Properties

Label 1008.2.k.c
Level $1008$
Weight $2$
Character orbit 1008.k
Analytic conductor $8.049$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{5} + ( -1 + \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} +O(q^{10})\) \( q + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) q^{5} + ( -1 + \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{7} + ( \zeta_{16}^{2} + 2 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{11} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{13} + ( 2 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{17} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{19} + ( 3 \zeta_{16}^{2} + 2 \zeta_{16}^{4} + 3 \zeta_{16}^{6} ) q^{23} + ( 3 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{25} + ( -\zeta_{16}^{2} + 4 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{29} + ( 4 \zeta_{16} + 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{31} + ( 4 \zeta_{16} + 2 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{35} + 4 q^{37} + ( -2 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{41} + ( -2 + 6 \zeta_{16}^{2} - 6 \zeta_{16}^{6} ) q^{43} + ( 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) q^{47} + ( -1 + 2 \zeta_{16} + 4 \zeta_{16}^{2} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{49} + ( \zeta_{16}^{2} - 4 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{53} + ( -2 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( 8 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{65} + ( 4 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{67} + ( -\zeta_{16}^{2} - 6 \zeta_{16}^{4} - \zeta_{16}^{6} ) q^{71} + ( 6 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{73} + ( 2 \zeta_{16} - 3 \zeta_{16}^{2} - 4 \zeta_{16}^{4} - 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{77} + ( 8 + 4 \zeta_{16}^{2} - 4 \zeta_{16}^{6} ) q^{79} + ( -8 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 8 \zeta_{16}^{7} ) q^{83} + ( -8 - 12 \zeta_{16}^{2} + 12 \zeta_{16}^{6} ) q^{85} + ( 2 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{89} + ( 2 \zeta_{16} - 4 \zeta_{16}^{2} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{91} + 8 \zeta_{16}^{4} q^{95} + ( 2 \zeta_{16} - 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} + 24q^{25} + 32q^{37} - 16q^{43} - 8q^{49} + 32q^{67} + 64q^{79} - 64q^{85} + O(q^{100}) \)

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
0.923880 + 0.382683i
0.923880 0.382683i
0.382683 0.923880i
0.382683 + 0.923880i
−0.382683 0.923880i
−0.382683 + 0.923880i
−0.923880 + 0.382683i
−0.923880 0.382683i
0 0 0 −3.69552 0 −2.41421 1.08239i 0 0 0
881.2 0 0 0 −3.69552 0 −2.41421 + 1.08239i 0 0 0
881.3 0 0 0 −1.53073 0 0.414214 2.61313i 0 0 0
881.4 0 0 0 −1.53073 0 0.414214 + 2.61313i 0 0 0
881.5 0 0 0 1.53073 0 0.414214 2.61313i 0 0 0
881.6 0 0 0 1.53073 0 0.414214 + 2.61313i 0 0 0
881.7 0 0 0 3.69552 0 −2.41421 1.08239i 0 0 0
881.8 0 0 0 3.69552 0 −2.41421 + 1.08239i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
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.c 8
3.b odd 2 1 inner 1008.2.k.c 8
4.b odd 2 1 504.2.k.a 8
7.b odd 2 1 inner 1008.2.k.c 8
8.b even 2 1 4032.2.k.e 8
8.d odd 2 1 4032.2.k.f 8
12.b even 2 1 504.2.k.a 8
21.c even 2 1 inner 1008.2.k.c 8
24.f even 2 1 4032.2.k.f 8
24.h odd 2 1 4032.2.k.e 8
28.d even 2 1 504.2.k.a 8
28.f even 6 2 3528.2.bl.b 16
28.g odd 6 2 3528.2.bl.b 16
56.e even 2 1 4032.2.k.f 8
56.h odd 2 1 4032.2.k.e 8
84.h odd 2 1 504.2.k.a 8
84.j odd 6 2 3528.2.bl.b 16
84.n even 6 2 3528.2.bl.b 16
168.e odd 2 1 4032.2.k.f 8
168.i even 2 1 4032.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.k.a 8 4.b odd 2 1
504.2.k.a 8 12.b even 2 1
504.2.k.a 8 28.d even 2 1
504.2.k.a 8 84.h odd 2 1
1008.2.k.c 8 1.a even 1 1 trivial
1008.2.k.c 8 3.b odd 2 1 inner
1008.2.k.c 8 7.b odd 2 1 inner
1008.2.k.c 8 21.c even 2 1 inner
3528.2.bl.b 16 28.f even 6 2
3528.2.bl.b 16 28.g odd 6 2
3528.2.bl.b 16 84.j odd 6 2
3528.2.bl.b 16 84.n even 6 2
4032.2.k.e 8 8.b even 2 1
4032.2.k.e 8 24.h odd 2 1
4032.2.k.e 8 56.h odd 2 1
4032.2.k.e 8 168.i even 2 1
4032.2.k.f 8 8.d odd 2 1
4032.2.k.f 8 24.f even 2 1
4032.2.k.f 8 56.e even 2 1
4032.2.k.f 8 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 16 T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 32 - 16 T^{2} + T^{4} )^{2} \)
$7$ \( ( 49 + 28 T + 10 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$11$ \( ( 4 + 12 T^{2} + T^{4} )^{2} \)
$13$ \( ( 128 + 32 T^{2} + T^{4} )^{2} \)
$17$ \( ( 1568 - 80 T^{2} + T^{4} )^{2} \)
$19$ \( ( 128 + 32 T^{2} + T^{4} )^{2} \)
$23$ \( ( 196 + 44 T^{2} + T^{4} )^{2} \)
$29$ \( ( 196 + 36 T^{2} + T^{4} )^{2} \)
$31$ \( ( 2048 + 128 T^{2} + T^{4} )^{2} \)
$37$ \( ( -4 + T )^{8} \)
$41$ \( ( 32 - 80 T^{2} + T^{4} )^{2} \)
$43$ \( ( -68 + 4 T + T^{2} )^{4} \)
$47$ \( ( 512 - 64 T^{2} + T^{4} )^{2} \)
$53$ \( ( 196 + 36 T^{2} + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( -16 - 8 T + T^{2} )^{4} \)
$71$ \( ( 1156 + 76 T^{2} + T^{4} )^{2} \)
$73$ \( ( 128 + 160 T^{2} + T^{4} )^{2} \)
$79$ \( ( 32 - 16 T + T^{2} )^{4} \)
$83$ \( ( 25088 - 320 T^{2} + T^{4} )^{2} \)
$89$ \( ( 1568 - 80 T^{2} + T^{4} )^{2} \)
$97$ \( ( 6272 + 160 T^{2} + T^{4} )^{2} \)
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