# Properties

 Label 3024.2.b.d Level $3024$ Weight $2$ Character orbit 3024.b 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.b (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + ( -2 + 4 \zeta_{6} ) q^{11} + ( -2 + 4 \zeta_{6} ) q^{13} + ( 1 - 2 \zeta_{6} ) q^{17} + 2 q^{19} + 2 q^{25} + 6 q^{29} -2 q^{31} + ( -5 + 4 \zeta_{6} ) q^{35} + q^{37} + ( 3 - 6 \zeta_{6} ) q^{41} + ( -1 + 2 \zeta_{6} ) q^{43} + 3 q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + 12 q^{53} + 6 q^{55} -3 q^{59} + ( 6 - 12 \zeta_{6} ) q^{61} + 6 q^{65} + ( -6 + 12 \zeta_{6} ) q^{67} + ( 8 - 16 \zeta_{6} ) q^{71} + ( 10 - 8 \zeta_{6} ) q^{77} + ( 1 - 2 \zeta_{6} ) q^{79} + 9 q^{83} -3 q^{85} + ( 4 - 8 \zeta_{6} ) q^{89} + ( 10 - 8 \zeta_{6} ) q^{91} + ( 2 - 4 \zeta_{6} ) q^{95} + ( -2 + 4 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{7} + O(q^{10})$$ $$2q - 4q^{7} + 4q^{19} + 4q^{25} + 12q^{29} - 4q^{31} - 6q^{35} + 2q^{37} + 6q^{47} + 2q^{49} + 24q^{53} + 12q^{55} - 6q^{59} + 12q^{65} + 12q^{77} + 18q^{83} - 6q^{85} + 12q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.73205i 0 −2.00000 1.73205i 0 0 0
1567.2 0 0 0 1.73205i 0 −2.00000 + 1.73205i 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
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.b.d yes 2
3.b odd 2 1 3024.2.b.b 2
4.b odd 2 1 3024.2.b.o yes 2
7.b odd 2 1 3024.2.b.o yes 2
12.b even 2 1 3024.2.b.m yes 2
21.c even 2 1 3024.2.b.m yes 2
28.d even 2 1 inner 3024.2.b.d yes 2
84.h odd 2 1 3024.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.2.b.b 2 3.b odd 2 1
3024.2.b.b 2 84.h odd 2 1
3024.2.b.d yes 2 1.a even 1 1 trivial
3024.2.b.d yes 2 28.d even 2 1 inner
3024.2.b.m yes 2 12.b even 2 1
3024.2.b.m yes 2 21.c even 2 1
3024.2.b.o yes 2 4.b odd 2 1
3024.2.b.o yes 2 7.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 3$$ $$T_{11}^{2} + 12$$ $$T_{13}^{2} + 12$$ $$T_{17}^{2} + 3$$ $$T_{19} - 2$$ $$T_{29} - 6$$ $$T_{47} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 7 T^{2} + 25 T^{4}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 - 10 T^{2} + 121 T^{4}$$
$13$ $$1 - 14 T^{2} + 169 T^{4}$$
$17$ $$1 - 31 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 2 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - T + 37 T^{2} )^{2}$$
$41$ $$1 - 55 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 13 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$( 1 - 3 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$( 1 + 3 T + 59 T^{2} )^{2}$$
$61$ $$1 - 14 T^{2} + 3721 T^{4}$$
$67$ $$1 - 26 T^{2} + 4489 T^{4}$$
$71$ $$1 + 50 T^{2} + 5041 T^{4}$$
$73$ $$( 1 - 73 T^{2} )^{2}$$
$79$ $$1 - 155 T^{2} + 6241 T^{4}$$
$83$ $$( 1 - 9 T + 83 T^{2} )^{2}$$
$89$ $$1 - 130 T^{2} + 7921 T^{4}$$
$97$ $$1 - 182 T^{2} + 9409 T^{4}$$