# Properties

 Label 728.2.q.a Level $728$ Weight $2$ Character orbit 728.q Analytic conductor $5.813$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [728,2,Mod(289,728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 2]))

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

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

chi := DirichletCharacter("728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$728 = 2^{3} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 728.q (of order $$3$$, degree $$2$$, 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: $$5.81310926715$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} - 3 \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 2) q^{9} +O(q^{10})$$ q + z * q^3 - 3*z * q^5 + (2*z + 1) * q^7 + (-2*z + 2) * q^9 $$q + \zeta_{6} q^{3} - 3 \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} + ( - 2 \zeta_{6} + 2) q^{9} + \zeta_{6} q^{11} + ( - 4 \zeta_{6} + 1) q^{13} + ( - 3 \zeta_{6} + 3) q^{15} - 2 q^{17} + ( - 3 \zeta_{6} + 3) q^{19} + (3 \zeta_{6} - 2) q^{21} + (4 \zeta_{6} - 4) q^{25} + 5 q^{27} + ( - 9 \zeta_{6} + 9) q^{29} + (\zeta_{6} - 1) q^{31} + (\zeta_{6} - 1) q^{33} + ( - 9 \zeta_{6} + 6) q^{35} + 10 q^{37} + ( - 3 \zeta_{6} + 4) q^{39} + (3 \zeta_{6} - 3) q^{41} - 3 \zeta_{6} q^{43} - 6 q^{45} - 11 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} - 2 \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} + ( - 3 \zeta_{6} + 3) q^{55} + 3 q^{57} + 12 q^{59} + ( - 5 \zeta_{6} + 5) q^{61} + ( - 2 \zeta_{6} + 6) q^{63} + (9 \zeta_{6} - 12) q^{65} + 9 \zeta_{6} q^{67} + \zeta_{6} q^{71} + (11 \zeta_{6} - 11) q^{73} - 4 q^{75} + (3 \zeta_{6} - 2) q^{77} + 9 \zeta_{6} q^{79} - \zeta_{6} q^{81} - 8 q^{83} + 6 \zeta_{6} q^{85} + 9 q^{87} - 10 q^{89} + ( - 10 \zeta_{6} + 9) q^{91} - q^{93} - 9 q^{95} + 13 \zeta_{6} q^{97} + 2 q^{99} +O(q^{100})$$ q + z * q^3 - 3*z * q^5 + (2*z + 1) * q^7 + (-2*z + 2) * q^9 + z * q^11 + (-4*z + 1) * q^13 + (-3*z + 3) * q^15 - 2 * q^17 + (-3*z + 3) * q^19 + (3*z - 2) * q^21 + (4*z - 4) * q^25 + 5 * q^27 + (-9*z + 9) * q^29 + (z - 1) * q^31 + (z - 1) * q^33 + (-9*z + 6) * q^35 + 10 * q^37 + (-3*z + 4) * q^39 + (3*z - 3) * q^41 - 3*z * q^43 - 6 * q^45 - 11*z * q^47 + (8*z - 3) * q^49 - 2*z * q^51 + (3*z - 3) * q^53 + (-3*z + 3) * q^55 + 3 * q^57 + 12 * q^59 + (-5*z + 5) * q^61 + (-2*z + 6) * q^63 + (9*z - 12) * q^65 + 9*z * q^67 + z * q^71 + (11*z - 11) * q^73 - 4 * q^75 + (3*z - 2) * q^77 + 9*z * q^79 - z * q^81 - 8 * q^83 + 6*z * q^85 + 9 * q^87 - 10 * q^89 + (-10*z + 9) * q^91 - q^93 - 9 * q^95 + 13*z * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 + 4 * q^7 + 2 * q^9 $$2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} - 4 q^{17} + 3 q^{19} - q^{21} - 4 q^{25} + 10 q^{27} + 9 q^{29} - q^{31} - q^{33} + 3 q^{35} + 20 q^{37} + 5 q^{39} - 3 q^{41} - 3 q^{43} - 12 q^{45} - 11 q^{47} + 2 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{55} + 6 q^{57} + 24 q^{59} + 5 q^{61} + 10 q^{63} - 15 q^{65} + 9 q^{67} + q^{71} - 11 q^{73} - 8 q^{75} - q^{77} + 9 q^{79} - q^{81} - 16 q^{83} + 6 q^{85} + 18 q^{87} - 20 q^{89} + 8 q^{91} - 2 q^{93} - 18 q^{95} + 13 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 + 4 * q^7 + 2 * q^9 + q^11 - 2 * q^13 + 3 * q^15 - 4 * q^17 + 3 * q^19 - q^21 - 4 * q^25 + 10 * q^27 + 9 * q^29 - q^31 - q^33 + 3 * q^35 + 20 * q^37 + 5 * q^39 - 3 * q^41 - 3 * q^43 - 12 * q^45 - 11 * q^47 + 2 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^55 + 6 * q^57 + 24 * q^59 + 5 * q^61 + 10 * q^63 - 15 * q^65 + 9 * q^67 + q^71 - 11 * q^73 - 8 * q^75 - q^77 + 9 * q^79 - q^81 - 16 * q^83 + 6 * q^85 + 18 * q^87 - 20 * q^89 + 8 * q^91 - 2 * q^93 - 18 * q^95 + 13 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/728\mathbb{Z}\right)^\times$$.

 $$n$$ $$183$$ $$365$$ $$521$$ $$561$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 −1.50000 + 2.59808i 0 2.00000 1.73205i 0 1.00000 + 1.73205i 0
529.1 0 0.500000 + 0.866025i 0 −1.50000 2.59808i 0 2.00000 + 1.73205i 0 1.00000 1.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 728.2.q.a 2
7.c even 3 1 728.2.t.a yes 2
13.c even 3 1 728.2.t.a yes 2
91.h even 3 1 inner 728.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.q.a 2 1.a even 1 1 trivial
728.2.q.a 2 91.h even 3 1 inner
728.2.t.a yes 2 7.c even 3 1
728.2.t.a yes 2 13.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(728, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 3T + 9$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$T^{2} + T + 1$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + 3T + 9$$
$47$ $$T^{2} + 11T + 121$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 9T + 81$$
$71$ $$T^{2} - T + 1$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} - 9T + 81$$
$83$ $$(T + 8)^{2}$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 13T + 169$$
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