# Properties

 Label 1568.2.a.f Level $1568$ Weight $2$ Character orbit 1568.a Self dual yes Analytic conductor $12.521$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1568.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5205430369$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{5} - 3q^{9} + O(q^{10})$$ $$q + 4q^{5} - 3q^{9} - 4q^{13} + 8q^{17} + 11q^{25} + 10q^{29} + 2q^{37} + 8q^{41} - 12q^{45} + 14q^{53} - 12q^{61} - 16q^{65} - 16q^{73} + 9q^{81} + 32q^{85} - 16q^{89} - 8q^{97} + O(q^{100})$$

## 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 0 0 −3.00000 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
$$2$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.f yes 1
4.b odd 2 1 CM 1568.2.a.f yes 1
7.b odd 2 1 1568.2.a.d 1
7.c even 3 2 1568.2.i.e 2
7.d odd 6 2 1568.2.i.h 2
8.b even 2 1 3136.2.a.l 1
8.d odd 2 1 3136.2.a.l 1
28.d even 2 1 1568.2.a.d 1
28.f even 6 2 1568.2.i.h 2
28.g odd 6 2 1568.2.i.e 2
56.e even 2 1 3136.2.a.r 1
56.h odd 2 1 3136.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.d 1 7.b odd 2 1
1568.2.a.d 1 28.d even 2 1
1568.2.a.f yes 1 1.a even 1 1 trivial
1568.2.a.f yes 1 4.b odd 2 1 CM
1568.2.i.e 2 7.c even 3 2
1568.2.i.e 2 28.g odd 6 2
1568.2.i.h 2 7.d odd 6 2
1568.2.i.h 2 28.f even 6 2
3136.2.a.l 1 8.b even 2 1
3136.2.a.l 1 8.d odd 2 1
3136.2.a.r 1 56.e even 2 1
3136.2.a.r 1 56.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1568))$$:

 $$T_{3}$$ $$T_{5} - 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$-8 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-10 + T$$
$31$ $$T$$
$37$ $$-2 + T$$
$41$ $$-8 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$-14 + T$$
$59$ $$T$$
$61$ $$12 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$16 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$16 + T$$
$97$ $$8 + T$$