Properties

 Label 784.6.a.d 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 4) 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)

 $$f(q)$$ $$=$$ $$q - 12 q^{3} - 54 q^{5} - 99 q^{9}+O(q^{10})$$ q - 12 * q^3 - 54 * q^5 - 99 * q^9 $$q - 12 q^{3} - 54 q^{5} - 99 q^{9} - 540 q^{11} + 418 q^{13} + 648 q^{15} - 594 q^{17} + 836 q^{19} + 4104 q^{23} - 209 q^{25} + 4104 q^{27} - 594 q^{29} + 4256 q^{31} + 6480 q^{33} - 298 q^{37} - 5016 q^{39} - 17226 q^{41} + 12100 q^{43} + 5346 q^{45} - 1296 q^{47} + 7128 q^{51} + 19494 q^{53} + 29160 q^{55} - 10032 q^{57} - 7668 q^{59} + 34738 q^{61} - 22572 q^{65} - 21812 q^{67} - 49248 q^{69} + 46872 q^{71} - 67562 q^{73} + 2508 q^{75} + 76912 q^{79} - 25191 q^{81} + 67716 q^{83} + 32076 q^{85} + 7128 q^{87} - 29754 q^{89} - 51072 q^{93} - 45144 q^{95} + 122398 q^{97} + 53460 q^{99}+O(q^{100})$$ q - 12 * q^3 - 54 * q^5 - 99 * q^9 - 540 * q^11 + 418 * q^13 + 648 * q^15 - 594 * q^17 + 836 * q^19 + 4104 * q^23 - 209 * q^25 + 4104 * q^27 - 594 * q^29 + 4256 * q^31 + 6480 * q^33 - 298 * q^37 - 5016 * q^39 - 17226 * q^41 + 12100 * q^43 + 5346 * q^45 - 1296 * q^47 + 7128 * q^51 + 19494 * q^53 + 29160 * q^55 - 10032 * q^57 - 7668 * q^59 + 34738 * q^61 - 22572 * q^65 - 21812 * q^67 - 49248 * q^69 + 46872 * q^71 - 67562 * q^73 + 2508 * q^75 + 76912 * q^79 - 25191 * q^81 + 67716 * q^83 + 32076 * q^85 + 7128 * q^87 - 29754 * q^89 - 51072 * q^93 - 45144 * q^95 + 122398 * q^97 + 53460 * 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 −12.0000 0 −54.0000 0 0 0 −99.0000 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.d 1
4.b odd 2 1 196.6.a.e 1
7.b odd 2 1 16.6.a.b 1
21.c even 2 1 144.6.a.c 1
28.d even 2 1 4.6.a.a 1
28.f even 6 2 196.6.e.g 2
28.g odd 6 2 196.6.e.d 2
35.c odd 2 1 400.6.a.d 1
35.f even 4 2 400.6.c.f 2
56.e even 2 1 64.6.a.f 1
56.h odd 2 1 64.6.a.b 1
84.h odd 2 1 36.6.a.a 1
112.j even 4 2 256.6.b.g 2
112.l odd 4 2 256.6.b.c 2
140.c even 2 1 100.6.a.b 1
140.j odd 4 2 100.6.c.b 2
168.e odd 2 1 576.6.a.bc 1
168.i even 2 1 576.6.a.bd 1
252.s odd 6 2 324.6.e.d 2
252.bi even 6 2 324.6.e.a 2
308.g odd 2 1 484.6.a.a 1
364.h even 2 1 676.6.a.a 1
364.p odd 4 2 676.6.d.a 2
420.o odd 2 1 900.6.a.h 1
420.w even 4 2 900.6.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 28.d even 2 1
16.6.a.b 1 7.b odd 2 1
36.6.a.a 1 84.h odd 2 1
64.6.a.b 1 56.h odd 2 1
64.6.a.f 1 56.e even 2 1
100.6.a.b 1 140.c even 2 1
100.6.c.b 2 140.j odd 4 2
144.6.a.c 1 21.c even 2 1
196.6.a.e 1 4.b odd 2 1
196.6.e.d 2 28.g odd 6 2
196.6.e.g 2 28.f even 6 2
256.6.b.c 2 112.l odd 4 2
256.6.b.g 2 112.j even 4 2
324.6.e.a 2 252.bi even 6 2
324.6.e.d 2 252.s odd 6 2
400.6.a.d 1 35.c odd 2 1
400.6.c.f 2 35.f even 4 2
484.6.a.a 1 308.g odd 2 1
576.6.a.bc 1 168.e odd 2 1
576.6.a.bd 1 168.i even 2 1
676.6.a.a 1 364.h even 2 1
676.6.d.a 2 364.p odd 4 2
784.6.a.d 1 1.a even 1 1 trivial
900.6.a.h 1 420.o odd 2 1
900.6.d.a 2 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 12$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 12$$
$5$ $$T + 54$$
$7$ $$T$$
$11$ $$T + 540$$
$13$ $$T - 418$$
$17$ $$T + 594$$
$19$ $$T - 836$$
$23$ $$T - 4104$$
$29$ $$T + 594$$
$31$ $$T - 4256$$
$37$ $$T + 298$$
$41$ $$T + 17226$$
$43$ $$T - 12100$$
$47$ $$T + 1296$$
$53$ $$T - 19494$$
$59$ $$T + 7668$$
$61$ $$T - 34738$$
$67$ $$T + 21812$$
$71$ $$T - 46872$$
$73$ $$T + 67562$$
$79$ $$T - 76912$$
$83$ $$T - 67716$$
$89$ $$T + 29754$$
$97$ $$T - 122398$$