# Properties

 Label 1584.2.a.s 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 66) 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 $$q + 4 q^{5} + 2 q^{7} + q^{11} + 4 q^{13} + 2 q^{17} - 6 q^{23} + 11 q^{25} - 10 q^{29} + 8 q^{31} + 8 q^{35} - 2 q^{37} - 2 q^{41} - 4 q^{43} - 2 q^{47} - 3 q^{49} - 4 q^{53} + 4 q^{55} - 8 q^{61} + 16 q^{65} + 12 q^{67} + 2 q^{71} - 6 q^{73} + 2 q^{77} - 10 q^{79} + 4 q^{83} + 8 q^{85} - 10 q^{89} + 8 q^{91} - 2 q^{97}+O(q^{100})$$ q + 4 * q^5 + 2 * q^7 + q^11 + 4 * q^13 + 2 * q^17 - 6 * q^23 + 11 * q^25 - 10 * q^29 + 8 * q^31 + 8 * q^35 - 2 * q^37 - 2 * q^41 - 4 * q^43 - 2 * q^47 - 3 * q^49 - 4 * q^53 + 4 * q^55 - 8 * q^61 + 16 * q^65 + 12 * q^67 + 2 * q^71 - 6 * q^73 + 2 * q^77 - 10 * q^79 + 4 * q^83 + 8 * q^85 - 10 * q^89 + 8 * q^91 - 2 * q^97

## 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.s 1
3.b odd 2 1 528.2.a.a 1
4.b odd 2 1 198.2.a.c 1
8.b even 2 1 6336.2.a.d 1
8.d odd 2 1 6336.2.a.c 1
12.b even 2 1 66.2.a.c 1
20.d odd 2 1 4950.2.a.bo 1
20.e even 4 2 4950.2.c.d 2
24.f even 2 1 2112.2.a.n 1
24.h odd 2 1 2112.2.a.bd 1
28.d even 2 1 9702.2.a.a 1
33.d even 2 1 5808.2.a.b 1
36.f odd 6 2 1782.2.e.n 2
36.h even 6 2 1782.2.e.l 2
44.c even 2 1 2178.2.a.m 1
60.h even 2 1 1650.2.a.c 1
60.l odd 4 2 1650.2.c.m 2
84.h odd 2 1 3234.2.a.s 1
132.d odd 2 1 726.2.a.d 1
132.n odd 10 4 726.2.e.m 4
132.o even 10 4 726.2.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.c 1 12.b even 2 1
198.2.a.c 1 4.b odd 2 1
528.2.a.a 1 3.b odd 2 1
726.2.a.d 1 132.d odd 2 1
726.2.e.e 4 132.o even 10 4
726.2.e.m 4 132.n odd 10 4
1584.2.a.s 1 1.a even 1 1 trivial
1650.2.a.c 1 60.h even 2 1
1650.2.c.m 2 60.l odd 4 2
1782.2.e.l 2 36.h even 6 2
1782.2.e.n 2 36.f odd 6 2
2112.2.a.n 1 24.f even 2 1
2112.2.a.bd 1 24.h odd 2 1
2178.2.a.m 1 44.c even 2 1
3234.2.a.s 1 84.h odd 2 1
4950.2.a.bo 1 20.d odd 2 1
4950.2.c.d 2 20.e even 4 2
5808.2.a.b 1 33.d even 2 1
6336.2.a.c 1 8.d odd 2 1
6336.2.a.d 1 8.b even 2 1
9702.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1584))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 2$$ T7 - 2 $$T_{13} - 4$$ T13 - 4 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

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