# Properties

 Label 728.2.r.c Level $728$ Weight $2$ Character orbit 728.r 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(417,728)]

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, 4, 0]))

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

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

chi := DirichletCharacter("728.417");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$728 = 2^{3} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 728.r (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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + 3*z * q^5 + (2*z - 3) * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} - q^{13} + 3 q^{15} + (7 \zeta_{6} - 7) q^{17} + \zeta_{6} q^{19} + (3 \zeta_{6} - 1) q^{21} - \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 5 q^{27} - 2 q^{29} + (9 \zeta_{6} - 9) q^{31} - 3 \zeta_{6} q^{33} + ( - 3 \zeta_{6} - 6) q^{35} + 3 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{39} + 10 q^{41} + 4 q^{43} + (6 \zeta_{6} - 6) q^{45} - 3 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + 7 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{53} + 9 q^{55} + q^{57} + ( - 11 \zeta_{6} + 11) q^{59} + \zeta_{6} q^{61} + ( - 2 \zeta_{6} - 4) q^{63} - 3 \zeta_{6} q^{65} + ( - 7 \zeta_{6} + 7) q^{67} - q^{69} + 8 q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + 4 \zeta_{6} q^{75} + (9 \zeta_{6} - 3) q^{77} + 11 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 4 q^{83} - 21 q^{85} + (2 \zeta_{6} - 2) q^{87} - \zeta_{6} q^{89} + ( - 2 \zeta_{6} + 3) q^{91} + 9 \zeta_{6} q^{93} + (3 \zeta_{6} - 3) q^{95} + 2 q^{97} + 6 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + 3*z * q^5 + (2*z - 3) * q^7 + 2*z * q^9 + (-3*z + 3) * q^11 - q^13 + 3 * q^15 + (7*z - 7) * q^17 + z * q^19 + (3*z - 1) * q^21 - z * q^23 + (4*z - 4) * q^25 + 5 * q^27 - 2 * q^29 + (9*z - 9) * q^31 - 3*z * q^33 + (-3*z - 6) * q^35 + 3*z * q^37 + (z - 1) * q^39 + 10 * q^41 + 4 * q^43 + (6*z - 6) * q^45 - 3*z * q^47 + (-8*z + 5) * q^49 + 7*z * q^51 + (-z + 1) * q^53 + 9 * q^55 + q^57 + (-11*z + 11) * q^59 + z * q^61 + (-2*z - 4) * q^63 - 3*z * q^65 + (-7*z + 7) * q^67 - q^69 + 8 * q^71 + (-7*z + 7) * q^73 + 4*z * q^75 + (9*z - 3) * q^77 + 11*z * q^79 + (z - 1) * q^81 - 4 * q^83 - 21 * q^85 + (2*z - 2) * q^87 - z * q^89 + (-2*z + 3) * q^91 + 9*z * q^93 + (3*z - 3) * q^95 + 2 * q^97 + 6 * 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 $$2 q + q^{3} + 3 q^{5} - 4 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} + 6 q^{15} - 7 q^{17} + q^{19} + q^{21} - q^{23} - 4 q^{25} + 10 q^{27} - 4 q^{29} - 9 q^{31} - 3 q^{33} - 15 q^{35} + 3 q^{37} - q^{39} + 20 q^{41} + 8 q^{43} - 6 q^{45} - 3 q^{47} + 2 q^{49} + 7 q^{51} + q^{53} + 18 q^{55} + 2 q^{57} + 11 q^{59} + q^{61} - 10 q^{63} - 3 q^{65} + 7 q^{67} - 2 q^{69} + 16 q^{71} + 7 q^{73} + 4 q^{75} + 3 q^{77} + 11 q^{79} - q^{81} - 8 q^{83} - 42 q^{85} - 2 q^{87} - q^{89} + 4 q^{91} + 9 q^{93} - 3 q^{95} + 4 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 3 * q^5 - 4 * q^7 + 2 * q^9 + 3 * q^11 - 2 * q^13 + 6 * q^15 - 7 * q^17 + q^19 + q^21 - q^23 - 4 * q^25 + 10 * q^27 - 4 * q^29 - 9 * q^31 - 3 * q^33 - 15 * q^35 + 3 * q^37 - q^39 + 20 * q^41 + 8 * q^43 - 6 * q^45 - 3 * q^47 + 2 * q^49 + 7 * q^51 + q^53 + 18 * q^55 + 2 * q^57 + 11 * q^59 + q^61 - 10 * q^63 - 3 * q^65 + 7 * q^67 - 2 * q^69 + 16 * q^71 + 7 * q^73 + 4 * q^75 + 3 * q^77 + 11 * q^79 - q^81 - 8 * q^83 - 42 * q^85 - 2 * q^87 - q^89 + 4 * q^91 + 9 * q^93 - 3 * q^95 + 4 * q^97 + 12 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 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
625.1 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
7.c 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.r.c 2
4.b odd 2 1 1456.2.r.e 2
7.c even 3 1 inner 728.2.r.c 2
7.c even 3 1 5096.2.a.b 1
7.d odd 6 1 5096.2.a.j 1
28.g odd 6 1 1456.2.r.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.r.c 2 1.a even 1 1 trivial
728.2.r.c 2 7.c even 3 1 inner
1456.2.r.e 2 4.b odd 2 1
1456.2.r.e 2 28.g odd 6 1
5096.2.a.b 1 7.c even 3 1
5096.2.a.j 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(728, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 7T + 49$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} + T + 1$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} + 9T + 81$$
$37$ $$T^{2} - 3T + 9$$
$41$ $$(T - 10)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$T^{2} - T + 1$$
$59$ $$T^{2} - 11T + 121$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + T + 1$$
$97$ $$(T - 2)^{2}$$
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