# Properties

 Label 2160.4.a.bi 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: 3.3.1772.1 Defining polynomial: $$x^{3} - x^{2} - 12x + 8$$ x^3 - x^2 - 12*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 135) 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.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + (\beta_{2} + 3 \beta_1) q^{7}+O(q^{10})$$ q - 5 * q^5 + (b2 + 3*b1) * q^7 $$q - 5 q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + (\beta_{2} - 5 \beta_1 + 3) q^{11} + ( - \beta_{2} + 2 \beta_1 + 2) q^{13} + (2 \beta_{2} + 5 \beta_1 - 54) q^{17} + ( - 8 \beta_{2} + \beta_1 + 19) q^{19} + (5 \beta_{2} - 5 \beta_1 + 95) q^{23} + 25 q^{25} + ( - 6 \beta_{2} - 25 \beta_1 - 28) q^{29} + ( - 9 \beta_{2} - 17 \beta_1 + 47) q^{31} + ( - 5 \beta_{2} - 15 \beta_1) q^{35} + (20 \beta_{2} + 25 \beta_1 - 143) q^{37} + ( - 9 \beta_{2} - 10 \beta_1 - 187) q^{41} + (3 \beta_{2} + 29 \beta_1 + 255) q^{43} + ( - 32 \beta_{2} - 5 \beta_1 - 36) q^{47} + (3 \beta_{2} - 41 \beta_1 + 205) q^{49} + ( - 17 \beta_{2} - 30 \beta_1 + 149) q^{53} + ( - 5 \beta_{2} + 25 \beta_1 - 15) q^{55} + (17 \beta_{2} + 121) q^{59} + ( - 3 \beta_{2} + 31 \beta_1 - 60) q^{61} + (5 \beta_{2} - 10 \beta_1 - 10) q^{65} + (31 \beta_{2} + 3 \beta_1 - 12) q^{67} + ( - 53 \beta_{2} - 30 \beta_1 + 41) q^{71} + ( - 14 \beta_{2} + 3 \beta_1 - 323) q^{73} + (46 \beta_{2} + 80 \beta_1 - 692) q^{77} + ( - 23 \beta_{2} - 54 \beta_1 - 312) q^{79} + (21 \beta_{2} + 60 \beta_1 + 63) q^{83} + ( - 10 \beta_{2} - 25 \beta_1 + 270) q^{85} + ( - 15 \beta_{2} - 60 \beta_1 - 315) q^{89} + ( - 26 \beta_{2} - 23 \beta_1 + 227) q^{91} + (40 \beta_{2} - 5 \beta_1 - 95) q^{95} + (28 \beta_{2} - \beta_1 + 231) q^{97}+O(q^{100})$$ q - 5 * q^5 + (b2 + 3*b1) * q^7 + (b2 - 5*b1 + 3) * q^11 + (-b2 + 2*b1 + 2) * q^13 + (2*b2 + 5*b1 - 54) * q^17 + (-8*b2 + b1 + 19) * q^19 + (5*b2 - 5*b1 + 95) * q^23 + 25 * q^25 + (-6*b2 - 25*b1 - 28) * q^29 + (-9*b2 - 17*b1 + 47) * q^31 + (-5*b2 - 15*b1) * q^35 + (20*b2 + 25*b1 - 143) * q^37 + (-9*b2 - 10*b1 - 187) * q^41 + (3*b2 + 29*b1 + 255) * q^43 + (-32*b2 - 5*b1 - 36) * q^47 + (3*b2 - 41*b1 + 205) * q^49 + (-17*b2 - 30*b1 + 149) * q^53 + (-5*b2 + 25*b1 - 15) * q^55 + (17*b2 + 121) * q^59 + (-3*b2 + 31*b1 - 60) * q^61 + (5*b2 - 10*b1 - 10) * q^65 + (31*b2 + 3*b1 - 12) * q^67 + (-53*b2 - 30*b1 + 41) * q^71 + (-14*b2 + 3*b1 - 323) * q^73 + (46*b2 + 80*b1 - 692) * q^77 + (-23*b2 - 54*b1 - 312) * q^79 + (21*b2 + 60*b1 + 63) * q^83 + (-10*b2 - 25*b1 + 270) * q^85 + (-15*b2 - 60*b1 - 315) * q^89 + (-26*b2 - 23*b1 + 227) * q^91 + (40*b2 - 5*b1 - 95) * q^95 + (28*b2 - b1 + 231) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} + 4 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 + 4 * q^7 $$3 q - 15 q^{5} + 4 q^{7} + 5 q^{11} + 7 q^{13} - 155 q^{17} + 50 q^{19} + 285 q^{23} + 75 q^{25} - 115 q^{29} + 115 q^{31} - 20 q^{35} - 384 q^{37} - 580 q^{41} + 797 q^{43} - 145 q^{47} + 577 q^{49} + 400 q^{53} - 25 q^{55} + 380 q^{59} - 152 q^{61} - 35 q^{65} - 2 q^{67} + 40 q^{71} - 980 q^{73} - 1950 q^{77} - 1013 q^{79} + 270 q^{83} + 775 q^{85} - 1020 q^{89} + 632 q^{91} - 250 q^{95} + 720 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 + 4 * q^7 + 5 * q^11 + 7 * q^13 - 155 * q^17 + 50 * q^19 + 285 * q^23 + 75 * q^25 - 115 * q^29 + 115 * q^31 - 20 * q^35 - 384 * q^37 - 580 * q^41 + 797 * q^43 - 145 * q^47 + 577 * q^49 + 400 * q^53 - 25 * q^55 + 380 * q^59 - 152 * q^61 - 35 * q^65 - 2 * q^67 + 40 * q^71 - 980 * q^73 - 1950 * q^77 - 1013 * q^79 + 270 * q^83 + 775 * q^85 - 1020 * q^89 + 632 * q^91 - 250 * q^95 + 720 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 12x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{2} + 2\nu + 8$$ -v^2 + 2*v + 8 $$\beta_{2}$$ $$=$$ $$2\nu^{2} + 2\nu - 17$$ 2*v^2 + 2*v - 17
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 1 ) / 6$$ (b2 + 2*b1 + 1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{2} - \beta _1 + 25 ) / 3$$ (b2 - b1 + 25) / 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
 −3.32803 0.654334 3.67370
0 0 0 −5.00000 0 −30.7000 0 0 0
1.2 0 0 0 −5.00000 0 11.8065 0 0 0
1.3 0 0 0 −5.00000 0 22.8935 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.bi 3
3.b odd 2 1 2160.4.a.bq 3
4.b odd 2 1 135.4.a.e 3
12.b even 2 1 135.4.a.h yes 3
20.d odd 2 1 675.4.a.s 3
20.e even 4 2 675.4.b.m 6
36.f odd 6 2 405.4.e.v 6
36.h even 6 2 405.4.e.q 6
60.h even 2 1 675.4.a.p 3
60.l odd 4 2 675.4.b.n 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 4.b odd 2 1
135.4.a.h yes 3 12.b even 2 1
405.4.e.q 6 36.h even 6 2
405.4.e.v 6 36.f odd 6 2
675.4.a.p 3 60.h even 2 1
675.4.a.s 3 20.d odd 2 1
675.4.b.m 6 20.e even 4 2
675.4.b.n 6 60.l odd 4 2
2160.4.a.bi 3 1.a even 1 1 trivial
2160.4.a.bq 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{3} - 4T_{7}^{2} - 795T_{7} + 8298$$ T7^3 - 4*T7^2 - 795*T7 + 8298 $$T_{11}^{3} - 5T_{11}^{2} - 2888T_{11} + 31260$$ T11^3 - 5*T11^2 - 2888*T11 + 31260

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} - 4 T^{2} - 795 T + 8298$$
$11$ $$T^{3} - 5 T^{2} - 2888 T + 31260$$
$13$ $$T^{3} - 7 T^{2} - 769 T - 6425$$
$17$ $$T^{3} + 155 T^{2} + 5608 T + 41760$$
$19$ $$T^{3} - 50 T^{2} - 16663 T + 368012$$
$23$ $$T^{3} - 285 T^{2} + 16200 T + 553500$$
$29$ $$T^{3} + 115 T^{2} - 47408 T - 6440340$$
$31$ $$T^{3} - 115 T^{2} - 29232 T + 938304$$
$37$ $$T^{3} + 384 T^{2} + \cdots - 22667198$$
$41$ $$T^{3} + 580 T^{2} + 89812 T + 3917280$$
$43$ $$T^{3} - 797 T^{2} + 142520 T + 5357936$$
$47$ $$T^{3} + 145 T^{2} + \cdots + 14388240$$
$53$ $$T^{3} - 400 T^{2} + \cdots + 12658320$$
$59$ $$T^{3} - 380 T^{2} - 27392 T + 5205120$$
$61$ $$T^{3} + 152 T^{2} - 87475 T - 5069066$$
$67$ $$T^{3} + 2 T^{2} - 243999 T - 20769300$$
$71$ $$T^{3} - 40 T^{2} + \cdots + 216071280$$
$73$ $$T^{3} + 980 T^{2} + \cdots + 16447954$$
$79$ $$T^{3} + 1013 T^{2} + \cdots - 90596925$$
$83$ $$T^{3} - 270 T^{2} + \cdots + 84539160$$
$89$ $$T^{3} + 1020 T^{2} + \cdots - 125064000$$
$97$ $$T^{3} - 720 T^{2} + \cdots + 27430558$$