# Properties

 Label 1008.2.s.j Level $1008$ Weight $2$ Character orbit 1008.s Analytic conductor $8.049$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.s (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 - 3 \zeta_{6} ) q^{7} -7 q^{13} -7 \zeta_{6} q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -7 + 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{37} -5 q^{43} + ( -5 - 3 \zeta_{6} ) q^{49} -14 \zeta_{6} q^{61} + ( 11 - 11 \zeta_{6} ) q^{67} + ( 7 - 7 \zeta_{6} ) q^{73} -13 \zeta_{6} q^{79} + ( -14 + 21 \zeta_{6} ) q^{91} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{7} + O(q^{10})$$ $$2q + q^{7} - 14q^{13} - 7q^{19} + 5q^{25} - 7q^{31} + q^{37} - 10q^{43} - 13q^{49} - 14q^{61} + 11q^{67} + 7q^{73} - 13q^{79} - 7q^{91} + 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$757$$ $$785$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 0.500000 + 2.59808i 0 0 0
865.1 0 0 0 0 0 0.500000 2.59808i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.s.j 2
3.b odd 2 1 CM 1008.2.s.j 2
4.b odd 2 1 63.2.e.a 2
7.c even 3 1 inner 1008.2.s.j 2
7.c even 3 1 7056.2.a.y 1
7.d odd 6 1 7056.2.a.bf 1
12.b even 2 1 63.2.e.a 2
21.g even 6 1 7056.2.a.bf 1
21.h odd 6 1 inner 1008.2.s.j 2
21.h odd 6 1 7056.2.a.y 1
28.d even 2 1 441.2.e.c 2
28.f even 6 1 441.2.a.e 1
28.f even 6 1 441.2.e.c 2
28.g odd 6 1 63.2.e.a 2
28.g odd 6 1 441.2.a.d 1
36.f odd 6 1 567.2.g.d 2
36.f odd 6 1 567.2.h.c 2
36.h even 6 1 567.2.g.d 2
36.h even 6 1 567.2.h.c 2
84.h odd 2 1 441.2.e.c 2
84.j odd 6 1 441.2.a.e 1
84.j odd 6 1 441.2.e.c 2
84.n even 6 1 63.2.e.a 2
84.n even 6 1 441.2.a.d 1
252.o even 6 1 567.2.h.c 2
252.u odd 6 1 567.2.g.d 2
252.bb even 6 1 567.2.g.d 2
252.bl odd 6 1 567.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 4.b odd 2 1
63.2.e.a 2 12.b even 2 1
63.2.e.a 2 28.g odd 6 1
63.2.e.a 2 84.n even 6 1
441.2.a.d 1 28.g odd 6 1
441.2.a.d 1 84.n even 6 1
441.2.a.e 1 28.f even 6 1
441.2.a.e 1 84.j odd 6 1
441.2.e.c 2 28.d even 2 1
441.2.e.c 2 28.f even 6 1
441.2.e.c 2 84.h odd 2 1
441.2.e.c 2 84.j odd 6 1
567.2.g.d 2 36.f odd 6 1
567.2.g.d 2 36.h even 6 1
567.2.g.d 2 252.u odd 6 1
567.2.g.d 2 252.bb even 6 1
567.2.h.c 2 36.f odd 6 1
567.2.h.c 2 36.h even 6 1
567.2.h.c 2 252.o even 6 1
567.2.h.c 2 252.bl odd 6 1
1008.2.s.j 2 1.a even 1 1 trivial
1008.2.s.j 2 3.b odd 2 1 CM
1008.2.s.j 2 7.c even 3 1 inner
1008.2.s.j 2 21.h odd 6 1 inner
7056.2.a.y 1 7.c even 3 1
7056.2.a.y 1 21.h odd 6 1
7056.2.a.bf 1 7.d odd 6 1
7056.2.a.bf 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

 $$T_{5}$$ $$T_{11}$$ $$T_{13} + 7$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 7 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$49 + 7 T + T^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$121 - 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$169 + 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$