# Properties

 Label 1008.2.s.q Level $1008$ Weight $2$ Character orbit 1008.s Analytic conductor $8.049$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 4 \zeta_{6} q^{5} + ( -2 - \zeta_{6} ) q^{7} -3 q^{13} + ( -4 + 4 \zeta_{6} ) q^{17} + 7 \zeta_{6} q^{19} -4 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 8 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( 4 - 12 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} -8 q^{41} -11 q^{43} -4 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 4 - 4 \zeta_{6} ) q^{53} + ( -12 + 12 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} -12 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} + 12 q^{71} + ( -1 + \zeta_{6} ) q^{73} + \zeta_{6} q^{79} + 12 q^{83} -16 q^{85} + 8 \zeta_{6} q^{89} + ( 6 + 3 \zeta_{6} ) q^{91} + ( -28 + 28 \zeta_{6} ) q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} - 5q^{7} + O(q^{10})$$ $$2q + 4q^{5} - 5q^{7} - 6q^{13} - 4q^{17} + 7q^{19} - 4q^{23} - 11q^{25} + 16q^{29} - 5q^{31} - 4q^{35} - 3q^{37} - 16q^{41} - 22q^{43} - 4q^{47} + 11q^{49} + 4q^{53} - 12q^{59} + 2q^{61} - 12q^{65} - 3q^{67} + 24q^{71} - q^{73} + q^{79} + 24q^{83} - 32q^{85} + 8q^{89} + 15q^{91} - 28q^{95} - 4q^{97} + O(q^{100})$$

## 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 2.00000 3.46410i 0 −2.50000 + 0.866025i 0 0 0
865.1 0 0 0 2.00000 + 3.46410i 0 −2.50000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.s.q 2
3.b odd 2 1 1008.2.s.a 2
4.b odd 2 1 504.2.s.h yes 2
7.c even 3 1 inner 1008.2.s.q 2
7.c even 3 1 7056.2.a.b 1
7.d odd 6 1 7056.2.a.cc 1
12.b even 2 1 504.2.s.a 2
21.g even 6 1 7056.2.a.d 1
21.h odd 6 1 1008.2.s.a 2
21.h odd 6 1 7056.2.a.cb 1
28.d even 2 1 3528.2.s.b 2
28.f even 6 1 3528.2.a.ba 1
28.f even 6 1 3528.2.s.b 2
28.g odd 6 1 504.2.s.h yes 2
28.g odd 6 1 3528.2.a.a 1
84.h odd 2 1 3528.2.s.bb 2
84.j odd 6 1 3528.2.a.c 1
84.j odd 6 1 3528.2.s.bb 2
84.n even 6 1 504.2.s.a 2
84.n even 6 1 3528.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.a 2 12.b even 2 1
504.2.s.a 2 84.n even 6 1
504.2.s.h yes 2 4.b odd 2 1
504.2.s.h yes 2 28.g odd 6 1
1008.2.s.a 2 3.b odd 2 1
1008.2.s.a 2 21.h odd 6 1
1008.2.s.q 2 1.a even 1 1 trivial
1008.2.s.q 2 7.c even 3 1 inner
3528.2.a.a 1 28.g odd 6 1
3528.2.a.c 1 84.j odd 6 1
3528.2.a.z 1 84.n even 6 1
3528.2.a.ba 1 28.f even 6 1
3528.2.s.b 2 28.d even 2 1
3528.2.s.b 2 28.f even 6 1
3528.2.s.bb 2 84.h odd 2 1
3528.2.s.bb 2 84.j odd 6 1
7056.2.a.b 1 7.c even 3 1
7056.2.a.d 1 21.g even 6 1
7056.2.a.cb 1 21.h odd 6 1
7056.2.a.cc 1 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{11}$$ $$T_{13} + 3$$ $$T_{17}^{2} + 4 T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$9 + 3 T + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$( 11 + T )^{2}$$
$47$ $$16 + 4 T + T^{2}$$
$53$ $$16 - 4 T + T^{2}$$
$59$ $$144 + 12 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$1 + T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$64 - 8 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$