# Properties

 Label 1008.2.cs.j Level $1008$ Weight $2$ Character orbit 1008.cs Analytic conductor $8.049$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.cs (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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 2 + \zeta_{6} ) q^{7} + ( -3 + 6 \zeta_{6} ) q^{13} + \zeta_{6} q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( 11 - 11 \zeta_{6} ) q^{31} + 11 \zeta_{6} q^{37} + ( -1 + 2 \zeta_{6} ) q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -8 + 4 \zeta_{6} ) q^{61} + ( 9 + 9 \zeta_{6} ) q^{67} + ( 1 + \zeta_{6} ) q^{73} + ( 6 - 3 \zeta_{6} ) q^{79} + ( -12 + 15 \zeta_{6} ) q^{91} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 5q^{7} + O(q^{10})$$ $$2q + 5q^{7} + q^{19} - 5q^{25} + 11q^{31} + 11q^{37} + 11q^{49} - 12q^{61} + 27q^{67} + 3q^{73} + 9q^{79} - 9q^{91} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 2.50000 0.866025i 0 0 0
703.1 0 0 0 0 0 2.50000 + 0.866025i 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
28.f even 6 1 inner
84.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.cs.j yes 2
3.b odd 2 1 CM 1008.2.cs.j yes 2
4.b odd 2 1 1008.2.cs.g 2
7.c even 3 1 7056.2.b.g 2
7.d odd 6 1 1008.2.cs.g 2
7.d odd 6 1 7056.2.b.h 2
12.b even 2 1 1008.2.cs.g 2
21.g even 6 1 1008.2.cs.g 2
21.g even 6 1 7056.2.b.h 2
21.h odd 6 1 7056.2.b.g 2
28.f even 6 1 inner 1008.2.cs.j yes 2
28.f even 6 1 7056.2.b.g 2
28.g odd 6 1 7056.2.b.h 2
84.j odd 6 1 inner 1008.2.cs.j yes 2
84.j odd 6 1 7056.2.b.g 2
84.n even 6 1 7056.2.b.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cs.g 2 4.b odd 2 1
1008.2.cs.g 2 7.d odd 6 1
1008.2.cs.g 2 12.b even 2 1
1008.2.cs.g 2 21.g even 6 1
1008.2.cs.j yes 2 1.a even 1 1 trivial
1008.2.cs.j yes 2 3.b odd 2 1 CM
1008.2.cs.j yes 2 28.f even 6 1 inner
1008.2.cs.j yes 2 84.j odd 6 1 inner
7056.2.b.g 2 7.c even 3 1
7056.2.b.g 2 21.h odd 6 1
7056.2.b.g 2 28.f even 6 1
7056.2.b.g 2 84.j odd 6 1
7056.2.b.h 2 7.d odd 6 1
7056.2.b.h 2 21.g even 6 1
7056.2.b.h 2 28.g odd 6 1
7056.2.b.h 2 84.n even 6 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}$$ $$T_{13}^{2} + 27$$ $$T_{17}$$ $$T_{19}^{2} - T_{19} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 5 T^{2} + 25 T^{4}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2} + 121 T^{4}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$1 + 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 + 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} )$$
$37$ $$( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} )$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$( 1 - 16 T + 67 T^{2} )( 1 - 11 T + 67 T^{2} )$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$
$79$ $$( 1 - 13 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$1 + 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$
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