# Properties

 Label 1008.2.s.c 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 84) 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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} + ( -2 + 2 \zeta_{6} ) q^{11} -3 q^{13} + ( 8 - 8 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} -8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} -4 q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + ( 6 - 2 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} -6 q^{41} -11 q^{43} -6 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -12 + 12 \zeta_{6} ) q^{53} + 4 q^{55} + ( -4 + 4 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + ( 13 - 13 \zeta_{6} ) q^{67} -10 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -2 - 4 \zeta_{6} ) q^{77} -3 \zeta_{6} q^{79} + 2 q^{83} -16 q^{85} + ( 6 - 9 \zeta_{6} ) q^{91} + ( -2 + 2 \zeta_{6} ) q^{95} + 10 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - q^{7} + O(q^{10})$$ $$2q - 2q^{5} - q^{7} - 2q^{11} - 6q^{13} + 8q^{17} - q^{19} - 8q^{23} + q^{25} - 8q^{29} + 3q^{31} + 10q^{35} + q^{37} - 12q^{41} - 22q^{43} - 6q^{47} - 13q^{49} - 12q^{53} + 8q^{55} - 4q^{59} + 6q^{61} + 6q^{65} + 13q^{67} - 20q^{71} + 11q^{73} - 8q^{77} - 3q^{79} + 4q^{83} - 32q^{85} + 3q^{91} - 2q^{95} + 20q^{97} + O(q^{100})$$

## 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 −1.00000 + 1.73205i 0 −0.500000 2.59808i 0 0 0
865.1 0 0 0 −1.00000 1.73205i 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.c 2
3.b odd 2 1 336.2.q.c 2
4.b odd 2 1 252.2.k.a 2
7.c even 3 1 inner 1008.2.s.c 2
7.c even 3 1 7056.2.a.bs 1
7.d odd 6 1 7056.2.a.o 1
12.b even 2 1 84.2.i.a 2
21.c even 2 1 2352.2.q.q 2
21.g even 6 1 2352.2.a.k 1
21.g even 6 1 2352.2.q.q 2
21.h odd 6 1 336.2.q.c 2
21.h odd 6 1 2352.2.a.o 1
24.f even 2 1 1344.2.q.b 2
24.h odd 2 1 1344.2.q.n 2
28.d even 2 1 1764.2.k.j 2
28.f even 6 1 1764.2.a.c 1
28.f even 6 1 1764.2.k.j 2
28.g odd 6 1 252.2.k.a 2
28.g odd 6 1 1764.2.a.h 1
36.f odd 6 1 2268.2.i.b 2
36.f odd 6 1 2268.2.l.g 2
36.h even 6 1 2268.2.i.g 2
36.h even 6 1 2268.2.l.b 2
60.h even 2 1 2100.2.q.b 2
60.l odd 4 2 2100.2.bc.a 4
84.h odd 2 1 588.2.i.b 2
84.j odd 6 1 588.2.a.f 1
84.j odd 6 1 588.2.i.b 2
84.n even 6 1 84.2.i.a 2
84.n even 6 1 588.2.a.a 1
168.s odd 6 1 1344.2.q.n 2
168.s odd 6 1 9408.2.a.bi 1
168.v even 6 1 1344.2.q.b 2
168.v even 6 1 9408.2.a.cx 1
168.ba even 6 1 9408.2.a.bx 1
168.be odd 6 1 9408.2.a.i 1
252.o even 6 1 2268.2.i.g 2
252.u odd 6 1 2268.2.l.g 2
252.bb even 6 1 2268.2.l.b 2
252.bl odd 6 1 2268.2.i.b 2
420.ba even 6 1 2100.2.q.b 2
420.bp odd 12 2 2100.2.bc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a 2 12.b even 2 1
84.2.i.a 2 84.n even 6 1
252.2.k.a 2 4.b odd 2 1
252.2.k.a 2 28.g odd 6 1
336.2.q.c 2 3.b odd 2 1
336.2.q.c 2 21.h odd 6 1
588.2.a.a 1 84.n even 6 1
588.2.a.f 1 84.j odd 6 1
588.2.i.b 2 84.h odd 2 1
588.2.i.b 2 84.j odd 6 1
1008.2.s.c 2 1.a even 1 1 trivial
1008.2.s.c 2 7.c even 3 1 inner
1344.2.q.b 2 24.f even 2 1
1344.2.q.b 2 168.v even 6 1
1344.2.q.n 2 24.h odd 2 1
1344.2.q.n 2 168.s odd 6 1
1764.2.a.c 1 28.f even 6 1
1764.2.a.h 1 28.g odd 6 1
1764.2.k.j 2 28.d even 2 1
1764.2.k.j 2 28.f even 6 1
2100.2.q.b 2 60.h even 2 1
2100.2.q.b 2 420.ba even 6 1
2100.2.bc.a 4 60.l odd 4 2
2100.2.bc.a 4 420.bp odd 12 2
2268.2.i.b 2 36.f odd 6 1
2268.2.i.b 2 252.bl odd 6 1
2268.2.i.g 2 36.h even 6 1
2268.2.i.g 2 252.o even 6 1
2268.2.l.b 2 36.h even 6 1
2268.2.l.b 2 252.bb even 6 1
2268.2.l.g 2 36.f odd 6 1
2268.2.l.g 2 252.u odd 6 1
2352.2.a.k 1 21.g even 6 1
2352.2.a.o 1 21.h odd 6 1
2352.2.q.q 2 21.c even 2 1
2352.2.q.q 2 21.g even 6 1
7056.2.a.o 1 7.d odd 6 1
7056.2.a.bs 1 7.c even 3 1
9408.2.a.i 1 168.be odd 6 1
9408.2.a.bi 1 168.s odd 6 1
9408.2.a.bx 1 168.ba even 6 1
9408.2.a.cx 1 168.v even 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13} + 3$$ $$T_{17}^{2} - 8 T_{17} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$64 - 8 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$9 - 3 T + T^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 11 + T )^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$36 - 6 T + T^{2}$$
$67$ $$169 - 13 T + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$9 + 3 T + T^{2}$$
$83$ $$( -2 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$( -10 + T )^{2}$$