# Properties

 Label 2160.4.a.be 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: 3.3.5637.1 Defining polynomial: $$x^{3} - x^{2} - 23x + 6$$ x^3 - x^2 - 23*x + 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + ( - \beta_1 - 15) q^{7}+O(q^{10})$$ q - 5 * q^5 + (-b1 - 15) * q^7 $$q - 5 q^{5} + ( - \beta_1 - 15) q^{7} + (\beta_{2} - \beta_1 + 12) q^{11} + ( - \beta_{2} - 5 \beta_1 + 8) q^{13} + (2 \beta_{2} + \beta_1 + 6) q^{17} + (\beta_{2} - 7 \beta_1 - 65) q^{19} + (\beta_{2} + 16 \beta_1 - 22) q^{23} + 25 q^{25} + (2 \beta_{2} - 21 \beta_1 - 61) q^{29} + ( - 2 \beta_{2} + 15 \beta_1 - 70) q^{31} + (5 \beta_1 + 75) q^{35} + ( - 2 \beta_{2} - 32 \beta_1 + 16) q^{37} + ( - 4 \beta_{2} - 21 \beta_1 + 107) q^{41} + (2 \beta_{2} - 39 \beta_1 - 21) q^{43} + (\beta_{2} + 14 \beta_1 - 153) q^{47} + (\beta_{2} + 30 \beta_1 - 56) q^{49} + ( - 3 \beta_{2} - 7 \beta_1 - 175) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 - 60) q^{55} + ( - 3 \beta_{2} - 41 \beta_1 + 34) q^{59} + (4 \beta_{2} + 30 \beta_1 + 207) q^{61} + (5 \beta_{2} + 25 \beta_1 - 40) q^{65} + ( - 6 \beta_{2} + 26 \beta_1 - 282) q^{67} + ( - 15 \beta_{2} + 17 \beta_1 - 190) q^{71} + (15 \beta_{2} + 11 \beta_1 + 430) q^{73} + ( - 13 \beta_{2} - 28 \beta_1 - 101) q^{77} + (3 \beta_{2} + 11 \beta_1 - 207) q^{79} + (15 \beta_{2} + 60 \beta_1 - 414) q^{83} + ( - 10 \beta_{2} - 5 \beta_1 - 30) q^{85} + (12 \beta_{2} + 33 \beta_1 - 711) q^{89} + (19 \beta_{2} + 98 \beta_1 + 173) q^{91} + ( - 5 \beta_{2} + 35 \beta_1 + 325) q^{95} + (2 \beta_{2} + 32 \beta_1 + 474) q^{97}+O(q^{100})$$ q - 5 * q^5 + (-b1 - 15) * q^7 + (b2 - b1 + 12) * q^11 + (-b2 - 5*b1 + 8) * q^13 + (2*b2 + b1 + 6) * q^17 + (b2 - 7*b1 - 65) * q^19 + (b2 + 16*b1 - 22) * q^23 + 25 * q^25 + (2*b2 - 21*b1 - 61) * q^29 + (-2*b2 + 15*b1 - 70) * q^31 + (5*b1 + 75) * q^35 + (-2*b2 - 32*b1 + 16) * q^37 + (-4*b2 - 21*b1 + 107) * q^41 + (2*b2 - 39*b1 - 21) * q^43 + (b2 + 14*b1 - 153) * q^47 + (b2 + 30*b1 - 56) * q^49 + (-3*b2 - 7*b1 - 175) * q^53 + (-5*b2 + 5*b1 - 60) * q^55 + (-3*b2 - 41*b1 + 34) * q^59 + (4*b2 + 30*b1 + 207) * q^61 + (5*b2 + 25*b1 - 40) * q^65 + (-6*b2 + 26*b1 - 282) * q^67 + (-15*b2 + 17*b1 - 190) * q^71 + (15*b2 + 11*b1 + 430) * q^73 + (-13*b2 - 28*b1 - 101) * q^77 + (3*b2 + 11*b1 - 207) * q^79 + (15*b2 + 60*b1 - 414) * q^83 + (-10*b2 - 5*b1 - 30) * q^85 + (12*b2 + 33*b1 - 711) * q^89 + (19*b2 + 98*b1 + 173) * q^91 + (-5*b2 + 35*b1 + 325) * q^95 + (2*b2 + 32*b1 + 474) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} - 44 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 - 44 * q^7 $$3 q - 15 q^{5} - 44 q^{7} + 38 q^{11} + 28 q^{13} + 19 q^{17} - 187 q^{19} - 81 q^{23} + 75 q^{25} - 160 q^{29} - 227 q^{31} + 220 q^{35} + 78 q^{37} + 338 q^{41} - 22 q^{43} - 472 q^{47} - 197 q^{49} - 521 q^{53} - 190 q^{55} + 140 q^{59} + 595 q^{61} - 140 q^{65} - 878 q^{67} - 602 q^{71} + 1294 q^{73} - 288 q^{77} - 629 q^{79} - 1287 q^{83} - 95 q^{85} - 2154 q^{89} + 440 q^{91} + 935 q^{95} + 1392 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 - 44 * q^7 + 38 * q^11 + 28 * q^13 + 19 * q^17 - 187 * q^19 - 81 * q^23 + 75 * q^25 - 160 * q^29 - 227 * q^31 + 220 * q^35 + 78 * q^37 + 338 * q^41 - 22 * q^43 - 472 * q^47 - 197 * q^49 - 521 * q^53 - 190 * q^55 + 140 * q^59 + 595 * q^61 - 140 * q^65 - 878 * q^67 - 602 * q^71 + 1294 * q^73 - 288 * q^77 - 629 * q^79 - 1287 * q^83 - 95 * q^85 - 2154 * q^89 + 440 * q^91 + 935 * q^95 + 1392 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 23x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$4\nu^{2} - 4\nu - 61$$ 4*v^2 - 4*v - 61
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 63 ) / 4$$ (b2 + 2*b1 + 63) / 4

## 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
 5.20067 0.258712 −4.45938
0 0 0 −5.00000 0 −24.4013 0 0 0
1.2 0 0 0 −5.00000 0 −14.5174 0 0 0
1.3 0 0 0 −5.00000 0 −5.08123 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.be 3
3.b odd 2 1 2160.4.a.bm 3
4.b odd 2 1 135.4.a.g yes 3
12.b even 2 1 135.4.a.f 3
20.d odd 2 1 675.4.a.q 3
20.e even 4 2 675.4.b.k 6
36.f odd 6 2 405.4.e.r 6
36.h even 6 2 405.4.e.t 6
60.h even 2 1 675.4.a.r 3
60.l odd 4 2 675.4.b.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 12.b even 2 1
135.4.a.g yes 3 4.b odd 2 1
405.4.e.r 6 36.f odd 6 2
405.4.e.t 6 36.h even 6 2
675.4.a.q 3 20.d odd 2 1
675.4.a.r 3 60.h even 2 1
675.4.b.k 6 20.e even 4 2
675.4.b.l 6 60.l odd 4 2
2160.4.a.be 3 1.a even 1 1 trivial
2160.4.a.bm 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{3} + 44T_{7}^{2} + 552T_{7} + 1800$$ T7^3 + 44*T7^2 + 552*T7 + 1800 $$T_{11}^{3} - 38T_{11}^{2} - 2612T_{11} + 83280$$ T11^3 - 38*T11^2 - 2612*T11 + 83280

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} + 44 T^{2} + 552 T + 1800$$
$11$ $$T^{3} - 38 T^{2} - 2612 T + 83280$$
$13$ $$T^{3} - 28 T^{2} - 4576 T + 100120$$
$17$ $$T^{3} - 19 T^{2} - 11477 T + 553887$$
$19$ $$T^{3} + 187 T^{2} + 3587 T - 525871$$
$23$ $$T^{3} + 81 T^{2} - 23301 T - 2043981$$
$29$ $$T^{3} + 160 T^{2} - 47768 T - 7892760$$
$31$ $$T^{3} + 227 T^{2} - 17973 T + 246321$$
$37$ $$T^{3} - 78 T^{2} - 99924 T + 13637080$$
$41$ $$T^{3} - 338 T^{2} + \cdots + 12116640$$
$43$ $$T^{3} + 22 T^{2} - 159916 T - 18464560$$
$47$ $$T^{3} + 472 T^{2} + 54208 T - 283200$$
$53$ $$T^{3} + 521 T^{2} + 61387 T - 939789$$
$59$ $$T^{3} - 140 T^{2} + \cdots + 34131480$$
$61$ $$T^{3} - 595 T^{2} - 2749 T + 1782607$$
$67$ $$T^{3} + 878 T^{2} + \cdots - 11295000$$
$71$ $$T^{3} + 602 T^{2} + \cdots - 280550880$$
$73$ $$T^{3} - 1294 T^{2} + \cdots + 404091280$$
$79$ $$T^{3} + 629 T^{2} + 97059 T + 2010303$$
$83$ $$T^{3} + 1287 T^{2} + \cdots - 346404411$$
$89$ $$T^{3} + 2154 T^{2} + \cdots + 74325600$$
$97$ $$T^{3} - 1392 T^{2} + \cdots - 63595520$$