# Properties

 Label 1080.6.a.h Level $1080$ Weight $6$ Character orbit 1080.a Self dual yes Analytic conductor $173.215$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1080.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$173.214525398$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ Defining polynomial: $$x^{3} - 460x - 1125$$ x^3 - 460*x - 1125 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 25 q^{5} + (\beta_1 + 47) q^{7}+O(q^{10})$$ q + 25 * q^5 + (b1 + 47) * q^7 $$q + 25 q^{5} + (\beta_1 + 47) q^{7} + (2 \beta_{2} + \beta_1 - 54) q^{11} + (3 \beta_{2} + 3 \beta_1 + 18) q^{13} + ( - 5 \beta_{2} + 8 \beta_1 - 199) q^{17} + ( - 7 \beta_{2} - 10 \beta_1 - 544) q^{19} + (2 \beta_{2} - 7 \beta_1 + 546) q^{23} + 625 q^{25} + ( - 15 \beta_{2} - 112 \beta_1 - 223) q^{29} + ( - 28 \beta_{2} - 79 \beta_1 - 962) q^{31} + (25 \beta_1 + 1175) q^{35} + (13 \beta_{2} - 88 \beta_1 - 2076) q^{37} + ( - 57 \beta_{2} + 119 \beta_1 - 4547) q^{41} + ( - 10 \beta_{2} + 137 \beta_1 - 744) q^{43} + (53 \beta_{2} + 74 \beta_1 - 4945) q^{47} + ( - 8 \beta_{2} + 55 \beta_1 - 11320) q^{49} + (3 \beta_{2} + 119 \beta_1 + 361) q^{53} + (50 \beta_{2} + 25 \beta_1 - 1350) q^{55} + ( - 83 \beta_{2} + 39 \beta_1 - 16067) q^{59} + (140 \beta_{2} - 509 \beta_1 + 1929) q^{61} + (75 \beta_{2} + 75 \beta_1 + 450) q^{65} + ( - 168 \beta_{2} - 335 \beta_1 + 1497) q^{67} + (13 \beta_{2} - 743 \beta_1 - 9577) q^{71} + (219 \beta_{2} + 532 \beta_1 - 11932) q^{73} + (162 \beta_{2} - 90 \beta_1 - 2340) q^{77} + (67 \beta_{2} + 399 \beta_1 + 8564) q^{79} + (541 \beta_{2} + 171 \beta_1 - 16025) q^{83} + ( - 125 \beta_{2} + 200 \beta_1 - 4975) q^{85} + ( - 217 \beta_{2} - 159 \beta_1 + 13683) q^{89} + (231 \beta_{2} - 24 \beta_1 + 6060) q^{91} + ( - 175 \beta_{2} - 250 \beta_1 - 13600) q^{95} + ( - 603 \beta_{2} + 1542 \beta_1 - 15874) q^{97}+O(q^{100})$$ q + 25 * q^5 + (b1 + 47) * q^7 + (2*b2 + b1 - 54) * q^11 + (3*b2 + 3*b1 + 18) * q^13 + (-5*b2 + 8*b1 - 199) * q^17 + (-7*b2 - 10*b1 - 544) * q^19 + (2*b2 - 7*b1 + 546) * q^23 + 625 * q^25 + (-15*b2 - 112*b1 - 223) * q^29 + (-28*b2 - 79*b1 - 962) * q^31 + (25*b1 + 1175) * q^35 + (13*b2 - 88*b1 - 2076) * q^37 + (-57*b2 + 119*b1 - 4547) * q^41 + (-10*b2 + 137*b1 - 744) * q^43 + (53*b2 + 74*b1 - 4945) * q^47 + (-8*b2 + 55*b1 - 11320) * q^49 + (3*b2 + 119*b1 + 361) * q^53 + (50*b2 + 25*b1 - 1350) * q^55 + (-83*b2 + 39*b1 - 16067) * q^59 + (140*b2 - 509*b1 + 1929) * q^61 + (75*b2 + 75*b1 + 450) * q^65 + (-168*b2 - 335*b1 + 1497) * q^67 + (13*b2 - 743*b1 - 9577) * q^71 + (219*b2 + 532*b1 - 11932) * q^73 + (162*b2 - 90*b1 - 2340) * q^77 + (67*b2 + 399*b1 + 8564) * q^79 + (541*b2 + 171*b1 - 16025) * q^83 + (-125*b2 + 200*b1 - 4975) * q^85 + (-217*b2 - 159*b1 + 13683) * q^89 + (231*b2 - 24*b1 + 6060) * q^91 + (-175*b2 - 250*b1 - 13600) * q^95 + (-603*b2 + 1542*b1 - 15874) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 75 q^{5} + 140 q^{7}+O(q^{10})$$ 3 * q + 75 * q^5 + 140 * q^7 $$3 q + 75 q^{5} + 140 q^{7} - 163 q^{11} + 51 q^{13} - 605 q^{17} - 1622 q^{19} + 1645 q^{23} + 1875 q^{25} - 557 q^{29} - 2807 q^{31} + 3500 q^{35} - 6140 q^{37} - 13760 q^{41} - 2369 q^{43} - 14909 q^{47} - 34015 q^{49} + 964 q^{53} - 4075 q^{55} - 48240 q^{59} + 6296 q^{61} + 1275 q^{65} + 4826 q^{67} - 27988 q^{71} - 36328 q^{73} - 6930 q^{77} + 25293 q^{79} - 48246 q^{83} - 15125 q^{85} + 41208 q^{89} + 18204 q^{91} - 40550 q^{95} - 49164 q^{97}+O(q^{100})$$ 3 * q + 75 * q^5 + 140 * q^7 - 163 * q^11 + 51 * q^13 - 605 * q^17 - 1622 * q^19 + 1645 * q^23 + 1875 * q^25 - 557 * q^29 - 2807 * q^31 + 3500 * q^35 - 6140 * q^37 - 13760 * q^41 - 2369 * q^43 - 14909 * q^47 - 34015 * q^49 + 964 * q^53 - 4075 * q^55 - 48240 * q^59 + 6296 * q^61 + 1275 * q^65 + 4826 * q^67 - 27988 * q^71 - 36328 * q^73 - 6930 * q^77 + 25293 * q^79 - 48246 * q^83 - 15125 * q^85 + 41208 * q^89 + 18204 * q^91 - 40550 * q^95 - 49164 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 460x - 1125$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{2} + 15\nu + 305 ) / 5$$ (-v^2 + 15*v + 305) / 5 $$\beta_{2}$$ $$=$$ $$( 3\nu^{2} + 15\nu - 920 ) / 5$$ (3*v^2 + 15*v - 920) / 5
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta _1 + 1 ) / 12$$ (b2 + 3*b1 + 1) / 12 $$\nu^{2}$$ $$=$$ $$( 5\beta_{2} - 5\beta _1 + 1225 ) / 4$$ (5*b2 - 5*b1 + 1225) / 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
 −20.1005 22.5793 −2.47876
0 0 0 25.0000 0 −33.1079 0 0 0
1.2 0 0 0 25.0000 0 73.7730 0 0 0
1.3 0 0 0 25.0000 0 99.3349 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 1080.6.a.h yes 3
3.b odd 2 1 1080.6.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.6.a.f 3 3.b odd 2 1
1080.6.a.h yes 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(1080))$$:

 $$T_{7}^{3} - 140T_{7}^{2} + 1597T_{7} + 242622$$ T7^3 - 140*T7^2 + 1597*T7 + 242622 $$T_{11}^{3} + 163T_{11}^{2} - 129312T_{11} - 18306324$$ T11^3 + 163*T11^2 - 129312*T11 - 18306324

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 25)^{3}$$
$7$ $$T^{3} - 140 T^{2} + 1597 T + 242622$$
$11$ $$T^{3} + 163 T^{2} + \cdots - 18306324$$
$13$ $$T^{3} - 51 T^{2} - 322641 T - 59054589$$
$17$ $$T^{3} + 605 T^{2} + \cdots - 897539376$$
$19$ $$T^{3} + 1622 T^{2} + \cdots + 142738684$$
$23$ $$T^{3} - 1645 T^{2} + \cdots + 159386316$$
$29$ $$T^{3} + 557 T^{2} + \cdots - 174014485380$$
$31$ $$T^{3} + 2807 T^{2} + \cdots + 7296447744$$
$37$ $$T^{3} + 6140 T^{2} + \cdots - 88793131806$$
$41$ $$T^{3} + 13760 T^{2} + \cdots - 2105614536000$$
$43$ $$T^{3} + 2369 T^{2} + \cdots + 99827900592$$
$47$ $$T^{3} + 14909 T^{2} + \cdots - 857156082128$$
$53$ $$T^{3} - 964 T^{2} + \cdots + 227607713808$$
$59$ $$T^{3} + 48240 T^{2} + \cdots - 911884153536$$
$61$ $$T^{3} - 6296 T^{2} + \cdots + 32138860458870$$
$67$ $$T^{3} - 4826 T^{2} + \cdots + 17472859220516$$
$71$ $$T^{3} + 27988 T^{2} + \cdots - 68617099591632$$
$73$ $$T^{3} + 36328 T^{2} + \cdots - 60737955374814$$
$79$ $$T^{3} - 25293 T^{2} + \cdots + 12677984881325$$
$83$ $$T^{3} + \cdots - 304428707967672$$
$89$ $$T^{3} - 41208 T^{2} + \cdots + 39239500298432$$
$97$ $$T^{3} + 49164 T^{2} + \cdots - 21\!\cdots\!10$$