# Properties

 Label 3024.1.cg.a Level $3024$ Weight $1$ Character orbit 3024.cg Analytic conductor $1.509$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3024.cg (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 756) Projective image $$D_{6}$$ Projective field Galois closure of 6.0.196036848.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{7} +O(q^{10})$$ $$q + \zeta_{6}^{2} q^{7} + ( -1 + \zeta_{6}^{2} ) q^{19} + \zeta_{6}^{2} q^{25} + ( 1 + \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{37} - q^{43} -\zeta_{6} q^{49} + ( -1 + \zeta_{6}^{2} ) q^{61} -2 \zeta_{6}^{2} q^{67} + ( -1 - \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{79} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{7} + O(q^{10})$$ $$2q - q^{7} - 3q^{19} - q^{25} + 3q^{31} + 2q^{37} - 2q^{43} - q^{49} - 3q^{61} + 2q^{67} - 3q^{73} + 2q^{79} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2161.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −0.500000 0.866025i 0 0 0
2593.1 0 0 0 0 0 −0.500000 + 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 $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.cg.a 2
3.b odd 2 1 CM 3024.1.cg.a 2
4.b odd 2 1 756.1.z.a 2
7.d odd 6 1 inner 3024.1.cg.a 2
12.b even 2 1 756.1.z.a 2
21.g even 6 1 inner 3024.1.cg.a 2
28.f even 6 1 756.1.z.a 2
36.f odd 6 1 2268.1.p.a 2
36.f odd 6 1 2268.1.bd.b 2
36.h even 6 1 2268.1.p.a 2
36.h even 6 1 2268.1.bd.b 2
84.j odd 6 1 756.1.z.a 2
252.n even 6 1 2268.1.bd.b 2
252.r odd 6 1 2268.1.p.a 2
252.bj even 6 1 2268.1.p.a 2
252.bn odd 6 1 2268.1.bd.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.z.a 2 4.b odd 2 1
756.1.z.a 2 12.b even 2 1
756.1.z.a 2 28.f even 6 1
756.1.z.a 2 84.j odd 6 1
2268.1.p.a 2 36.f odd 6 1
2268.1.p.a 2 36.h even 6 1
2268.1.p.a 2 252.r odd 6 1
2268.1.p.a 2 252.bj even 6 1
2268.1.bd.b 2 36.f odd 6 1
2268.1.bd.b 2 36.h even 6 1
2268.1.bd.b 2 252.n even 6 1
2268.1.bd.b 2 252.bn odd 6 1
3024.1.cg.a 2 1.a even 1 1 trivial
3024.1.cg.a 2 3.b odd 2 1 CM
3024.1.cg.a 2 7.d odd 6 1 inner
3024.1.cg.a 2 21.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}$$ acting on $$S_{1}^{\mathrm{new}}(3024, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$7$ $$1 + T + T^{2}$$
$11$ $$1 - T^{2} + T^{4}$$
$13$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$17$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$19$ $$( 1 + T )^{2}( 1 + T + T^{2} )$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$( 1 + T^{2} )^{2}$$
$31$ $$( 1 - T )^{2}( 1 - T + T^{2} )$$
$37$ $$( 1 - T + T^{2} )^{2}$$
$41$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$43$ $$( 1 + T + T^{2} )^{2}$$
$47$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$53$ $$1 - T^{2} + T^{4}$$
$59$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$61$ $$( 1 + T )^{2}( 1 + T + T^{2} )$$
$67$ $$( 1 - T + T^{2} )^{2}$$
$71$ $$( 1 + T^{2} )^{2}$$
$73$ $$( 1 + T )^{2}( 1 + T + T^{2} )$$
$79$ $$( 1 - T + T^{2} )^{2}$$
$83$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$89$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$97$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$