# Properties

 Label 1620.2.i.c 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: yes 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} -6 q^{17} -7 q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( 3 - 3 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} + 2 q^{35} -4 q^{37} + 3 \zeta_{6} q^{41} + ( -8 + 8 \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{49} + 6 q^{53} -3 q^{55} -3 \zeta_{6} q^{59} + ( -14 + 14 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -15 q^{71} -10 q^{73} + 6 \zeta_{6} q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} + 6 \zeta_{6} q^{85} -15 q^{89} -8 q^{91} + 7 \zeta_{6} q^{95} + ( -8 + 8 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} - 2q^{7} + O(q^{10})$$ $$2q - q^{5} - 2q^{7} + 3q^{11} + 4q^{13} - 12q^{17} - 14q^{19} + 6q^{23} - q^{25} + 3q^{29} - 5q^{31} + 4q^{35} - 8q^{37} + 3q^{41} - 8q^{43} + 3q^{49} + 12q^{53} - 6q^{55} - 3q^{59} - 14q^{61} + 4q^{65} - 2q^{67} - 30q^{71} - 20q^{73} + 6q^{77} - 8q^{79} + 6q^{85} - 30q^{89} - 16q^{91} + 7q^{95} - 8q^{97} + O(q^{100})$$

## 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 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 −0.500000 + 0.866025i 0 −1.00000 1.73205i 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.c 2
3.b odd 2 1 1620.2.i.g 2
9.c even 3 1 1620.2.a.f yes 1
9.c even 3 1 inner 1620.2.i.c 2
9.d odd 6 1 1620.2.a.c 1
9.d odd 6 1 1620.2.i.g 2
36.f odd 6 1 6480.2.a.p 1
36.h even 6 1 6480.2.a.b 1
45.h odd 6 1 8100.2.a.e 1
45.j even 6 1 8100.2.a.b 1
45.k odd 12 2 8100.2.d.d 2
45.l even 12 2 8100.2.d.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.c 1 9.d odd 6 1
1620.2.a.f yes 1 9.c even 3 1
1620.2.i.c 2 1.a even 1 1 trivial
1620.2.i.c 2 9.c even 3 1 inner
1620.2.i.g 2 3.b odd 2 1
1620.2.i.g 2 9.d odd 6 1
6480.2.a.b 1 36.h even 6 1
6480.2.a.p 1 36.f odd 6 1
8100.2.a.b 1 45.j even 6 1
8100.2.a.e 1 45.h odd 6 1
8100.2.d.d 2 45.k odd 12 2
8100.2.d.i 2 45.l even 12 2

## Hecke kernels

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

 $$T_{7}^{2} + 2 T_{7} + 4$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$4 + 2 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$( 7 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$9 - 3 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( 15 + T )^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 15 + T )^{2}$$
$97$ $$64 + 8 T + T^{2}$$