Properties

Label 1008.2.k.b
Level $1008$
Weight $2$
Character orbit 1008.k
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
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

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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + ( 1 + \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} + ( 1 + \beta_{1} ) q^{7} -\beta_{2} q^{11} -2 \beta_{1} q^{13} -\beta_{3} q^{17} -2 \beta_{1} q^{19} + \beta_{2} q^{23} + 7 q^{25} + \beta_{2} q^{29} + ( 2 \beta_{2} - \beta_{3} ) q^{35} -8 q^{37} + \beta_{3} q^{41} + 2 q^{43} -2 \beta_{3} q^{47} + ( -5 + 2 \beta_{1} ) q^{49} + 3 \beta_{2} q^{53} -6 \beta_{1} q^{55} -4 \beta_{3} q^{59} -4 \beta_{1} q^{61} -4 \beta_{2} q^{65} -8 q^{67} + \beta_{2} q^{71} + 2 \beta_{1} q^{73} + ( -\beta_{2} - 3 \beta_{3} ) q^{77} + 4 q^{79} + 2 \beta_{3} q^{83} + 12 q^{85} + 3 \beta_{3} q^{89} + ( 12 - 2 \beta_{1} ) q^{91} -4 \beta_{2} q^{95} -2 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 28q^{25} - 32q^{37} + 8q^{43} - 20q^{49} - 32q^{67} + 16q^{79} + 48q^{85} + 48q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{2}\)\(=\)\( -3 \nu^{3} - 9 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} - 9 \beta_{1}\)\()/6\)

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
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 \(S_{2}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -12 + T^{2} )^{2} \)
$7$ \( ( 7 - 2 T + T^{2} )^{2} \)
$11$ \( ( 18 + T^{2} )^{2} \)
$13$ \( ( 24 + T^{2} )^{2} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 24 + T^{2} )^{2} \)
$23$ \( ( 18 + T^{2} )^{2} \)
$29$ \( ( 18 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 8 + T )^{4} \)
$41$ \( ( -12 + T^{2} )^{2} \)
$43$ \( ( -2 + T )^{4} \)
$47$ \( ( -48 + T^{2} )^{2} \)
$53$ \( ( 162 + T^{2} )^{2} \)
$59$ \( ( -192 + T^{2} )^{2} \)
$61$ \( ( 96 + T^{2} )^{2} \)
$67$ \( ( 8 + T )^{4} \)
$71$ \( ( 18 + T^{2} )^{2} \)
$73$ \( ( 24 + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( -48 + T^{2} )^{2} \)
$89$ \( ( -108 + T^{2} )^{2} \)
$97$ \( ( 24 + T^{2} )^{2} \)
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