# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta q^{5} -3 q^{9} +O(q^{10})$$ $$q + 3 \beta q^{5} -3 q^{9} + 5 \beta q^{13} -3 \beta q^{17} + 13 q^{25} + 4 q^{29} + 12 q^{37} -\beta q^{41} -9 \beta q^{45} -14 q^{53} + \beta q^{61} + 30 q^{65} -5 \beta q^{73} + 9 q^{81} -18 q^{85} + 13 \beta q^{89} -13 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{9} + O(q^{10})$$ $$2q - 6q^{9} + 26q^{25} + 8q^{29} + 24q^{37} - 28q^{53} + 60q^{65} + 18q^{81} - 36q^{85} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 0 0 −4.24264 0 0 0 −3.00000 0
1.2 0 0 0 4.24264 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 $$\Q(\sqrt{-1})$$
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

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

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

 $$T_{3}$$ $$T_{5}^{2} - 18$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-18 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$-50 + T^{2}$$
$17$ $$-18 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( -12 + T )^{2}$$
$41$ $$-2 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$( 14 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$-2 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-50 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$-338 + T^{2}$$
$97$ $$-338 + T^{2}$$