# Properties

 Label 2160.4.a.bu Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

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## 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: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 141x^{2} + 200x + 3500$$ x^4 - x^3 - 141*x^2 + 200*x + 3500 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + (\beta_{2} - 3) q^{7}+O(q^{10})$$ q - 5 * q^5 + (b2 - 3) * q^7 $$q - 5 q^{5} + (\beta_{2} - 3) q^{7} + (\beta_1 - 1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 + 7) q^{13} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{17} + (\beta_{3} + 3 \beta_{2} - 18) q^{19} + ( - 2 \beta_{3} - 3 \beta_{2} + 44) q^{23} + 25 q^{25} + ( - \beta_{3} - \beta_{2} + 4 \beta_1 - 51) q^{29} + ( - 2 \beta_{3} - 3 \beta_1 + 19) q^{31} + ( - 5 \beta_{2} + 15) q^{35} + ( - 3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 74) q^{37} + (\beta_{3} + 2 \beta_1 - 95) q^{41} + (2 \beta_{3} - 7 \beta_{2} + 2 \beta_1 - 48) q^{43} + ( - \beta_{3} + 5 \beta_{2} - 4 \beta_1 + 31) q^{47} + ( - 4 \beta_{3} - 7 \beta_{2} - 10 \beta_1 + 236) q^{49} + (5 \beta_{3} - 10 \beta_{2} - 8 \beta_1 + 59) q^{53} + ( - 5 \beta_1 + 5) q^{55} + (11 \beta_{3} - 4 \beta_{2} - 53) q^{59} + (4 \beta_{3} + 23 \beta_{2} + 4 \beta_1 + 203) q^{61} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 35) q^{65} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 - 83) q^{67} + (3 \beta_{3} - 16 \beta_{2} + 2 \beta_1 - 273) q^{71} + (7 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 358) q^{73} + (6 \beta_{3} - 36 \beta_{2} - 12 \beta_1 - 72) q^{77} + ( - 7 \beta_{3} - 31 \beta_{2} + 9 \beta_1 - 201) q^{79} + ( - 9 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 193) q^{83} + (5 \beta_{3} + 10 \beta_{2} + 5 \beta_1 - 30) q^{85} + (\beta_{3} + 6 \beta_{2} + 10 \beta_1 + 103) q^{89} + (3 \beta_{3} + 21 \beta_{2} + 30 \beta_1 - 756) q^{91} + ( - 5 \beta_{3} - 15 \beta_{2} + 90) q^{95} + ( - 7 \beta_{3} + 19 \beta_{2} + 16 \beta_1 + 844) q^{97}+O(q^{100})$$ q - 5 * q^5 + (b2 - 3) * q^7 + (b1 - 1) * q^11 + (b3 - b2 - b1 + 7) * q^13 + (-b3 - 2*b2 - b1 + 6) * q^17 + (b3 + 3*b2 - 18) * q^19 + (-2*b3 - 3*b2 + 44) * q^23 + 25 * q^25 + (-b3 - b2 + 4*b1 - 51) * q^29 + (-2*b3 - 3*b1 + 19) * q^31 + (-5*b2 + 15) * q^35 + (-3*b3 - 3*b2 + 4*b1 + 74) * q^37 + (b3 + 2*b1 - 95) * q^41 + (2*b3 - 7*b2 + 2*b1 - 48) * q^43 + (-b3 + 5*b2 - 4*b1 + 31) * q^47 + (-4*b3 - 7*b2 - 10*b1 + 236) * q^49 + (5*b3 - 10*b2 - 8*b1 + 59) * q^53 + (-5*b1 + 5) * q^55 + (11*b3 - 4*b2 - 53) * q^59 + (4*b3 + 23*b2 + 4*b1 + 203) * q^61 + (-5*b3 + 5*b2 + 5*b1 - 35) * q^65 + (-4*b3 - b2 - 2*b1 - 83) * q^67 + (3*b3 - 16*b2 + 2*b1 - 273) * q^71 + (7*b3 - 5*b2 - 4*b1 + 358) * q^73 + (6*b3 - 36*b2 - 12*b1 - 72) * q^77 + (-7*b3 - 31*b2 + 9*b1 - 201) * q^79 + (-9*b3 - 2*b2 + 2*b1 - 193) * q^83 + (5*b3 + 10*b2 + 5*b1 - 30) * q^85 + (b3 + 6*b2 + 10*b1 + 103) * q^89 + (3*b3 + 21*b2 + 30*b1 - 756) * q^91 + (-5*b3 - 15*b2 + 90) * q^95 + (-7*b3 + 19*b2 + 16*b1 + 844) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{5} - 14 q^{7}+O(q^{10})$$ 4 * q - 20 * q^5 - 14 * q^7 $$4 q - 20 q^{5} - 14 q^{7} - 4 q^{11} + 30 q^{13} + 28 q^{17} - 78 q^{19} + 182 q^{23} + 100 q^{25} - 202 q^{29} + 76 q^{31} + 70 q^{35} + 302 q^{37} - 380 q^{41} - 178 q^{43} + 114 q^{47} + 958 q^{49} + 256 q^{53} + 20 q^{55} - 204 q^{59} + 766 q^{61} - 150 q^{65} - 330 q^{67} - 1060 q^{71} + 1442 q^{73} - 216 q^{77} - 742 q^{79} - 768 q^{83} - 140 q^{85} + 400 q^{89} - 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100})$$ 4 * q - 20 * q^5 - 14 * q^7 - 4 * q^11 + 30 * q^13 + 28 * q^17 - 78 * q^19 + 182 * q^23 + 100 * q^25 - 202 * q^29 + 76 * q^31 + 70 * q^35 + 302 * q^37 - 380 * q^41 - 178 * q^43 + 114 * q^47 + 958 * q^49 + 256 * q^53 + 20 * q^55 - 204 * q^59 + 766 * q^61 - 150 * q^65 - 330 * q^67 - 1060 * q^71 + 1442 * q^73 - 216 * q^77 - 742 * q^79 - 768 * q^83 - 140 * q^85 + 400 * q^89 - 3066 * q^91 + 390 * q^95 + 3338 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 141x^{2} + 200x + 3500$$ :

 $$\beta_{1}$$ $$=$$ $$( 7\nu^{3} + 3\nu^{2} - 617\nu + 250 ) / 40$$ (7*v^3 + 3*v^2 - 617*v + 250) / 40 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 9\nu^{2} + 131\nu + 550 ) / 20$$ (-v^3 - 9*v^2 + 131*v + 550) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 9\nu^{2} - 71\nu - 575 ) / 5$$ (v^3 + 9*v^2 - 71*v - 575) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + 4\beta_{2} + 5 ) / 12$$ (b3 + 4*b2 + 5) / 12 $$\nu^{2}$$ $$=$$ $$( 5\beta_{3} - 8\beta_{2} - 8\beta _1 + 845 ) / 12$$ (5*b3 - 8*b2 - 8*b1 + 845) / 12 $$\nu^{3}$$ $$=$$ $$( 43\beta_{3} + 178\beta_{2} + 36\beta _1 - 175 ) / 6$$ (43*b3 + 178*b2 + 36*b1 - 175) / 6

## 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
 −10.9094 −4.73555 9.55367 7.09129
0 0 0 −5.00000 0 −35.5942 0 0 0
1.2 0 0 0 −5.00000 0 −11.2995 0 0 0
1.3 0 0 0 −5.00000 0 2.40438 0 0 0
1.4 0 0 0 −5.00000 0 30.4893 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.bu 4
3.b odd 2 1 2160.4.a.bv 4
4.b odd 2 1 1080.4.a.o 4
12.b even 2 1 1080.4.a.p yes 4

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

 $$T_{7}^{4} + 14T_{7}^{3} - 1067T_{7}^{2} - 9792T_{7} + 29484$$ T7^4 + 14*T7^3 - 1067*T7^2 - 9792*T7 + 29484 $$T_{11}^{4} + 4T_{11}^{3} - 3749T_{11}^{2} - 45756T_{11} + 1807596$$ T11^4 + 4*T11^3 - 3749*T11^2 - 45756*T11 + 1807596

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T + 5)^{4}$$
$7$ $$T^{4} + 14 T^{3} - 1067 T^{2} + \cdots + 29484$$
$11$ $$T^{4} + 4 T^{3} - 3749 T^{2} + \cdots + 1807596$$
$13$ $$T^{4} - 30 T^{3} - 8928 T^{2} + \cdots + 18230751$$
$17$ $$T^{4} - 28 T^{3} - 13553 T^{2} + \cdots + 4462704$$
$19$ $$T^{4} + 78 T^{3} - 11003 T^{2} + \cdots + 21407584$$
$23$ $$T^{4} - 182 T^{3} + \cdots - 146544156$$
$29$ $$T^{4} + 202 T^{3} + \cdots - 561688740$$
$31$ $$T^{4} - 76 T^{3} + \cdots + 901737216$$
$37$ $$T^{4} - 302 T^{3} - 57743 T^{2} + \cdots + 9791964$$
$41$ $$T^{4} + 380 T^{3} + 29260 T^{2} + \cdots + 1440000$$
$43$ $$T^{4} + 178 T^{3} + \cdots - 372768336$$
$47$ $$T^{4} - 114 T^{3} + \cdots + 465218416$$
$53$ $$T^{4} - 256 T^{3} + \cdots + 26357772096$$
$59$ $$T^{4} + 204 T^{3} + \cdots + 3841737984$$
$61$ $$T^{4} - 766 T^{3} + \cdots - 50264811540$$
$67$ $$T^{4} + 330 T^{3} + \cdots + 415421296$$
$71$ $$T^{4} + 1060 T^{3} + \cdots - 897110784$$
$73$ $$T^{4} - 1442 T^{3} + \cdots + 1120468716$$
$79$ $$T^{4} + 742 T^{3} + \cdots + 364066942735$$
$83$ $$T^{4} + 768 T^{3} + \cdots + 4935879504$$
$89$ $$T^{4} - 400 T^{3} + \cdots + 16886941696$$
$97$ $$T^{4} - 3338 T^{3} + \cdots - 650133282260$$
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