# Properties

 Label 2160.4.a.bl Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.697.1 Defining polynomial: $$x^{3} - 7x - 5$$ x^3 - 7*x - 5 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.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + (\beta_{2} + 8) q^{7}+O(q^{10})$$ q - 5 * q^5 + (b2 + 8) * q^7 $$q - 5 q^{5} + (\beta_{2} + 8) q^{7} + (3 \beta_{2} - \beta_1 + 2) q^{11} + ( - 3 \beta_{2} - \beta_1 + 16) q^{13} + (\beta_{2} - 4 \beta_1 + 9) q^{17} + ( - 3 \beta_{2} - 3 \beta_1 + 65) q^{19} + (10 \beta_{2} - 3 \beta_1 - 9) q^{23} + 25 q^{25} + (3 \beta_{2} - 2 \beta_1 - 20) q^{29} + (\beta_{2} - 2 \beta_1 + 93) q^{31} + ( - 5 \beta_{2} - 40) q^{35} + (8 \beta_{2} + 2 \beta_1 - 46) q^{37} + ( - \beta_{2} + 12 \beta_1 - 22) q^{41} + (11 \beta_{2} + 10 \beta_1 - 74) q^{43} + ( - 24 \beta_{2} + 7 \beta_1 - 88) q^{47} + (8 \beta_{2} - 3 \beta_1 - 79) q^{49} + ( - 5 \beta_{2} + \beta_1 + 169) q^{53} + ( - 15 \beta_{2} + 5 \beta_1 - 10) q^{55} + (15 \beta_{2} + 3 \beta_1 - 320) q^{59} + (42 \beta_{2} + 2 \beta_1 + 181) q^{61} + (15 \beta_{2} + 5 \beta_1 - 80) q^{65} + (46 \beta_{2} + 6 \beta_1 + 362) q^{67} + ( - 3 \beta_{2} - 5 \beta_1 - 606) q^{71} + ( - 23 \beta_{2} - 9 \beta_1 + 454) q^{73} + (14 \beta_{2} - 25 \beta_1 + 592) q^{77} + ( - 17 \beta_{2} - 15 \beta_1 + 43) q^{79} + ( - 2 \beta_{2} + 13 \beta_1 - 523) q^{83} + ( - 5 \beta_{2} + 20 \beta_1 - 45) q^{85} + ( - 55 \beta_{2} + 16 \beta_1 + 590) q^{89} + (28 \beta_{2} - 7 \beta_1 - 496) q^{91} + (15 \beta_{2} + 15 \beta_1 - 325) q^{95} + ( - 56 \beta_{2} + 26 \beta_1 - 112) q^{97}+O(q^{100})$$ q - 5 * q^5 + (b2 + 8) * q^7 + (3*b2 - b1 + 2) * q^11 + (-3*b2 - b1 + 16) * q^13 + (b2 - 4*b1 + 9) * q^17 + (-3*b2 - 3*b1 + 65) * q^19 + (10*b2 - 3*b1 - 9) * q^23 + 25 * q^25 + (3*b2 - 2*b1 - 20) * q^29 + (b2 - 2*b1 + 93) * q^31 + (-5*b2 - 40) * q^35 + (8*b2 + 2*b1 - 46) * q^37 + (-b2 + 12*b1 - 22) * q^41 + (11*b2 + 10*b1 - 74) * q^43 + (-24*b2 + 7*b1 - 88) * q^47 + (8*b2 - 3*b1 - 79) * q^49 + (-5*b2 + b1 + 169) * q^53 + (-15*b2 + 5*b1 - 10) * q^55 + (15*b2 + 3*b1 - 320) * q^59 + (42*b2 + 2*b1 + 181) * q^61 + (15*b2 + 5*b1 - 80) * q^65 + (46*b2 + 6*b1 + 362) * q^67 + (-3*b2 - 5*b1 - 606) * q^71 + (-23*b2 - 9*b1 + 454) * q^73 + (14*b2 - 25*b1 + 592) * q^77 + (-17*b2 - 15*b1 + 43) * q^79 + (-2*b2 + 13*b1 - 523) * q^83 + (-5*b2 + 20*b1 - 45) * q^85 + (-55*b2 + 16*b1 + 590) * q^89 + (28*b2 - 7*b1 - 496) * q^91 + (15*b2 + 15*b1 - 325) * q^95 + (-56*b2 + 26*b1 - 112) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} + 24 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 + 24 * q^7 $$3 q - 15 q^{5} + 24 q^{7} + 6 q^{11} + 48 q^{13} + 27 q^{17} + 195 q^{19} - 27 q^{23} + 75 q^{25} - 60 q^{29} + 279 q^{31} - 120 q^{35} - 138 q^{37} - 66 q^{41} - 222 q^{43} - 264 q^{47} - 237 q^{49} + 507 q^{53} - 30 q^{55} - 960 q^{59} + 543 q^{61} - 240 q^{65} + 1086 q^{67} - 1818 q^{71} + 1362 q^{73} + 1776 q^{77} + 129 q^{79} - 1569 q^{83} - 135 q^{85} + 1770 q^{89} - 1488 q^{91} - 975 q^{95} - 336 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 + 24 * q^7 + 6 * q^11 + 48 * q^13 + 27 * q^17 + 195 * q^19 - 27 * q^23 + 75 * q^25 - 60 * q^29 + 279 * q^31 - 120 * q^35 - 138 * q^37 - 66 * q^41 - 222 * q^43 - 264 * q^47 - 237 * q^49 + 507 * q^53 - 30 * q^55 - 960 * q^59 + 543 * q^61 - 240 * q^65 + 1086 * q^67 - 1818 * q^71 + 1362 * q^73 + 1776 * q^77 + 129 * q^79 - 1569 * q^83 - 135 * q^85 + 1770 * q^89 - 1488 * q^91 - 975 * q^95 - 336 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 7x - 5$$ :

 $$\beta_{1}$$ $$=$$ $$12\nu$$ 12*v $$\beta_{2}$$ $$=$$ $$6\nu^{2} - 6\nu - 28$$ 6*v^2 - 6*v - 28
 $$\nu$$ $$=$$ $$( \beta_1 ) / 12$$ (b1) / 12 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 56 ) / 12$$ (2*b2 + b1 + 56) / 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
 −0.782816 2.94883 −2.16601
0 0 0 −5.00000 0 −11.6263 0 0 0
1.2 0 0 0 −5.00000 0 14.4806 0 0 0
1.3 0 0 0 −5.00000 0 21.1457 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.bl 3
3.b odd 2 1 2160.4.a.bt 3
4.b odd 2 1 1080.4.a.c 3
12.b even 2 1 1080.4.a.i yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.c 3 4.b odd 2 1
1080.4.a.i yes 3 12.b even 2 1
2160.4.a.bl 3 1.a even 1 1 trivial
2160.4.a.bt 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{3} - 24T_{7}^{2} - 108T_{7} + 3560$$ T7^3 - 24*T7^2 - 108*T7 + 3560 $$T_{11}^{3} - 6T_{11}^{2} - 3480T_{11} - 44648$$ T11^3 - 6*T11^2 - 3480*T11 - 44648

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} - 24 T^{2} - 108 T + 3560$$
$11$ $$T^{3} - 6 T^{2} - 3480 T - 44648$$
$13$ $$T^{3} - 48 T^{2} - 3156 T + 8360$$
$17$ $$T^{3} - 27 T^{2} - 15897 T + 428443$$
$19$ $$T^{3} - 195 T^{2} + 255 T + 954091$$
$23$ $$T^{3} + 27 T^{2} - 36669 T - 1787159$$
$29$ $$T^{3} + 60 T^{2} - 5100 T - 306136$$
$31$ $$T^{3} - 279 T^{2} + 21759 T - 418385$$
$37$ $$T^{3} + 138 T^{2} - 18036 T + 122040$$
$41$ $$T^{3} + 66 T^{2} - 143136 T - 15824376$$
$43$ $$T^{3} + 222 T^{2} + \cdots - 25516184$$
$47$ $$T^{3} + 264 T^{2} - 186864 T + 774208$$
$53$ $$T^{3} - 507 T^{2} + 77535 T - 3425517$$
$59$ $$T^{3} + 960 T^{2} + \cdots + 15048440$$
$61$ $$T^{3} - 543 T^{2} + \cdots + 236291075$$
$67$ $$T^{3} - 1086 T^{2} + \cdots + 422390264$$
$71$ $$T^{3} + 1818 T^{2} + \cdots + 206864712$$
$73$ $$T^{3} - 1362 T^{2} + \cdots + 5163048$$
$79$ $$T^{3} - 129 T^{2} + \cdots + 64805697$$
$83$ $$T^{3} + 1569 T^{2} + \cdots + 40821251$$
$89$ $$T^{3} - 1770 T^{2} + \cdots + 666862200$$
$97$ $$T^{3} + 336 T^{2} + \cdots + 503900672$$