# Properties

 Label 1728.2.i.c Level $1728$ Weight $2$ Character orbit 1728.i Analytic conductor $13.798$ Analytic rank $1$ Dimension $2$ 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 # Warning: the index may be different

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

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} +O(q^{10})$$ q - 3*z * q^5 + (z - 1) * q^7 $$q - 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} + ( - 3 \zeta_{6} + 3) q^{11} - \zeta_{6} q^{13} - 6 q^{17} - 4 q^{19} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + (3 \zeta_{6} - 3) q^{29} + 5 \zeta_{6} q^{31} + 3 q^{35} - 2 q^{37} + 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} + ( - 9 \zeta_{6} + 9) q^{47} + 6 \zeta_{6} q^{49} - 6 q^{53} - 9 q^{55} - 3 \zeta_{6} q^{59} + (13 \zeta_{6} - 13) q^{61} + (3 \zeta_{6} - 3) q^{65} + 7 \zeta_{6} q^{67} - 12 q^{71} - 10 q^{73} + 3 \zeta_{6} q^{77} + ( - 11 \zeta_{6} + 11) q^{79} + (9 \zeta_{6} - 9) q^{83} + 18 \zeta_{6} q^{85} - 6 q^{89} + q^{91} + 12 \zeta_{6} q^{95} + (11 \zeta_{6} - 11) q^{97} +O(q^{100})$$ q - 3*z * q^5 + (z - 1) * q^7 + (-3*z + 3) * q^11 - z * q^13 - 6 * q^17 - 4 * q^19 + 3*z * q^23 + (4*z - 4) * q^25 + (3*z - 3) * q^29 + 5*z * q^31 + 3 * q^35 - 2 * q^37 + 3*z * q^41 + (-z + 1) * q^43 + (-9*z + 9) * q^47 + 6*z * q^49 - 6 * q^53 - 9 * q^55 - 3*z * q^59 + (13*z - 13) * q^61 + (3*z - 3) * q^65 + 7*z * q^67 - 12 * q^71 - 10 * q^73 + 3*z * q^77 + (-11*z + 11) * q^79 + (9*z - 9) * q^83 + 18*z * q^85 - 6 * q^89 + q^91 + 12*z * q^95 + (11*z - 11) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - q^7 $$2 q - 3 q^{5} - q^{7} + 3 q^{11} - q^{13} - 12 q^{17} - 8 q^{19} + 3 q^{23} - 4 q^{25} - 3 q^{29} + 5 q^{31} + 6 q^{35} - 4 q^{37} + 3 q^{41} + q^{43} + 9 q^{47} + 6 q^{49} - 12 q^{53} - 18 q^{55} - 3 q^{59} - 13 q^{61} - 3 q^{65} + 7 q^{67} - 24 q^{71} - 20 q^{73} + 3 q^{77} + 11 q^{79} - 9 q^{83} + 18 q^{85} - 12 q^{89} + 2 q^{91} + 12 q^{95} - 11 q^{97}+O(q^{100})$$ 2 * q - 3 * q^5 - q^7 + 3 * q^11 - q^13 - 12 * q^17 - 8 * q^19 + 3 * q^23 - 4 * q^25 - 3 * q^29 + 5 * q^31 + 6 * q^35 - 4 * q^37 + 3 * q^41 + q^43 + 9 * q^47 + 6 * q^49 - 12 * q^53 - 18 * q^55 - 3 * q^59 - 13 * q^61 - 3 * q^65 + 7 * q^67 - 24 * q^71 - 20 * q^73 + 3 * q^77 + 11 * q^79 - 9 * q^83 + 18 * q^85 - 12 * q^89 + 2 * q^91 + 12 * q^95 - 11 * q^97

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.50000 + 2.59808i 0 −0.500000 0.866025i 0 0 0
1153.1 0 0 0 −1.50000 2.59808i 0 −0.500000 + 0.866025i 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.c 2
3.b odd 2 1 576.2.i.e 2
4.b odd 2 1 1728.2.i.d 2
8.b even 2 1 432.2.i.c 2
8.d odd 2 1 108.2.e.a 2
9.c even 3 1 inner 1728.2.i.c 2
9.c even 3 1 5184.2.a.bb 1
9.d odd 6 1 576.2.i.e 2
9.d odd 6 1 5184.2.a.f 1
12.b even 2 1 576.2.i.f 2
24.f even 2 1 36.2.e.a 2
24.h odd 2 1 144.2.i.a 2
36.f odd 6 1 1728.2.i.d 2
36.f odd 6 1 5184.2.a.ba 1
36.h even 6 1 576.2.i.f 2
36.h even 6 1 5184.2.a.e 1
40.e odd 2 1 2700.2.i.b 2
40.k even 4 2 2700.2.s.b 4
56.e even 2 1 5292.2.j.a 2
56.k odd 6 1 5292.2.i.c 2
56.k odd 6 1 5292.2.l.a 2
56.m even 6 1 5292.2.i.a 2
56.m even 6 1 5292.2.l.c 2
72.j odd 6 1 144.2.i.a 2
72.j odd 6 1 1296.2.a.k 1
72.l even 6 1 36.2.e.a 2
72.l even 6 1 324.2.a.c 1
72.n even 6 1 432.2.i.c 2
72.n even 6 1 1296.2.a.b 1
72.p odd 6 1 108.2.e.a 2
72.p odd 6 1 324.2.a.a 1
120.m even 2 1 900.2.i.b 2
120.q odd 4 2 900.2.s.b 4
168.e odd 2 1 1764.2.j.b 2
168.v even 6 1 1764.2.i.a 2
168.v even 6 1 1764.2.l.c 2
168.be odd 6 1 1764.2.i.c 2
168.be odd 6 1 1764.2.l.a 2
360.z odd 6 1 2700.2.i.b 2
360.z odd 6 1 8100.2.a.g 1
360.bd even 6 1 900.2.i.b 2
360.bd even 6 1 8100.2.a.j 1
360.bo even 12 2 2700.2.s.b 4
360.bo even 12 2 8100.2.d.c 2
360.bt odd 12 2 900.2.s.b 4
360.bt odd 12 2 8100.2.d.h 2
504.u odd 6 1 1764.2.i.c 2
504.ba odd 6 1 5292.2.i.c 2
504.be even 6 1 5292.2.j.a 2
504.bf even 6 1 5292.2.l.c 2
504.bt even 6 1 1764.2.l.c 2
504.ce odd 6 1 5292.2.l.a 2
504.cm odd 6 1 1764.2.l.a 2
504.co odd 6 1 1764.2.j.b 2
504.cy even 6 1 1764.2.i.a 2
504.cz even 6 1 5292.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 24.f even 2 1
36.2.e.a 2 72.l even 6 1
108.2.e.a 2 8.d odd 2 1
108.2.e.a 2 72.p odd 6 1
144.2.i.a 2 24.h odd 2 1
144.2.i.a 2 72.j odd 6 1
324.2.a.a 1 72.p odd 6 1
324.2.a.c 1 72.l even 6 1
432.2.i.c 2 8.b even 2 1
432.2.i.c 2 72.n even 6 1
576.2.i.e 2 3.b odd 2 1
576.2.i.e 2 9.d odd 6 1
576.2.i.f 2 12.b even 2 1
576.2.i.f 2 36.h even 6 1
900.2.i.b 2 120.m even 2 1
900.2.i.b 2 360.bd even 6 1
900.2.s.b 4 120.q odd 4 2
900.2.s.b 4 360.bt odd 12 2
1296.2.a.b 1 72.n even 6 1
1296.2.a.k 1 72.j odd 6 1
1728.2.i.c 2 1.a even 1 1 trivial
1728.2.i.c 2 9.c even 3 1 inner
1728.2.i.d 2 4.b odd 2 1
1728.2.i.d 2 36.f odd 6 1
1764.2.i.a 2 168.v even 6 1
1764.2.i.a 2 504.cy even 6 1
1764.2.i.c 2 168.be odd 6 1
1764.2.i.c 2 504.u odd 6 1
1764.2.j.b 2 168.e odd 2 1
1764.2.j.b 2 504.co odd 6 1
1764.2.l.a 2 168.be odd 6 1
1764.2.l.a 2 504.cm odd 6 1
1764.2.l.c 2 168.v even 6 1
1764.2.l.c 2 504.bt even 6 1
2700.2.i.b 2 40.e odd 2 1
2700.2.i.b 2 360.z odd 6 1
2700.2.s.b 4 40.k even 4 2
2700.2.s.b 4 360.bo even 12 2
5184.2.a.e 1 36.h even 6 1
5184.2.a.f 1 9.d odd 6 1
5184.2.a.ba 1 36.f odd 6 1
5184.2.a.bb 1 9.c even 3 1
5292.2.i.a 2 56.m even 6 1
5292.2.i.a 2 504.cz even 6 1
5292.2.i.c 2 56.k odd 6 1
5292.2.i.c 2 504.ba odd 6 1
5292.2.j.a 2 56.e even 2 1
5292.2.j.a 2 504.be even 6 1
5292.2.l.a 2 56.k odd 6 1
5292.2.l.a 2 504.ce odd 6 1
5292.2.l.c 2 56.m even 6 1
5292.2.l.c 2 504.bf even 6 1
8100.2.a.g 1 360.z odd 6 1
8100.2.a.j 1 360.bd even 6 1
8100.2.d.c 2 360.bo even 12 2
8100.2.d.h 2 360.bt odd 12 2

## Hecke kernels

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

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} - 5T + 25$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 9T + 81$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} + 13T + 169$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 11T + 121$$