# Properties

 Label 1728.4.a.bd Level $1728$ Weight $4$ Character orbit 1728.a Self dual yes Analytic conductor $101.955$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,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, 4, names="a")

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

chi := DirichletCharacter("1728.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$101.955300490$$ Analytic rank: $$0$$ 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: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 15 q^{5} + 25 q^{7}+O(q^{10})$$ q + 15 * q^5 + 25 * q^7 $$q + 15 q^{5} + 25 q^{7} + 15 q^{11} - 20 q^{13} - 72 q^{17} + 2 q^{19} + 114 q^{23} + 100 q^{25} + 30 q^{29} - 101 q^{31} + 375 q^{35} + 430 q^{37} + 30 q^{41} + 110 q^{43} - 330 q^{47} + 282 q^{49} + 621 q^{53} + 225 q^{55} + 660 q^{59} + 376 q^{61} - 300 q^{65} - 250 q^{67} - 360 q^{71} + 785 q^{73} + 375 q^{77} - 488 q^{79} - 489 q^{83} - 1080 q^{85} + 450 q^{89} - 500 q^{91} + 30 q^{95} - 1105 q^{97}+O(q^{100})$$ q + 15 * q^5 + 25 * q^7 + 15 * q^11 - 20 * q^13 - 72 * q^17 + 2 * q^19 + 114 * q^23 + 100 * q^25 + 30 * q^29 - 101 * q^31 + 375 * q^35 + 430 * q^37 + 30 * q^41 + 110 * q^43 - 330 * q^47 + 282 * q^49 + 621 * q^53 + 225 * q^55 + 660 * q^59 + 376 * q^61 - 300 * q^65 - 250 * q^67 - 360 * q^71 + 785 * q^73 + 375 * q^77 - 488 * q^79 - 489 * q^83 - 1080 * q^85 + 450 * q^89 - 500 * q^91 + 30 * q^95 - 1105 * 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 15.0000 0 25.0000 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

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 1728.4.a.bd 1
3.b odd 2 1 1728.4.a.d 1
4.b odd 2 1 1728.4.a.bc 1
8.b even 2 1 432.4.a.a 1
8.d odd 2 1 27.4.a.a 1
12.b even 2 1 1728.4.a.c 1
24.f even 2 1 27.4.a.b yes 1
24.h odd 2 1 432.4.a.n 1
40.e odd 2 1 675.4.a.j 1
40.k even 4 2 675.4.b.b 2
56.e even 2 1 1323.4.a.d 1
72.l even 6 2 81.4.c.a 2
72.p odd 6 2 81.4.c.c 2
120.m even 2 1 675.4.a.a 1
120.q odd 4 2 675.4.b.a 2
168.e odd 2 1 1323.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 8.d odd 2 1
27.4.a.b yes 1 24.f even 2 1
81.4.c.a 2 72.l even 6 2
81.4.c.c 2 72.p odd 6 2
432.4.a.a 1 8.b even 2 1
432.4.a.n 1 24.h odd 2 1
675.4.a.a 1 120.m even 2 1
675.4.a.j 1 40.e odd 2 1
675.4.b.a 2 120.q odd 4 2
675.4.b.b 2 40.k even 4 2
1323.4.a.d 1 56.e even 2 1
1323.4.a.k 1 168.e odd 2 1
1728.4.a.c 1 12.b even 2 1
1728.4.a.d 1 3.b odd 2 1
1728.4.a.bc 1 4.b odd 2 1
1728.4.a.bd 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5} - 15$$ T5 - 15 $$T_{7} - 25$$ T7 - 25 $$T_{11} - 15$$ T11 - 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 15$$
$7$ $$T - 25$$
$11$ $$T - 15$$
$13$ $$T + 20$$
$17$ $$T + 72$$
$19$ $$T - 2$$
$23$ $$T - 114$$
$29$ $$T - 30$$
$31$ $$T + 101$$
$37$ $$T - 430$$
$41$ $$T - 30$$
$43$ $$T - 110$$
$47$ $$T + 330$$
$53$ $$T - 621$$
$59$ $$T - 660$$
$61$ $$T - 376$$
$67$ $$T + 250$$
$71$ $$T + 360$$
$73$ $$T - 785$$
$79$ $$T + 488$$
$83$ $$T + 489$$
$89$ $$T - 450$$
$97$ $$T + 1105$$