# Properties

 Label 1728.2.c.f Level $1728$ Weight $2$ Character orbit 1728.c Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(1727,1728)]

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([1, 0, 1]))

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

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

chi := DirichletCharacter("1728.1727");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7}+O(q^{10})$$ q + b2 * q^5 + (b3 + b1) * q^7 $$q + \beta_{2} q^{5} + (\beta_{3} + \beta_1) q^{7} - \beta_{6} q^{11} + ( - \beta_{4} - 1) q^{13} - \beta_{7} q^{17} + \beta_{3} q^{19} + (2 \beta_{6} + \beta_{5}) q^{23} + ( - 2 \beta_{4} - 3) q^{25} + 2 \beta_{2} q^{29} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} + ( - \beta_{6} - 2 \beta_{5}) q^{35} + ( - \beta_{4} + 3) q^{37} - 2 \beta_{2} q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + (2 \beta_{6} - \beta_{5}) q^{47} - 2 \beta_{4} q^{49} + 2 \beta_{7} q^{53} - 2 \beta_1 q^{55} + ( - \beta_{6} + 2 \beta_{5}) q^{59} + ( - 3 \beta_{4} - 1) q^{61} + (\beta_{7} - 4 \beta_{2}) q^{65} + (\beta_{3} - 4 \beta_1) q^{67} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{71} + ( - 2 \beta_{4} + 3) q^{73} - \beta_{2} q^{77} + ( - 3 \beta_{3} + 5 \beta_1) q^{79} - 2 \beta_{6} q^{83} - 2 \beta_{4} q^{85} - \beta_{7} q^{89} + ( - 5 \beta_{3} - 4 \beta_1) q^{91} - \beta_{5} q^{95} + 7 q^{97}+O(q^{100})$$ q + b2 * q^5 + (b3 + b1) * q^7 - b6 * q^11 + (-b4 - 1) * q^13 - b7 * q^17 + b3 * q^19 + (2*b6 + b5) * q^23 + (-2*b4 - 3) * q^25 + 2*b2 * q^29 + (-2*b3 - 2*b1) * q^31 + (-b6 - 2*b5) * q^35 + (-b4 + 3) * q^37 - 2*b2 * q^41 + (-2*b3 + 2*b1) * q^43 + (2*b6 - b5) * q^47 - 2*b4 * q^49 + 2*b7 * q^53 - 2*b1 * q^55 + (-b6 + 2*b5) * q^59 + (-3*b4 - 1) * q^61 + (b7 - 4*b2) * q^65 + (b3 - 4*b1) * q^67 + (-2*b6 + 2*b5) * q^71 + (-2*b4 + 3) * q^73 - b2 * q^77 + (-3*b3 + 5*b1) * q^79 - 2*b6 * q^83 - 2*b4 * q^85 - b7 * q^89 + (-5*b3 - 4*b1) * q^91 - b5 * q^95 + 7 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{13} - 24 q^{25} + 24 q^{37} - 8 q^{61} + 24 q^{73} + 56 q^{97}+O(q^{100})$$ 8 * q - 8 * q^13 - 24 * q^25 + 24 * q^37 - 8 * q^61 + 24 * q^73 + 56 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{24}^{6}$$ 2*v^6 $$\beta_{2}$$ $$=$$ $$2\zeta_{24}^{7} + 2\zeta_{24}^{5}$$ 2*v^7 + 2*v^5 $$\beta_{3}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{4}$$ $$=$$ $$-2\zeta_{24}^{6} + 4\zeta_{24}^{2}$$ -2*v^6 + 4*v^2 $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24}$$ -2*v^7 - 2*v^5 + 4*v^3 + 4*v $$\beta_{6}$$ $$=$$ $$-2\zeta_{24}^{7} + 2\zeta_{24}^{5}$$ -2*v^7 + 2*v^5 $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} - 2\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24}$$ 2*v^7 - 2*v^5 - 4*v^3 + 4*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} ) / 8$$ (b7 + b6 + b5 + b2) / 8 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} + \beta_1 ) / 4$$ (b4 + b1) / 4 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} ) / 8$$ (-b7 - b6 + b5 + b2) / 8 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{3} + 1 ) / 2$$ (b3 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{6} + \beta_{2} ) / 4$$ (b6 + b2) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{6} + \beta_{2} ) / 4$$ (-b6 + b2) / 4

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1727.1
 0.965926 − 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i
0 0 0 3.86370i 0 3.73205i 0 0 0
1727.2 0 0 0 3.86370i 0 3.73205i 0 0 0
1727.3 0 0 0 1.03528i 0 0.267949i 0 0 0
1727.4 0 0 0 1.03528i 0 0.267949i 0 0 0
1727.5 0 0 0 1.03528i 0 0.267949i 0 0 0
1727.6 0 0 0 1.03528i 0 0.267949i 0 0 0
1727.7 0 0 0 3.86370i 0 3.73205i 0 0 0
1727.8 0 0 0 3.86370i 0 3.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.c.f 8
3.b odd 2 1 inner 1728.2.c.f 8
4.b odd 2 1 inner 1728.2.c.f 8
8.b even 2 1 864.2.c.b 8
8.d odd 2 1 864.2.c.b 8
12.b even 2 1 inner 1728.2.c.f 8
24.f even 2 1 864.2.c.b 8
24.h odd 2 1 864.2.c.b 8
72.j odd 6 1 2592.2.s.c 8
72.j odd 6 1 2592.2.s.g 8
72.l even 6 1 2592.2.s.c 8
72.l even 6 1 2592.2.s.g 8
72.n even 6 1 2592.2.s.c 8
72.n even 6 1 2592.2.s.g 8
72.p odd 6 1 2592.2.s.c 8
72.p odd 6 1 2592.2.s.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.b 8 8.b even 2 1
864.2.c.b 8 8.d odd 2 1
864.2.c.b 8 24.f even 2 1
864.2.c.b 8 24.h odd 2 1
1728.2.c.f 8 1.a even 1 1 trivial
1728.2.c.f 8 3.b odd 2 1 inner
1728.2.c.f 8 4.b odd 2 1 inner
1728.2.c.f 8 12.b even 2 1 inner
2592.2.s.c 8 72.j odd 6 1
2592.2.s.c 8 72.l even 6 1
2592.2.s.c 8 72.n even 6 1
2592.2.s.c 8 72.p odd 6 1
2592.2.s.g 8 72.j odd 6 1
2592.2.s.g 8 72.l even 6 1
2592.2.s.g 8 72.n even 6 1
2592.2.s.g 8 72.p odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 16T_{5}^{2} + 16$$ T5^4 + 16*T5^2 + 16 $$T_{7}^{4} + 14T_{7}^{2} + 1$$ T7^4 + 14*T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 16 T^{2} + 16)^{2}$$
$7$ $$(T^{4} + 14 T^{2} + 1)^{2}$$
$11$ $$(T^{4} - 16 T^{2} + 16)^{2}$$
$13$ $$(T^{2} + 2 T - 11)^{4}$$
$17$ $$(T^{4} + 48 T^{2} + 144)^{2}$$
$19$ $$(T^{2} + 3)^{4}$$
$23$ $$(T^{4} - 112 T^{2} + 2704)^{2}$$
$29$ $$(T^{4} + 64 T^{2} + 256)^{2}$$
$31$ $$(T^{4} + 56 T^{2} + 16)^{2}$$
$37$ $$(T^{2} - 6 T - 3)^{4}$$
$41$ $$(T^{4} + 64 T^{2} + 256)^{2}$$
$43$ $$(T^{4} + 56 T^{2} + 16)^{2}$$
$47$ $$(T^{4} - 112 T^{2} + 1936)^{2}$$
$53$ $$(T^{4} + 192 T^{2} + 2304)^{2}$$
$59$ $$(T^{4} - 208 T^{2} + 8464)^{2}$$
$61$ $$(T^{2} + 2 T - 107)^{4}$$
$67$ $$(T^{4} + 134 T^{2} + 3721)^{2}$$
$71$ $$(T^{2} - 128)^{4}$$
$73$ $$(T^{2} - 6 T - 39)^{4}$$
$79$ $$(T^{4} + 254 T^{2} + 5329)^{2}$$
$83$ $$(T^{4} - 64 T^{2} + 256)^{2}$$
$89$ $$(T^{4} + 48 T^{2} + 144)^{2}$$
$97$ $$(T - 7)^{8}$$