# Properties

 Label 3024.2.cz.a Level $3024$ Weight $2$ Character orbit 3024.cz Analytic conductor $24.147$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1008) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 3 \zeta_{6} ) q^{7} + ( 4 - 2 \zeta_{6} ) q^{11} + ( -2 + \zeta_{6} ) q^{13} + ( -1 - \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} + ( -2 - 2 \zeta_{6} ) q^{23} -5 \zeta_{6} q^{25} + ( 3 - 3 \zeta_{6} ) q^{29} + q^{31} -7 \zeta_{6} q^{37} + ( 2 - \zeta_{6} ) q^{41} + ( 1 + \zeta_{6} ) q^{43} + 9 q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -9 + 9 \zeta_{6} ) q^{53} + 15 q^{59} + ( 1 - 2 \zeta_{6} ) q^{61} + ( 9 - 18 \zeta_{6} ) q^{67} + ( -6 + 12 \zeta_{6} ) q^{71} + ( 1 + \zeta_{6} ) q^{73} + ( -2 + 10 \zeta_{6} ) q^{77} + ( 1 - 2 \zeta_{6} ) q^{79} + ( 9 - 9 \zeta_{6} ) q^{83} + ( -2 + \zeta_{6} ) q^{89} + ( 1 - 5 \zeta_{6} ) q^{91} + ( -1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{7} + O(q^{10})$$ $$2 q - q^{7} + 6 q^{11} - 3 q^{13} - 3 q^{17} - 5 q^{19} - 6 q^{23} - 5 q^{25} + 3 q^{29} + 2 q^{31} - 7 q^{37} + 3 q^{41} + 3 q^{43} + 18 q^{47} - 13 q^{49} - 9 q^{53} + 30 q^{59} + 3 q^{73} + 6 q^{77} + 9 q^{83} - 3 q^{89} - 3 q^{91} - 3 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1279.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −0.500000 2.59808i 0 0 0
2719.1 0 0 0 0 0 −0.500000 + 2.59808i 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
252.bj even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.cz.a 2
3.b odd 2 1 1008.2.cz.a yes 2
4.b odd 2 1 3024.2.cz.b 2
7.d odd 6 1 3024.2.bf.c 2
9.c even 3 1 3024.2.bf.b 2
9.d odd 6 1 1008.2.bf.b 2
12.b even 2 1 1008.2.cz.d yes 2
21.g even 6 1 1008.2.bf.c yes 2
28.f even 6 1 3024.2.bf.b 2
36.f odd 6 1 3024.2.bf.c 2
36.h even 6 1 1008.2.bf.c yes 2
63.i even 6 1 1008.2.cz.d yes 2
63.t odd 6 1 3024.2.cz.b 2
84.j odd 6 1 1008.2.bf.b 2
252.r odd 6 1 1008.2.cz.a yes 2
252.bj even 6 1 inner 3024.2.cz.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.bf.b 2 9.d odd 6 1
1008.2.bf.b 2 84.j odd 6 1
1008.2.bf.c yes 2 21.g even 6 1
1008.2.bf.c yes 2 36.h even 6 1
1008.2.cz.a yes 2 3.b odd 2 1
1008.2.cz.a yes 2 252.r odd 6 1
1008.2.cz.d yes 2 12.b even 2 1
1008.2.cz.d yes 2 63.i even 6 1
3024.2.bf.b 2 9.c even 3 1
3024.2.bf.b 2 28.f even 6 1
3024.2.bf.c 2 7.d odd 6 1
3024.2.bf.c 2 36.f odd 6 1
3024.2.cz.a 2 1.a even 1 1 trivial
3024.2.cz.a 2 252.bj even 6 1 inner
3024.2.cz.b 2 4.b odd 2 1
3024.2.cz.b 2 63.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} - 6 T_{11} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$12 - 6 T + T^{2}$$
$13$ $$3 + 3 T + T^{2}$$
$17$ $$3 + 3 T + T^{2}$$
$19$ $$25 + 5 T + T^{2}$$
$23$ $$12 + 6 T + T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$3 - 3 T + T^{2}$$
$43$ $$3 - 3 T + T^{2}$$
$47$ $$( -9 + T )^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$( -15 + T )^{2}$$
$61$ $$3 + T^{2}$$
$67$ $$243 + T^{2}$$
$71$ $$108 + T^{2}$$
$73$ $$3 - 3 T + T^{2}$$
$79$ $$3 + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$3 + 3 T + T^{2}$$
$97$ $$3 + 3 T + T^{2}$$