# Properties

 Label 6384.2.a.bj Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.9764966504$$ Analytic rank: $$0$$ 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 1596) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -1 + \beta ) q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -1 + \beta ) q^{5} + q^{7} + q^{9} + 2 \beta q^{11} -4 q^{13} + ( 1 - \beta ) q^{15} + ( -1 - 3 \beta ) q^{17} + q^{19} - q^{21} + ( 4 - 2 \beta ) q^{23} + ( -1 - 2 \beta ) q^{25} - q^{27} + ( -1 + 3 \beta ) q^{29} + ( 2 + 4 \beta ) q^{31} -2 \beta q^{33} + ( -1 + \beta ) q^{35} + ( -4 + 2 \beta ) q^{37} + 4 q^{39} + ( -6 + 2 \beta ) q^{41} -2 \beta q^{43} + ( -1 + \beta ) q^{45} + ( -1 + \beta ) q^{47} + q^{49} + ( 1 + 3 \beta ) q^{51} + ( 7 - \beta ) q^{53} + ( 6 - 2 \beta ) q^{55} - q^{57} + ( 4 - 4 \beta ) q^{59} + ( -4 + 6 \beta ) q^{61} + q^{63} + ( 4 - 4 \beta ) q^{65} + ( 2 + 2 \beta ) q^{67} + ( -4 + 2 \beta ) q^{69} + ( -5 + 3 \beta ) q^{71} + ( -2 + 8 \beta ) q^{73} + ( 1 + 2 \beta ) q^{75} + 2 \beta q^{77} + ( 10 - 2 \beta ) q^{79} + q^{81} + ( 3 + \beta ) q^{83} + ( -8 + 2 \beta ) q^{85} + ( 1 - 3 \beta ) q^{87} + ( 6 - 2 \beta ) q^{89} -4 q^{91} + ( -2 - 4 \beta ) q^{93} + ( -1 + \beta ) q^{95} -6 q^{97} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 14 q^{53} + 12 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} + 2 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 10 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 16 q^{85} + 2 q^{87} + 12 q^{89} - 8 q^{91} - 4 q^{93} - 2 q^{95} - 12 q^{97} + O(q^{100})$$

## 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 −2.73205 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 0.732051 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 6384.2.a.bj 2
4.b odd 2 1 1596.2.a.i 2
12.b even 2 1 4788.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1596.2.a.i 2 4.b odd 2 1
4788.2.a.j 2 12.b even 2 1
6384.2.a.bj 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}^{2} + 2 T_{5} - 2$$ $$T_{11}^{2} - 12$$ $$T_{13} + 4$$ $$T_{17}^{2} + 2 T_{17} - 26$$ $$T_{23}^{2} - 8 T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-2 + 2 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$-26 + 2 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$4 - 8 T + T^{2}$$
$29$ $$-26 + 2 T + T^{2}$$
$31$ $$-44 - 4 T + T^{2}$$
$37$ $$4 + 8 T + T^{2}$$
$41$ $$24 + 12 T + T^{2}$$
$43$ $$-12 + T^{2}$$
$47$ $$-2 + 2 T + T^{2}$$
$53$ $$46 - 14 T + T^{2}$$
$59$ $$-32 - 8 T + T^{2}$$
$61$ $$-92 + 8 T + T^{2}$$
$67$ $$-8 - 4 T + T^{2}$$
$71$ $$-2 + 10 T + T^{2}$$
$73$ $$-188 + 4 T + T^{2}$$
$79$ $$88 - 20 T + T^{2}$$
$83$ $$6 - 6 T + T^{2}$$
$89$ $$24 - 12 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$