# Properties

 Label 1584.2.a.r Level $1584$ Weight $2$ Character orbit 1584.a Self dual yes Analytic conductor $12.648$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4 q^{5} + 2 q^{7} + O(q^{10})$$ $$q + 4 q^{5} + 2 q^{7} - q^{11} - 2 q^{13} - 2 q^{17} + 6 q^{19} + 4 q^{23} + 11 q^{25} + 6 q^{29} - 4 q^{31} + 8 q^{35} - 6 q^{37} + 10 q^{41} - 6 q^{43} - 8 q^{47} - 3 q^{49} - 4 q^{55} + 4 q^{59} - 6 q^{61} - 8 q^{65} - 8 q^{67} - 2 q^{73} - 2 q^{77} + 10 q^{79} + 12 q^{83} - 8 q^{85} - 4 q^{91} + 24 q^{95} + 2 q^{97} + O(q^{100})$$

## 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
 0
0 0 0 4.00000 0 2.00000 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.r 1
3.b odd 2 1 1584.2.a.b 1
4.b odd 2 1 99.2.a.c yes 1
8.b even 2 1 6336.2.a.f 1
8.d odd 2 1 6336.2.a.b 1
12.b even 2 1 99.2.a.a 1
20.d odd 2 1 2475.2.a.c 1
20.e even 4 2 2475.2.c.g 2
24.f even 2 1 6336.2.a.cl 1
24.h odd 2 1 6336.2.a.cm 1
28.d even 2 1 4851.2.a.o 1
36.f odd 6 2 891.2.e.c 2
36.h even 6 2 891.2.e.j 2
44.c even 2 1 1089.2.a.d 1
60.h even 2 1 2475.2.a.j 1
60.l odd 4 2 2475.2.c.b 2
84.h odd 2 1 4851.2.a.g 1
132.d odd 2 1 1089.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 12.b even 2 1
99.2.a.c yes 1 4.b odd 2 1
891.2.e.c 2 36.f odd 6 2
891.2.e.j 2 36.h even 6 2
1089.2.a.d 1 44.c even 2 1
1089.2.a.h 1 132.d odd 2 1
1584.2.a.b 1 3.b odd 2 1
1584.2.a.r 1 1.a even 1 1 trivial
2475.2.a.c 1 20.d odd 2 1
2475.2.a.j 1 60.h even 2 1
2475.2.c.b 2 60.l odd 4 2
2475.2.c.g 2 20.e even 4 2
4851.2.a.g 1 84.h odd 2 1
4851.2.a.o 1 28.d even 2 1
6336.2.a.b 1 8.d odd 2 1
6336.2.a.f 1 8.b even 2 1
6336.2.a.cl 1 24.f even 2 1
6336.2.a.cm 1 24.h odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$-2 + T$$
$11$ $$1 + T$$
$13$ $$2 + T$$
$17$ $$2 + T$$
$19$ $$-6 + T$$
$23$ $$-4 + T$$
$29$ $$-6 + T$$
$31$ $$4 + T$$
$37$ $$6 + T$$
$41$ $$-10 + T$$
$43$ $$6 + T$$
$47$ $$8 + T$$
$53$ $$T$$
$59$ $$-4 + T$$
$61$ $$6 + T$$
$67$ $$8 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$-10 + T$$
$83$ $$-12 + T$$
$89$ $$T$$
$97$ $$-2 + T$$