# Properties

 Label 4864.2.a.v Level $4864$ Weight $2$ Character orbit 4864.a Self dual yes Analytic conductor $38.839$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2432) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} -2 q^{5} + ( -1 + \beta ) q^{7} -2 q^{9} +O(q^{10})$$ $$q + q^{3} -2 q^{5} + ( -1 + \beta ) q^{7} -2 q^{9} -4 q^{11} + ( 1 - \beta ) q^{13} -2 q^{15} + ( -1 + 2 \beta ) q^{17} + q^{19} + ( -1 + \beta ) q^{21} + ( 3 - \beta ) q^{23} - q^{25} -5 q^{27} + ( 1 + \beta ) q^{29} + ( 2 + 2 \beta ) q^{31} -4 q^{33} + ( 2 - 2 \beta ) q^{35} -2 q^{37} + ( 1 - \beta ) q^{39} + 2 \beta q^{41} + ( -2 - 2 \beta ) q^{43} + 4 q^{45} -2 \beta q^{47} + ( 6 - 2 \beta ) q^{49} + ( -1 + 2 \beta ) q^{51} + ( 9 - \beta ) q^{53} + 8 q^{55} + q^{57} + ( -3 + 2 \beta ) q^{59} + ( -2 + 2 \beta ) q^{61} + ( 2 - 2 \beta ) q^{63} + ( -2 + 2 \beta ) q^{65} + ( 1 + 2 \beta ) q^{67} + ( 3 - \beta ) q^{69} + 14 q^{71} -7 q^{73} - q^{75} + ( 4 - 4 \beta ) q^{77} -2 q^{79} + q^{81} + ( -4 - 2 \beta ) q^{83} + ( 2 - 4 \beta ) q^{85} + ( 1 + \beta ) q^{87} + ( 6 + 2 \beta ) q^{89} + ( -13 + 2 \beta ) q^{91} + ( 2 + 2 \beta ) q^{93} -2 q^{95} + ( 6 - 2 \beta ) q^{97} + 8 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 4q^{5} - 2q^{7} - 4q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 4q^{5} - 2q^{7} - 4q^{9} - 8q^{11} + 2q^{13} - 4q^{15} - 2q^{17} + 2q^{19} - 2q^{21} + 6q^{23} - 2q^{25} - 10q^{27} + 2q^{29} + 4q^{31} - 8q^{33} + 4q^{35} - 4q^{37} + 2q^{39} - 4q^{43} + 8q^{45} + 12q^{49} - 2q^{51} + 18q^{53} + 16q^{55} + 2q^{57} - 6q^{59} - 4q^{61} + 4q^{63} - 4q^{65} + 2q^{67} + 6q^{69} + 28q^{71} - 14q^{73} - 2q^{75} + 8q^{77} - 4q^{79} + 2q^{81} - 8q^{83} + 4q^{85} + 2q^{87} + 12q^{89} - 26q^{91} + 4q^{93} - 4q^{95} + 12q^{97} + 16q^{99} + O(q^{100})$$

## 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 1.00000 0 −2.00000 0 −4.46410 0 −2.00000 0
1.2 0 1.00000 0 −2.00000 0 2.46410 0 −2.00000 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

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 4864.2.a.v 2
4.b odd 2 1 4864.2.a.s 2
8.b even 2 1 4864.2.a.u 2
8.d odd 2 1 4864.2.a.x 2
16.e even 4 2 2432.2.c.d yes 4
16.f odd 4 2 2432.2.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.c 4 16.f odd 4 2
2432.2.c.d yes 4 16.e even 4 2
4864.2.a.s 2 4.b odd 2 1
4864.2.a.u 2 8.b even 2 1
4864.2.a.v 2 1.a even 1 1 trivial
4864.2.a.x 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ $$T_{5} + 2$$ $$T_{7}^{2} + 2 T_{7} - 11$$ $$T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$-11 + 2 T + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-11 - 2 T + T^{2}$$
$17$ $$-47 + 2 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-3 - 6 T + T^{2}$$
$29$ $$-11 - 2 T + T^{2}$$
$31$ $$-44 - 4 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$-48 + T^{2}$$
$43$ $$-44 + 4 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$69 - 18 T + T^{2}$$
$59$ $$-39 + 6 T + T^{2}$$
$61$ $$-44 + 4 T + T^{2}$$
$67$ $$-47 - 2 T + T^{2}$$
$71$ $$( -14 + T )^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$( 2 + T )^{2}$$
$83$ $$-32 + 8 T + T^{2}$$
$89$ $$-12 - 12 T + T^{2}$$
$97$ $$-12 - 12 T + T^{2}$$