# 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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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