# Properties

 Label 1536.2.a.k Level $1536$ Weight $2$ Character orbit 1536.a Self dual yes Analytic conductor $12.265$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 2 + \beta ) q^{5} -\beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 2 + \beta ) q^{5} -\beta q^{7} + q^{9} + ( -2 - 2 \beta ) q^{11} + ( 2 - 2 \beta ) q^{13} + ( 2 + \beta ) q^{15} + ( 2 + 2 \beta ) q^{17} + 2 \beta q^{19} -\beta q^{21} + ( 4 - 2 \beta ) q^{23} + ( 1 + 4 \beta ) q^{25} + q^{27} + ( 6 + \beta ) q^{29} + 5 \beta q^{31} + ( -2 - 2 \beta ) q^{33} + ( -2 - 2 \beta ) q^{35} + ( 6 + 4 \beta ) q^{37} + ( 2 - 2 \beta ) q^{39} + ( -2 - 6 \beta ) q^{41} + ( 4 + 2 \beta ) q^{43} + ( 2 + \beta ) q^{45} + ( 4 + 6 \beta ) q^{47} -5 q^{49} + ( 2 + 2 \beta ) q^{51} + ( 6 - 3 \beta ) q^{53} + ( -8 - 6 \beta ) q^{55} + 2 \beta q^{57} + ( -4 + 4 \beta ) q^{59} + ( 6 - 4 \beta ) q^{61} -\beta q^{63} -2 \beta q^{65} -4 \beta q^{67} + ( 4 - 2 \beta ) q^{69} -6 \beta q^{71} -8 \beta q^{73} + ( 1 + 4 \beta ) q^{75} + ( 4 + 2 \beta ) q^{77} + ( -16 - \beta ) q^{79} + q^{81} + ( -6 - 2 \beta ) q^{83} + ( 8 + 6 \beta ) q^{85} + ( 6 + \beta ) q^{87} + ( -6 + 8 \beta ) q^{89} + ( 4 - 2 \beta ) q^{91} + 5 \beta q^{93} + ( 4 + 4 \beta ) q^{95} + ( -2 - 4 \beta ) q^{97} + ( -2 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} - 4q^{11} + 4q^{13} + 4q^{15} + 4q^{17} + 8q^{23} + 2q^{25} + 2q^{27} + 12q^{29} - 4q^{33} - 4q^{35} + 12q^{37} + 4q^{39} - 4q^{41} + 8q^{43} + 4q^{45} + 8q^{47} - 10q^{49} + 4q^{51} + 12q^{53} - 16q^{55} - 8q^{59} + 12q^{61} + 8q^{69} + 2q^{75} + 8q^{77} - 32q^{79} + 2q^{81} - 12q^{83} + 16q^{85} + 12q^{87} - 12q^{89} + 8q^{91} + 8q^{95} - 4q^{97} - 4q^{99} + O(q^{100})$$

## 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 0.585786 0 1.41421 0 1.00000 0
1.2 0 1.00000 0 3.41421 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

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 1536.2.a.k yes 2
3.b odd 2 1 4608.2.a.d 2
4.b odd 2 1 1536.2.a.f yes 2
8.b even 2 1 1536.2.a.a 2
8.d odd 2 1 1536.2.a.h yes 2
12.b even 2 1 4608.2.a.b 2
16.e even 4 2 1536.2.d.e 4
16.f odd 4 2 1536.2.d.b 4
24.f even 2 1 4608.2.a.q 2
24.h odd 2 1 4608.2.a.o 2
48.i odd 4 2 4608.2.d.h 4
48.k even 4 2 4608.2.d.f 4

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

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 2$$ $$T_{7}^{2} - 2$$ $$T_{11}^{2} + 4 T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$2 - 4 T + T^{2}$$
$7$ $$-2 + T^{2}$$
$11$ $$-4 + 4 T + T^{2}$$
$13$ $$-4 - 4 T + T^{2}$$
$17$ $$-4 - 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$8 - 8 T + T^{2}$$
$29$ $$34 - 12 T + T^{2}$$
$31$ $$-50 + T^{2}$$
$37$ $$4 - 12 T + T^{2}$$
$41$ $$-68 + 4 T + T^{2}$$
$43$ $$8 - 8 T + T^{2}$$
$47$ $$-56 - 8 T + T^{2}$$
$53$ $$18 - 12 T + T^{2}$$
$59$ $$-16 + 8 T + T^{2}$$
$61$ $$4 - 12 T + T^{2}$$
$67$ $$-32 + T^{2}$$
$71$ $$-72 + T^{2}$$
$73$ $$-128 + T^{2}$$
$79$ $$254 + 32 T + T^{2}$$
$83$ $$28 + 12 T + T^{2}$$
$89$ $$-92 + 12 T + T^{2}$$
$97$ $$-28 + 4 T + T^{2}$$