# Properties

 Label 1008.2.s.e 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( 1 - \zeta_{6} ) q^{11} + 2 q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + 3 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + 6 q^{29} + ( -1 + \zeta_{6} ) q^{31} + ( 2 + \zeta_{6} ) q^{35} + 5 \zeta_{6} q^{37} + 10 q^{41} + 4 q^{43} -\zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -9 + 9 \zeta_{6} ) q^{53} - q^{55} + ( -3 + 3 \zeta_{6} ) q^{59} -3 \zeta_{6} q^{61} -2 \zeta_{6} q^{65} + ( 11 - 11 \zeta_{6} ) q^{67} + 16 q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} + ( -1 + 3 \zeta_{6} ) q^{77} -11 \zeta_{6} q^{79} -4 q^{83} -3 q^{85} -9 \zeta_{6} q^{89} + ( -6 + 4 \zeta_{6} ) q^{91} + ( 5 - 5 \zeta_{6} ) q^{95} + 6 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} - 4q^{7} + O(q^{10})$$ $$2q - q^{5} - 4q^{7} + q^{11} + 4q^{13} + 3q^{17} + 5q^{19} + 3q^{23} + 4q^{25} + 12q^{29} - q^{31} + 5q^{35} + 5q^{37} + 20q^{41} + 8q^{43} - q^{47} + 2q^{49} - 9q^{53} - 2q^{55} - 3q^{59} - 3q^{61} - 2q^{65} + 11q^{67} + 32q^{71} - 7q^{73} + q^{77} - 11q^{79} - 8q^{83} - 6q^{85} - 9q^{89} - 8q^{91} + 5q^{95} + 12q^{97} + O(q^{100})$$

## 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 −0.500000 + 0.866025i 0 −2.00000 1.73205i 0 0 0
865.1 0 0 0 −0.500000 0.866025i 0 −2.00000 + 1.73205i 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.e 2
3.b odd 2 1 112.2.i.c 2
4.b odd 2 1 504.2.s.e 2
7.c even 3 1 inner 1008.2.s.e 2
7.c even 3 1 7056.2.a.bi 1
7.d odd 6 1 7056.2.a.s 1
12.b even 2 1 56.2.i.a 2
21.c even 2 1 784.2.i.a 2
21.g even 6 1 784.2.a.j 1
21.g even 6 1 784.2.i.a 2
21.h odd 6 1 112.2.i.c 2
21.h odd 6 1 784.2.a.a 1
24.f even 2 1 448.2.i.f 2
24.h odd 2 1 448.2.i.a 2
28.d even 2 1 3528.2.s.o 2
28.f even 6 1 3528.2.a.k 1
28.f even 6 1 3528.2.s.o 2
28.g odd 6 1 504.2.s.e 2
28.g odd 6 1 3528.2.a.r 1
60.h even 2 1 1400.2.q.g 2
60.l odd 4 2 1400.2.bh.f 4
84.h odd 2 1 392.2.i.f 2
84.j odd 6 1 392.2.a.a 1
84.j odd 6 1 392.2.i.f 2
84.n even 6 1 56.2.i.a 2
84.n even 6 1 392.2.a.f 1
168.s odd 6 1 448.2.i.a 2
168.s odd 6 1 3136.2.a.bc 1
168.v even 6 1 448.2.i.f 2
168.v even 6 1 3136.2.a.b 1
168.ba even 6 1 3136.2.a.a 1
168.be odd 6 1 3136.2.a.bb 1
420.ba even 6 1 1400.2.q.g 2
420.ba even 6 1 9800.2.a.b 1
420.be odd 6 1 9800.2.a.bp 1
420.bp odd 12 2 1400.2.bh.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 12.b even 2 1
56.2.i.a 2 84.n even 6 1
112.2.i.c 2 3.b odd 2 1
112.2.i.c 2 21.h odd 6 1
392.2.a.a 1 84.j odd 6 1
392.2.a.f 1 84.n even 6 1
392.2.i.f 2 84.h odd 2 1
392.2.i.f 2 84.j odd 6 1
448.2.i.a 2 24.h odd 2 1
448.2.i.a 2 168.s odd 6 1
448.2.i.f 2 24.f even 2 1
448.2.i.f 2 168.v even 6 1
504.2.s.e 2 4.b odd 2 1
504.2.s.e 2 28.g odd 6 1
784.2.a.a 1 21.h odd 6 1
784.2.a.j 1 21.g even 6 1
784.2.i.a 2 21.c even 2 1
784.2.i.a 2 21.g even 6 1
1008.2.s.e 2 1.a even 1 1 trivial
1008.2.s.e 2 7.c even 3 1 inner
1400.2.q.g 2 60.h even 2 1
1400.2.q.g 2 420.ba even 6 1
1400.2.bh.f 4 60.l odd 4 2
1400.2.bh.f 4 420.bp odd 12 2
3136.2.a.a 1 168.ba even 6 1
3136.2.a.b 1 168.v even 6 1
3136.2.a.bb 1 168.be odd 6 1
3136.2.a.bc 1 168.s odd 6 1
3528.2.a.k 1 28.f even 6 1
3528.2.a.r 1 28.g odd 6 1
3528.2.s.o 2 28.d even 2 1
3528.2.s.o 2 28.f even 6 1
7056.2.a.s 1 7.d odd 6 1
7056.2.a.bi 1 7.c even 3 1
9800.2.a.b 1 420.ba even 6 1
9800.2.a.bp 1 420.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + T_{5} + 1$$ $$T_{11}^{2} - T_{11} + 1$$ $$T_{13} - 2$$ $$T_{17}^{2} - 3 T_{17} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$9 - 3 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$1 + T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$1 + T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$9 + 3 T + T^{2}$$
$67$ $$121 - 11 T + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$49 + 7 T + T^{2}$$
$79$ $$121 + 11 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$81 + 9 T + T^{2}$$
$97$ $$( -6 + T )^{2}$$