# Properties

 Label 1568.2.a.l Level $1568$ Weight $2$ Character orbit 1568.a Self dual yes Analytic conductor $12.521$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + ( 1 - 2 \beta ) q^{5} -2 \beta q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + ( 1 - 2 \beta ) q^{5} -2 \beta q^{9} + ( 1 + \beta ) q^{11} + 2 \beta q^{13} + ( -5 + 3 \beta ) q^{15} + ( -3 + 2 \beta ) q^{17} + ( -5 - \beta ) q^{19} + ( 1 + 3 \beta ) q^{23} + ( 4 - 4 \beta ) q^{25} + ( -1 - \beta ) q^{27} -2 \beta q^{29} + ( -7 + \beta ) q^{31} + q^{33} + ( 3 + 4 \beta ) q^{37} + ( 4 - 2 \beta ) q^{39} + ( -4 - 2 \beta ) q^{41} + ( -4 - 4 \beta ) q^{43} + ( 8 - 2 \beta ) q^{45} + ( -9 - \beta ) q^{47} + ( 7 - 5 \beta ) q^{51} - q^{53} + ( -3 - \beta ) q^{55} + ( 3 - 4 \beta ) q^{57} + ( -1 - 7 \beta ) q^{59} + ( 3 + 4 \beta ) q^{61} + ( -8 + 2 \beta ) q^{65} + ( 7 - 3 \beta ) q^{67} + ( 5 - 2 \beta ) q^{69} + ( -8 - 4 \beta ) q^{71} + ( -9 - 4 \beta ) q^{73} + ( -12 + 8 \beta ) q^{75} + ( 1 - 5 \beta ) q^{79} + ( -1 + 6 \beta ) q^{81} + ( -4 + 8 \beta ) q^{83} + ( -11 + 8 \beta ) q^{85} + ( -4 + 2 \beta ) q^{87} -9 q^{89} + ( 9 - 8 \beta ) q^{93} + ( -1 + 9 \beta ) q^{95} + ( -4 + 2 \beta ) q^{97} + ( -4 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} + 2q^{11} - 10q^{15} - 6q^{17} - 10q^{19} + 2q^{23} + 8q^{25} - 2q^{27} - 14q^{31} + 2q^{33} + 6q^{37} + 8q^{39} - 8q^{41} - 8q^{43} + 16q^{45} - 18q^{47} + 14q^{51} - 2q^{53} - 6q^{55} + 6q^{57} - 2q^{59} + 6q^{61} - 16q^{65} + 14q^{67} + 10q^{69} - 16q^{71} - 18q^{73} - 24q^{75} + 2q^{79} - 2q^{81} - 8q^{83} - 22q^{85} - 8q^{87} - 18q^{89} + 18q^{93} - 2q^{95} - 8q^{97} - 8q^{99} + O(q^{100})$$

## 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 −2.41421 0 3.82843 0 0 0 2.82843 0
1.2 0 0.414214 0 −1.82843 0 0 0 −2.82843 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

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 1568.2.a.l 2
4.b odd 2 1 1568.2.a.w 2
7.b odd 2 1 1568.2.a.u 2
7.c even 3 2 224.2.i.d yes 4
7.d odd 6 2 1568.2.i.o 4
8.b even 2 1 3136.2.a.bw 2
8.d odd 2 1 3136.2.a.bd 2
21.h odd 6 2 2016.2.s.s 4
28.d even 2 1 1568.2.a.j 2
28.f even 6 2 1568.2.i.x 4
28.g odd 6 2 224.2.i.a 4
56.e even 2 1 3136.2.a.bx 2
56.h odd 2 1 3136.2.a.be 2
56.k odd 6 2 448.2.i.j 4
56.p even 6 2 448.2.i.g 4
84.n even 6 2 2016.2.s.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 28.g odd 6 2
224.2.i.d yes 4 7.c even 3 2
448.2.i.g 4 56.p even 6 2
448.2.i.j 4 56.k odd 6 2
1568.2.a.j 2 28.d even 2 1
1568.2.a.l 2 1.a even 1 1 trivial
1568.2.a.u 2 7.b odd 2 1
1568.2.a.w 2 4.b odd 2 1
1568.2.i.o 4 7.d odd 6 2
1568.2.i.x 4 28.f even 6 2
2016.2.s.q 4 84.n even 6 2
2016.2.s.s 4 21.h odd 6 2
3136.2.a.bd 2 8.d odd 2 1
3136.2.a.be 2 56.h odd 2 1
3136.2.a.bw 2 8.b even 2 1
3136.2.a.bx 2 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1568))$$:

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{5}^{2} - 2 T_{5} - 7$$ $$T_{11}^{2} - 2 T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + 2 T + T^{2}$$
$5$ $$-7 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1 - 2 T + T^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$1 + 6 T + T^{2}$$
$19$ $$23 + 10 T + T^{2}$$
$23$ $$-17 - 2 T + T^{2}$$
$29$ $$-8 + T^{2}$$
$31$ $$47 + 14 T + T^{2}$$
$37$ $$-23 - 6 T + T^{2}$$
$41$ $$8 + 8 T + T^{2}$$
$43$ $$-16 + 8 T + T^{2}$$
$47$ $$79 + 18 T + T^{2}$$
$53$ $$( 1 + T )^{2}$$
$59$ $$-97 + 2 T + T^{2}$$
$61$ $$-23 - 6 T + T^{2}$$
$67$ $$31 - 14 T + T^{2}$$
$71$ $$32 + 16 T + T^{2}$$
$73$ $$49 + 18 T + T^{2}$$
$79$ $$-49 - 2 T + T^{2}$$
$83$ $$-112 + 8 T + T^{2}$$
$89$ $$( 9 + T )^{2}$$
$97$ $$8 + 8 T + T^{2}$$