# Properties

 Label 3024.2.b.f Level 3024 Weight 2 Character orbit 3024.b Analytic conductor 24.147 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 3 \zeta_{6} ) q^{7} + ( 3 - 6 \zeta_{6} ) q^{13} -7 q^{19} + 5 q^{25} + 4 q^{31} -11 q^{37} + ( 6 - 12 \zeta_{6} ) q^{43} + ( -5 - 3 \zeta_{6} ) q^{49} + ( 9 - 18 \zeta_{6} ) q^{61} + ( -9 + 18 \zeta_{6} ) q^{67} + ( 9 - 18 \zeta_{6} ) q^{73} + ( 3 - 6 \zeta_{6} ) q^{79} + ( 12 + 3 \zeta_{6} ) q^{91} + ( 3 - 6 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{7} + O(q^{10})$$ $$2q - q^{7} - 14q^{19} + 10q^{25} + 8q^{31} - 22q^{37} - 13q^{49} + 27q^{91} + O(q^{100})$$

## 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 0 0 −0.500000 2.59808i 0 0 0
1567.2 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
28.d even 2 1 inner
84.h odd 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.f 2
3.b odd 2 1 CM 3024.2.b.f 2
4.b odd 2 1 3024.2.b.k yes 2
7.b odd 2 1 3024.2.b.k yes 2
12.b even 2 1 3024.2.b.k yes 2
21.c even 2 1 3024.2.b.k yes 2
28.d even 2 1 inner 3024.2.b.f 2
84.h odd 2 1 inner 3024.2.b.f 2

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}$$ $$T_{13}^{2} + 27$$ $$T_{17}$$ $$T_{19} + 7$$ $$T_{29}$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 5 T^{2} )^{2}$$
$7$ $$1 + T + 7 T^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$( 1 - 17 T^{2} )^{2}$$
$19$ $$( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 11 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - T + 61 T^{2} )( 1 + T + 61 T^{2} )$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} )$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$( 1 - 7 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} )$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} )$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 - 89 T^{2} )^{2}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$