Properties

Label 1008.4.k.a
Level $1008$
Weight $4$
Character orbit 1008.k
Analytic conductor $59.474$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q -7 \beta_{2} q^{7} +O(q^{10})\) \( q -7 \beta_{2} q^{7} + ( -6 \beta_{1} - 7 \beta_{3} ) q^{11} + ( -16 \beta_{1} + 9 \beta_{3} ) q^{23} -125 q^{25} + ( 13 \beta_{1} - 24 \beta_{3} ) q^{29} -4 \beta_{2} q^{37} + 202 \beta_{2} q^{43} + 343 q^{49} + ( -9 \beta_{1} + 76 \beta_{3} ) q^{53} -740 q^{67} + ( -44 \beta_{1} + 97 \beta_{3} ) q^{71} + ( -133 \beta_{1} - 28 \beta_{3} ) q^{77} + 1384 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 500q^{25} + 1372q^{49} - 2960q^{67} + 5536q^{79} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 17 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{3} - 31 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 5 \beta_{1}\)\()/18\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\((\)\(-17 \beta_{3} - 31 \beta_{1}\)\()/18\)

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 −18.5203 0 0 0
881.2 0 0 0 0 0 −18.5203 0 0 0
881.3 0 0 0 0 0 18.5203 0 0 0
881.4 0 0 0 0 0 18.5203 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 1008.4.k.a 4
3.b odd 2 1 inner 1008.4.k.a 4
4.b odd 2 1 63.4.c.a 4
7.b odd 2 1 CM 1008.4.k.a 4
12.b even 2 1 63.4.c.a 4
21.c even 2 1 inner 1008.4.k.a 4
28.d even 2 1 63.4.c.a 4
28.f even 6 2 441.4.p.b 8
28.g odd 6 2 441.4.p.b 8
84.h odd 2 1 63.4.c.a 4
84.j odd 6 2 441.4.p.b 8
84.n even 6 2 441.4.p.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.c.a 4 4.b odd 2 1
63.4.c.a 4 12.b even 2 1
63.4.c.a 4 28.d even 2 1
63.4.c.a 4 84.h odd 2 1
441.4.p.b 8 28.f even 6 2
441.4.p.b 8 28.g odd 6 2
441.4.p.b 8 84.j odd 6 2
441.4.p.b 8 84.n even 6 2
1008.4.k.a 4 1.a even 1 1 trivial
1008.4.k.a 4 3.b odd 2 1 inner
1008.4.k.a 4 7.b odd 2 1 CM
1008.4.k.a 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{4}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -343 + T^{2} )^{2} \)
$11$ \( 3849444 + 5324 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 516834756 + 48668 T^{2} + T^{4} \)
$29$ \( 450373284 + 97556 T^{2} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -112 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( -285628 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( 2534719716 + 595508 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 740 + T )^{4} \)
$71$ \( 58795580484 + 1431644 T^{2} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( -1384 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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