# Properties

 Label 1620.1.l.b Level $1620$ Weight $1$ Character orbit 1620.l Analytic conductor $0.808$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1620.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.808485320465$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.0.40500.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{5} + (i - 1) q^{7} +O(q^{10})$$ q - z * q^5 + (z - 1) * q^7 $$q - i q^{5} + (i - 1) q^{7} + q^{11} + ( - i + 1) q^{17} - i q^{19} + (i + 1) q^{23} - q^{25} - i q^{29} + q^{31} + (i + 1) q^{35} + q^{41} - i q^{49} + ( - i - 1) q^{53} - i q^{55} - i q^{59} + (i - 1) q^{67} - q^{71} + (i + 1) q^{73} + (i - 1) q^{77} + ( - i - 1) q^{85} + i q^{89} - q^{95} +O(q^{100})$$ q - z * q^5 + (z - 1) * q^7 + q^11 + (-z + 1) * q^17 - z * q^19 + (z + 1) * q^23 - q^25 - z * q^29 + q^31 + (z + 1) * q^35 + q^41 - z * q^49 + (-z - 1) * q^53 - z * q^55 - z * q^59 + (z - 1) * q^67 - q^71 + (z + 1) * q^73 + (z - 1) * q^77 + (-z - 1) * q^85 + z * q^89 - q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} + 2 q^{11} + 2 q^{17} + 2 q^{23} - 2 q^{25} + 2 q^{31} + 2 q^{35} + 2 q^{41} - 2 q^{53} - 2 q^{67} - 2 q^{71} + 2 q^{73} - 2 q^{77} - 2 q^{85} - 2 q^{95}+O(q^{100})$$ 2 * q - 2 * q^7 + 2 * q^11 + 2 * q^17 + 2 * q^23 - 2 * q^25 + 2 * q^31 + 2 * q^35 + 2 * q^41 - 2 * q^53 - 2 * q^67 - 2 * q^71 + 2 * q^73 - 2 * q^77 - 2 * q^85 - 2 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
973.1
 1.00000i − 1.00000i
0 0 0 1.00000i 0 −1.00000 + 1.00000i 0 0 0
1297.1 0 0 0 1.00000i 0 −1.00000 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.l.b yes 2
3.b odd 2 1 1620.1.l.a 2
5.c odd 4 1 inner 1620.1.l.b yes 2
9.c even 3 2 1620.1.v.a 4
9.d odd 6 2 1620.1.v.b 4
15.e even 4 1 1620.1.l.a 2
45.k odd 12 2 1620.1.v.a 4
45.l even 12 2 1620.1.v.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.1.l.a 2 3.b odd 2 1
1620.1.l.a 2 15.e even 4 1
1620.1.l.b yes 2 1.a even 1 1 trivial
1620.1.l.b yes 2 5.c odd 4 1 inner
1620.1.v.a 4 9.c even 3 2
1620.1.v.a 4 45.k odd 12 2
1620.1.v.b 4 9.d odd 6 2
1620.1.v.b 4 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

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