# Properties

 Label 1728.4.a.bk Level $1728$ Weight $4$ Character orbit 1728.a Self dual yes Analytic conductor $101.955$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 11 q^{7}+O(q^{10})$$ q + b * q^5 - 11 * q^7 $$q + \beta q^{5} - 11 q^{7} + \beta q^{11} - 29 q^{13} - 3 \beta q^{17} + 29 q^{19} - 5 \beta q^{23} + 163 q^{25} - 16 \beta q^{29} + 268 q^{31} - 11 \beta q^{35} - 83 q^{37} - 16 \beta q^{41} - 232 q^{43} + 23 \beta q^{47} - 222 q^{49} - 18 \beta q^{53} + 288 q^{55} + 17 \beta q^{59} - 767 q^{61} - 29 \beta q^{65} - 511 q^{67} - 42 \beta q^{71} + 137 q^{73} - 11 \beta q^{77} + 475 q^{79} + 34 \beta q^{83} - 864 q^{85} - 15 \beta q^{89} + 319 q^{91} + 29 \beta q^{95} + 821 q^{97} +O(q^{100})$$ q + b * q^5 - 11 * q^7 + b * q^11 - 29 * q^13 - 3*b * q^17 + 29 * q^19 - 5*b * q^23 + 163 * q^25 - 16*b * q^29 + 268 * q^31 - 11*b * q^35 - 83 * q^37 - 16*b * q^41 - 232 * q^43 + 23*b * q^47 - 222 * q^49 - 18*b * q^53 + 288 * q^55 + 17*b * q^59 - 767 * q^61 - 29*b * q^65 - 511 * q^67 - 42*b * q^71 + 137 * q^73 - 11*b * q^77 + 475 * q^79 + 34*b * q^83 - 864 * q^85 - 15*b * q^89 + 319 * q^91 + 29*b * q^95 + 821 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 22 q^{7}+O(q^{10})$$ 2 * q - 22 * q^7 $$2 q - 22 q^{7} - 58 q^{13} + 58 q^{19} + 326 q^{25} + 536 q^{31} - 166 q^{37} - 464 q^{43} - 444 q^{49} + 576 q^{55} - 1534 q^{61} - 1022 q^{67} + 274 q^{73} + 950 q^{79} - 1728 q^{85} + 638 q^{91} + 1642 q^{97}+O(q^{100})$$ 2 * q - 22 * q^7 - 58 * q^13 + 58 * q^19 + 326 * q^25 + 536 * q^31 - 166 * q^37 - 464 * q^43 - 444 * q^49 + 576 * q^55 - 1534 * q^61 - 1022 * q^67 + 274 * q^73 + 950 * q^79 - 1728 * q^85 + 638 * q^91 + 1642 * 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
 −1.41421 1.41421
0 0 0 −16.9706 0 −11.0000 0 0 0
1.2 0 0 0 16.9706 0 −11.0000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.a.bk 2
3.b odd 2 1 inner 1728.4.a.bk 2
4.b odd 2 1 1728.4.a.bp 2
8.b even 2 1 432.4.a.q 2
8.d odd 2 1 27.4.a.c 2
12.b even 2 1 1728.4.a.bp 2
24.f even 2 1 27.4.a.c 2
24.h odd 2 1 432.4.a.q 2
40.e odd 2 1 675.4.a.n 2
40.k even 4 2 675.4.b.i 4
56.e even 2 1 1323.4.a.t 2
72.l even 6 2 81.4.c.e 4
72.p odd 6 2 81.4.c.e 4
120.m even 2 1 675.4.a.n 2
120.q odd 4 2 675.4.b.i 4
168.e odd 2 1 1323.4.a.t 2

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

## Hecke kernels

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

 $$T_{5}^{2} - 288$$ T5^2 - 288 $$T_{7} + 11$$ T7 + 11 $$T_{11}^{2} - 288$$ T11^2 - 288

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 288$$
$7$ $$(T + 11)^{2}$$
$11$ $$T^{2} - 288$$
$13$ $$(T + 29)^{2}$$
$17$ $$T^{2} - 2592$$
$19$ $$(T - 29)^{2}$$
$23$ $$T^{2} - 7200$$
$29$ $$T^{2} - 73728$$
$31$ $$(T - 268)^{2}$$
$37$ $$(T + 83)^{2}$$
$41$ $$T^{2} - 73728$$
$43$ $$(T + 232)^{2}$$
$47$ $$T^{2} - 152352$$
$53$ $$T^{2} - 93312$$
$59$ $$T^{2} - 83232$$
$61$ $$(T + 767)^{2}$$
$67$ $$(T + 511)^{2}$$
$71$ $$T^{2} - 508032$$
$73$ $$(T - 137)^{2}$$
$79$ $$(T - 475)^{2}$$
$83$ $$T^{2} - 332928$$
$89$ $$T^{2} - 64800$$
$97$ $$(T - 821)^{2}$$