# Properties

 Label 8712.2.a.r Level 8712 Weight 2 Character orbit 8712.a Self dual yes Analytic conductor 69.566 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8712 = 2^{3} \cdot 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8712.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.5656702409$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 264) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{5} - 4q^{7} + O(q^{10})$$ $$q + 2q^{5} - 4q^{7} - 6q^{13} + 6q^{17} + 8q^{19} - q^{25} - 6q^{29} - 8q^{35} + 6q^{37} - 10q^{41} + 8q^{43} + 9q^{49} - 6q^{53} - 4q^{59} + 2q^{61} - 12q^{65} - 12q^{67} + 8q^{71} - 2q^{73} + 4q^{79} - 12q^{83} + 12q^{85} + 6q^{89} + 24q^{91} + 16q^{95} + 2q^{97} + O(q^{100})$$

## 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

## 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 8712.2.a.r 1
3.b odd 2 1 2904.2.a.i 1
11.b odd 2 1 792.2.a.f 1
12.b even 2 1 5808.2.a.f 1
33.d even 2 1 264.2.a.b 1
44.c even 2 1 1584.2.a.n 1
88.b odd 2 1 6336.2.a.v 1
88.g even 2 1 6336.2.a.o 1
132.d odd 2 1 528.2.a.b 1
165.d even 2 1 6600.2.a.a 1
165.l odd 4 2 6600.2.d.n 2
264.m even 2 1 2112.2.a.m 1
264.p odd 2 1 2112.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.b 1 33.d even 2 1
528.2.a.b 1 132.d odd 2 1
792.2.a.f 1 11.b odd 2 1
1584.2.a.n 1 44.c even 2 1
2112.2.a.m 1 264.m even 2 1
2112.2.a.y 1 264.p odd 2 1
2904.2.a.i 1 3.b odd 2 1
5808.2.a.f 1 12.b even 2 1
6336.2.a.o 1 88.g even 2 1
6336.2.a.v 1 88.b odd 2 1
6600.2.a.a 1 165.d even 2 1
6600.2.d.n 2 165.l odd 4 2
8712.2.a.r 1 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8712))$$:

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

## Hecke characteristic polynomials

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