# Properties

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

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

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$1 + 2 T + 3 T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 2 T + 21 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 + 18 T^{2} + 169 T^{4}$$
$17$ $$1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 10 T + 61 T^{2} + 190 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 29 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 + 50 T^{2} + 841 T^{4}$$
$31$ $$1 + 14 T + 109 T^{2} + 434 T^{3} + 961 T^{4}$$
$37$ $$1 - 6 T + 51 T^{2} - 222 T^{3} + 1369 T^{4}$$
$41$ $$1 - 8 T + 90 T^{2} - 328 T^{3} + 1681 T^{4}$$
$43$ $$1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4}$$
$47$ $$1 + 18 T + 173 T^{2} + 846 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + T + 53 T^{2} )^{2}$$
$59$ $$1 + 2 T + 21 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T + 99 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 + 14 T + 165 T^{2} + 938 T^{3} + 4489 T^{4}$$
$71$ $$1 - 16 T + 174 T^{2} - 1136 T^{3} + 5041 T^{4}$$
$73$ $$1 - 18 T + 195 T^{2} - 1314 T^{3} + 5329 T^{4}$$
$79$ $$1 + 2 T + 109 T^{2} + 158 T^{3} + 6241 T^{4}$$
$83$ $$1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{2}$$
$97$ $$1 - 8 T + 202 T^{2} - 776 T^{3} + 9409 T^{4}$$