# Properties

 Label 784.6.a.f 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 28) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 96 q^{5} - 239 q^{9}+O(q^{10})$$ q - 2 * q^3 + 96 * q^5 - 239 * q^9 $$q - 2 q^{3} + 96 q^{5} - 239 q^{9} + 720 q^{11} - 572 q^{13} - 192 q^{15} - 1254 q^{17} - 94 q^{19} - 96 q^{23} + 6091 q^{25} + 964 q^{27} - 4374 q^{29} - 6244 q^{31} - 1440 q^{33} - 10798 q^{37} + 1144 q^{39} - 12006 q^{41} + 9160 q^{43} - 22944 q^{45} - 25836 q^{47} + 2508 q^{51} + 1014 q^{53} + 69120 q^{55} + 188 q^{57} + 1242 q^{59} - 7592 q^{61} - 54912 q^{65} - 41132 q^{67} + 192 q^{69} + 37632 q^{71} + 13438 q^{73} - 12182 q^{75} - 6248 q^{79} + 56149 q^{81} - 25254 q^{83} - 120384 q^{85} + 8748 q^{87} + 45126 q^{89} + 12488 q^{93} - 9024 q^{95} - 107222 q^{97} - 172080 q^{99}+O(q^{100})$$ q - 2 * q^3 + 96 * q^5 - 239 * q^9 + 720 * q^11 - 572 * q^13 - 192 * q^15 - 1254 * q^17 - 94 * q^19 - 96 * q^23 + 6091 * q^25 + 964 * q^27 - 4374 * q^29 - 6244 * q^31 - 1440 * q^33 - 10798 * q^37 + 1144 * q^39 - 12006 * q^41 + 9160 * q^43 - 22944 * q^45 - 25836 * q^47 + 2508 * q^51 + 1014 * q^53 + 69120 * q^55 + 188 * q^57 + 1242 * q^59 - 7592 * q^61 - 54912 * q^65 - 41132 * q^67 + 192 * q^69 + 37632 * q^71 + 13438 * q^73 - 12182 * q^75 - 6248 * q^79 + 56149 * q^81 - 25254 * q^83 - 120384 * q^85 + 8748 * q^87 + 45126 * q^89 + 12488 * q^93 - 9024 * q^95 - 107222 * q^97 - 172080 * 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 −2.00000 0 96.0000 0 0 0 −239.000 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.f 1
4.b odd 2 1 196.6.a.d 1
7.b odd 2 1 112.6.a.e 1
21.c even 2 1 1008.6.a.bb 1
28.d even 2 1 28.6.a.a 1
28.f even 6 2 196.6.e.f 2
28.g odd 6 2 196.6.e.e 2
56.e even 2 1 448.6.a.i 1
56.h odd 2 1 448.6.a.h 1
84.h odd 2 1 252.6.a.d 1
140.c even 2 1 700.6.a.d 1
140.j odd 4 2 700.6.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.a 1 28.d even 2 1
112.6.a.e 1 7.b odd 2 1
196.6.a.d 1 4.b odd 2 1
196.6.e.e 2 28.g odd 6 2
196.6.e.f 2 28.f even 6 2
252.6.a.d 1 84.h odd 2 1
448.6.a.h 1 56.h odd 2 1
448.6.a.i 1 56.e even 2 1
700.6.a.d 1 140.c even 2 1
700.6.e.d 2 140.j odd 4 2
784.6.a.f 1 1.a even 1 1 trivial
1008.6.a.bb 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 96$$
$7$ $$T$$
$11$ $$T - 720$$
$13$ $$T + 572$$
$17$ $$T + 1254$$
$19$ $$T + 94$$
$23$ $$T + 96$$
$29$ $$T + 4374$$
$31$ $$T + 6244$$
$37$ $$T + 10798$$
$41$ $$T + 12006$$
$43$ $$T - 9160$$
$47$ $$T + 25836$$
$53$ $$T - 1014$$
$59$ $$T - 1242$$
$61$ $$T + 7592$$
$67$ $$T + 41132$$
$71$ $$T - 37632$$
$73$ $$T - 13438$$
$79$ $$T + 6248$$
$83$ $$T + 25254$$
$89$ $$T - 45126$$
$97$ $$T + 107222$$