# Properties

 Label 1568.2.a.t Level $1568$ Weight $2$ Character orbit 1568.a Self dual yes Analytic conductor $12.521$ Analytic rank $0$ Dimension $2$ CM no 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: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 3 q^{5} + 4 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 3 q^{5} + 4 q^{9} + \beta q^{11} + 4 q^{13} + 3 \beta q^{15} - q^{17} -3 \beta q^{19} + \beta q^{23} + 4 q^{25} + \beta q^{27} -4 q^{29} -\beta q^{31} + 7 q^{33} -5 q^{37} + 4 \beta q^{39} -8 q^{41} -4 \beta q^{43} + 12 q^{45} + \beta q^{47} -\beta q^{51} + 7 q^{53} + 3 \beta q^{55} -21 q^{57} + \beta q^{59} + 5 q^{61} + 12 q^{65} -\beta q^{67} + 7 q^{69} + 9 q^{73} + 4 \beta q^{75} + \beta q^{79} -5 q^{81} -4 \beta q^{83} -3 q^{85} -4 \beta q^{87} + 9 q^{89} -7 q^{93} -9 \beta q^{95} + 8 q^{97} + 4 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{5} + 8q^{9} + O(q^{10})$$ $$2q + 6q^{5} + 8q^{9} + 8q^{13} - 2q^{17} + 8q^{25} - 8q^{29} + 14q^{33} - 10q^{37} - 16q^{41} + 24q^{45} + 14q^{53} - 42q^{57} + 10q^{61} + 24q^{65} + 14q^{69} + 18q^{73} - 10q^{81} - 6q^{85} + 18q^{89} - 14q^{93} + 16q^{97} + O(q^{100})$$

## 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
 −2.64575 2.64575
0 −2.64575 0 3.00000 0 0 0 4.00000 0
1.2 0 2.64575 0 3.00000 0 0 0 4.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.t 2
4.b odd 2 1 inner 1568.2.a.t 2
7.b odd 2 1 1568.2.a.m 2
7.c even 3 2 1568.2.i.p 4
7.d odd 6 2 224.2.i.c 4
8.b even 2 1 3136.2.a.bg 2
8.d odd 2 1 3136.2.a.bg 2
21.g even 6 2 2016.2.s.o 4
28.d even 2 1 1568.2.a.m 2
28.f even 6 2 224.2.i.c 4
28.g odd 6 2 1568.2.i.p 4
56.e even 2 1 3136.2.a.bv 2
56.h odd 2 1 3136.2.a.bv 2
56.j odd 6 2 448.2.i.h 4
56.m even 6 2 448.2.i.h 4
84.j odd 6 2 2016.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 7.d odd 6 2
224.2.i.c 4 28.f even 6 2
448.2.i.h 4 56.j odd 6 2
448.2.i.h 4 56.m even 6 2
1568.2.a.m 2 7.b odd 2 1
1568.2.a.m 2 28.d even 2 1
1568.2.a.t 2 1.a even 1 1 trivial
1568.2.a.t 2 4.b odd 2 1 inner
1568.2.i.p 4 7.c even 3 2
1568.2.i.p 4 28.g odd 6 2
2016.2.s.o 4 21.g even 6 2
2016.2.s.o 4 84.j odd 6 2
3136.2.a.bg 2 8.b even 2 1
3136.2.a.bg 2 8.d odd 2 1
3136.2.a.bv 2 56.e even 2 1
3136.2.a.bv 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1568))$$:

 $$T_{3}^{2} - 7$$ $$T_{5} - 3$$ $$T_{11}^{2} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-7 + T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-7 + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$( 1 + T )^{2}$$
$19$ $$-63 + T^{2}$$
$23$ $$-7 + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$-7 + T^{2}$$
$37$ $$( 5 + T )^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$-112 + T^{2}$$
$47$ $$-7 + T^{2}$$
$53$ $$( -7 + T )^{2}$$
$59$ $$-7 + T^{2}$$
$61$ $$( -5 + T )^{2}$$
$67$ $$-7 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -9 + T )^{2}$$
$79$ $$-7 + T^{2}$$
$83$ $$-112 + T^{2}$$
$89$ $$( -9 + T )^{2}$$
$97$ $$( -8 + T )^{2}$$