# Properties

 Label 1008.2.bf.b Level $1008$ Weight $2$ Character orbit 1008.bf 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.bf (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} + ( 1 + \zeta_{6} ) q^{13} + ( 1 + \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( 2 - 4 \zeta_{6} ) q^{23} + 5 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} -3 \zeta_{6} q^{29} -\zeta_{6} q^{31} + 6 q^{33} -7 \zeta_{6} q^{37} + ( -3 + 3 \zeta_{6} ) q^{39} + ( 1 + \zeta_{6} ) q^{41} + ( -2 + \zeta_{6} ) q^{43} + ( 9 - 9 \zeta_{6} ) q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -3 + 3 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} + ( 10 - 5 \zeta_{6} ) q^{57} + 15 \zeta_{6} q^{59} + ( 1 + \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + ( -18 + 9 \zeta_{6} ) q^{67} + 6 q^{69} + ( 6 - 12 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + ( -10 + 8 \zeta_{6} ) q^{77} + ( 1 + \zeta_{6} ) q^{79} + 9 q^{81} -9 \zeta_{6} q^{83} + ( 6 - 3 \zeta_{6} ) q^{87} + ( 2 - \zeta_{6} ) q^{89} + ( 1 - 5 \zeta_{6} ) q^{91} + ( 2 - \zeta_{6} ) q^{93} + ( 2 - \zeta_{6} ) q^{97} + ( -6 + 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} - 6 q^{9} + O(q^{10})$$ $$2 q - 4 q^{7} - 6 q^{9} + 3 q^{13} + 3 q^{17} - 5 q^{19} + 6 q^{21} + 10 q^{25} - 3 q^{29} - q^{31} + 12 q^{33} - 7 q^{37} - 3 q^{39} + 3 q^{41} - 3 q^{43} + 9 q^{47} + 2 q^{49} - 3 q^{51} + 9 q^{53} + 15 q^{57} + 15 q^{59} + 3 q^{61} + 12 q^{63} - 27 q^{67} + 12 q^{69} + 3 q^{73} - 12 q^{77} + 3 q^{79} + 18 q^{81} - 9 q^{83} + 9 q^{87} + 3 q^{89} - 3 q^{91} + 3 q^{93} + 3 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
943.1 0 1.73205i 0 0 0 −2.00000 + 1.73205i 0 −3.00000 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
252.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.bf.b 2
3.b odd 2 1 3024.2.bf.b 2
4.b odd 2 1 1008.2.bf.c yes 2
7.d odd 6 1 1008.2.cz.d yes 2
9.c even 3 1 1008.2.cz.a yes 2
9.d odd 6 1 3024.2.cz.a 2
12.b even 2 1 3024.2.bf.c 2
21.g even 6 1 3024.2.cz.b 2
28.f even 6 1 1008.2.cz.a yes 2
36.f odd 6 1 1008.2.cz.d yes 2
36.h even 6 1 3024.2.cz.b 2
63.k odd 6 1 1008.2.bf.c yes 2
63.s even 6 1 3024.2.bf.c 2
84.j odd 6 1 3024.2.cz.a 2
252.n even 6 1 inner 1008.2.bf.b 2
252.bn odd 6 1 3024.2.bf.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.b 2 1.a even 1 1 trivial
1008.2.bf.b 2 252.n even 6 1 inner
1008.2.bf.c yes 2 4.b odd 2 1
1008.2.bf.c yes 2 63.k odd 6 1
1008.2.cz.a yes 2 9.c even 3 1
1008.2.cz.a yes 2 28.f even 6 1
1008.2.cz.d yes 2 7.d odd 6 1
1008.2.cz.d yes 2 36.f odd 6 1
3024.2.bf.b 2 3.b odd 2 1
3024.2.bf.b 2 252.bn odd 6 1
3024.2.bf.c 2 12.b even 2 1
3024.2.bf.c 2 63.s even 6 1
3024.2.cz.a 2 9.d odd 6 1
3024.2.cz.a 2 84.j odd 6 1
3024.2.cz.b 2 21.g even 6 1
3024.2.cz.b 2 36.h even 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{19}^{2} + 5 T_{19} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$3 - 3 T + T^{2}$$
$17$ $$3 - 3 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$12 + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$1 + T + T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$3 - 3 T + T^{2}$$
$43$ $$3 + 3 T + T^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$81 - 9 T + T^{2}$$
$59$ $$225 - 15 T + T^{2}$$
$61$ $$3 - 3 T + T^{2}$$
$67$ $$243 + 27 T + T^{2}$$
$71$ $$108 + T^{2}$$
$73$ $$3 - 3 T + T^{2}$$
$79$ $$3 - 3 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$3 - 3 T + T^{2}$$
$97$ $$3 - 3 T + T^{2}$$