# Properties

 Label 1008.2.cz.b Level $1008$ Weight $2$ Character orbit 1008.cz Analytic conductor $8.049$ Analytic rank $1$ 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.cz (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.04892052375$$ Analytic rank: $$1$$ 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: 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -6 + 3 \zeta_{6} ) q^{11} + ( 6 - 3 \zeta_{6} ) q^{13} + ( -3 + 3 \zeta_{6} ) q^{15} + ( 1 + \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} + ( 1 + \zeta_{6} ) q^{23} -2 \zeta_{6} q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -9 + 9 \zeta_{6} ) q^{29} -8 q^{31} + 9 \zeta_{6} q^{33} + ( -1 + 5 \zeta_{6} ) q^{35} + 11 \zeta_{6} q^{37} -9 \zeta_{6} q^{39} + ( 2 - \zeta_{6} ) q^{41} + ( -5 - 5 \zeta_{6} ) q^{43} + ( 3 + 3 \zeta_{6} ) q^{45} + 12 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{51} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} + ( 2 - \zeta_{6} ) q^{57} -12 q^{59} + ( 3 + 6 \zeta_{6} ) q^{63} -9 q^{65} + ( 2 - 4 \zeta_{6} ) q^{67} + ( 3 - 3 \zeta_{6} ) q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( -7 - 7 \zeta_{6} ) q^{73} + ( -4 + 2 \zeta_{6} ) q^{75} + ( 12 + 3 \zeta_{6} ) q^{77} + ( 6 - 12 \zeta_{6} ) q^{79} + 9 q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} -3 \zeta_{6} q^{85} + ( 9 + 9 \zeta_{6} ) q^{87} + ( -6 + 3 \zeta_{6} ) q^{89} + ( -12 - 3 \zeta_{6} ) q^{91} + ( -8 + 16 \zeta_{6} ) q^{93} + ( 1 - 2 \zeta_{6} ) q^{95} + ( 1 + \zeta_{6} ) q^{97} + ( 18 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{5} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 3q^{5} - 4q^{7} - 6q^{9} - 9q^{11} + 9q^{13} - 3q^{15} + 3q^{17} + q^{19} - 6q^{21} + 3q^{23} - 2q^{25} - 9q^{29} - 16q^{31} + 9q^{33} + 3q^{35} + 11q^{37} - 9q^{39} + 3q^{41} - 15q^{43} + 9q^{45} + 24q^{47} + 2q^{49} + 3q^{51} + 3q^{53} + 18q^{55} + 3q^{57} - 24q^{59} + 12q^{63} - 18q^{65} + 3q^{69} - 21q^{73} - 6q^{75} + 27q^{77} + 18q^{81} - 9q^{83} - 3q^{85} + 27q^{87} - 9q^{89} - 27q^{91} + 3q^{97} + 27q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
367.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 −1.50000 0.866025i 0 −2.00000 1.73205i 0 −3.00000 0
607.1 0 1.73205i 0 −1.50000 + 0.866025i 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.bj 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.cz.b yes 2
3.b odd 2 1 3024.2.cz.c 2
4.b odd 2 1 1008.2.cz.c yes 2
7.d odd 6 1 1008.2.bf.a 2
9.c even 3 1 1008.2.bf.d yes 2
9.d odd 6 1 3024.2.bf.d 2
12.b even 2 1 3024.2.cz.d 2
21.g even 6 1 3024.2.bf.a 2
28.f even 6 1 1008.2.bf.d yes 2
36.f odd 6 1 1008.2.bf.a 2
36.h even 6 1 3024.2.bf.a 2
63.i even 6 1 3024.2.cz.d 2
63.t odd 6 1 1008.2.cz.c yes 2
84.j odd 6 1 3024.2.bf.d 2
252.r odd 6 1 3024.2.cz.c 2
252.bj even 6 1 inner 1008.2.cz.b yes 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 3 T_{5} + 3$$ $$T_{11}^{2} + 9 T_{11} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$3 + 3 T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$27 + 9 T + T^{2}$$
$13$ $$27 - 9 T + T^{2}$$
$17$ $$3 - 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$3 - 3 T + T^{2}$$
$29$ $$81 + 9 T + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$121 - 11 T + T^{2}$$
$41$ $$3 - 3 T + T^{2}$$
$43$ $$75 + 15 T + T^{2}$$
$47$ $$( -12 + T )^{2}$$
$53$ $$9 - 3 T + T^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$12 + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$147 + 21 T + T^{2}$$
$79$ $$108 + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$27 + 9 T + T^{2}$$
$97$ $$3 - 3 T + T^{2}$$