# Properties

 Label 728.2.a.b Level $728$ Weight $2$ Character orbit 728.a Self dual yes Analytic conductor $5.813$ Analytic rank $0$ 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] = [728,2,Mod(1,728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$728 = 2^{3} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 728.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: $$5.81310926715$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + q^7 - 2 * q^9 $$q - q^{3} + q^{7} - 2 q^{9} + 3 q^{11} - q^{13} + 4 q^{17} + 2 q^{19} - q^{21} + q^{23} - 5 q^{25} + 5 q^{27} + 4 q^{29} + 9 q^{31} - 3 q^{33} + 3 q^{37} + q^{39} - 5 q^{41} + 4 q^{43} + 9 q^{47} + q^{49} - 4 q^{51} - 4 q^{53} - 2 q^{57} + 10 q^{59} + 5 q^{61} - 2 q^{63} + 11 q^{67} - q^{69} - 16 q^{71} + 11 q^{73} + 5 q^{75} + 3 q^{77} - 5 q^{79} + q^{81} - 4 q^{83} - 4 q^{87} - 2 q^{89} - q^{91} - 9 q^{93} - 7 q^{97} - 6 q^{99}+O(q^{100})$$ q - q^3 + q^7 - 2 * q^9 + 3 * q^11 - q^13 + 4 * q^17 + 2 * q^19 - q^21 + q^23 - 5 * q^25 + 5 * q^27 + 4 * q^29 + 9 * q^31 - 3 * q^33 + 3 * q^37 + q^39 - 5 * q^41 + 4 * q^43 + 9 * q^47 + q^49 - 4 * q^51 - 4 * q^53 - 2 * q^57 + 10 * q^59 + 5 * q^61 - 2 * q^63 + 11 * q^67 - q^69 - 16 * q^71 + 11 * q^73 + 5 * q^75 + 3 * q^77 - 5 * q^79 + q^81 - 4 * q^83 - 4 * q^87 - 2 * q^89 - q^91 - 9 * q^93 - 7 * q^97 - 6 * q^99

## 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 −1.00000 0 0 0 1.00000 0 −2.00000 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
$$2$$ $$1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 728.2.a.b 1
3.b odd 2 1 6552.2.a.m 1
4.b odd 2 1 1456.2.a.j 1
7.b odd 2 1 5096.2.a.i 1
8.b even 2 1 5824.2.a.y 1
8.d odd 2 1 5824.2.a.j 1
13.b even 2 1 9464.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.a.b 1 1.a even 1 1 trivial
1456.2.a.j 1 4.b odd 2 1
5096.2.a.i 1 7.b odd 2 1
5824.2.a.j 1 8.d odd 2 1
5824.2.a.y 1 8.b even 2 1
6552.2.a.m 1 3.b odd 2 1
9464.2.a.b 1 13.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(728))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T - 3$$
$13$ $$T + 1$$
$17$ $$T - 4$$
$19$ $$T - 2$$
$23$ $$T - 1$$
$29$ $$T - 4$$
$31$ $$T - 9$$
$37$ $$T - 3$$
$41$ $$T + 5$$
$43$ $$T - 4$$
$47$ $$T - 9$$
$53$ $$T + 4$$
$59$ $$T - 10$$
$61$ $$T - 5$$
$67$ $$T - 11$$
$71$ $$T + 16$$
$73$ $$T - 11$$
$79$ $$T + 5$$
$83$ $$T + 4$$
$89$ $$T + 2$$
$97$ $$T + 7$$