# Properties

 Label 1620.2.a.c Level $1620$ Weight $2$ Character orbit 1620.a Self dual yes Analytic conductor $12.936$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + 2q^{7} + O(q^{10})$$ $$q - q^{5} + 2q^{7} + 3q^{11} - 4q^{13} + 6q^{17} - 7q^{19} + 6q^{23} + q^{25} + 3q^{29} + 5q^{31} - 2q^{35} - 4q^{37} + 3q^{41} + 8q^{43} - 3q^{49} - 6q^{53} - 3q^{55} - 3q^{59} + 14q^{61} + 4q^{65} + 2q^{67} + 15q^{71} - 10q^{73} + 6q^{77} + 8q^{79} - 6q^{85} + 15q^{89} - 8q^{91} + 7q^{95} + 8q^{97} + O(q^{100})$$

## 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 2.00000 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.c 1
3.b odd 2 1 1620.2.a.f yes 1
4.b odd 2 1 6480.2.a.b 1
5.b even 2 1 8100.2.a.e 1
5.c odd 4 2 8100.2.d.i 2
9.c even 3 2 1620.2.i.g 2
9.d odd 6 2 1620.2.i.c 2
12.b even 2 1 6480.2.a.p 1
15.d odd 2 1 8100.2.a.b 1
15.e even 4 2 8100.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.c 1 1.a even 1 1 trivial
1620.2.a.f yes 1 3.b odd 2 1
1620.2.i.c 2 9.d odd 6 2
1620.2.i.g 2 9.c even 3 2
6480.2.a.b 1 4.b odd 2 1
6480.2.a.p 1 12.b even 2 1
8100.2.a.b 1 15.d odd 2 1
8100.2.a.e 1 5.b even 2 1
8100.2.d.d 2 15.e even 4 2
8100.2.d.i 2 5.c odd 4 2

## Hecke kernels

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

 $$T_{7} - 2$$ $$T_{11} - 3$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$-2 + T$$
$11$ $$-3 + T$$
$13$ $$4 + T$$
$17$ $$-6 + T$$
$19$ $$7 + T$$
$23$ $$-6 + T$$
$29$ $$-3 + T$$
$31$ $$-5 + T$$
$37$ $$4 + T$$
$41$ $$-3 + T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$3 + T$$
$61$ $$-14 + T$$
$67$ $$-2 + T$$
$71$ $$-15 + T$$
$73$ $$10 + T$$
$79$ $$-8 + T$$
$83$ $$T$$
$89$ $$-15 + T$$
$97$ $$-8 + T$$