# Properties

 Label 784.6.a.j 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 196) 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)

 $$f(q)$$ $$=$$ $$q + 16 q^{3} + 16 q^{5} + 13 q^{9}+O(q^{10})$$ q + 16 * q^3 + 16 * q^5 + 13 * q^9 $$q + 16 q^{3} + 16 q^{5} + 13 q^{9} + 76 q^{11} + 880 q^{13} + 256 q^{15} - 1056 q^{17} - 1936 q^{19} - 936 q^{23} - 2869 q^{25} - 3680 q^{27} - 3982 q^{29} - 1568 q^{31} + 1216 q^{33} + 4938 q^{37} + 14080 q^{39} - 15840 q^{41} + 16412 q^{43} + 208 q^{45} + 20768 q^{47} - 16896 q^{51} - 37402 q^{53} + 1216 q^{55} - 30976 q^{57} - 21136 q^{59} - 2992 q^{61} + 14080 q^{65} + 45836 q^{67} - 14976 q^{69} + 49840 q^{71} - 56320 q^{73} - 45904 q^{75} - 40744 q^{79} - 62039 q^{81} - 112464 q^{83} - 16896 q^{85} - 63712 q^{87} + 64256 q^{89} - 25088 q^{93} - 30976 q^{95} - 2272 q^{97} + 988 q^{99}+O(q^{100})$$ q + 16 * q^3 + 16 * q^5 + 13 * q^9 + 76 * q^11 + 880 * q^13 + 256 * q^15 - 1056 * q^17 - 1936 * q^19 - 936 * q^23 - 2869 * q^25 - 3680 * q^27 - 3982 * q^29 - 1568 * q^31 + 1216 * q^33 + 4938 * q^37 + 14080 * q^39 - 15840 * q^41 + 16412 * q^43 + 208 * q^45 + 20768 * q^47 - 16896 * q^51 - 37402 * q^53 + 1216 * q^55 - 30976 * q^57 - 21136 * q^59 - 2992 * q^61 + 14080 * q^65 + 45836 * q^67 - 14976 * q^69 + 49840 * q^71 - 56320 * q^73 - 45904 * q^75 - 40744 * q^79 - 62039 * q^81 - 112464 * q^83 - 16896 * q^85 - 63712 * q^87 + 64256 * q^89 - 25088 * q^93 - 30976 * q^95 - 2272 * q^97 + 988 * 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 16.0000 0 16.0000 0 0 0 13.0000 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.j 1
4.b odd 2 1 196.6.a.c 1
7.b odd 2 1 784.6.a.b 1
28.d even 2 1 196.6.a.f yes 1
28.f even 6 2 196.6.e.c 2
28.g odd 6 2 196.6.e.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.6.a.c 1 4.b odd 2 1
196.6.a.f yes 1 28.d even 2 1
196.6.e.c 2 28.f even 6 2
196.6.e.h 2 28.g odd 6 2
784.6.a.b 1 7.b odd 2 1
784.6.a.j 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 16$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 16$$
$5$ $$T - 16$$
$7$ $$T$$
$11$ $$T - 76$$
$13$ $$T - 880$$
$17$ $$T + 1056$$
$19$ $$T + 1936$$
$23$ $$T + 936$$
$29$ $$T + 3982$$
$31$ $$T + 1568$$
$37$ $$T - 4938$$
$41$ $$T + 15840$$
$43$ $$T - 16412$$
$47$ $$T - 20768$$
$53$ $$T + 37402$$
$59$ $$T + 21136$$
$61$ $$T + 2992$$
$67$ $$T - 45836$$
$71$ $$T - 49840$$
$73$ $$T + 56320$$
$79$ $$T + 40744$$
$83$ $$T + 112464$$
$89$ $$T - 64256$$
$97$ $$T + 2272$$