# Properties

 Label 6384.2.a.g Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 798) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
0 −1.00000 0 −2.00000 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.g 1
4.b odd 2 1 798.2.a.b 1
12.b even 2 1 2394.2.a.m 1
28.d even 2 1 5586.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.b 1 4.b odd 2 1
2394.2.a.m 1 12.b even 2 1
5586.2.a.j 1 28.d even 2 1
6384.2.a.g 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5} + 2$$ $$T_{11} - 2$$ $$T_{13} - 2$$ $$T_{17} + 4$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$2 + T$$
$7$ $$-1 + T$$
$11$ $$-2 + T$$
$13$ $$-2 + T$$
$17$ $$4 + T$$
$19$ $$1 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$-10 + T$$
$37$ $$T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$6 + T$$
$53$ $$6 + T$$
$59$ $$-12 + T$$
$61$ $$-10 + T$$
$67$ $$-2 + T$$
$71$ $$8 + T$$
$73$ $$6 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$-10 + T$$
$97$ $$12 + T$$