# Properties

 Label 784.6.a.n 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)

 $$f(q)$$ $$=$$ $$q + 30 q^{3} - 32 q^{5} + 657 q^{9}+O(q^{10})$$ q + 30 * q^3 - 32 * q^5 + 657 * q^9 $$q + 30 q^{3} - 32 q^{5} + 657 q^{9} + 624 q^{11} + 708 q^{13} - 960 q^{15} - 934 q^{17} + 1858 q^{19} + 1120 q^{23} - 2101 q^{25} + 12420 q^{27} - 1174 q^{29} + 2908 q^{31} + 18720 q^{33} - 12462 q^{37} + 21240 q^{39} - 2662 q^{41} + 7144 q^{43} - 21024 q^{45} - 7468 q^{47} - 28020 q^{51} - 27274 q^{53} - 19968 q^{55} + 55740 q^{57} + 2490 q^{59} + 11096 q^{61} - 22656 q^{65} - 39756 q^{67} + 33600 q^{69} + 69888 q^{71} - 16450 q^{73} - 63030 q^{75} - 78376 q^{79} + 212949 q^{81} + 109818 q^{83} + 29888 q^{85} - 35220 q^{87} + 56966 q^{89} + 87240 q^{93} - 59456 q^{95} + 115946 q^{97} + 409968 q^{99}+O(q^{100})$$ q + 30 * q^3 - 32 * q^5 + 657 * q^9 + 624 * q^11 + 708 * q^13 - 960 * q^15 - 934 * q^17 + 1858 * q^19 + 1120 * q^23 - 2101 * q^25 + 12420 * q^27 - 1174 * q^29 + 2908 * q^31 + 18720 * q^33 - 12462 * q^37 + 21240 * q^39 - 2662 * q^41 + 7144 * q^43 - 21024 * q^45 - 7468 * q^47 - 28020 * q^51 - 27274 * q^53 - 19968 * q^55 + 55740 * q^57 + 2490 * q^59 + 11096 * q^61 - 22656 * q^65 - 39756 * q^67 + 33600 * q^69 + 69888 * q^71 - 16450 * q^73 - 63030 * q^75 - 78376 * q^79 + 212949 * q^81 + 109818 * q^83 + 29888 * q^85 - 35220 * q^87 + 56966 * q^89 + 87240 * q^93 - 59456 * q^95 + 115946 * q^97 + 409968 * 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 30.0000 0 −32.0000 0 0 0 657.000 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.n 1
4.b odd 2 1 392.6.a.a 1
7.b odd 2 1 112.6.a.a 1
21.c even 2 1 1008.6.a.h 1
28.d even 2 1 56.6.a.b 1
28.f even 6 2 392.6.i.a 2
28.g odd 6 2 392.6.i.f 2
56.e even 2 1 448.6.a.a 1
56.h odd 2 1 448.6.a.p 1
84.h odd 2 1 504.6.a.b 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 30$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 30$$
$5$ $$T + 32$$
$7$ $$T$$
$11$ $$T - 624$$
$13$ $$T - 708$$
$17$ $$T + 934$$
$19$ $$T - 1858$$
$23$ $$T - 1120$$
$29$ $$T + 1174$$
$31$ $$T - 2908$$
$37$ $$T + 12462$$
$41$ $$T + 2662$$
$43$ $$T - 7144$$
$47$ $$T + 7468$$
$53$ $$T + 27274$$
$59$ $$T - 2490$$
$61$ $$T - 11096$$
$67$ $$T + 39756$$
$71$ $$T - 69888$$
$73$ $$T + 16450$$
$79$ $$T + 78376$$
$83$ $$T - 109818$$
$89$ $$T - 56966$$
$97$ $$T - 115946$$