# Properties

 Label 2160.4.a.bk 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, 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("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.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: $$127.444125612$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

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.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + ( - \beta_{2} - \beta_1 + 4) q^{7}+O(q^{10})$$ 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.

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
 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.bk 3
3.b odd 2 1 2160.4.a.bs 3
4.b odd 2 1 1080.4.a.d 3
12.b even 2 1 1080.4.a.j yes 3

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

 $$T_{7}^{3} - 10T_{7}^{2} - 704T_{7} + 2808$$ 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

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