# Properties

 Label 2160.4.a.br 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: 3.3.4281.1 Defining polynomial: $$x^{3} - x^{2} - 12x + 15$$ x^3 - x^2 - 12*x + 15 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.

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + ( - \beta_1 + 3) q^{7}+O(q^{10})$$ q + 5 * q^5 + (-b1 + 3) * q^7 $$q + 5 q^{5} + ( - \beta_1 + 3) q^{7} + (\beta_{2} - \beta_1 + 4) q^{11} + ( - \beta_{2} - \beta_1 + 16) q^{13} + (\beta_1 + 12) q^{17} + (\beta_{2} - \beta_1 - 9) q^{19} + ( - \beta_{2} - 2 \beta_1 + 4) q^{23} + 25 q^{25} + (2 \beta_{2} + 3 \beta_1 + 9) q^{29} + ( - 2 \beta_{2} - \beta_1 - 14) q^{31} + ( - 5 \beta_1 + 15) q^{35} + (2 \beta_{2} - 8 \beta_1 + 80) q^{37} + (4 \beta_{2} - \beta_1 + 125) q^{41} + ( - 6 \beta_{2} - 11 \beta_1 + 45) q^{43} + (9 \beta_{2} - 4 \beta_1 + 23) q^{47} + ( - 3 \beta_{2} - 20 \beta_1 + 188) q^{49} + ( - \beta_{2} - 9 \beta_1 + 271) q^{53} + (5 \beta_{2} - 5 \beta_1 + 20) q^{55} + ( - 11 \beta_{2} + 19 \beta_1 - 202) q^{59} + (6 \beta_{2} + 22 \beta_1 - 91) q^{61} + ( - 5 \beta_{2} - 5 \beta_1 + 80) q^{65} + ( - 2 \beta_{2} + 18 \beta_1 - 2) q^{67} + (5 \beta_{2} + 9 \beta_1 - 414) q^{71} + ( - \beta_{2} + 3 \beta_1 - 134) q^{73} + ( - 15 \beta_{2} - 6 \beta_1 + 429) q^{77} + (17 \beta_{2} + 5 \beta_1 + 111) q^{79} + ( - 13 \beta_{2} + 26 \beta_1 - 242) q^{83} + (5 \beta_1 + 60) q^{85} + ( - 15 \beta_1 + 791) q^{89} + (9 \beta_{2} - 48 \beta_1 + 675) q^{91} + (5 \beta_{2} - 5 \beta_1 - 45) q^{95} + ( - 14 \beta_{2} - 8 \beta_1 + 190) q^{97}+O(q^{100})$$ q + 5 * q^5 + (-b1 + 3) * q^7 + (b2 - b1 + 4) * q^11 + (-b2 - b1 + 16) * q^13 + (b1 + 12) * q^17 + (b2 - b1 - 9) * q^19 + (-b2 - 2*b1 + 4) * q^23 + 25 * q^25 + (2*b2 + 3*b1 + 9) * q^29 + (-2*b2 - b1 - 14) * q^31 + (-5*b1 + 15) * q^35 + (2*b2 - 8*b1 + 80) * q^37 + (4*b2 - b1 + 125) * q^41 + (-6*b2 - 11*b1 + 45) * q^43 + (9*b2 - 4*b1 + 23) * q^47 + (-3*b2 - 20*b1 + 188) * q^49 + (-b2 - 9*b1 + 271) * q^53 + (5*b2 - 5*b1 + 20) * q^55 + (-11*b2 + 19*b1 - 202) * q^59 + (6*b2 + 22*b1 - 91) * q^61 + (-5*b2 - 5*b1 + 80) * q^65 + (-2*b2 + 18*b1 - 2) * q^67 + (5*b2 + 9*b1 - 414) * q^71 + (-b2 + 3*b1 - 134) * q^73 + (-15*b2 - 6*b1 + 429) * q^77 + (17*b2 + 5*b1 + 111) * q^79 + (-13*b2 + 26*b1 - 242) * q^83 + (5*b1 + 60) * q^85 + (-15*b1 + 791) * q^89 + (9*b2 - 48*b1 + 675) * q^91 + (5*b2 - 5*b1 - 45) * q^95 + (-14*b2 - 8*b1 + 190) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} + 8 q^{7}+O(q^{10})$$ 3 * q + 15 * q^5 + 8 * q^7 $$3 q + 15 q^{5} + 8 q^{7} + 10 q^{11} + 48 q^{13} + 37 q^{17} - 29 q^{19} + 11 q^{23} + 75 q^{25} + 28 q^{29} - 41 q^{31} + 40 q^{35} + 230 q^{37} + 370 q^{41} + 130 q^{43} + 56 q^{47} + 547 q^{49} + 805 q^{53} + 50 q^{55} - 576 q^{59} - 257 q^{61} + 240 q^{65} + 14 q^{67} - 1238 q^{71} - 398 q^{73} + 1296 q^{77} + 321 q^{79} - 687 q^{83} + 185 q^{85} + 2358 q^{89} + 1968 q^{91} - 145 q^{95} + 576 q^{97}+O(q^{100})$$ 3 * q + 15 * q^5 + 8 * q^7 + 10 * q^11 + 48 * q^13 + 37 * q^17 - 29 * q^19 + 11 * q^23 + 75 * q^25 + 28 * q^29 - 41 * q^31 + 40 * q^35 + 230 * q^37 + 370 * q^41 + 130 * q^43 + 56 * q^47 + 547 * q^49 + 805 * q^53 + 50 * q^55 - 576 * q^59 - 257 * q^61 + 240 * q^65 + 14 * q^67 - 1238 * q^71 - 398 * q^73 + 1296 * q^77 + 321 * q^79 - 687 * q^83 + 185 * q^85 + 2358 * q^89 + 1968 * q^91 - 145 * q^95 + 576 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 12x + 15$$ :

 $$\beta_{1}$$ $$=$$ $$-2\nu^{2} + 6\nu + 15$$ -2*v^2 + 6*v + 15 $$\beta_{2}$$ $$=$$ $$8\nu^{2} + 12\nu - 71$$ 8*v^2 + 12*v - 71
 $$\nu$$ $$=$$ $$( \beta_{2} + 4\beta _1 + 11 ) / 36$$ (b2 + 4*b1 + 11) / 36 $$\nu^{2}$$ $$=$$ $$( \beta_{2} - 2\beta _1 + 101 ) / 12$$ (b2 - 2*b1 + 101) / 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
 1.29027 3.26757 −3.55784
0 0 0 5.00000 0 −16.4120 0 0 0
1.2 0 0 0 5.00000 0 −10.2514 0 0 0
1.3 0 0 0 5.00000 0 34.6634 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.br 3
3.b odd 2 1 2160.4.a.bj 3
4.b odd 2 1 1080.4.a.k yes 3
12.b even 2 1 1080.4.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.e 3 12.b even 2 1
1080.4.a.k yes 3 4.b odd 2 1
2160.4.a.bj 3 3.b odd 2 1
2160.4.a.br 3 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{7}^{3} - 8T_{7}^{2} - 756T_{7} - 5832$$ T7^3 - 8*T7^2 - 756*T7 - 5832 $$T_{11}^{3} - 10T_{11}^{2} - 2864T_{11} + 59400$$ T11^3 - 10*T11^2 - 2864*T11 + 59400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 5)^{3}$$
$7$ $$T^{3} - 8 T^{2} - 756 T - 5832$$
$11$ $$T^{3} - 10 T^{2} - 2864 T + 59400$$
$13$ $$T^{3} - 48 T^{2} - 2700 T + 118584$$
$17$ $$T^{3} - 37 T^{2} - 321 T + 15597$$
$19$ $$T^{3} + 29 T^{2} - 2617 T + 22675$$
$23$ $$T^{3} - 11 T^{2} - 6045 T + 44775$$
$29$ $$T^{3} - 28 T^{2} - 18068 T - 296760$$
$31$ $$T^{3} + 41 T^{2} - 10409 T + 291591$$
$37$ $$T^{3} - 230 T^{2} - 37172 T + 4001784$$
$41$ $$T^{3} - 370 T^{2} + 7512 T + 2205432$$
$43$ $$T^{3} - 130 T^{2} + \cdots + 16735752$$
$47$ $$T^{3} - 56 T^{2} - 195952 T + 7432640$$
$53$ $$T^{3} - 805 T^{2} + 148071 T - 7723035$$
$59$ $$T^{3} + 576 T^{2} + \cdots - 227290536$$
$61$ $$T^{3} + 257 T^{2} + \cdots + 37318467$$
$67$ $$T^{3} - 14 T^{2} - 251140 T + 30539000$$
$71$ $$T^{3} + 1238 T^{2} + \cdots + 9117432$$
$73$ $$T^{3} + 398 T^{2} + 44256 T + 1074888$$
$79$ $$T^{3} - 321 T^{2} + \cdots - 143418599$$
$83$ $$T^{3} + 687 T^{2} + \cdots - 435938355$$
$89$ $$T^{3} - 2358 T^{2} + \cdots - 374731256$$
$97$ $$T^{3} - 576 T^{2} + \cdots + 257454592$$