# Properties

 Label 1008.2.q.f Level $1008$ Weight $2$ Character orbit 1008.q 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.q (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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 2 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{5} + ( 2 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + 4 \zeta_{6} q^{11} -3 \zeta_{6} q^{13} + ( -2 + 4 \zeta_{6} ) q^{15} + ( -7 + 7 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + ( 5 - \zeta_{6} ) q^{21} + ( 4 - 4 \zeta_{6} ) q^{23} + \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 - \zeta_{6} ) q^{29} + 3 q^{31} + ( 4 + 4 \zeta_{6} ) q^{33} + ( -6 + 4 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + ( -3 - 3 \zeta_{6} ) q^{39} + 9 \zeta_{6} q^{41} + ( 5 - 5 \zeta_{6} ) q^{43} + 6 \zeta_{6} q^{45} -3 q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -7 + 14 \zeta_{6} ) q^{51} + ( -3 + 3 \zeta_{6} ) q^{53} -8 q^{55} + ( 5 + 5 \zeta_{6} ) q^{57} + 7 q^{59} + 3 q^{61} + ( 9 - 6 \zeta_{6} ) q^{63} + 6 q^{65} -13 q^{67} + ( 4 - 8 \zeta_{6} ) q^{69} + 8 q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} + ( 1 + \zeta_{6} ) q^{75} + ( -4 + 12 \zeta_{6} ) q^{77} + 9 q^{79} -9 \zeta_{6} q^{81} + ( 1 - \zeta_{6} ) q^{83} -14 \zeta_{6} q^{85} + ( 1 - 2 \zeta_{6} ) q^{87} -15 \zeta_{6} q^{89} + ( 3 - 9 \zeta_{6} ) q^{91} + ( 6 - 3 \zeta_{6} ) q^{93} -10 q^{95} + ( 17 - 17 \zeta_{6} ) q^{97} + 12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 2q^{5} + 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 2q^{5} + 5q^{7} + 3q^{9} + 4q^{11} - 3q^{13} - 7q^{17} + 5q^{19} + 9q^{21} + 4q^{23} + q^{25} + q^{29} + 6q^{31} + 12q^{33} - 8q^{35} - 11q^{37} - 9q^{39} + 9q^{41} + 5q^{43} + 6q^{45} - 6q^{47} + 11q^{49} - 3q^{53} - 16q^{55} + 15q^{57} + 14q^{59} + 6q^{61} + 12q^{63} + 12q^{65} - 26q^{67} + 16q^{71} - 7q^{73} + 3q^{75} + 4q^{77} + 18q^{79} - 9q^{81} + q^{83} - 14q^{85} - 15q^{89} - 3q^{91} + 9q^{93} - 20q^{95} + 17q^{97} + 24q^{99} + O(q^{100})$$

## 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 1.50000 0.866025i 0 −1.00000 + 1.73205i 0 2.50000 + 0.866025i 0 1.50000 2.59808i 0
625.1 0 1.50000 + 0.866025i 0 −1.00000 1.73205i 0 2.50000 0.866025i 0 1.50000 + 2.59808i 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.f 2
3.b odd 2 1 3024.2.q.e 2
4.b odd 2 1 252.2.i.a 2
7.c even 3 1 1008.2.t.b 2
9.c even 3 1 1008.2.t.b 2
9.d odd 6 1 3024.2.t.b 2
12.b even 2 1 756.2.i.a 2
21.h odd 6 1 3024.2.t.b 2
28.d even 2 1 1764.2.i.b 2
28.f even 6 1 1764.2.j.a 2
28.f even 6 1 1764.2.l.b 2
28.g odd 6 1 252.2.l.a yes 2
28.g odd 6 1 1764.2.j.c 2
36.f odd 6 1 252.2.l.a yes 2
36.f odd 6 1 2268.2.k.a 2
36.h even 6 1 756.2.l.a 2
36.h even 6 1 2268.2.k.b 2
63.h even 3 1 inner 1008.2.q.f 2
63.j odd 6 1 3024.2.q.e 2
84.h odd 2 1 5292.2.i.b 2
84.j odd 6 1 5292.2.j.b 2
84.j odd 6 1 5292.2.l.b 2
84.n even 6 1 756.2.l.a 2
84.n even 6 1 5292.2.j.c 2
252.n even 6 1 1764.2.j.a 2
252.o even 6 1 2268.2.k.b 2
252.o even 6 1 5292.2.j.c 2
252.r odd 6 1 5292.2.i.b 2
252.s odd 6 1 5292.2.l.b 2
252.u odd 6 1 252.2.i.a 2
252.bb even 6 1 756.2.i.a 2
252.bi even 6 1 1764.2.l.b 2
252.bj even 6 1 1764.2.i.b 2
252.bl odd 6 1 1764.2.j.c 2
252.bl odd 6 1 2268.2.k.a 2
252.bn odd 6 1 5292.2.j.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 4.b odd 2 1
252.2.i.a 2 252.u odd 6 1
252.2.l.a yes 2 28.g odd 6 1
252.2.l.a yes 2 36.f odd 6 1
756.2.i.a 2 12.b even 2 1
756.2.i.a 2 252.bb even 6 1
756.2.l.a 2 36.h even 6 1
756.2.l.a 2 84.n even 6 1
1008.2.q.f 2 1.a even 1 1 trivial
1008.2.q.f 2 63.h even 3 1 inner
1008.2.t.b 2 7.c even 3 1
1008.2.t.b 2 9.c even 3 1
1764.2.i.b 2 28.d even 2 1
1764.2.i.b 2 252.bj even 6 1
1764.2.j.a 2 28.f even 6 1
1764.2.j.a 2 252.n even 6 1
1764.2.j.c 2 28.g odd 6 1
1764.2.j.c 2 252.bl odd 6 1
1764.2.l.b 2 28.f even 6 1
1764.2.l.b 2 252.bi even 6 1
2268.2.k.a 2 36.f odd 6 1
2268.2.k.a 2 252.bl odd 6 1
2268.2.k.b 2 36.h even 6 1
2268.2.k.b 2 252.o even 6 1
3024.2.q.e 2 3.b odd 2 1
3024.2.q.e 2 63.j odd 6 1
3024.2.t.b 2 9.d odd 6 1
3024.2.t.b 2 21.h odd 6 1
5292.2.i.b 2 84.h odd 2 1
5292.2.i.b 2 252.r odd 6 1
5292.2.j.b 2 84.j odd 6 1
5292.2.j.b 2 252.bn odd 6 1
5292.2.j.c 2 84.n even 6 1
5292.2.j.c 2 252.o even 6 1
5292.2.l.b 2 84.j odd 6 1
5292.2.l.b 2 252.s odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} - 4 T_{11} + 16$$