# Properties

 Label 1620.2.i.k Level $1620$ Weight $2$ Character orbit 1620.i Analytic conductor $12.936$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} -3 q^{17} -7 q^{19} + 9 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 7 \zeta_{6} q^{31} + 4 q^{35} + 2 q^{37} -6 \zeta_{6} q^{41} + ( -2 + 2 \zeta_{6} ) q^{43} -9 \zeta_{6} q^{49} + 9 q^{53} -6 q^{55} + 12 \zeta_{6} q^{59} + ( 7 - 7 \zeta_{6} ) q^{61} + ( -4 + 4 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + 6 q^{71} + 2 q^{73} + 24 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{83} -3 \zeta_{6} q^{85} -6 q^{89} + 16 q^{91} -7 \zeta_{6} q^{95} + ( -8 + 8 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} + 4q^{7} + O(q^{10})$$ $$2q + q^{5} + 4q^{7} - 6q^{11} + 4q^{13} - 6q^{17} - 14q^{19} + 9q^{23} - q^{25} + 7q^{31} + 8q^{35} + 4q^{37} - 6q^{41} - 2q^{43} - 9q^{49} + 18q^{53} - 12q^{55} + 12q^{59} + 7q^{61} - 4q^{65} - 2q^{67} + 12q^{71} + 4q^{73} + 24q^{77} + q^{79} - 9q^{83} - 3q^{85} - 12q^{89} + 32q^{91} - 7q^{95} - 8q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0.500000 + 0.866025i 0 2.00000 3.46410i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 2.00000 + 3.46410i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.k 2
3.b odd 2 1 1620.2.i.f 2
9.c even 3 1 540.2.a.a 1
9.c even 3 1 inner 1620.2.i.k 2
9.d odd 6 1 540.2.a.d yes 1
9.d odd 6 1 1620.2.i.f 2
36.f odd 6 1 2160.2.a.k 1
36.h even 6 1 2160.2.a.x 1
45.h odd 6 1 2700.2.a.r 1
45.j even 6 1 2700.2.a.t 1
45.k odd 12 2 2700.2.d.l 2
45.l even 12 2 2700.2.d.b 2
72.j odd 6 1 8640.2.a.b 1
72.l even 6 1 8640.2.a.bc 1
72.n even 6 1 8640.2.a.be 1
72.p odd 6 1 8640.2.a.ch 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.a 1 9.c even 3 1
540.2.a.d yes 1 9.d odd 6 1
1620.2.i.f 2 3.b odd 2 1
1620.2.i.f 2 9.d odd 6 1
1620.2.i.k 2 1.a even 1 1 trivial
1620.2.i.k 2 9.c even 3 1 inner
2160.2.a.k 1 36.f odd 6 1
2160.2.a.x 1 36.h even 6 1
2700.2.a.r 1 45.h odd 6 1
2700.2.a.t 1 45.j even 6 1
2700.2.d.b 2 45.l even 12 2
2700.2.d.l 2 45.k odd 12 2
8640.2.a.b 1 72.j odd 6 1
8640.2.a.bc 1 72.l even 6 1
8640.2.a.be 1 72.n even 6 1
8640.2.a.ch 1 72.p odd 6 1

## Hecke kernels

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

 $$T_{7}^{2} - 4 T_{7} + 16$$ $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$16 - 4 T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$81 - 9 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$36 + 6 T + T^{2}$$
$43$ $$4 + 2 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -9 + T )^{2}$$
$59$ $$144 - 12 T + T^{2}$$
$61$ $$49 - 7 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( -6 + T )^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$64 + 8 T + T^{2}$$