# Properties

 Label 4864.2.a.z Level $4864$ Weight $2$ Character orbit 4864.a Self dual yes Analytic conductor $38.839$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1216) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} -\beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} -\beta q^{5} + \beta q^{7} + q^{9} -3 q^{11} -2 \beta q^{15} -3 q^{17} + q^{19} + 2 \beta q^{21} + 2 \beta q^{23} -2 q^{25} -4 q^{27} + 2 \beta q^{29} -4 \beta q^{31} -6 q^{33} -3 q^{35} + 6 \beta q^{37} -6 q^{41} - q^{43} -\beta q^{45} -3 \beta q^{47} -4 q^{49} -6 q^{51} + 3 \beta q^{55} + 2 q^{57} -6 q^{59} -3 \beta q^{61} + \beta q^{63} + 4 q^{67} + 4 \beta q^{69} -2 \beta q^{71} + q^{73} -4 q^{75} -3 \beta q^{77} -11 q^{81} -12 q^{83} + 3 \beta q^{85} + 4 \beta q^{87} -8 \beta q^{93} -\beta q^{95} + 4 q^{97} -3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 2q^{9} - 6q^{11} - 6q^{17} + 2q^{19} - 4q^{25} - 8q^{27} - 12q^{33} - 6q^{35} - 12q^{41} - 2q^{43} - 8q^{49} - 12q^{51} + 4q^{57} - 12q^{59} + 8q^{67} + 2q^{73} - 8q^{75} - 22q^{81} - 24q^{83} + 8q^{97} - 6q^{99} + O(q^{100})$$

## 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
 1.73205 −1.73205
0 2.00000 0 −1.73205 0 1.73205 0 1.00000 0
1.2 0 2.00000 0 1.73205 0 −1.73205 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
$$2$$ $$-1$$
$$19$$ $$-1$$

## 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 4864.2.a.z 2
4.b odd 2 1 4864.2.a.q 2
8.b even 2 1 4864.2.a.q 2
8.d odd 2 1 inner 4864.2.a.z 2
16.e even 4 2 1216.2.c.e 4
16.f odd 4 2 1216.2.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.c.e 4 16.e even 4 2
1216.2.c.e 4 16.f odd 4 2
4864.2.a.q 2 4.b odd 2 1
4864.2.a.q 2 8.b even 2 1
4864.2.a.z 2 1.a even 1 1 trivial
4864.2.a.z 2 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4864))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -2 + T )^{2}$$
$5$ $$-3 + T^{2}$$
$7$ $$-3 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-12 + T^{2}$$
$29$ $$-12 + T^{2}$$
$31$ $$-48 + T^{2}$$
$37$ $$-108 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$-27 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$-27 + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$-12 + T^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$( -4 + T )^{2}$$