# Properties

 Label 3024.1.o.a Level $3024$ Weight $1$ Character orbit 3024.o Self dual yes Analytic conductor $1.509$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -84 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3024.o (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.2.16003008.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} - q^{7} +O(q^{10})$$ $$q -\beta q^{5} - q^{7} -\beta q^{11} + q^{19} + \beta q^{23} + 2 q^{25} - q^{31} + \beta q^{35} + q^{37} + \beta q^{41} + q^{49} + 3 q^{55} -\beta q^{71} + \beta q^{77} + \beta q^{89} -\beta q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{7} + O(q^{10})$$ $$2q - 2q^{7} + 2q^{19} + 4q^{25} - 2q^{31} + 2q^{37} + 2q^{49} + 6q^{55} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3023.1
 1.73205 −1.73205
0 0 0 −1.73205 0 −1.00000 0 0 0
3023.2 0 0 0 1.73205 0 −1.00000 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
84.h odd 2 1 CM by $$\Q(\sqrt{-21})$$
3.b odd 2 1 inner
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 3024.1.o.a 2
3.b odd 2 1 inner 3024.1.o.a 2
4.b odd 2 1 3024.1.o.d yes 2
7.b odd 2 1 3024.1.o.d yes 2
12.b even 2 1 3024.1.o.d yes 2
21.c even 2 1 3024.1.o.d yes 2
28.d even 2 1 inner 3024.1.o.a 2
84.h odd 2 1 CM 3024.1.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.1.o.a 2 1.a even 1 1 trivial
3024.1.o.a 2 3.b odd 2 1 inner
3024.1.o.a 2 28.d even 2 1 inner
3024.1.o.a 2 84.h odd 2 1 CM
3024.1.o.d yes 2 4.b odd 2 1
3024.1.o.d yes 2 7.b odd 2 1
3024.1.o.d yes 2 12.b even 2 1
3024.1.o.d 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 $$S_{1}^{\mathrm{new}}(3024, [\chi])$$:

 $$T_{5}^{2} - 3$$ $$T_{19} - 1$$

## Hecke characteristic polynomials

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