# Properties

 Label 1620.2.a.g Level $1620$ Weight $2$ Character orbit 1620.a Self dual yes Analytic conductor $12.936$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (\beta - 1) q^{7}+O(q^{10})$$ q - q^5 + (b - 1) * q^7 $$q - q^{5} + (\beta - 1) q^{7} - \beta q^{11} + ( - 2 \beta + 2) q^{13} + (\beta - 3) q^{17} + (2 \beta - 1) q^{19} - 2 \beta q^{23} + q^{25} + (\beta - 6) q^{29} + ( - 4 \beta - 1) q^{31} + ( - \beta + 1) q^{35} + (3 \beta - 1) q^{37} + (3 \beta - 6) q^{41} + (\beta + 5) q^{43} + ( - \beta - 3) q^{47} + ( - 2 \beta - 3) q^{49} + ( - \beta - 9) q^{53} + \beta q^{55} + (\beta - 6) q^{59} - 4 q^{61} + (2 \beta - 2) q^{65} + ( - 6 \beta - 4) q^{67} + (3 \beta - 6) q^{71} + (3 \beta + 5) q^{73} + (\beta - 3) q^{77} + (6 \beta - 4) q^{79} + (7 \beta - 3) q^{83} + ( - \beta + 3) q^{85} - 3 \beta q^{89} + (4 \beta - 8) q^{91} + ( - 2 \beta + 1) q^{95} + ( - \beta - 1) q^{97} +O(q^{100})$$ q - q^5 + (b - 1) * q^7 - b * q^11 + (-2*b + 2) * q^13 + (b - 3) * q^17 + (2*b - 1) * q^19 - 2*b * q^23 + q^25 + (b - 6) * q^29 + (-4*b - 1) * q^31 + (-b + 1) * q^35 + (3*b - 1) * q^37 + (3*b - 6) * q^41 + (b + 5) * q^43 + (-b - 3) * q^47 + (-2*b - 3) * q^49 + (-b - 9) * q^53 + b * q^55 + (b - 6) * q^59 - 4 * q^61 + (2*b - 2) * q^65 + (-6*b - 4) * q^67 + (3*b - 6) * q^71 + (3*b + 5) * q^73 + (b - 3) * q^77 + (6*b - 4) * q^79 + (7*b - 3) * q^83 + (-b + 3) * q^85 - 3*b * q^89 + (4*b - 8) * q^91 + (-2*b + 1) * q^95 + (-b - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^7 $$2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65} - 8 q^{67} - 12 q^{71} + 10 q^{73} - 6 q^{77} - 8 q^{79} - 6 q^{83} + 6 q^{85} - 16 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 - 6 * q^17 - 2 * q^19 + 2 * q^25 - 12 * q^29 - 2 * q^31 + 2 * q^35 - 2 * q^37 - 12 * q^41 + 10 * q^43 - 6 * q^47 - 6 * q^49 - 18 * q^53 - 12 * q^59 - 8 * q^61 - 4 * q^65 - 8 * q^67 - 12 * q^71 + 10 * q^73 - 6 * q^77 - 8 * q^79 - 6 * q^83 + 6 * q^85 - 16 * q^91 + 2 * q^95 - 2 * q^97

## 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
 −1.73205 1.73205
0 0 0 −1.00000 0 −2.73205 0 0 0
1.2 0 0 0 −1.00000 0 0.732051 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.g 2
3.b odd 2 1 1620.2.a.h yes 2
4.b odd 2 1 6480.2.a.bh 2
5.b even 2 1 8100.2.a.t 2
5.c odd 4 2 8100.2.d.m 4
9.c even 3 2 1620.2.i.n 4
9.d odd 6 2 1620.2.i.m 4
12.b even 2 1 6480.2.a.bp 2
15.d odd 2 1 8100.2.a.s 2
15.e even 4 2 8100.2.d.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 1.a even 1 1 trivial
1620.2.a.h yes 2 3.b odd 2 1
1620.2.i.m 4 9.d odd 6 2
1620.2.i.n 4 9.c even 3 2
6480.2.a.bh 2 4.b odd 2 1
6480.2.a.bp 2 12.b even 2 1
8100.2.a.s 2 15.d odd 2 1
8100.2.a.t 2 5.b even 2 1
8100.2.d.l 4 15.e even 4 2
8100.2.d.m 4 5.c odd 4 2

## Hecke kernels

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

 $$T_{7}^{2} + 2T_{7} - 2$$ T7^2 + 2*T7 - 2 $$T_{11}^{2} - 3$$ T11^2 - 3 $$T_{17}^{2} + 6T_{17} + 6$$ T17^2 + 6*T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 2T - 2$$
$11$ $$T^{2} - 3$$
$13$ $$T^{2} - 4T - 8$$
$17$ $$T^{2} + 6T + 6$$
$19$ $$T^{2} + 2T - 11$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} + 12T + 33$$
$31$ $$T^{2} + 2T - 47$$
$37$ $$T^{2} + 2T - 26$$
$41$ $$T^{2} + 12T + 9$$
$43$ $$T^{2} - 10T + 22$$
$47$ $$T^{2} + 6T + 6$$
$53$ $$T^{2} + 18T + 78$$
$59$ $$T^{2} + 12T + 33$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 8T - 92$$
$71$ $$T^{2} + 12T + 9$$
$73$ $$T^{2} - 10T - 2$$
$79$ $$T^{2} + 8T - 92$$
$83$ $$T^{2} + 6T - 138$$
$89$ $$T^{2} - 27$$
$97$ $$T^{2} + 2T - 2$$