# Properties

 Label 6384.2.a.q Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 399) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 4.00000 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.q 1
4.b odd 2 1 399.2.a.b 1
12.b even 2 1 1197.2.a.c 1
20.d odd 2 1 9975.2.a.j 1
28.d even 2 1 2793.2.a.c 1
76.d even 2 1 7581.2.a.e 1
84.h odd 2 1 8379.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.a.b 1 4.b odd 2 1
1197.2.a.c 1 12.b even 2 1
2793.2.a.c 1 28.d even 2 1
6384.2.a.q 1 1.a even 1 1 trivial
7581.2.a.e 1 76.d even 2 1
8379.2.a.o 1 84.h odd 2 1
9975.2.a.j 1 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5} - 4$$ $$T_{11} - 2$$ $$T_{13} - 4$$ $$T_{17}$$ $$T_{23} - 6$$

## Hecke characteristic polynomials

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