# Properties

 Label 1728.2.a.n Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(1,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.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: $$13.7981494693$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{7}+O(q^{10})$$ q - q^7 $$q - q^{7} - 5 q^{13} + 7 q^{19} - 5 q^{25} - 4 q^{31} - 11 q^{37} - 8 q^{43} - 6 q^{49} + q^{61} - 5 q^{67} - 7 q^{73} + 17 q^{79} + 5 q^{91} - 19 q^{97}+O(q^{100})$$ q - q^7 - 5 * q^13 + 7 * q^19 - 5 * q^25 - 4 * q^31 - 11 * q^37 - 8 * q^43 - 6 * q^49 + q^61 - 5 * q^67 - 7 * q^73 + 17 * q^79 + 5 * q^91 - 19 * q^97

## 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
0 0 0 0 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.n 1
3.b odd 2 1 CM 1728.2.a.n 1
4.b odd 2 1 1728.2.a.o 1
8.b even 2 1 27.2.a.a 1
8.d odd 2 1 432.2.a.e 1
12.b even 2 1 1728.2.a.o 1
24.f even 2 1 432.2.a.e 1
24.h odd 2 1 27.2.a.a 1
40.f even 2 1 675.2.a.e 1
40.i odd 4 2 675.2.b.f 2
56.h odd 2 1 1323.2.a.i 1
72.j odd 6 2 81.2.c.a 2
72.l even 6 2 1296.2.i.i 2
72.n even 6 2 81.2.c.a 2
72.p odd 6 2 1296.2.i.i 2
88.b odd 2 1 3267.2.a.f 1
104.e even 2 1 4563.2.a.e 1
120.i odd 2 1 675.2.a.e 1
120.w even 4 2 675.2.b.f 2
136.h even 2 1 7803.2.a.k 1
152.g odd 2 1 9747.2.a.f 1
168.i even 2 1 1323.2.a.i 1
216.t even 18 6 729.2.e.f 6
216.x odd 18 6 729.2.e.f 6
264.m even 2 1 3267.2.a.f 1
312.b odd 2 1 4563.2.a.e 1
408.b odd 2 1 7803.2.a.k 1
456.p even 2 1 9747.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 8.b even 2 1
27.2.a.a 1 24.h odd 2 1
81.2.c.a 2 72.j odd 6 2
81.2.c.a 2 72.n even 6 2
432.2.a.e 1 8.d odd 2 1
432.2.a.e 1 24.f even 2 1
675.2.a.e 1 40.f even 2 1
675.2.a.e 1 120.i odd 2 1
675.2.b.f 2 40.i odd 4 2
675.2.b.f 2 120.w even 4 2
729.2.e.f 6 216.t even 18 6
729.2.e.f 6 216.x odd 18 6
1296.2.i.i 2 72.l even 6 2
1296.2.i.i 2 72.p odd 6 2
1323.2.a.i 1 56.h odd 2 1
1323.2.a.i 1 168.i even 2 1
1728.2.a.n 1 1.a even 1 1 trivial
1728.2.a.n 1 3.b odd 2 1 CM
1728.2.a.o 1 4.b odd 2 1
1728.2.a.o 1 12.b even 2 1
3267.2.a.f 1 88.b odd 2 1
3267.2.a.f 1 264.m even 2 1
4563.2.a.e 1 104.e even 2 1
4563.2.a.e 1 312.b odd 2 1
7803.2.a.k 1 136.h even 2 1
7803.2.a.k 1 408.b odd 2 1
9747.2.a.f 1 152.g odd 2 1
9747.2.a.f 1 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5}$$ T5 $$T_{7} + 1$$ T7 + 1 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T + 5$$
$17$ $$T$$
$19$ $$T - 7$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T + 11$$
$41$ $$T$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 1$$
$67$ $$T + 5$$
$71$ $$T$$
$73$ $$T + 7$$
$79$ $$T - 17$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 19$$