# Properties

 Label 2160.4.a.bg Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.985.1 Defining polynomial: $$x^{3} - x^{2} - 6x + 1$$ x^3 - x^2 - 6*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1080) 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.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + (\beta_{2} - 2) q^{7}+O(q^{10})$$ q - 5 * q^5 + (b2 - 2) * q^7 $$q - 5 q^{5} + (\beta_{2} - 2) q^{7} + ( - \beta_{2} - \beta_1 + 4) q^{11} + ( - \beta_{2} - \beta_1 + 6) q^{13} + (3 \beta_{2} + 2 \beta_1 + 7) q^{17} + ( - 3 \beta_{2} + 3 \beta_1 - 19) q^{19} + ( - 2 \beta_{2} + 3 \beta_1 + 29) q^{23} + 25 q^{25} + (\beta_{2} - 8 \beta_1 - 46) q^{29} + (5 \beta_{2} - 8 \beta_1 - 39) q^{31} + ( - 5 \beta_{2} + 10) q^{35} + (4 \beta_{2} + 8 \beta_1 + 50) q^{37} + (5 \beta_{2} + 12 \beta_1) q^{41} + ( - 21 \beta_{2} + 4 \beta_1 - 60) q^{43} + (16 \beta_{2} + \beta_1 + 228) q^{47} + ( - 16 \beta_{2} - 3 \beta_1 - 27) q^{49} + ( - 19 \beta_{2} + \beta_1 - 29) q^{53} + (5 \beta_{2} + 5 \beta_1 - 20) q^{55} + (15 \beta_{2} - 3 \beta_1 + 238) q^{59} + ( - 22 \beta_{2} - 10 \beta_1 - 171) q^{61} + (5 \beta_{2} + 5 \beta_1 - 30) q^{65} + ( - 14 \beta_{2} + 12 \beta_1 + 58) q^{67} + ( - 11 \beta_{2} + 19 \beta_1 + 256) q^{71} + (7 \beta_{2} + 15 \beta_1 - 84) q^{73} + (22 \beta_{2} - 7 \beta_1 - 296) q^{77} + ( - 17 \beta_{2} - 21 \beta_1 - 69) q^{79} + ( - 10 \beta_{2} - 17 \beta_1 + 563) q^{83} + ( - 15 \beta_{2} - 10 \beta_1 - 35) q^{85} + ( - 33 \beta_{2} - 14 \beta_1 - 104) q^{89} + (24 \beta_{2} - 7 \beta_1 - 300) q^{91} + (15 \beta_{2} - 15 \beta_1 + 95) q^{95} + (60 \beta_{2} - 4 \beta_1 - 360) q^{97}+O(q^{100})$$ q - 5 * q^5 + (b2 - 2) * q^7 + (-b2 - b1 + 4) * q^11 + (-b2 - b1 + 6) * q^13 + (3*b2 + 2*b1 + 7) * q^17 + (-3*b2 + 3*b1 - 19) * q^19 + (-2*b2 + 3*b1 + 29) * q^23 + 25 * q^25 + (b2 - 8*b1 - 46) * q^29 + (5*b2 - 8*b1 - 39) * q^31 + (-5*b2 + 10) * q^35 + (4*b2 + 8*b1 + 50) * q^37 + (5*b2 + 12*b1) * q^41 + (-21*b2 + 4*b1 - 60) * q^43 + (16*b2 + b1 + 228) * q^47 + (-16*b2 - 3*b1 - 27) * q^49 + (-19*b2 + b1 - 29) * q^53 + (5*b2 + 5*b1 - 20) * q^55 + (15*b2 - 3*b1 + 238) * q^59 + (-22*b2 - 10*b1 - 171) * q^61 + (5*b2 + 5*b1 - 30) * q^65 + (-14*b2 + 12*b1 + 58) * q^67 + (-11*b2 + 19*b1 + 256) * q^71 + (7*b2 + 15*b1 - 84) * q^73 + (22*b2 - 7*b1 - 296) * q^77 + (-17*b2 - 21*b1 - 69) * q^79 + (-10*b2 - 17*b1 + 563) * q^83 + (-15*b2 - 10*b1 - 35) * q^85 + (-33*b2 - 14*b1 - 104) * q^89 + (24*b2 - 7*b1 - 300) * q^91 + (15*b2 - 15*b1 + 95) * q^95 + (60*b2 - 4*b1 - 360) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} - 6 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 - 6 * q^7 $$3 q - 15 q^{5} - 6 q^{7} + 12 q^{11} + 18 q^{13} + 21 q^{17} - 57 q^{19} + 87 q^{23} + 75 q^{25} - 138 q^{29} - 117 q^{31} + 30 q^{35} + 150 q^{37} - 180 q^{43} + 684 q^{47} - 81 q^{49} - 87 q^{53} - 60 q^{55} + 714 q^{59} - 513 q^{61} - 90 q^{65} + 174 q^{67} + 768 q^{71} - 252 q^{73} - 888 q^{77} - 207 q^{79} + 1689 q^{83} - 105 q^{85} - 312 q^{89} - 900 q^{91} + 285 q^{95} - 1080 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 - 6 * q^7 + 12 * q^11 + 18 * q^13 + 21 * q^17 - 57 * q^19 + 87 * q^23 + 75 * q^25 - 138 * q^29 - 117 * q^31 + 30 * q^35 + 150 * q^37 - 180 * q^43 + 684 * q^47 - 81 * q^49 - 87 * q^53 - 60 * q^55 + 714 * q^59 - 513 * q^61 - 90 * q^65 + 174 * q^67 + 768 * q^71 - 252 * q^73 - 888 * q^77 - 207 * q^79 + 1689 * q^83 - 105 * q^85 - 312 * q^89 - 900 * q^91 + 285 * q^95 - 1080 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$12\nu - 4$$ 12*v - 4 $$\beta_{2}$$ $$=$$ $$6\nu^{2} - 6\nu - 24$$ 6*v^2 - 6*v - 24
 $$\nu$$ $$=$$ $$( \beta _1 + 4 ) / 12$$ (b1 + 4) / 12 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 52 ) / 12$$ (2*b2 + b1 + 52) / 12

## 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
 0.162962 2.93080 −2.09376
0 0 0 −5.00000 0 −26.8184 0 0 0
1.2 0 0 0 −5.00000 0 7.95278 0 0 0
1.3 0 0 0 −5.00000 0 12.8657 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 2160.4.a.bg 3
3.b odd 2 1 2160.4.a.bo 3
4.b odd 2 1 1080.4.a.g 3
12.b even 2 1 1080.4.a.m yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.g 3 4.b odd 2 1
1080.4.a.m yes 3 12.b even 2 1
2160.4.a.bg 3 1.a even 1 1 trivial
2160.4.a.bo 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{3} + 6T_{7}^{2} - 456T_{7} + 2744$$ T7^3 + 6*T7^2 - 456*T7 + 2744 $$T_{11}^{3} - 12T_{11}^{2} - 1260T_{11} + 20920$$ T11^3 - 12*T11^2 - 1260*T11 + 20920

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} + 6 T^{2} - 456 T + 2744$$
$11$ $$T^{3} - 12 T^{2} - 1260 T + 20920$$
$13$ $$T^{3} - 18 T^{2} - 1200 T + 23384$$
$17$ $$T^{3} - 21 T^{2} - 7281 T - 47219$$
$19$ $$T^{3} + 57 T^{2} - 11985 T + 332479$$
$23$ $$T^{3} - 87 T^{2} - 7989 T + 655435$$
$29$ $$T^{3} + 138 T^{2} - 53064 T - 3137800$$
$31$ $$T^{3} + 117 T^{2} - 68385 T - 9394301$$
$37$ $$T^{3} - 150 T^{2} - 56052 T - 2749896$$
$41$ $$T^{3} - 138708 T - 17332488$$
$43$ $$T^{3} + 180 T^{2} + \cdots - 31951208$$
$47$ $$T^{3} - 684 T^{2} + \cdots + 31272640$$
$53$ $$T^{3} + 87 T^{2} - 168705 T - 28044495$$
$59$ $$T^{3} - 714 T^{2} + \cdots + 20643704$$
$61$ $$T^{3} + 513 T^{2} + \cdots - 57378877$$
$67$ $$T^{3} - 174 T^{2} + \cdots + 55705400$$
$71$ $$T^{3} - 768 T^{2} + \cdots + 166892184$$
$73$ $$T^{3} + 252 T^{2} + \cdots - 54480168$$
$79$ $$T^{3} + 207 T^{2} + \cdots + 102453849$$
$83$ $$T^{3} - 1689 T^{2} + \cdots + 51094085$$
$89$ $$T^{3} + 312 T^{2} + \cdots - 120696696$$
$97$ $$T^{3} + 1080 T^{2} + \cdots + 134280704$$