# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1620.v (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.808485320465$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{5} q^{5} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{7} +O(q^{10})$$ $$q -\zeta_{12}^{5} q^{5} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{7} -\zeta_{12}^{4} q^{11} + ( -1 + \zeta_{12}^{3} ) q^{17} -\zeta_{12}^{3} q^{19} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{23} -\zeta_{12}^{4} q^{25} -\zeta_{12} q^{29} -\zeta_{12}^{2} q^{31} + ( -1 - \zeta_{12}^{3} ) q^{35} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{49} + ( 1 + \zeta_{12}^{3} ) q^{53} -\zeta_{12}^{3} q^{55} -\zeta_{12}^{5} q^{59} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{67} + q^{71} + ( 1 + \zeta_{12}^{3} ) q^{73} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{77} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{85} -\zeta_{12}^{3} q^{89} -\zeta_{12}^{2} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{7} + O(q^{10})$$ $$4 q + 2 q^{7} + 2 q^{11} - 4 q^{17} + 2 q^{23} + 2 q^{25} - 2 q^{31} - 4 q^{35} + 2 q^{41} + 4 q^{53} + 2 q^{67} + 4 q^{71} + 4 q^{73} - 2 q^{77} + 2 q^{85} - 2 q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-\zeta_{12}^{3}$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
217.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0 0 0 −0.866025 + 0.500000i 0 1.36603 0.366025i 0 0 0
433.1 0 0 0 −0.866025 0.500000i 0 1.36603 + 0.366025i 0 0 0
757.1 0 0 0 0.866025 + 0.500000i 0 −0.366025 + 1.36603i 0 0 0
1513.1 0 0 0 0.866025 0.500000i 0 −0.366025 1.36603i 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
9.c even 3 1 inner
45.k odd 12 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} - T_{11} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1620, [\chi])$$.

## Hecke characteristic polynomials

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