# Properties

 Label 1620.2.a.e Level $1620$ Weight $2$ Character orbit 1620.a Self dual yes Analytic conductor $12.936$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9357651274$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7} + O(q^{10})$$ $$q + q^{5} - q^{7} - 4q^{13} - 6q^{17} + 2q^{19} - 3q^{23} + q^{25} + 3q^{29} - 10q^{31} - q^{35} - 10q^{37} + 9q^{41} - 4q^{43} + 9q^{47} - 6q^{49} - 6q^{53} - 6q^{59} - q^{61} - 4q^{65} + 11q^{67} + 12q^{71} - 4q^{73} - 10q^{79} - 9q^{83} - 6q^{85} + 9q^{89} + 4q^{91} + 2q^{95} - 10q^{97} + O(q^{100})$$

## 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 −1.00000 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 1620.2.a.e 1
3.b odd 2 1 1620.2.a.b 1
4.b odd 2 1 6480.2.a.t 1
5.b even 2 1 8100.2.a.i 1
5.c odd 4 2 8100.2.d.e 2
9.c even 3 2 180.2.i.a 2
9.d odd 6 2 540.2.i.a 2
12.b even 2 1 6480.2.a.h 1
15.d odd 2 1 8100.2.a.h 1
15.e even 4 2 8100.2.d.f 2
36.f odd 6 2 720.2.q.a 2
36.h even 6 2 2160.2.q.e 2
45.h odd 6 2 2700.2.i.a 2
45.j even 6 2 900.2.i.a 2
45.k odd 12 4 900.2.s.a 4
45.l even 12 4 2700.2.s.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.a 2 9.c even 3 2
540.2.i.a 2 9.d odd 6 2
720.2.q.a 2 36.f odd 6 2
900.2.i.a 2 45.j even 6 2
900.2.s.a 4 45.k odd 12 4
1620.2.a.b 1 3.b odd 2 1
1620.2.a.e 1 1.a even 1 1 trivial
2160.2.q.e 2 36.h even 6 2
2700.2.i.a 2 45.h odd 6 2
2700.2.s.a 4 45.l even 12 4
6480.2.a.h 1 12.b even 2 1
6480.2.a.t 1 4.b odd 2 1
8100.2.a.h 1 15.d odd 2 1
8100.2.a.i 1 5.b even 2 1
8100.2.d.e 2 5.c odd 4 2
8100.2.d.f 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1620))$$:

 $$T_{7} + 1$$ $$T_{11}$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$1 + T$$
$11$ $$T$$
$13$ $$4 + T$$
$17$ $$6 + T$$
$19$ $$-2 + T$$
$23$ $$3 + T$$
$29$ $$-3 + T$$
$31$ $$10 + T$$
$37$ $$10 + T$$
$41$ $$-9 + T$$
$43$ $$4 + T$$
$47$ $$-9 + T$$
$53$ $$6 + T$$
$59$ $$6 + T$$
$61$ $$1 + T$$
$67$ $$-11 + T$$
$71$ $$-12 + T$$
$73$ $$4 + T$$
$79$ $$10 + T$$
$83$ $$9 + T$$
$89$ $$-9 + T$$
$97$ $$10 + T$$