# Properties

 Label 6384.2.a.r 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 3192) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 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.r 1
4.b odd 2 1 3192.2.a.a 1
12.b even 2 1 9576.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.a 1 4.b odd 2 1
6384.2.a.r 1 1.a even 1 1 trivial
9576.2.a.ba 1 12.b even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$4 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$2 + T$$
$17$ $$-4 + T$$
$19$ $$1 + T$$
$23$ $$4 + T$$
$29$ $$-4 + T$$
$31$ $$T$$
$37$ $$10 + T$$
$41$ $$-10 + T$$
$43$ $$4 + T$$
$47$ $$-6 + T$$
$53$ $$T$$
$59$ $$4 + T$$
$61$ $$10 + T$$
$67$ $$-4 + T$$
$71$ $$-2 + T$$
$73$ $$2 + T$$
$79$ $$4 + T$$
$83$ $$-14 + T$$
$89$ $$6 + T$$
$97$ $$18 + T$$