# Properties

 Label 1536.2.a.n Level $1536$ Weight $2$ Character orbit 1536.a Self dual yes Analytic conductor $12.265$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_{1} q^{5} + \beta_{3} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta_{1} q^{5} + \beta_{3} q^{7} + q^{9} + ( 2 + \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( 2 - \beta_{2} ) q^{17} -\beta_{2} q^{19} + \beta_{3} q^{21} -2 \beta_{1} q^{23} + ( 5 + 2 \beta_{2} ) q^{25} + q^{27} + ( -\beta_{1} + 2 \beta_{3} ) q^{29} -\beta_{3} q^{31} + ( 2 + \beta_{2} ) q^{33} + ( 2 - 3 \beta_{2} ) q^{35} + ( -\beta_{1} - \beta_{3} ) q^{37} + ( \beta_{1} - \beta_{3} ) q^{39} + ( 6 - \beta_{2} ) q^{41} + ( -4 + 3 \beta_{2} ) q^{43} + \beta_{1} q^{45} -2 \beta_{1} q^{47} + ( 7 - 4 \beta_{2} ) q^{49} + ( 2 - \beta_{2} ) q^{51} + ( -\beta_{1} - 2 \beta_{3} ) q^{53} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{55} -\beta_{2} q^{57} + ( 4 + 2 \beta_{2} ) q^{59} + ( -\beta_{1} - \beta_{3} ) q^{61} + \beta_{3} q^{63} + ( 8 + 5 \beta_{2} ) q^{65} + ( -8 - 2 \beta_{2} ) q^{67} -2 \beta_{1} q^{69} -2 \beta_{3} q^{71} + 4 q^{73} + ( 5 + 2 \beta_{2} ) q^{75} -2 \beta_{1} q^{77} + \beta_{3} q^{79} + q^{81} + ( 6 + \beta_{2} ) q^{83} + 2 \beta_{3} q^{85} + ( -\beta_{1} + 2 \beta_{3} ) q^{87} + 10 q^{89} + ( -12 + \beta_{2} ) q^{91} -\beta_{3} q^{93} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 6 + 2 \beta_{2} ) q^{97} + ( 2 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 4q^{9} + 8q^{11} + 8q^{17} + 20q^{25} + 4q^{27} + 8q^{33} + 8q^{35} + 24q^{41} - 16q^{43} + 28q^{49} + 8q^{51} + 16q^{59} + 32q^{65} - 32q^{67} + 16q^{73} + 20q^{75} + 4q^{81} + 24q^{83} + 40q^{89} - 48q^{91} + 24q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 6 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} - 2 \nu^{2} - 8 \nu$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 3 \nu^{2} + 2 \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 4 \beta_{2} - 5 \beta_{1} + 6$$$$)/2$$

## 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
 2.68554 0.334904 −1.74912 −1.27133
0 1.00000 0 −3.95687 0 1.63899 0 1.00000 0
1.2 0 1.00000 0 −2.08402 0 −5.03127 0 1.00000 0
1.3 0 1.00000 0 2.08402 0 5.03127 0 1.00000 0
1.4 0 1.00000 0 3.95687 0 −1.63899 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.n yes 4
3.b odd 2 1 4608.2.a.t 4
4.b odd 2 1 1536.2.a.m 4
8.b even 2 1 1536.2.a.m 4
8.d odd 2 1 inner 1536.2.a.n yes 4
12.b even 2 1 4608.2.a.ba 4
16.e even 4 2 1536.2.d.g 8
16.f odd 4 2 1536.2.d.g 8
24.f even 2 1 4608.2.a.t 4
24.h odd 2 1 4608.2.a.ba 4
48.i odd 4 2 4608.2.d.p 8
48.k even 4 2 4608.2.d.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m 4 4.b odd 2 1
1536.2.a.m 4 8.b even 2 1
1536.2.a.n yes 4 1.a even 1 1 trivial
1536.2.a.n yes 4 8.d odd 2 1 inner
1536.2.d.g 8 16.e even 4 2
1536.2.d.g 8 16.f odd 4 2
4608.2.a.t 4 3.b odd 2 1
4608.2.a.t 4 24.f even 2 1
4608.2.a.ba 4 12.b even 2 1
4608.2.a.ba 4 24.h odd 2 1
4608.2.d.p 8 48.i odd 4 2
4608.2.d.p 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1536))$$:

 $$T_{5}^{4} - 20 T_{5}^{2} + 68$$ $$T_{7}^{4} - 28 T_{7}^{2} + 68$$ $$T_{11}^{2} - 4 T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$68 - 20 T^{2} + T^{4}$$
$7$ $$68 - 28 T^{2} + T^{4}$$
$11$ $$( -4 - 4 T + T^{2} )^{2}$$
$13$ $$272 - 40 T^{2} + T^{4}$$
$17$ $$( -4 - 4 T + T^{2} )^{2}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$1088 - 80 T^{2} + T^{4}$$
$29$ $$3332 - 116 T^{2} + T^{4}$$
$31$ $$68 - 28 T^{2} + T^{4}$$
$37$ $$272 - 56 T^{2} + T^{4}$$
$41$ $$( 28 - 12 T + T^{2} )^{2}$$
$43$ $$( -56 + 8 T + T^{2} )^{2}$$
$47$ $$1088 - 80 T^{2} + T^{4}$$
$53$ $$68 - 148 T^{2} + T^{4}$$
$59$ $$( -16 - 8 T + T^{2} )^{2}$$
$61$ $$272 - 56 T^{2} + T^{4}$$
$67$ $$( 32 + 16 T + T^{2} )^{2}$$
$71$ $$1088 - 112 T^{2} + T^{4}$$
$73$ $$( -4 + T )^{4}$$
$79$ $$68 - 28 T^{2} + T^{4}$$
$83$ $$( 28 - 12 T + T^{2} )^{2}$$
$89$ $$( -10 + T )^{4}$$
$97$ $$( 4 - 12 T + T^{2} )^{2}$$