Properties

 Label 1620.2.i.e 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -\zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + 5 q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( 9 - 9 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -4 q^{35} + 2 q^{37} + 9 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + ( 6 - 6 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} -12 q^{53} + 3 q^{55} -9 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} + ( 4 - 4 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} + 3 q^{71} -4 q^{73} + 12 \zeta_{6} q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -6 + 6 \zeta_{6} ) q^{83} -9 q^{89} + 16 q^{91} -5 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{5} + 4q^{7} + O(q^{10})$$ $$2q - q^{5} + 4q^{7} - 3q^{11} + 4q^{13} + 10q^{19} + 6q^{23} - q^{25} + 9q^{29} - 5q^{31} - 8q^{35} + 4q^{37} + 9q^{41} + 10q^{43} + 6q^{47} - 9q^{49} - 24q^{53} + 6q^{55} - 9q^{59} + 10q^{61} + 4q^{65} - 2q^{67} + 6q^{71} - 8q^{73} + 12q^{77} + 4q^{79} - 6q^{83} - 18q^{89} + 32q^{91} - 5q^{95} - 2q^{97} + O(q^{100})$$

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.e 2
3.b odd 2 1 1620.2.i.l 2
9.c even 3 1 1620.2.a.d yes 1
9.c even 3 1 inner 1620.2.i.e 2
9.d odd 6 1 1620.2.a.a 1
9.d odd 6 1 1620.2.i.l 2
36.f odd 6 1 6480.2.a.y 1
36.h even 6 1 6480.2.a.m 1
45.h odd 6 1 8100.2.a.m 1
45.j even 6 1 8100.2.a.n 1
45.k odd 12 2 8100.2.d.g 2
45.l even 12 2 8100.2.d.b 2

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

Hecke kernels

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

 $$T_{7}^{2} - 4 T_{7} + 16$$ $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{17}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$16 - 4 T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -5 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$81 - 9 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$100 - 10 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( 12 + T )^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$4 + 2 T + T^{2}$$
$71$ $$( -3 + T )^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$36 + 6 T + T^{2}$$
$89$ $$( 9 + T )^{2}$$
$97$ $$4 + 2 T + T^{2}$$