4 168.i even 2 1
4032.2.k.c 4 8.d odd 2 1
4032.2.k.c 4 24.f even 2 1
4032.2.k.c 4 56.e even 2 1
4032.2.k.c 4 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 36 + 44 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 324 + 92 T^{2} + T^{4} \)
$29$ \( 2916 + 116 T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -112 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -28 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 36 + 212 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( -4 + T )^{4} \)
$71$ \( 12996 + 284 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 8 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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