# Properties

 Label 1080.4.a.h Level $1080$ Weight $4$ Character orbit 1080.a Self dual yes Analytic conductor $63.722$ Analytic rank $0$ 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] = [1080,4,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, 4, names="a")

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

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$63.7220628062$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.47977.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 60x - 44$$ x^3 - x^2 - 60*x - 44 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + (\beta_{2} + 3) q^{7}+O(q^{10})$$ q - 5 * q^5 + (b2 + 3) * q^7 $$q - 5 q^{5} + (\beta_{2} + 3) q^{7} + (\beta_{2} - \beta_1 - 6) q^{11} + (2 \beta_{2} + \beta_1 - 7) q^{13} + (\beta_{2} + \beta_1 - 28) q^{17} + (3 \beta_{2} + 7) q^{19} + (\beta_{2} - 16) q^{23} + 25 q^{25} + ( - 5 \beta_{2} - 4 \beta_1 + 12) q^{29} + ( - 3 \beta_{2} - 5 \beta_1 + 108) q^{31} + ( - 5 \beta_{2} - 15) q^{35} + ( - 3 \beta_{2} + 4 \beta_1 + 11) q^{37} + ( - 2 \beta_{2} + 6 \beta_1 - 38) q^{41} + (7 \beta_{2} - 2 \beta_1 + 94) q^{43} + (17 \beta_{2} + 4 \beta_1 - 94) q^{47} + ( - 5 \beta_{2} - 6 \beta_1 + 76) q^{49} + ( - 6 \beta_{2} + 8 \beta_1 - 74) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 + 30) q^{55} + 92 q^{59} + (11 \beta_{2} + 4 \beta_1 + 101) q^{61} + ( - 10 \beta_{2} - 5 \beta_1 + 35) q^{65} + ( - 23 \beta_{2} + 18 \beta_1 + 345) q^{67} + ( - 10 \beta_{2} - 2 \beta_1 - 170) q^{71} + ( - 25 \beta_{2} - 12 \beta_1 + 149) q^{73} + ( - 20 \beta_1 + 526) q^{77} + (22 \beta_{2} - 9 \beta_1 + 259) q^{79} + (36 \beta_{2} - 2 \beta_1 - 26) q^{83} + ( - 5 \beta_{2} - 5 \beta_1 + 140) q^{85} + (20 \beta_{2} - 10 \beta_1 + 108) q^{89} + ( - 37 \beta_{2} + 2 \beta_1 + 665) q^{91} + ( - 15 \beta_{2} - 35) q^{95} + (19 \beta_{2} + 16 \beta_1 + 397) q^{97}+O(q^{100})$$ q - 5 * q^5 + (b2 + 3) * q^7 + (b2 - b1 - 6) * q^11 + (2*b2 + b1 - 7) * q^13 + (b2 + b1 - 28) * q^17 + (3*b2 + 7) * q^19 + (b2 - 16) * q^23 + 25 * q^25 + (-5*b2 - 4*b1 + 12) * q^29 + (-3*b2 - 5*b1 + 108) * q^31 + (-5*b2 - 15) * q^35 + (-3*b2 + 4*b1 + 11) * q^37 + (-2*b2 + 6*b1 - 38) * q^41 + (7*b2 - 2*b1 + 94) * q^43 + (17*b2 + 4*b1 - 94) * q^47 + (-5*b2 - 6*b1 + 76) * q^49 + (-6*b2 + 8*b1 - 74) * q^53 + (-5*b2 + 5*b1 + 30) * q^55 + 92 * q^59 + (11*b2 + 4*b1 + 101) * q^61 + (-10*b2 - 5*b1 + 35) * q^65 + (-23*b2 + 18*b1 + 345) * q^67 + (-10*b2 - 2*b1 - 170) * q^71 + (-25*b2 - 12*b1 + 149) * q^73 + (-20*b1 + 526) * q^77 + (22*b2 - 9*b1 + 259) * q^79 + (36*b2 - 2*b1 - 26) * q^83 + (-5*b2 - 5*b1 + 140) * q^85 + (20*b2 - 10*b1 + 108) * q^89 + (-37*b2 + 2*b1 + 665) * q^91 + (-15*b2 - 35) * q^95 + (19*b2 + 16*b1 + 397) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} + 9 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 + 9 * q^7 $$3 q - 15 q^{5} + 9 q^{7} - 18 q^{11} - 21 q^{13} - 84 q^{17} + 21 q^{19} - 48 q^{23} + 75 q^{25} + 36 q^{29} + 324 q^{31} - 45 q^{35} + 33 q^{37} - 114 q^{41} + 282 q^{43} - 282 q^{47} + 228 q^{49} - 222 q^{53} + 90 q^{55} + 276 q^{59} + 303 q^{61} + 105 q^{65} + 1035 q^{67} - 510 q^{71} + 447 q^{73} + 1578 q^{77} + 777 q^{79} - 78 q^{83} + 420 q^{85} + 324 q^{89} + 1995 q^{91} - 105 q^{95} + 1191 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 + 9 * q^7 - 18 * q^11 - 21 * q^13 - 84 * q^17 + 21 * q^19 - 48 * q^23 + 75 * q^25 + 36 * q^29 + 324 * q^31 - 45 * q^35 + 33 * q^37 - 114 * q^41 + 282 * q^43 - 282 * q^47 + 228 * q^49 - 222 * q^53 + 90 * q^55 + 276 * q^59 + 303 * q^61 + 105 * q^65 + 1035 * q^67 - 510 * q^71 + 447 * q^73 + 1578 * q^77 + 777 * q^79 - 78 * q^83 + 420 * q^85 + 324 * q^89 + 1995 * q^91 - 105 * q^95 + 1191 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 60x - 44$$ :

 $$\beta_{1}$$ $$=$$ $$6\nu - 2$$ 6*v - 2 $$\beta_{2}$$ $$=$$ $$( 3\nu^{2} - 9\nu - 118 ) / 4$$ (3*v^2 - 9*v - 118) / 4
 $$\nu$$ $$=$$ $$( \beta _1 + 2 ) / 6$$ (b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( 8\beta_{2} + 3\beta _1 + 242 ) / 6$$ (8*b2 + 3*b1 + 242) / 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.

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
 −0.749725 8.58548 −6.83575
0 0 0 −5.00000 0 −24.3916 0 0 0
1.2 0 0 0 −5.00000 0 9.46549 0 0 0
1.3 0 0 0 −5.00000 0 23.9261 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 1080.4.a.h 3
3.b odd 2 1 1080.4.a.n yes 3
4.b odd 2 1 2160.4.a.bf 3
12.b even 2 1 2160.4.a.bn 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.h 3 1.a even 1 1 trivial
1080.4.a.n yes 3 3.b odd 2 1
2160.4.a.bf 3 4.b odd 2 1
2160.4.a.bn 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1080))$$:

 $$T_{7}^{3} - 9T_{7}^{2} - 588T_{7} + 5524$$ T7^3 - 9*T7^2 - 588*T7 + 5524 $$T_{11}^{3} + 18T_{11}^{2} - 3081T_{11} - 76426$$ T11^3 + 18*T11^2 - 3081*T11 - 76426

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} - 9 T^{2} - 588 T + 5524$$
$11$ $$T^{3} + 18 T^{2} - 3081 T - 76426$$
$13$ $$T^{3} + 21 T^{2} - 3681 T - 30901$$
$17$ $$T^{3} + 84 T^{2} - 33 T - 86732$$
$19$ $$T^{3} - 21 T^{2} - 5388 T + 138464$$
$23$ $$T^{3} + 48 T^{2} + 153 T - 2038$$
$29$ $$T^{3} - 36 T^{2} - 41655 T + 3034586$$
$31$ $$T^{3} - 324 T^{2} - 18813 T + 9213176$$
$37$ $$T^{3} - 33 T^{2} - 44748 T + 2851956$$
$41$ $$T^{3} + 114 T^{2} - 81144 T - 1847232$$
$43$ $$T^{3} - 282 T^{2} - 17943 T + 1113236$$
$47$ $$T^{3} + 282 T^{2} + \cdots + 11238296$$
$53$ $$T^{3} + 222 T^{2} - 164016 T + 5908896$$
$59$ $$(T - 92)^{3}$$
$61$ $$T^{3} - 303 T^{2} + \cdots + 13330220$$
$67$ $$T^{3} - 1035 T^{2} + \cdots + 849676336$$
$71$ $$T^{3} + 510 T^{2} + \cdots - 11439360$$
$73$ $$T^{3} - 447 T^{2} + \cdots + 78574428$$
$79$ $$T^{3} - 777 T^{2} + \cdots - 14000175$$
$83$ $$T^{3} + 78 T^{2} - 832644 T + 87546568$$
$89$ $$T^{3} - 324 T^{2} + \cdots - 92454912$$
$97$ $$T^{3} - 1191 T^{2} + \cdots + 31736348$$