# Properties

 Label 3024.2.a.bk Level $3024$ Weight $2$ Character orbit 3024.a Self dual yes Analytic conductor $24.147$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1512) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + q^{7} +O(q^{10})$$ $$q + \beta q^{5} + q^{7} + ( 2 + \beta ) q^{11} -2 \beta q^{13} + q^{19} + ( 4 + \beta ) q^{23} + 2 q^{25} -6 q^{29} + ( 3 + 2 \beta ) q^{31} + \beta q^{35} + ( 3 + 2 \beta ) q^{37} + ( -6 + \beta ) q^{41} + ( 2 - 2 \beta ) q^{43} + ( 6 - 2 \beta ) q^{47} + q^{49} + ( 7 + 2 \beta ) q^{55} + ( 2 + 2 \beta ) q^{59} + ( 4 + 4 \beta ) q^{61} -14 q^{65} + ( -4 - 2 \beta ) q^{67} -5 \beta q^{71} + ( 6 + 2 \beta ) q^{73} + ( 2 + \beta ) q^{77} + ( 6 - 2 \beta ) q^{79} + 10 q^{83} + ( 6 - 3 \beta ) q^{89} -2 \beta q^{91} + \beta q^{95} + ( 8 - 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} + O(q^{10})$$ $$2 q + 2 q^{7} + 4 q^{11} + 2 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{29} + 6 q^{31} + 6 q^{37} - 12 q^{41} + 4 q^{43} + 12 q^{47} + 2 q^{49} + 14 q^{55} + 4 q^{59} + 8 q^{61} - 28 q^{65} - 8 q^{67} + 12 q^{73} + 4 q^{77} + 12 q^{79} + 20 q^{83} + 12 q^{89} + 16 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.64575 2.64575
0 0 0 −2.64575 0 1.00000 0 0 0
1.2 0 0 0 2.64575 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.a.bk 2
3.b odd 2 1 3024.2.a.bh 2
4.b odd 2 1 1512.2.a.o 2
12.b even 2 1 1512.2.a.p yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.a.o 2 4.b odd 2 1
1512.2.a.p yes 2 12.b even 2 1
3024.2.a.bh 2 3.b odd 2 1
3024.2.a.bk 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3024))$$:

 $$T_{5}^{2} - 7$$ $$T_{11}^{2} - 4 T_{11} - 3$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-7 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-3 - 4 T + T^{2}$$
$13$ $$-28 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$9 - 8 T + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$-19 - 6 T + T^{2}$$
$37$ $$-19 - 6 T + T^{2}$$
$41$ $$29 + 12 T + T^{2}$$
$43$ $$-24 - 4 T + T^{2}$$
$47$ $$8 - 12 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$-24 - 4 T + T^{2}$$
$61$ $$-96 - 8 T + T^{2}$$
$67$ $$-12 + 8 T + T^{2}$$
$71$ $$-175 + T^{2}$$
$73$ $$8 - 12 T + T^{2}$$
$79$ $$8 - 12 T + T^{2}$$
$83$ $$( -10 + T )^{2}$$
$89$ $$-27 - 12 T + T^{2}$$
$97$ $$-48 - 16 T + T^{2}$$