# Properties

 Label 1620.2.a.j Level $1620$ Weight $2$ Character orbit 1620.a Self dual yes Analytic conductor $12.936$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 180) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( 1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q + q^{5} + ( 1 - \beta_{1} ) q^{7} + \beta_{2} q^{11} + ( 2 + \beta_{2} ) q^{13} -\beta_{2} q^{17} + ( 2 - \beta_{2} ) q^{19} + ( 1 + \beta_{1} ) q^{23} + q^{25} + ( 1 - 2 \beta_{1} ) q^{29} + ( 2 - \beta_{2} ) q^{31} + ( 1 - \beta_{1} ) q^{35} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 + \beta_{2} ) q^{43} + ( -5 + \beta_{1} - \beta_{2} ) q^{47} + ( 6 + \beta_{2} ) q^{49} + ( 2 + 2 \beta_{1} ) q^{53} + \beta_{2} q^{55} + ( -2 - 2 \beta_{1} ) q^{59} + ( 7 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( 2 + \beta_{2} ) q^{65} + ( 3 + \beta_{1} + \beta_{2} ) q^{67} + ( -8 - 2 \beta_{1} + \beta_{2} ) q^{71} + 8 q^{73} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{77} + 2 q^{79} + ( 7 + \beta_{1} ) q^{83} -\beta_{2} q^{85} + 3 q^{89} + ( -4 \beta_{1} + 3 \beta_{2} ) q^{91} + ( 2 - \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{5} + 3q^{7} + O(q^{10})$$ $$3q + 3q^{5} + 3q^{7} + 6q^{13} + 6q^{19} + 3q^{23} + 3q^{25} + 3q^{29} + 6q^{31} + 3q^{35} + 12q^{37} - 3q^{41} + 6q^{43} - 15q^{47} + 18q^{49} + 6q^{53} - 6q^{59} + 21q^{61} + 6q^{65} + 9q^{67} - 24q^{71} + 24q^{73} - 6q^{77} + 6q^{79} + 21q^{83} + 9q^{89} + 6q^{95} + 18q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1} + 11$$$$)/3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.51414 −2.08613 0.571993
0 0 0 1.00000 0 −3.83502 0 0 0
1.2 0 0 0 1.00000 0 2.73419 0 0 0
1.3 0 0 0 1.00000 0 4.10083 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.a.j 3
3.b odd 2 1 1620.2.a.i 3
4.b odd 2 1 6480.2.a.bw 3
5.b even 2 1 8100.2.a.u 3
5.c odd 4 2 8100.2.d.o 6
9.c even 3 2 540.2.i.b 6
9.d odd 6 2 180.2.i.b 6
12.b even 2 1 6480.2.a.bt 3
15.d odd 2 1 8100.2.a.v 3
15.e even 4 2 8100.2.d.p 6
36.f odd 6 2 2160.2.q.i 6
36.h even 6 2 720.2.q.k 6
45.h odd 6 2 900.2.i.c 6
45.j even 6 2 2700.2.i.c 6
45.k odd 12 4 2700.2.s.c 12
45.l even 12 4 900.2.s.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 9.d odd 6 2
540.2.i.b 6 9.c even 3 2
720.2.q.k 6 36.h even 6 2
900.2.i.c 6 45.h odd 6 2
900.2.s.c 12 45.l even 12 4
1620.2.a.i 3 3.b odd 2 1
1620.2.a.j 3 1.a even 1 1 trivial
2160.2.q.i 6 36.f odd 6 2
2700.2.i.c 6 45.j even 6 2
2700.2.s.c 12 45.k odd 12 4
6480.2.a.bt 3 12.b even 2 1
6480.2.a.bw 3 4.b odd 2 1
8100.2.a.u 3 5.b even 2 1
8100.2.a.v 3 15.d odd 2 1
8100.2.d.o 6 5.c odd 4 2
8100.2.d.p 6 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1620))$$:

 $$T_{7}^{3} - 3 T_{7}^{2} - 15 T_{7} + 43$$ $$T_{11}^{3} - 24 T_{11} + 36$$ $$T_{17}^{3} - 24 T_{17} - 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$43 - 15 T - 3 T^{2} + T^{3}$$
$11$ $$36 - 24 T + T^{3}$$
$13$ $$76 - 12 T - 6 T^{2} + T^{3}$$
$17$ $$-36 - 24 T + T^{3}$$
$19$ $$4 - 12 T - 6 T^{2} + T^{3}$$
$23$ $$-9 - 15 T - 3 T^{2} + T^{3}$$
$29$ $$279 - 69 T - 3 T^{2} + T^{3}$$
$31$ $$4 - 12 T - 6 T^{2} + T^{3}$$
$37$ $$436 - 36 T - 12 T^{2} + T^{3}$$
$41$ $$81 - 81 T + 3 T^{2} + T^{3}$$
$43$ $$76 - 12 T - 6 T^{2} + T^{3}$$
$47$ $$27 + 39 T + 15 T^{2} + T^{3}$$
$53$ $$-72 - 60 T - 6 T^{2} + T^{3}$$
$59$ $$72 - 60 T + 6 T^{2} + T^{3}$$
$61$ $$409 + 63 T - 21 T^{2} + T^{3}$$
$67$ $$151 - 21 T - 9 T^{2} + T^{3}$$
$71$ $$-324 + 108 T + 24 T^{2} + T^{3}$$
$73$ $$( -8 + T )^{3}$$
$79$ $$( -2 + T )^{3}$$
$83$ $$-243 + 129 T - 21 T^{2} + T^{3}$$
$89$ $$( -3 + T )^{3}$$
$97$ $$424 + 36 T - 18 T^{2} + T^{3}$$