# Properties

 Label 784.6.a.e Level $784$ Weight $6$ Character orbit 784.a Self dual yes Analytic conductor $125.741$ 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] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{3} - 4 q^{5} - 207 q^{9}+O(q^{10})$$ q - 6 * q^3 - 4 * q^5 - 207 * q^9 $$q - 6 q^{3} - 4 q^{5} - 207 q^{9} + 240 q^{11} + 744 q^{13} + 24 q^{15} + 1042 q^{17} - 986 q^{19} - 184 q^{23} - 3109 q^{25} + 2700 q^{27} - 734 q^{29} + 5140 q^{31} - 1440 q^{33} - 6054 q^{37} - 4464 q^{39} - 7598 q^{41} - 13016 q^{43} + 828 q^{45} + 14668 q^{47} - 6252 q^{51} - 14522 q^{53} - 960 q^{55} + 5916 q^{57} - 13362 q^{59} - 9676 q^{61} - 2976 q^{65} + 62124 q^{67} + 1104 q^{69} + 2112 q^{71} + 28910 q^{73} + 18654 q^{75} + 101768 q^{79} + 34101 q^{81} - 23922 q^{83} - 4168 q^{85} + 4404 q^{87} - 141674 q^{89} - 30840 q^{93} + 3944 q^{95} - 99982 q^{97} - 49680 q^{99}+O(q^{100})$$ q - 6 * q^3 - 4 * q^5 - 207 * q^9 + 240 * q^11 + 744 * q^13 + 24 * q^15 + 1042 * q^17 - 986 * q^19 - 184 * q^23 - 3109 * q^25 + 2700 * q^27 - 734 * q^29 + 5140 * q^31 - 1440 * q^33 - 6054 * q^37 - 4464 * q^39 - 7598 * q^41 - 13016 * q^43 + 828 * q^45 + 14668 * q^47 - 6252 * q^51 - 14522 * q^53 - 960 * q^55 + 5916 * q^57 - 13362 * q^59 - 9676 * q^61 - 2976 * q^65 + 62124 * q^67 + 1104 * q^69 + 2112 * q^71 + 28910 * q^73 + 18654 * q^75 + 101768 * q^79 + 34101 * q^81 - 23922 * q^83 - 4168 * q^85 + 4404 * q^87 - 141674 * q^89 - 30840 * q^93 + 3944 * q^95 - 99982 * q^97 - 49680 * 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 −6.00000 0 −4.00000 0 0 0 −207.000 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

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 784.6.a.e 1
4.b odd 2 1 392.6.a.c 1
7.b odd 2 1 112.6.a.f 1
21.c even 2 1 1008.6.a.p 1
28.d even 2 1 56.6.a.a 1
28.f even 6 2 392.6.i.d 2
28.g odd 6 2 392.6.i.c 2
56.e even 2 1 448.6.a.j 1
56.h odd 2 1 448.6.a.g 1
84.h odd 2 1 504.6.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.a 1 28.d even 2 1
112.6.a.f 1 7.b odd 2 1
392.6.a.c 1 4.b odd 2 1
392.6.i.c 2 28.g odd 6 2
392.6.i.d 2 28.f even 6 2
448.6.a.g 1 56.h odd 2 1
448.6.a.j 1 56.e even 2 1
504.6.a.e 1 84.h odd 2 1
784.6.a.e 1 1.a even 1 1 trivial
1008.6.a.p 1 21.c even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 6$$
$5$ $$T + 4$$
$7$ $$T$$
$11$ $$T - 240$$
$13$ $$T - 744$$
$17$ $$T - 1042$$
$19$ $$T + 986$$
$23$ $$T + 184$$
$29$ $$T + 734$$
$31$ $$T - 5140$$
$37$ $$T + 6054$$
$41$ $$T + 7598$$
$43$ $$T + 13016$$
$47$ $$T - 14668$$
$53$ $$T + 14522$$
$59$ $$T + 13362$$
$61$ $$T + 9676$$
$67$ $$T - 62124$$
$71$ $$T - 2112$$
$73$ $$T - 28910$$
$79$ $$T - 101768$$
$83$ $$T + 23922$$
$89$ $$T + 141674$$
$97$ $$T + 99982$$