# Properties

 Label 6480.2.a.e Level $6480$ Weight $2$ Character orbit 6480.a Self dual yes Analytic conductor $51.743$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6480 = 2^{4} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6480.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.7430605098$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + O(q^{10})$$ $$q - q^{5} + 5q^{11} + 3q^{17} - 5q^{19} - 6q^{23} + q^{25} - 10q^{29} + 2q^{31} + 4q^{37} - 3q^{41} - 3q^{43} - 4q^{47} - 7q^{49} - 6q^{53} - 5q^{55} + 3q^{59} + 2q^{61} + 11q^{67} + 14q^{71} - 15q^{73} - 10q^{79} + 12q^{83} - 3q^{85} + 14q^{89} + 5q^{95} - 13q^{97} + O(q^{100})$$

## 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 0 0 −1.00000 0 0 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$

## 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 6480.2.a.e 1
3.b odd 2 1 6480.2.a.q 1
4.b odd 2 1 3240.2.a.b 1
9.c even 3 2 720.2.q.e 2
9.d odd 6 2 2160.2.q.c 2
12.b even 2 1 3240.2.a.f 1
36.f odd 6 2 360.2.q.a 2
36.h even 6 2 1080.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.a 2 36.f odd 6 2
720.2.q.e 2 9.c even 3 2
1080.2.q.a 2 36.h even 6 2
2160.2.q.c 2 9.d odd 6 2
3240.2.a.b 1 4.b odd 2 1
3240.2.a.f 1 12.b even 2 1
6480.2.a.e 1 1.a even 1 1 trivial
6480.2.a.q 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6480))$$:

 $$T_{7}$$ $$T_{11} - 5$$ $$T_{13}$$ $$T_{17} - 3$$ $$T_{19} + 5$$ $$T_{23} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$-5 + T$$
$13$ $$T$$
$17$ $$-3 + T$$
$19$ $$5 + T$$
$23$ $$6 + T$$
$29$ $$10 + T$$
$31$ $$-2 + T$$
$37$ $$-4 + T$$
$41$ $$3 + T$$
$43$ $$3 + T$$
$47$ $$4 + T$$
$53$ $$6 + T$$
$59$ $$-3 + T$$
$61$ $$-2 + T$$
$67$ $$-11 + T$$
$71$ $$-14 + T$$
$73$ $$15 + T$$
$79$ $$10 + T$$
$83$ $$-12 + T$$
$89$ $$-14 + T$$
$97$ $$13 + T$$