# Properties

 Label 1008.2.s.o 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} + 2 q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -9 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + ( -9 + 6 \zeta_{6} ) q^{35} + 10 \zeta_{6} q^{37} + 4 q^{43} -12 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( 3 - 3 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( 2 - 2 \zeta_{6} ) q^{67} + ( -2 + 2 \zeta_{6} ) q^{73} + ( 6 + 3 \zeta_{6} ) q^{77} + 5 \zeta_{6} q^{79} + 9 q^{83} + 18 q^{85} -6 \zeta_{6} q^{89} + ( -2 + 6 \zeta_{6} ) q^{91} + ( -6 + 6 \zeta_{6} ) q^{95} -13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{5} + q^{7} + O(q^{10})$$ $$2q + 3q^{5} + q^{7} + 3q^{11} + 4q^{13} + 6q^{17} + 2q^{19} + 6q^{23} - 4q^{25} - 18q^{29} - 7q^{31} - 12q^{35} + 10q^{37} + 8q^{43} - 12q^{47} - 13q^{49} - 3q^{53} + 18q^{55} + 3q^{59} + 4q^{61} + 6q^{65} + 2q^{67} - 2q^{73} + 15q^{77} + 5q^{79} + 18q^{83} + 36q^{85} - 6q^{89} + 2q^{91} - 6q^{95} - 26q^{97} + O(q^{100})$$

## 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 1.50000 2.59808i 0 0.500000 2.59808i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 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
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.o 2
3.b odd 2 1 1008.2.s.b 2
4.b odd 2 1 126.2.g.d yes 2
7.c even 3 1 inner 1008.2.s.o 2
7.c even 3 1 7056.2.a.e 1
7.d odd 6 1 7056.2.a.bx 1
12.b even 2 1 126.2.g.a 2
21.g even 6 1 7056.2.a.h 1
21.h odd 6 1 1008.2.s.b 2
21.h odd 6 1 7056.2.a.by 1
28.d even 2 1 882.2.g.g 2
28.f even 6 1 882.2.a.e 1
28.f even 6 1 882.2.g.g 2
28.g odd 6 1 126.2.g.d yes 2
28.g odd 6 1 882.2.a.a 1
36.f odd 6 1 1134.2.e.g 2
36.f odd 6 1 1134.2.h.j 2
36.h even 6 1 1134.2.e.k 2
36.h even 6 1 1134.2.h.f 2
84.h odd 2 1 882.2.g.e 2
84.j odd 6 1 882.2.a.h 1
84.j odd 6 1 882.2.g.e 2
84.n even 6 1 126.2.g.a 2
84.n even 6 1 882.2.a.j 1
252.o even 6 1 1134.2.e.k 2
252.u odd 6 1 1134.2.h.j 2
252.bb even 6 1 1134.2.h.f 2
252.bl odd 6 1 1134.2.e.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 12.b even 2 1
126.2.g.a 2 84.n even 6 1
126.2.g.d yes 2 4.b odd 2 1
126.2.g.d yes 2 28.g odd 6 1
882.2.a.a 1 28.g odd 6 1
882.2.a.e 1 28.f even 6 1
882.2.a.h 1 84.j odd 6 1
882.2.a.j 1 84.n even 6 1
882.2.g.e 2 84.h odd 2 1
882.2.g.e 2 84.j odd 6 1
882.2.g.g 2 28.d even 2 1
882.2.g.g 2 28.f even 6 1
1008.2.s.b 2 3.b odd 2 1
1008.2.s.b 2 21.h odd 6 1
1008.2.s.o 2 1.a even 1 1 trivial
1008.2.s.o 2 7.c even 3 1 inner
1134.2.e.g 2 36.f odd 6 1
1134.2.e.g 2 252.bl odd 6 1
1134.2.e.k 2 36.h even 6 1
1134.2.e.k 2 252.o even 6 1
1134.2.h.f 2 36.h even 6 1
1134.2.h.f 2 252.bb even 6 1
1134.2.h.j 2 36.f odd 6 1
1134.2.h.j 2 252.u odd 6 1
7056.2.a.e 1 7.c even 3 1
7056.2.a.h 1 21.g even 6 1
7056.2.a.bx 1 7.d odd 6 1
7056.2.a.by 1 21.h odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{13} - 2$$ $$T_{17}^{2} - 6 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$4 - 2 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$49 + 7 T + T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$25 - 5 T + T^{2}$$
$83$ $$( -9 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 13 + T )^{2}$$