# Properties

 Label 2160.3.bs.a Level $2160$ Weight $3$ Character orbit 2160.bs Analytic conductor $58.856$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(881,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2160.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2160.bs (of order $$6$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$58.8557371018$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{7}+O(q^{10})$$ q - b1 * q^5 + (-2*b3 - 4*b2 - 2*b1) * q^7 $$q - \beta_1 q^{5} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{7} + ( - 6 \beta_{3} - \beta_{2} + 6 \beta_1 + 2) q^{11} + ( - 4 \beta_{3} - 10 \beta_{2} + \cdots + 10) q^{13}+ \cdots + ( - 26 \beta_{3} + \beta_{2} - 26 \beta_1) q^{97}+O(q^{100})$$ q - b1 * q^5 + (-2*b3 - 4*b2 - 2*b1) * q^7 + (-6*b3 - b2 + 6*b1 + 2) * q^11 + (-4*b3 - 10*b2 + 2*b1 + 10) * q^13 + (6*b3 + 2*b2 - 1) * q^17 + (2*b3 - 4*b1 + 19) * q^19 + (4*b2 + 12*b1 + 4) * q^23 + 5*b2 * q^25 + (12*b3 - 10*b2 - 12*b1 + 20) * q^29 + (-28*b3 - 2*b2 + 14*b1 + 2) * q^31 + (4*b3 + 20*b2 - 10) * q^35 + 14 * q^37 + (15*b2 + 15) * q^41 + (-8*b3 + 11*b2 - 8*b1) * q^43 + (-6*b3 - 34*b2 + 6*b1 + 68) * q^47 + (32*b3 + 27*b2 - 16*b1 - 27) * q^49 + (6*b3 + 28*b2 - 14) * q^53 + (b3 - 2*b1 - 30) * q^55 + (-29*b2 - 6*b1 - 29) * q^59 + (-6*b3 + 22*b2 - 6*b1) * q^61 + (10*b3 + 10*b2 - 10*b1 - 20) * q^65 + (-48*b3 + 7*b2 + 24*b1 - 7) * q^67 + (-18*b3 + 72*b2 - 36) * q^71 + (6*b3 - 12*b1 - 37) * q^73 + (-64*b2 - 30*b1 - 64) * q^77 + (6*b3 + 80*b2 + 6*b1) * q^79 + (52*b2 - 104) * q^83 + (-2*b3 - 30*b2 + b1 + 30) * q^85 + (-24*b3 + 48*b2 - 24) * q^89 + (28*b3 - 56*b1 - 100) * q^91 + (10*b2 - 19*b1 + 10) * q^95 + (-26*b3 + b2 - 26*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7}+O(q^{10})$$ 4 * q - 8 * q^7 $$4 q - 8 q^{7} + 6 q^{11} + 20 q^{13} + 76 q^{19} + 24 q^{23} + 10 q^{25} + 60 q^{29} + 4 q^{31} + 56 q^{37} + 90 q^{41} + 22 q^{43} + 204 q^{47} - 54 q^{49} - 120 q^{55} - 174 q^{59} + 44 q^{61} - 60 q^{65} - 14 q^{67} - 148 q^{73} - 384 q^{77} + 160 q^{79} - 312 q^{83} + 60 q^{85} - 400 q^{91} + 60 q^{95} + 2 q^{97}+O(q^{100})$$ 4 * q - 8 * q^7 + 6 * q^11 + 20 * q^13 + 76 * q^19 + 24 * q^23 + 10 * q^25 + 60 * q^29 + 4 * q^31 + 56 * q^37 + 90 * q^41 + 22 * q^43 + 204 * q^47 - 54 * q^49 - 120 * q^55 - 174 * q^59 + 44 * q^61 - 60 * q^65 - 14 * q^67 - 148 * q^73 - 384 * q^77 + 160 * q^79 - 312 * q^83 + 60 * q^85 - 400 * q^91 + 60 * q^95 + 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 1.93649 + 1.11803i −1.93649 − 1.11803i 1.93649 − 1.11803i −1.93649 + 1.11803i
0 0 0 −1.93649 1.11803i 0 −5.87298 10.1723i 0 0 0
881.2 0 0 0 1.93649 + 1.11803i 0 1.87298 + 3.24410i 0 0 0
1601.1 0 0 0 −1.93649 + 1.11803i 0 −5.87298 + 10.1723i 0 0 0
1601.2 0 0 0 1.93649 1.11803i 0 1.87298 3.24410i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.3.bs.a 4
3.b odd 2 1 720.3.bs.a 4
4.b odd 2 1 540.3.o.a 4
9.c even 3 1 720.3.bs.a 4
9.d odd 6 1 inner 2160.3.bs.a 4
12.b even 2 1 180.3.o.a 4
20.d odd 2 1 2700.3.p.a 4
20.e even 4 2 2700.3.u.a 8
36.f odd 6 1 180.3.o.a 4
36.f odd 6 1 1620.3.g.a 4
36.h even 6 1 540.3.o.a 4
36.h even 6 1 1620.3.g.a 4
60.h even 2 1 900.3.p.b 4
60.l odd 4 2 900.3.u.b 8
180.n even 6 1 2700.3.p.a 4
180.p odd 6 1 900.3.p.b 4
180.v odd 12 2 2700.3.u.a 8
180.x even 12 2 900.3.u.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a 4 12.b even 2 1
180.3.o.a 4 36.f odd 6 1
540.3.o.a 4 4.b odd 2 1
540.3.o.a 4 36.h even 6 1
720.3.bs.a 4 3.b odd 2 1
720.3.bs.a 4 9.c even 3 1
900.3.p.b 4 60.h even 2 1
900.3.p.b 4 180.p odd 6 1
900.3.u.b 8 60.l odd 4 2
900.3.u.b 8 180.x even 12 2
1620.3.g.a 4 36.f odd 6 1
1620.3.g.a 4 36.h even 6 1
2160.3.bs.a 4 1.a even 1 1 trivial
2160.3.bs.a 4 9.d odd 6 1 inner
2700.3.p.a 4 20.d odd 2 1
2700.3.p.a 4 180.n even 6 1
2700.3.u.a 8 20.e even 4 2
2700.3.u.a 8 180.v odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 8T_{7}^{3} + 108T_{7}^{2} - 352T_{7} + 1936$$ acting on $$S_{3}^{\mathrm{new}}(2160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$T^{4} + 8 T^{3} + \cdots + 1936$$
$11$ $$T^{4} - 6 T^{3} + \cdots + 31329$$
$13$ $$T^{4} - 20 T^{3} + \cdots + 1600$$
$17$ $$T^{4} + 366 T^{2} + 31329$$
$19$ $$(T^{2} - 38 T + 301)^{2}$$
$23$ $$T^{4} - 24 T^{3} + \cdots + 451584$$
$29$ $$T^{4} - 60 T^{3} + \cdots + 176400$$
$31$ $$T^{4} - 4 T^{3} + \cdots + 8620096$$
$37$ $$(T - 14)^{4}$$
$41$ $$(T^{2} - 45 T + 675)^{2}$$
$43$ $$T^{4} - 22 T^{3} + \cdots + 703921$$
$47$ $$T^{4} - 204 T^{3} + \cdots + 10810944$$
$53$ $$T^{4} + 1536 T^{2} + 166464$$
$59$ $$T^{4} + 174 T^{3} + \cdots + 5489649$$
$61$ $$T^{4} - 44 T^{3} + \cdots + 3136$$
$67$ $$T^{4} + 14 T^{3} + \cdots + 73805281$$
$71$ $$T^{4} + 11016 T^{2} + 5143824$$
$73$ $$(T^{2} + 74 T + 829)^{2}$$
$79$ $$T^{4} - 160 T^{3} + \cdots + 34339600$$
$83$ $$(T^{2} + 156 T + 8112)^{2}$$
$89$ $$T^{4} + 9216 T^{2} + 1327104$$
$97$ $$T^{4} - 2 T^{3} + \cdots + 102799321$$