# Properties

 Label 1536.2.a.j Level $1536$ Weight $2$ Character orbit 1536.a Self dual yes Analytic conductor $12.265$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1536.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.2650217505$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta q^{5} + \beta q^{7} + q^{9} + 2 q^{11} + \beta q^{15} + 2 q^{17} + 4 q^{19} + \beta q^{21} + 2 \beta q^{23} -3 q^{25} + q^{27} -7 \beta q^{29} -5 \beta q^{31} + 2 q^{33} + 2 q^{35} + 6 \beta q^{37} + 6 q^{41} + 8 q^{43} + \beta q^{45} + 2 \beta q^{47} -5 q^{49} + 2 q^{51} + \beta q^{53} + 2 \beta q^{55} + 4 q^{57} + 12 q^{59} -10 \beta q^{61} + \beta q^{63} + 8 q^{67} + 2 \beta q^{69} -10 \beta q^{71} -8 q^{73} -3 q^{75} + 2 \beta q^{77} -3 \beta q^{79} + q^{81} + 6 q^{83} + 2 \beta q^{85} -7 \beta q^{87} + 2 q^{89} -5 \beta q^{93} + 4 \beta q^{95} -14 q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} + 4q^{11} + 4q^{17} + 8q^{19} - 6q^{25} + 2q^{27} + 4q^{33} + 4q^{35} + 12q^{41} + 16q^{43} - 10q^{49} + 4q^{51} + 8q^{57} + 24q^{59} + 16q^{67} - 16q^{73} - 6q^{75} + 2q^{81} + 12q^{83} + 4q^{89} - 28q^{97} + 4q^{99} + O(q^{100})$$

## 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 1.00000 0 −1.41421 0 −1.41421 0 1.00000 0
1.2 0 1.00000 0 1.41421 0 1.41421 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.a.j yes 2
3.b odd 2 1 4608.2.a.h 2
4.b odd 2 1 1536.2.a.c 2
8.b even 2 1 1536.2.a.c 2
8.d odd 2 1 inner 1536.2.a.j yes 2
12.b even 2 1 4608.2.a.j 2
16.e even 4 2 1536.2.d.d 4
16.f odd 4 2 1536.2.d.d 4
24.f even 2 1 4608.2.a.h 2
24.h odd 2 1 4608.2.a.j 2
48.i odd 4 2 4608.2.d.g 4
48.k even 4 2 4608.2.d.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.c 2 4.b odd 2 1
1536.2.a.c 2 8.b even 2 1
1536.2.a.j yes 2 1.a even 1 1 trivial
1536.2.a.j yes 2 8.d odd 2 1 inner
1536.2.d.d 4 16.e even 4 2
1536.2.d.d 4 16.f odd 4 2
4608.2.a.h 2 3.b odd 2 1
4608.2.a.h 2 24.f even 2 1
4608.2.a.j 2 12.b even 2 1
4608.2.a.j 2 24.h odd 2 1
4608.2.d.g 4 48.i odd 4 2
4608.2.d.g 4 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1536))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$-2 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-98 + T^{2}$$
$31$ $$-50 + T^{2}$$
$37$ $$-72 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$-2 + T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$-200 + T^{2}$$
$67$ $$( -8 + T )^{2}$$
$71$ $$-200 + T^{2}$$
$73$ $$( 8 + T )^{2}$$
$79$ $$-18 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$