# Properties

 Label 1728.2.i.i Level $1728$ Weight $2$ Character orbit 1728.i Analytic conductor $13.798$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(577,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1728.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.i (of order $$3$$, 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: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{7}+O(q^{10})$$ q + b3 * q^5 + (-b3 + b2 + 2*b1 - 1) * q^7 $$q + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_1 - 1) q^{11} + (\beta_{3} + 2 \beta_1) q^{13} + ( - \beta_{2} + 2) q^{17} + ( - \beta_{2} - 4) q^{19} + (\beta_{3} + 2 \beta_1) q^{23} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{25} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{29} + (\beta_{3} - 4 \beta_1) q^{31} + ( - \beta_{2} + 7) q^{35} + (2 \beta_{2} - 2) q^{37} + (2 \beta_{3} - 7 \beta_1) q^{41} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{43} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{47} + (3 \beta_{3} - 5 \beta_1) q^{49} + ( - 2 \beta_{2} - 6) q^{53} + ( - \beta_{2} - 1) q^{55} - 7 \beta_1 q^{59} + (\beta_{3} - \beta_{2} - 1) q^{61} + (3 \beta_{3} - 3 \beta_{2} + 8 \beta_1 - 11) q^{65} + (2 \beta_{3} + \beta_1) q^{67} + 4 q^{71} + (3 \beta_{2} - 2) q^{73} + (\beta_{3} - 2 \beta_1) q^{77} + (\beta_{3} - \beta_{2} - 4 \beta_1 + 3) q^{79} + (\beta_{3} - \beta_{2} + 12 \beta_1 - 13) q^{83} + (2 \beta_{3} - 8 \beta_1) q^{85} - 6 q^{89} + (\beta_{2} + 5) q^{91} + ( - 4 \beta_{3} - 8 \beta_1) q^{95} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{97}+O(q^{100})$$ q + b3 * q^5 + (-b3 + b2 + 2*b1 - 1) * q^7 + (b1 - 1) * q^11 + (b3 + 2*b1) * q^13 + (-b2 + 2) * q^17 + (-b2 - 4) * q^19 + (b3 + 2*b1) * q^23 + (b3 - b2 + 3*b1 - 4) * q^25 + (b3 - b2 - 2*b1 + 1) * q^29 + (b3 - 4*b1) * q^31 + (-b2 + 7) * q^35 + (2*b2 - 2) * q^37 + (2*b3 - 7*b1) * q^41 + (-2*b3 + 2*b2 - 3*b1 + 5) * q^43 + (b3 - b2 - 2*b1 + 1) * q^47 + (3*b3 - 5*b1) * q^49 + (-2*b2 - 6) * q^53 + (-b2 - 1) * q^55 - 7*b1 * q^59 + (b3 - b2 - 1) * q^61 + (3*b3 - 3*b2 + 8*b1 - 11) * q^65 + (2*b3 + b1) * q^67 + 4 * q^71 + (3*b2 - 2) * q^73 + (b3 - 2*b1) * q^77 + (b3 - b2 - 4*b1 + 3) * q^79 + (b3 - b2 + 12*b1 - 13) * q^83 + (2*b3 - 8*b1) * q^85 - 6 * q^89 + (b2 + 5) * q^91 + (-4*b3 - 8*b1) * q^95 + (-2*b3 + 2*b2 - 3*b1 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{5} - 3 q^{7}+O(q^{10})$$ 4 * q + q^5 - 3 * q^7 $$4 q + q^{5} - 3 q^{7} - 2 q^{11} + 5 q^{13} + 10 q^{17} - 14 q^{19} + 5 q^{23} - 7 q^{25} + 3 q^{29} - 7 q^{31} + 30 q^{35} - 12 q^{37} - 12 q^{41} + 8 q^{43} + 3 q^{47} - 7 q^{49} - 20 q^{53} - 2 q^{55} - 14 q^{59} - q^{61} - 19 q^{65} + 4 q^{67} + 16 q^{71} - 14 q^{73} - 3 q^{77} + 7 q^{79} - 25 q^{83} - 14 q^{85} - 24 q^{89} + 18 q^{91} - 20 q^{95} + 8 q^{97}+O(q^{100})$$ 4 * q + q^5 - 3 * q^7 - 2 * q^11 + 5 * q^13 + 10 * q^17 - 14 * q^19 + 5 * q^23 - 7 * q^25 + 3 * q^29 - 7 * q^31 + 30 * q^35 - 12 * q^37 - 12 * q^41 + 8 * q^43 + 3 * q^47 - 7 * q^49 - 20 * q^53 - 2 * q^55 - 14 * q^59 - q^61 - 19 * q^65 + 4 * q^67 + 16 * q^71 - 14 * q^73 - 3 * q^77 + 7 * q^79 - 25 * q^83 - 14 * q^85 - 24 * q^89 + 18 * q^91 - 20 * q^95 + 8 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 0 0 −1.18614 + 2.05446i 0 −2.18614 3.78651i 0 0 0
577.2 0 0 0 1.68614 2.92048i 0 0.686141 + 1.18843i 0 0 0
1153.1 0 0 0 −1.18614 2.05446i 0 −2.18614 + 3.78651i 0 0 0
1153.2 0 0 0 1.68614 + 2.92048i 0 0.686141 1.18843i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.i 4
3.b odd 2 1 576.2.i.j 4
4.b odd 2 1 1728.2.i.j 4
8.b even 2 1 216.2.i.b 4
8.d odd 2 1 432.2.i.d 4
9.c even 3 1 inner 1728.2.i.i 4
9.c even 3 1 5184.2.a.bp 2
9.d odd 6 1 576.2.i.j 4
9.d odd 6 1 5184.2.a.bt 2
12.b even 2 1 576.2.i.l 4
24.f even 2 1 144.2.i.d 4
24.h odd 2 1 72.2.i.b 4
36.f odd 6 1 1728.2.i.j 4
36.f odd 6 1 5184.2.a.bo 2
36.h even 6 1 576.2.i.l 4
36.h even 6 1 5184.2.a.bs 2
72.j odd 6 1 72.2.i.b 4
72.j odd 6 1 648.2.a.f 2
72.l even 6 1 144.2.i.d 4
72.l even 6 1 1296.2.a.n 2
72.n even 6 1 216.2.i.b 4
72.n even 6 1 648.2.a.g 2
72.p odd 6 1 432.2.i.d 4
72.p odd 6 1 1296.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b 4 24.h odd 2 1
72.2.i.b 4 72.j odd 6 1
144.2.i.d 4 24.f even 2 1
144.2.i.d 4 72.l even 6 1
216.2.i.b 4 8.b even 2 1
216.2.i.b 4 72.n even 6 1
432.2.i.d 4 8.d odd 2 1
432.2.i.d 4 72.p odd 6 1
576.2.i.j 4 3.b odd 2 1
576.2.i.j 4 9.d odd 6 1
576.2.i.l 4 12.b even 2 1
576.2.i.l 4 36.h even 6 1
648.2.a.f 2 72.j odd 6 1
648.2.a.g 2 72.n even 6 1
1296.2.a.n 2 72.l even 6 1
1296.2.a.p 2 72.p odd 6 1
1728.2.i.i 4 1.a even 1 1 trivial
1728.2.i.i 4 9.c even 3 1 inner
1728.2.i.j 4 4.b odd 2 1
1728.2.i.j 4 36.f odd 6 1
5184.2.a.bo 2 36.f odd 6 1
5184.2.a.bp 2 9.c even 3 1
5184.2.a.bs 2 36.h even 6 1
5184.2.a.bt 2 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} - T_{5}^{3} + 9T_{5}^{2} + 8T_{5} + 64$$ T5^4 - T5^3 + 9*T5^2 + 8*T5 + 64 $$T_{7}^{4} + 3T_{7}^{3} + 15T_{7}^{2} - 18T_{7} + 36$$ T7^4 + 3*T7^3 + 15*T7^2 - 18*T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - T^{3} + 9 T^{2} + 8 T + 64$$
$7$ $$T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36$$
$11$ $$(T^{2} + T + 1)^{2}$$
$13$ $$T^{4} - 5 T^{3} + 27 T^{2} + 10 T + 4$$
$17$ $$(T^{2} - 5 T - 2)^{2}$$
$19$ $$(T^{2} + 7 T + 4)^{2}$$
$23$ $$T^{4} - 5 T^{3} + 27 T^{2} + 10 T + 4$$
$29$ $$T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36$$
$31$ $$T^{4} + 7 T^{3} + 45 T^{2} + 28 T + 16$$
$37$ $$(T^{2} + 6 T - 24)^{2}$$
$41$ $$T^{4} + 12 T^{3} + 141 T^{2} + 36 T + 9$$
$43$ $$T^{4} - 8 T^{3} + 81 T^{2} + 136 T + 289$$
$47$ $$T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36$$
$53$ $$(T^{2} + 10 T - 8)^{2}$$
$59$ $$(T^{2} + 7 T + 49)^{2}$$
$61$ $$T^{4} + T^{3} + 9 T^{2} - 8 T + 64$$
$67$ $$T^{4} - 4 T^{3} + 45 T^{2} + 116 T + 841$$
$71$ $$(T - 4)^{4}$$
$73$ $$(T^{2} + 7 T - 62)^{2}$$
$79$ $$T^{4} - 7 T^{3} + 45 T^{2} - 28 T + 16$$
$83$ $$T^{4} + 25 T^{3} + 477 T^{2} + \cdots + 21904$$
$89$ $$(T + 6)^{4}$$
$97$ $$T^{4} - 8 T^{3} + 81 T^{2} + 136 T + 289$$