# Properties

 Label 1008.2.q.b Level $1008$ Weight $2$ Character orbit 1008.q 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.q (of order $$3$$, degree $$2$$, not 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: no (minimal twist has level 504) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} -3 \zeta_{6} q^{11} -\zeta_{6} q^{13} + ( 1 + \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} + ( 1 - \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -9 + 9 \zeta_{6} ) q^{29} -4 q^{31} + ( -6 + 3 \zeta_{6} ) q^{33} + ( 3 - \zeta_{6} ) q^{35} -5 \zeta_{6} q^{37} + ( -2 + \zeta_{6} ) q^{39} -7 \zeta_{6} q^{41} + ( 3 - 3 \zeta_{6} ) q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -8 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} + ( -9 + 9 \zeta_{6} ) q^{53} + 3 q^{55} + ( 10 - 5 \zeta_{6} ) q^{57} + 4 q^{59} + 2 q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + q^{65} -12 q^{67} + ( -1 - \zeta_{6} ) q^{69} -8 q^{71} + ( 13 - 13 \zeta_{6} ) q^{73} + ( 8 - 4 \zeta_{6} ) q^{75} + ( -6 + 9 \zeta_{6} ) q^{77} -8 q^{79} + 9 q^{81} + ( -13 + 13 \zeta_{6} ) q^{83} -3 \zeta_{6} q^{85} + ( 9 + 9 \zeta_{6} ) q^{87} + 9 \zeta_{6} q^{89} + ( -2 + 3 \zeta_{6} ) q^{91} + ( -4 + 8 \zeta_{6} ) q^{93} -5 q^{95} + ( 17 - 17 \zeta_{6} ) q^{97} + 9 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - q^{5} - 4q^{7} - 6q^{9} - 3q^{11} - q^{13} + 3q^{15} - 3q^{17} + 5q^{19} - 6q^{21} + q^{23} + 4q^{25} - 9q^{29} - 8q^{31} - 9q^{33} + 5q^{35} - 5q^{37} - 3q^{39} - 7q^{41} + 3q^{43} + 3q^{45} - 16q^{47} + 2q^{49} + 9q^{51} - 9q^{53} + 6q^{55} + 15q^{57} + 8q^{59} + 4q^{61} + 12q^{63} + 2q^{65} - 24q^{67} - 3q^{69} - 16q^{71} + 13q^{73} + 12q^{75} - 3q^{77} - 16q^{79} + 18q^{81} - 13q^{83} - 3q^{85} + 27q^{87} + 9q^{89} - q^{91} - 10q^{95} + 17q^{97} + 9q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 −0.500000 + 0.866025i 0 −2.00000 1.73205i 0 −3.00000 0
625.1 0 1.73205i 0 −0.500000 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
63.h 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.q.b 2
3.b odd 2 1 3024.2.q.d 2
4.b odd 2 1 504.2.q.a 2
7.c even 3 1 1008.2.t.e 2
9.c even 3 1 1008.2.t.e 2
9.d odd 6 1 3024.2.t.c 2
12.b even 2 1 1512.2.q.b 2
21.h odd 6 1 3024.2.t.c 2
28.g odd 6 1 504.2.t.a yes 2
36.f odd 6 1 504.2.t.a yes 2
36.h even 6 1 1512.2.t.a 2
63.h even 3 1 inner 1008.2.q.b 2
63.j odd 6 1 3024.2.q.d 2
84.n even 6 1 1512.2.t.a 2
252.u odd 6 1 504.2.q.a 2
252.bb even 6 1 1512.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.a 2 4.b odd 2 1
504.2.q.a 2 252.u odd 6 1
504.2.t.a yes 2 28.g odd 6 1
504.2.t.a yes 2 36.f odd 6 1
1008.2.q.b 2 1.a even 1 1 trivial
1008.2.q.b 2 63.h even 3 1 inner
1008.2.t.e 2 7.c even 3 1
1008.2.t.e 2 9.c even 3 1
1512.2.q.b 2 12.b even 2 1
1512.2.q.b 2 252.bb even 6 1
1512.2.t.a 2 36.h even 6 1
1512.2.t.a 2 84.n even 6 1
3024.2.q.d 2 3.b odd 2 1
3024.2.q.d 2 63.j odd 6 1
3024.2.t.c 2 9.d odd 6 1
3024.2.t.c 2 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 $$S_{2}^{\mathrm{new}}(1008, [\chi])$$:

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