# Properties

 Label 784.6.a.g Level $784$ Weight $6$ Character orbit 784.a Self dual yes Analytic conductor $125.741$ Analytic rank $1$ Dimension $1$ CM discriminant -7 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$125.740914733$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$+1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 243 q^{9}+O(q^{10})$$ q - 243 * q^9 $$q - 243 q^{9} + 76 q^{11} + 4952 q^{23} - 3125 q^{25} + 7282 q^{29} - 8886 q^{37} - 11748 q^{43} + 24550 q^{53} - 69364 q^{67} + 2224 q^{71} - 80168 q^{79} + 59049 q^{81} - 18468 q^{99}+O(q^{100})$$ q - 243 * q^9 + 76 * q^11 + 4952 * q^23 - 3125 * q^25 + 7282 * q^29 - 8886 * q^37 - 11748 * q^43 + 24550 * q^53 - 69364 * q^67 + 2224 * q^71 - 80168 * q^79 + 59049 * q^81 - 18468 * 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 0 0 0 0 0 0 −243.000 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
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.a.g 1
4.b odd 2 1 49.6.a.b 1
7.b odd 2 1 CM 784.6.a.g 1
12.b even 2 1 441.6.a.a 1
28.d even 2 1 49.6.a.b 1
28.f even 6 2 49.6.c.a 2
28.g odd 6 2 49.6.c.a 2
84.h odd 2 1 441.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.b 1 4.b odd 2 1
49.6.a.b 1 28.d even 2 1
49.6.c.a 2 28.f even 6 2
49.6.c.a 2 28.g odd 6 2
441.6.a.a 1 12.b even 2 1
441.6.a.a 1 84.h odd 2 1
784.6.a.g 1 1.a even 1 1 trivial
784.6.a.g 1 7.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 76$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 4952$$
$29$ $$T - 7282$$
$31$ $$T$$
$37$ $$T + 8886$$
$41$ $$T$$
$43$ $$T + 11748$$
$47$ $$T$$
$53$ $$T - 24550$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T + 69364$$
$71$ $$T - 2224$$
$73$ $$T$$
$79$ $$T + 80168$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$