# Properties

 Label 2160.4.a.bs 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: 3.3.1257.1 Defining polynomial: $$x^{3} - x^{2} - 8x + 9$$ x^3 - x^2 - 8*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ 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.

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + ( - \beta_{2} - \beta_1 + 4) q^{7}+O(q^{10})$$ q + 5 * q^5 + (-b2 - b1 + 4) * q^7 $$q + 5 q^{5} + ( - \beta_{2} - \beta_1 + 4) q^{7} + (5 \beta_1 - 11) q^{11} + (3 \beta_1 - 27) q^{13} + ( - 3 \beta_{2} - \beta_1 + 5) q^{17} + (2 \beta_{2} - 3 \beta_1 + 24) q^{19} + (7 \beta_{2} - 12 \beta_1 + 10) q^{23} + 25 q^{25} + (\beta_{2} - 23 \beta_1 - 32) q^{29} + (11 \beta_{2} + 3 \beta_1 + 31) q^{31} + ( - 5 \beta_{2} - 5 \beta_1 + 20) q^{35} + (12 \beta_{2} + 4 \beta_1 - 142) q^{37} + (\beta_{2} + 13 \beta_1 - 202) q^{41} + ( - 7 \beta_{2} + 21 \beta_1 - 22) q^{43} + (7 \beta_{2} - 30 \beta_1 + 201) q^{47} + (5 \beta_{2} - 42 \beta_1 + 172) q^{49} + (4 \beta_{2} - 17 \beta_1 - 52) q^{53} + (25 \beta_1 - 55) q^{55} + ( - 14 \beta_{2} + 41 \beta_1 - 87) q^{59} + (26 \beta_1 - 233) q^{61} + (15 \beta_1 - 135) q^{65} + (26 \beta_{2} - 34 \beta_1 - 126) q^{67} + ( - 22 \beta_{2} + 3 \beta_1 + 1) q^{71} + ( - 8 \beta_{2} + 91 \beta_1 - 325) q^{73} + (21 \beta_{2} + 96 \beta_1 - 639) q^{77} + ( - 28 \beta_{2} - 53 \beta_1 - 394) q^{79} + ( - 17 \beta_{2} + 68 \beta_1 + 588) q^{83} + ( - 15 \beta_{2} - 5 \beta_1 + 25) q^{85} + ( - 35 \beta_{2} + 11 \beta_1 - 592) q^{89} + (33 \beta_{2} + 78 \beta_1 - 465) q^{91} + (10 \beta_{2} - 15 \beta_1 + 120) q^{95} + (48 \beta_{2} - 12 \beta_1 - 292) q^{97}+O(q^{100})$$ q + 5 * q^5 + (-b2 - b1 + 4) * q^7 + (5*b1 - 11) * q^11 + (3*b1 - 27) * q^13 + (-3*b2 - b1 + 5) * q^17 + (2*b2 - 3*b1 + 24) * q^19 + (7*b2 - 12*b1 + 10) * q^23 + 25 * q^25 + (b2 - 23*b1 - 32) * q^29 + (11*b2 + 3*b1 + 31) * q^31 + (-5*b2 - 5*b1 + 20) * q^35 + (12*b2 + 4*b1 - 142) * q^37 + (b2 + 13*b1 - 202) * q^41 + (-7*b2 + 21*b1 - 22) * q^43 + (7*b2 - 30*b1 + 201) * q^47 + (5*b2 - 42*b1 + 172) * q^49 + (4*b2 - 17*b1 - 52) * q^53 + (25*b1 - 55) * q^55 + (-14*b2 + 41*b1 - 87) * q^59 + (26*b1 - 233) * q^61 + (15*b1 - 135) * q^65 + (26*b2 - 34*b1 - 126) * q^67 + (-22*b2 + 3*b1 + 1) * q^71 + (-8*b2 + 91*b1 - 325) * q^73 + (21*b2 + 96*b1 - 639) * q^77 + (-28*b2 - 53*b1 - 394) * q^79 + (-17*b2 + 68*b1 + 588) * q^83 + (-15*b2 - 5*b1 + 25) * q^85 + (-35*b2 + 11*b1 - 592) * q^89 + (33*b2 + 78*b1 - 465) * q^91 + (10*b2 - 15*b1 + 120) * q^95 + (48*b2 - 12*b1 - 292) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} + 10 q^{7}+O(q^{10})$$ 3 * q + 15 * q^5 + 10 * q^7 $$3 q + 15 q^{5} + 10 q^{7} - 28 q^{11} - 78 q^{13} + 11 q^{17} + 71 q^{19} + 25 q^{23} + 75 q^{25} - 118 q^{29} + 107 q^{31} + 50 q^{35} - 410 q^{37} - 592 q^{41} - 52 q^{43} + 580 q^{47} + 479 q^{49} - 169 q^{53} - 140 q^{55} - 234 q^{59} - 673 q^{61} - 390 q^{65} - 386 q^{67} - 16 q^{71} - 892 q^{73} - 1800 q^{77} - 1263 q^{79} + 1815 q^{83} + 55 q^{85} - 1800 q^{89} - 1284 q^{91} + 355 q^{95} - 840 q^{97}+O(q^{100})$$ 3 * q + 15 * q^5 + 10 * q^7 - 28 * q^11 - 78 * q^13 + 11 * q^17 + 71 * q^19 + 25 * q^23 + 75 * q^25 - 118 * q^29 + 107 * q^31 + 50 * q^35 - 410 * q^37 - 592 * q^41 - 52 * q^43 + 580 * q^47 + 479 * q^49 - 169 * q^53 - 140 * q^55 - 234 * q^59 - 673 * q^61 - 390 * q^65 - 386 * q^67 - 16 * q^71 - 892 * q^73 - 1800 * q^77 - 1263 * q^79 + 1815 * q^83 + 55 * q^85 - 1800 * q^89 - 1284 * q^91 + 355 * q^95 - 840 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 8x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$-2\nu^{2} + 2\nu + 11$$ -2*v^2 + 2*v + 11 $$\beta_{2}$$ $$=$$ $$4\nu^{2} + 8\nu - 25$$ 4*v^2 + 8*v - 25
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 3 ) / 12$$ (b2 + 2*b1 + 3) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{2} - 4\beta _1 + 69 ) / 12$$ (b2 - 4*b1 + 69) / 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
 2.72396 1.14974 −2.87370
0 0 0 5.00000 0 −24.0794 0 0 0
1.2 0 0 0 5.00000 0 3.85875 0 0 0
1.3 0 0 0 5.00000 0 30.2207 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.bs 3
3.b odd 2 1 2160.4.a.bk 3
4.b odd 2 1 1080.4.a.j yes 3
12.b even 2 1 1080.4.a.d 3

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

 $$T_{7}^{3} - 10T_{7}^{2} - 704T_{7} + 2808$$ T7^3 - 10*T7^2 - 704*T7 + 2808 $$T_{11}^{3} + 28T_{11}^{2} - 2772T_{11} - 8424$$ T11^3 + 28*T11^2 - 2772*T11 - 8424

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 5)^{3}$$
$7$ $$T^{3} - 10 T^{2} - 704 T + 2808$$
$11$ $$T^{3} + 28 T^{2} - 2772 T - 8424$$
$13$ $$T^{3} + 78 T^{2} + 936 T - 6696$$
$17$ $$T^{3} - 11 T^{2} - 5033 T + 120315$$
$19$ $$T^{3} - 71 T^{2} - 889 T + 58295$$
$23$ $$T^{3} - 25 T^{2} - 34325 T + 1363653$$
$29$ $$T^{3} + 118 T^{2} - 57792 T - 2593368$$
$31$ $$T^{3} - 107 T^{2} - 63129 T - 2882997$$
$37$ $$T^{3} + 410 T^{2} + \cdots - 15053832$$
$41$ $$T^{3} + 592 T^{2} + 94516 T + 4157400$$
$43$ $$T^{3} + 52 T^{2} - 63452 T - 7351224$$
$47$ $$T^{3} - 580 T^{2} + \cdots + 28223296$$
$53$ $$T^{3} + 169 T^{2} - 27113 T + 581991$$
$59$ $$T^{3} + 234 T^{2} + \cdots - 66081528$$
$61$ $$T^{3} + 673 T^{2} + 68955 T - 4428477$$
$67$ $$T^{3} + 386 T^{2} + \cdots - 51000008$$
$71$ $$T^{3} + 16 T^{2} - 244884 T + 45142200$$
$73$ $$T^{3} + 892 T^{2} + \cdots - 350087736$$
$79$ $$T^{3} + 1263 T^{2} + \cdots - 504123839$$
$83$ $$T^{3} - 1815 T^{2} + \cdots - 28157301$$
$89$ $$T^{3} + 1800 T^{2} + \cdots + 30835240$$
$97$ $$T^{3} + 840 T^{2} + \cdots - 775886336$$