# Properties

 Label 1728.4.a.z Level $1728$ Weight $4$ Character orbit 1728.a Self dual yes Analytic conductor $101.955$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{5} + q^{7}+O(q^{10})$$ q + 9 * q^5 + q^7 $$q + 9 q^{5} + q^{7} - 63 q^{11} + 28 q^{13} - 72 q^{17} + 98 q^{19} + 126 q^{23} - 44 q^{25} - 126 q^{29} + 259 q^{31} + 9 q^{35} - 386 q^{37} + 450 q^{41} - 34 q^{43} - 54 q^{47} - 342 q^{49} - 693 q^{53} - 567 q^{55} - 180 q^{59} + 280 q^{61} + 252 q^{65} - 586 q^{67} + 504 q^{71} + 161 q^{73} - 63 q^{77} - 440 q^{79} - 999 q^{83} - 648 q^{85} - 882 q^{89} + 28 q^{91} + 882 q^{95} - 721 q^{97}+O(q^{100})$$ q + 9 * q^5 + q^7 - 63 * q^11 + 28 * q^13 - 72 * q^17 + 98 * q^19 + 126 * q^23 - 44 * q^25 - 126 * q^29 + 259 * q^31 + 9 * q^35 - 386 * q^37 + 450 * q^41 - 34 * q^43 - 54 * q^47 - 342 * q^49 - 693 * q^53 - 567 * q^55 - 180 * q^59 + 280 * q^61 + 252 * q^65 - 586 * q^67 + 504 * q^71 + 161 * q^73 - 63 * q^77 - 440 * q^79 - 999 * q^83 - 648 * q^85 - 882 * q^89 + 28 * q^91 + 882 * q^95 - 721 * 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 9.00000 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

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.z 1
3.b odd 2 1 1728.4.a.h 1
4.b odd 2 1 1728.4.a.y 1
8.b even 2 1 432.4.a.c 1
8.d odd 2 1 108.4.a.a 1
12.b even 2 1 1728.4.a.g 1
24.f even 2 1 108.4.a.d yes 1
24.h odd 2 1 432.4.a.l 1
72.l even 6 2 324.4.e.b 2
72.p odd 6 2 324.4.e.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.a 1 8.d odd 2 1
108.4.a.d yes 1 24.f even 2 1
324.4.e.b 2 72.l even 6 2
324.4.e.g 2 72.p odd 6 2
432.4.a.c 1 8.b even 2 1
432.4.a.l 1 24.h odd 2 1
1728.4.a.g 1 12.b even 2 1
1728.4.a.h 1 3.b odd 2 1
1728.4.a.y 1 4.b odd 2 1
1728.4.a.z 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5} - 9$$ T5 - 9 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 63$$ T11 + 63

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T - 1$$
$11$ $$T + 63$$
$13$ $$T - 28$$
$17$ $$T + 72$$
$19$ $$T - 98$$
$23$ $$T - 126$$
$29$ $$T + 126$$
$31$ $$T - 259$$
$37$ $$T + 386$$
$41$ $$T - 450$$
$43$ $$T + 34$$
$47$ $$T + 54$$
$53$ $$T + 693$$
$59$ $$T + 180$$
$61$ $$T - 280$$
$67$ $$T + 586$$
$71$ $$T - 504$$
$73$ $$T - 161$$
$79$ $$T + 440$$
$83$ $$T + 999$$
$89$ $$T + 882$$
$97$ $$T + 721$$