# Properties

 Label 1008.2.r.a Level $1008$ Weight $2$ Character orbit 1008.r 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.r (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 126) 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i 0 −1.00000 + 1.73205i 0 −0.500000 0.866025i 0 1.50000 2.59808i 0
673.1 0 −1.50000 0.866025i 0 −1.00000 1.73205i 0 −0.500000 + 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
9.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.r.a 2
3.b odd 2 1 3024.2.r.c 2
4.b odd 2 1 126.2.f.b 2
9.c even 3 1 inner 1008.2.r.a 2
9.c even 3 1 9072.2.a.t 1
9.d odd 6 1 3024.2.r.c 2
9.d odd 6 1 9072.2.a.f 1
12.b even 2 1 378.2.f.b 2
28.d even 2 1 882.2.f.f 2
28.f even 6 1 882.2.e.e 2
28.f even 6 1 882.2.h.g 2
28.g odd 6 1 882.2.e.a 2
28.g odd 6 1 882.2.h.h 2
36.f odd 6 1 126.2.f.b 2
36.f odd 6 1 1134.2.a.c 1
36.h even 6 1 378.2.f.b 2
36.h even 6 1 1134.2.a.f 1
84.h odd 2 1 2646.2.f.b 2
84.j odd 6 1 2646.2.e.h 2
84.j odd 6 1 2646.2.h.c 2
84.n even 6 1 2646.2.e.i 2
84.n even 6 1 2646.2.h.b 2
252.n even 6 1 882.2.e.e 2
252.o even 6 1 2646.2.e.i 2
252.r odd 6 1 2646.2.h.c 2
252.s odd 6 1 2646.2.f.b 2
252.s odd 6 1 7938.2.a.bb 1
252.u odd 6 1 882.2.h.h 2
252.bb even 6 1 2646.2.h.b 2
252.bi even 6 1 882.2.f.f 2
252.bi even 6 1 7938.2.a.e 1
252.bj even 6 1 882.2.h.g 2
252.bl odd 6 1 882.2.e.a 2
252.bn odd 6 1 2646.2.e.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 4.b odd 2 1
126.2.f.b 2 36.f odd 6 1
378.2.f.b 2 12.b even 2 1
378.2.f.b 2 36.h even 6 1
882.2.e.a 2 28.g odd 6 1
882.2.e.a 2 252.bl odd 6 1
882.2.e.e 2 28.f even 6 1
882.2.e.e 2 252.n even 6 1
882.2.f.f 2 28.d even 2 1
882.2.f.f 2 252.bi even 6 1
882.2.h.g 2 28.f even 6 1
882.2.h.g 2 252.bj even 6 1
882.2.h.h 2 28.g odd 6 1
882.2.h.h 2 252.u odd 6 1
1008.2.r.a 2 1.a even 1 1 trivial
1008.2.r.a 2 9.c even 3 1 inner
1134.2.a.c 1 36.f odd 6 1
1134.2.a.f 1 36.h even 6 1
2646.2.e.h 2 84.j odd 6 1
2646.2.e.h 2 252.bn odd 6 1
2646.2.e.i 2 84.n even 6 1
2646.2.e.i 2 252.o even 6 1
2646.2.f.b 2 84.h odd 2 1
2646.2.f.b 2 252.s odd 6 1
2646.2.h.b 2 84.n even 6 1
2646.2.h.b 2 252.bb even 6 1
2646.2.h.c 2 84.j odd 6 1
2646.2.h.c 2 252.r odd 6 1
3024.2.r.c 2 3.b odd 2 1
3024.2.r.c 2 9.d odd 6 1
7938.2.a.e 1 252.bi even 6 1
7938.2.a.bb 1 252.s odd 6 1
9072.2.a.f 1 9.d odd 6 1
9072.2.a.t 1 9.c even 3 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} - T_{11} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4}$$
$13$ $$1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 5 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}$$
$23$ $$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$
$43$ $$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - T + 73 T^{2} )^{2}$$
$79$ $$1 + 6 T - 43 T^{2} + 474 T^{3} + 6241 T^{4}$$
$83$ $$1 - 16 T + 173 T^{2} - 1328 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$