# Properties

 Label 1080.6.a.g 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,6,Mod(1,1080)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$173.214525398$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.15881.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 29x - 55$$ x^3 - 29*x - 55 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}\cdot 11$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + 25 q^{5} + (\beta_{2} - 10) q^{7}+O(q^{10})$$ q + 25 * q^5 + (b2 - 10) * q^7 $$q + 25 q^{5} + (\beta_{2} - 10) q^{7} + (3 \beta_{2} - \beta_1 - 220) q^{11} + ( - 3 \beta_{2} + \beta_1 - 426) q^{13} + ( - 7 \beta_{2} - 2 \beta_1 + 577) q^{17} + (3 \beta_{2} + 9 \beta_1 + 337) q^{19} + ( - 8 \beta_{2} - 21 \beta_1 + 811) q^{23} + 625 q^{25} + (3 \beta_{2} + 32 \beta_1 + 1262) q^{29} + ( - 11 \beta_{2} + 16 \beta_1 + 1377) q^{31} + (25 \beta_{2} - 250) q^{35} + ( - 68 \beta_{2} - 32 \beta_1 - 2374) q^{37} + ( - 77 \beta_{2} + 12 \beta_1 + 5784) q^{41} + ( - \beta_{2} - 44 \beta_1 + 3852) q^{43} + ( - 6 \beta_{2} + 37 \beta_1 - 6516) q^{47} + ( - 98 \beta_{2} + 15 \beta_1 - 2619) q^{49} + ( - 151 \beta_{2} - 103 \beta_1 - 1655) q^{53} + (75 \beta_{2} - 25 \beta_1 - 5500) q^{55} + ( - 153 \beta_{2} + 45 \beta_1 - 10702) q^{59} + (126 \beta_{2} + 142 \beta_1 - 5439) q^{61} + ( - 75 \beta_{2} + 25 \beta_1 - 10650) q^{65} + (250 \beta_{2} + 84 \beta_1 - 17086) q^{67} + (273 \beta_{2} + 127 \beta_1 - 28024) q^{71} + ( - 175 \beta_{2} - 75 \beta_1 - 53964) q^{73} + ( - 548 \beta_{2} - 23 \beta_1 + 47656) q^{77} + (553 \beta_{2} - 135 \beta_1 + 12147) q^{79} + (112 \beta_{2} - 497 \beta_1 - 21527) q^{83} + ( - 175 \beta_{2} - 50 \beta_1 + 14425) q^{85} + (637 \beta_{2} + 194 \beta_1 - 20528) q^{89} + ( - 98 \beta_{2} + 23 \beta_1 - 41196) q^{91} + (75 \beta_{2} + 225 \beta_1 + 8425) q^{95} + (284 \beta_{2} - 92 \beta_1 + 4920) q^{97}+O(q^{100})$$ q + 25 * q^5 + (b2 - 10) * q^7 + (3*b2 - b1 - 220) * q^11 + (-3*b2 + b1 - 426) * q^13 + (-7*b2 - 2*b1 + 577) * q^17 + (3*b2 + 9*b1 + 337) * q^19 + (-8*b2 - 21*b1 + 811) * q^23 + 625 * q^25 + (3*b2 + 32*b1 + 1262) * q^29 + (-11*b2 + 16*b1 + 1377) * q^31 + (25*b2 - 250) * q^35 + (-68*b2 - 32*b1 - 2374) * q^37 + (-77*b2 + 12*b1 + 5784) * q^41 + (-b2 - 44*b1 + 3852) * q^43 + (-6*b2 + 37*b1 - 6516) * q^47 + (-98*b2 + 15*b1 - 2619) * q^49 + (-151*b2 - 103*b1 - 1655) * q^53 + (75*b2 - 25*b1 - 5500) * q^55 + (-153*b2 + 45*b1 - 10702) * q^59 + (126*b2 + 142*b1 - 5439) * q^61 + (-75*b2 + 25*b1 - 10650) * q^65 + (250*b2 + 84*b1 - 17086) * q^67 + (273*b2 + 127*b1 - 28024) * q^71 + (-175*b2 - 75*b1 - 53964) * q^73 + (-548*b2 - 23*b1 + 47656) * q^77 + (553*b2 - 135*b1 + 12147) * q^79 + (112*b2 - 497*b1 - 21527) * q^83 + (-175*b2 - 50*b1 + 14425) * q^85 + (637*b2 + 194*b1 - 20528) * q^89 + (-98*b2 + 23*b1 - 41196) * q^91 + (75*b2 + 225*b1 + 8425) * q^95 + (284*b2 - 92*b1 + 4920) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 75 q^{5} - 30 q^{7}+O(q^{10})$$ 3 * q + 75 * q^5 - 30 * q^7 $$3 q + 75 q^{5} - 30 q^{7} - 660 q^{11} - 1278 q^{13} + 1731 q^{17} + 1011 q^{19} + 2433 q^{23} + 1875 q^{25} + 3786 q^{29} + 4131 q^{31} - 750 q^{35} - 7122 q^{37} + 17352 q^{41} + 11556 q^{43} - 19548 q^{47} - 7857 q^{49} - 4965 q^{53} - 16500 q^{55} - 32106 q^{59} - 16317 q^{61} - 31950 q^{65} - 51258 q^{67} - 84072 q^{71} - 161892 q^{73} + 142968 q^{77} + 36441 q^{79} - 64581 q^{83} + 43275 q^{85} - 61584 q^{89} - 123588 q^{91} + 25275 q^{95} + 14760 q^{97}+O(q^{100})$$ 3 * q + 75 * q^5 - 30 * q^7 - 660 * q^11 - 1278 * q^13 + 1731 * q^17 + 1011 * q^19 + 2433 * q^23 + 1875 * q^25 + 3786 * q^29 + 4131 * q^31 - 750 * q^35 - 7122 * q^37 + 17352 * q^41 + 11556 * q^43 - 19548 * q^47 - 7857 * q^49 - 4965 * q^53 - 16500 * q^55 - 32106 * q^59 - 16317 * q^61 - 31950 * q^65 - 51258 * q^67 - 84072 * q^71 - 161892 * q^73 + 142968 * q^77 + 36441 * q^79 - 64581 * q^83 + 43275 * q^85 - 61584 * q^89 - 123588 * q^91 + 25275 * q^95 + 14760 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 29x - 55$$ :

 $$\beta_{1}$$ $$=$$ $$-12\nu^{2} + 232$$ -12*v^2 + 232 $$\beta_{2}$$ $$=$$ $$-18\nu^{2} + 66\nu + 348$$ -18*v^2 + 66*v + 348
 $$\nu$$ $$=$$ $$( 2\beta_{2} - 3\beta_1 ) / 132$$ (2*b2 - 3*b1) / 132 $$\nu^{2}$$ $$=$$ $$( -\beta _1 + 232 ) / 12$$ (-b1 + 232) / 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.82250 6.15876 −2.33626
0 0 0 25.0000 0 −177.292 0 0 0
1.2 0 0 0 25.0000 0 61.7318 0 0 0
1.3 0 0 0 25.0000 0 85.5605 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.g yes 3
3.b odd 2 1 1080.6.a.e 3

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

 $$T_{7}^{3} + 30T_{7}^{2} - 20832T_{7} + 936424$$ T7^3 + 30*T7^2 - 20832*T7 + 936424 $$T_{11}^{3} + 660T_{11}^{2} - 114084T_{11} - 16969720$$ T11^3 + 660*T11^2 - 114084*T11 - 16969720

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T - 25)^{3}$$
$7$ $$T^{3} + 30 T^{2} - 20832 T + 936424$$
$11$ $$T^{3} + 660 T^{2} + \cdots - 16969720$$
$13$ $$T^{3} + 1278 T^{2} + \cdots - 62570968$$
$17$ $$T^{3} - 1731 T^{2} + \cdots + 362051059$$
$19$ $$T^{3} - 1011 T^{2} + \cdots + 1066478795$$
$23$ $$T^{3} - 2433 T^{2} + \cdots + 16297632125$$
$29$ $$T^{3} - 3786 T^{2} + \cdots + 100231188440$$
$31$ $$T^{3} - 4131 T^{2} + \cdots + 36775432739$$
$37$ $$T^{3} + 7122 T^{2} + \cdots - 11280290472$$
$41$ $$T^{3} - 17352 T^{2} + \cdots + 19821902472$$
$43$ $$T^{3} - 11556 T^{2} + \cdots + 74089212824$$
$47$ $$T^{3} + 19548 T^{2} + \cdots + 48232251200$$
$53$ $$T^{3} + 4965 T^{2} + \cdots + 6200694992835$$
$59$ $$T^{3} + 32106 T^{2} + \cdots - 9865368652616$$
$61$ $$T^{3} + 16317 T^{2} + \cdots - 15693428262193$$
$67$ $$T^{3} + 51258 T^{2} + \cdots - 20150295569000$$
$71$ $$T^{3} + 84072 T^{2} + \cdots - 46987871536152$$
$73$ $$T^{3} + \cdots + 119813275420344$$
$79$ $$T^{3} + \cdots + 306616858223481$$
$83$ $$T^{3} + \cdots - 623768894560265$$
$89$ $$T^{3} + \cdots - 152901460706184$$
$97$ $$T^{3} - 14760 T^{2} + \cdots + 36671348002304$$