# Properties

 Label 1584.2.a.f Level $1584$ Weight $2$ Character orbit 1584.a Self dual yes Analytic conductor $12.648$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1584.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.6483036802$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{5} + 4q^{7} + O(q^{10})$$ $$q - 2q^{5} + 4q^{7} - q^{11} - 6q^{13} - 2q^{17} - 4q^{19} + 4q^{23} - q^{25} - 6q^{29} - 8q^{35} + 6q^{37} + 6q^{41} - 4q^{43} - 12q^{47} + 9q^{49} - 2q^{53} + 2q^{55} + 12q^{59} - 14q^{61} + 12q^{65} - 4q^{67} - 12q^{71} - 6q^{73} - 4q^{77} + 4q^{79} + 4q^{83} + 4q^{85} - 10q^{89} - 24q^{91} + 8q^{95} - 14q^{97} + O(q^{100})$$

## 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
 0
0 0 0 −2.00000 0 4.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$1$$

## 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 1584.2.a.f 1
3.b odd 2 1 528.2.a.j 1
4.b odd 2 1 198.2.a.a 1
8.b even 2 1 6336.2.a.cj 1
8.d odd 2 1 6336.2.a.bw 1
12.b even 2 1 66.2.a.b 1
20.d odd 2 1 4950.2.a.bu 1
20.e even 4 2 4950.2.c.p 2
24.f even 2 1 2112.2.a.r 1
24.h odd 2 1 2112.2.a.e 1
28.d even 2 1 9702.2.a.x 1
33.d even 2 1 5808.2.a.bc 1
36.f odd 6 2 1782.2.e.v 2
36.h even 6 2 1782.2.e.e 2
44.c even 2 1 2178.2.a.g 1
60.h even 2 1 1650.2.a.k 1
60.l odd 4 2 1650.2.c.e 2
84.h odd 2 1 3234.2.a.t 1
132.d odd 2 1 726.2.a.c 1
132.n odd 10 4 726.2.e.o 4
132.o even 10 4 726.2.e.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 12.b even 2 1
198.2.a.a 1 4.b odd 2 1
528.2.a.j 1 3.b odd 2 1
726.2.a.c 1 132.d odd 2 1
726.2.e.g 4 132.o even 10 4
726.2.e.o 4 132.n odd 10 4
1584.2.a.f 1 1.a even 1 1 trivial
1650.2.a.k 1 60.h even 2 1
1650.2.c.e 2 60.l odd 4 2
1782.2.e.e 2 36.h even 6 2
1782.2.e.v 2 36.f odd 6 2
2112.2.a.e 1 24.h odd 2 1
2112.2.a.r 1 24.f even 2 1
2178.2.a.g 1 44.c even 2 1
3234.2.a.t 1 84.h odd 2 1
4950.2.a.bu 1 20.d odd 2 1
4950.2.c.p 2 20.e even 4 2
5808.2.a.bc 1 33.d even 2 1
6336.2.a.bw 1 8.d odd 2 1
6336.2.a.cj 1 8.b even 2 1
9702.2.a.x 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1584))$$:

 $$T_{5} + 2$$ $$T_{7} - 4$$ $$T_{13} + 6$$ $$T_{17} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$2 + T$$
$7$ $$-4 + T$$
$11$ $$1 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$-4 + T$$
$29$ $$6 + T$$
$31$ $$T$$
$37$ $$-6 + T$$
$41$ $$-6 + T$$
$43$ $$4 + T$$
$47$ $$12 + T$$
$53$ $$2 + T$$
$59$ $$-12 + T$$
$61$ $$14 + T$$
$67$ $$4 + T$$
$71$ $$12 + T$$
$73$ $$6 + T$$
$79$ $$-4 + T$$
$83$ $$-4 + T$$
$89$ $$10 + T$$
$97$ $$14 + T$$