Properties

 Label 784.6.a.l 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 8) 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)

 $$f(q)$$ $$=$$ $$q + 20 q^{3} + 74 q^{5} + 157 q^{9}+O(q^{10})$$ q + 20 * q^3 + 74 * q^5 + 157 * q^9 $$q + 20 q^{3} + 74 q^{5} + 157 q^{9} - 124 q^{11} - 478 q^{13} + 1480 q^{15} + 1198 q^{17} + 3044 q^{19} - 184 q^{23} + 2351 q^{25} - 1720 q^{27} - 3282 q^{29} - 5728 q^{31} - 2480 q^{33} + 10326 q^{37} - 9560 q^{39} + 8886 q^{41} + 9188 q^{43} + 11618 q^{45} + 23664 q^{47} + 23960 q^{51} + 11686 q^{53} - 9176 q^{55} + 60880 q^{57} + 16876 q^{59} + 18482 q^{61} - 35372 q^{65} + 15532 q^{67} - 3680 q^{69} + 31960 q^{71} + 4886 q^{73} + 47020 q^{75} - 44560 q^{79} - 72551 q^{81} + 67364 q^{83} + 88652 q^{85} - 65640 q^{87} - 71994 q^{89} - 114560 q^{93} + 225256 q^{95} - 48866 q^{97} - 19468 q^{99}+O(q^{100})$$ q + 20 * q^3 + 74 * q^5 + 157 * q^9 - 124 * q^11 - 478 * q^13 + 1480 * q^15 + 1198 * q^17 + 3044 * q^19 - 184 * q^23 + 2351 * q^25 - 1720 * q^27 - 3282 * q^29 - 5728 * q^31 - 2480 * q^33 + 10326 * q^37 - 9560 * q^39 + 8886 * q^41 + 9188 * q^43 + 11618 * q^45 + 23664 * q^47 + 23960 * q^51 + 11686 * q^53 - 9176 * q^55 + 60880 * q^57 + 16876 * q^59 + 18482 * q^61 - 35372 * q^65 + 15532 * q^67 - 3680 * q^69 + 31960 * q^71 + 4886 * q^73 + 47020 * q^75 - 44560 * q^79 - 72551 * q^81 + 67364 * q^83 + 88652 * q^85 - 65640 * q^87 - 71994 * q^89 - 114560 * q^93 + 225256 * q^95 - 48866 * q^97 - 19468 * 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 20.0000 0 74.0000 0 0 0 157.000 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.l 1
4.b odd 2 1 392.6.a.b 1
7.b odd 2 1 16.6.a.a 1
21.c even 2 1 144.6.a.k 1
28.d even 2 1 8.6.a.a 1
28.f even 6 2 392.6.i.b 2
28.g odd 6 2 392.6.i.e 2
35.c odd 2 1 400.6.a.l 1
35.f even 4 2 400.6.c.d 2
56.e even 2 1 64.6.a.a 1
56.h odd 2 1 64.6.a.g 1
84.h odd 2 1 72.6.a.f 1
112.j even 4 2 256.6.b.f 2
112.l odd 4 2 256.6.b.d 2
140.c even 2 1 200.6.a.a 1
140.j odd 4 2 200.6.c.a 2
168.e odd 2 1 576.6.a.g 1
168.i even 2 1 576.6.a.h 1
308.g odd 2 1 968.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 28.d even 2 1
16.6.a.a 1 7.b odd 2 1
64.6.a.a 1 56.e even 2 1
64.6.a.g 1 56.h odd 2 1
72.6.a.f 1 84.h odd 2 1
144.6.a.k 1 21.c even 2 1
200.6.a.a 1 140.c even 2 1
200.6.c.a 2 140.j odd 4 2
256.6.b.d 2 112.l odd 4 2
256.6.b.f 2 112.j even 4 2
392.6.a.b 1 4.b odd 2 1
392.6.i.b 2 28.f even 6 2
392.6.i.e 2 28.g odd 6 2
400.6.a.l 1 35.c odd 2 1
400.6.c.d 2 35.f even 4 2
576.6.a.g 1 168.e odd 2 1
576.6.a.h 1 168.i even 2 1
784.6.a.l 1 1.a even 1 1 trivial
968.6.a.a 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 20$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 20$$
$5$ $$T - 74$$
$7$ $$T$$
$11$ $$T + 124$$
$13$ $$T + 478$$
$17$ $$T - 1198$$
$19$ $$T - 3044$$
$23$ $$T + 184$$
$29$ $$T + 3282$$
$31$ $$T + 5728$$
$37$ $$T - 10326$$
$41$ $$T - 8886$$
$43$ $$T - 9188$$
$47$ $$T - 23664$$
$53$ $$T - 11686$$
$59$ $$T - 16876$$
$61$ $$T - 18482$$
$67$ $$T - 15532$$
$71$ $$T - 31960$$
$73$ $$T - 4886$$
$79$ $$T + 44560$$
$83$ $$T - 67364$$
$89$ $$T + 71994$$
$97$ $$T + 48866$$