# Properties

 Label 784.6.a.h Level $784$ Weight $6$ Character orbit 784.a Self dual yes Analytic conductor $125.741$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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)

 $$f(q)$$ $$=$$ $$q + 8 q^{3} - 10 q^{5} - 179 q^{9}+O(q^{10})$$ q + 8 * q^3 - 10 * q^5 - 179 * q^9 $$q + 8 q^{3} - 10 q^{5} - 179 q^{9} + 340 q^{11} + 294 q^{13} - 80 q^{15} - 1226 q^{17} + 2432 q^{19} - 2000 q^{23} - 3025 q^{25} - 3376 q^{27} - 6746 q^{29} + 8856 q^{31} + 2720 q^{33} + 9182 q^{37} + 2352 q^{39} + 14574 q^{41} - 8108 q^{43} + 1790 q^{45} - 312 q^{47} - 9808 q^{51} - 14634 q^{53} - 3400 q^{55} + 19456 q^{57} - 27656 q^{59} - 34338 q^{61} - 2940 q^{65} - 12316 q^{67} - 16000 q^{69} - 36920 q^{71} + 61718 q^{73} - 24200 q^{75} + 64752 q^{79} + 16489 q^{81} - 77056 q^{83} + 12260 q^{85} - 53968 q^{87} + 8166 q^{89} + 70848 q^{93} - 24320 q^{95} - 20650 q^{97} - 60860 q^{99}+O(q^{100})$$ q + 8 * q^3 - 10 * q^5 - 179 * q^9 + 340 * q^11 + 294 * q^13 - 80 * q^15 - 1226 * q^17 + 2432 * q^19 - 2000 * q^23 - 3025 * q^25 - 3376 * q^27 - 6746 * q^29 + 8856 * q^31 + 2720 * q^33 + 9182 * q^37 + 2352 * q^39 + 14574 * q^41 - 8108 * q^43 + 1790 * q^45 - 312 * q^47 - 9808 * q^51 - 14634 * q^53 - 3400 * q^55 + 19456 * q^57 - 27656 * q^59 - 34338 * q^61 - 2940 * q^65 - 12316 * q^67 - 16000 * q^69 - 36920 * q^71 + 61718 * q^73 - 24200 * q^75 + 64752 * q^79 + 16489 * q^81 - 77056 * q^83 + 12260 * q^85 - 53968 * q^87 + 8166 * q^89 + 70848 * q^93 - 24320 * q^95 - 20650 * q^97 - 60860 * 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 8.00000 0 −10.0000 0 0 0 −179.000 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.h 1
4.b odd 2 1 98.6.a.b 1
7.b odd 2 1 112.6.a.d 1
12.b even 2 1 882.6.a.g 1
21.c even 2 1 1008.6.a.n 1
28.d even 2 1 14.6.a.b 1
28.f even 6 2 98.6.c.a 2
28.g odd 6 2 98.6.c.b 2
56.e even 2 1 448.6.a.f 1
56.h odd 2 1 448.6.a.k 1
84.h odd 2 1 126.6.a.c 1
140.c even 2 1 350.6.a.b 1
140.j odd 4 2 350.6.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 28.d even 2 1
98.6.a.b 1 4.b odd 2 1
98.6.c.a 2 28.f even 6 2
98.6.c.b 2 28.g odd 6 2
112.6.a.d 1 7.b odd 2 1
126.6.a.c 1 84.h odd 2 1
350.6.a.b 1 140.c even 2 1
350.6.c.f 2 140.j odd 4 2
448.6.a.f 1 56.e even 2 1
448.6.a.k 1 56.h odd 2 1
784.6.a.h 1 1.a even 1 1 trivial
882.6.a.g 1 12.b even 2 1
1008.6.a.n 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T + 10$$
$7$ $$T$$
$11$ $$T - 340$$
$13$ $$T - 294$$
$17$ $$T + 1226$$
$19$ $$T - 2432$$
$23$ $$T + 2000$$
$29$ $$T + 6746$$
$31$ $$T - 8856$$
$37$ $$T - 9182$$
$41$ $$T - 14574$$
$43$ $$T + 8108$$
$47$ $$T + 312$$
$53$ $$T + 14634$$
$59$ $$T + 27656$$
$61$ $$T + 34338$$
$67$ $$T + 12316$$
$71$ $$T + 36920$$
$73$ $$T - 61718$$
$79$ $$T - 64752$$
$83$ $$T + 77056$$
$89$ $$T - 8166$$
$97$ $$T + 20650$$