# Properties

 Label 784.4.a.y Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$46.2574974445$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{3} - 14 \beta q^{5} + 23 q^{9} +O(q^{10})$$ q + 5*b * q^3 - 14*b * q^5 + 23 * q^9 $$q + 5 \beta q^{3} - 14 \beta q^{5} + 23 q^{9} + 14 q^{11} + 36 \beta q^{13} - 140 q^{15} + \beta q^{17} + \beta q^{19} - 140 q^{23} + 267 q^{25} - 20 \beta q^{27} - 286 q^{29} + 66 \beta q^{31} + 70 \beta q^{33} - 38 q^{37} + 360 q^{39} - 89 \beta q^{41} + 34 q^{43} - 322 \beta q^{45} - 370 \beta q^{47} + 10 q^{51} - 74 q^{53} - 196 \beta q^{55} + 10 q^{57} - 307 \beta q^{59} + 10 \beta q^{61} - 1008 q^{65} - 684 q^{67} - 700 \beta q^{69} - 588 q^{71} - 191 \beta q^{73} + 1335 \beta q^{75} - 1220 q^{79} - 821 q^{81} - 299 \beta q^{83} - 28 q^{85} - 1430 \beta q^{87} + 437 \beta q^{89} + 660 q^{93} - 28 q^{95} + 1049 \beta q^{97} + 322 q^{99} +O(q^{100})$$ q + 5*b * q^3 - 14*b * q^5 + 23 * q^9 + 14 * q^11 + 36*b * q^13 - 140 * q^15 + b * q^17 + b * q^19 - 140 * q^23 + 267 * q^25 - 20*b * q^27 - 286 * q^29 + 66*b * q^31 + 70*b * q^33 - 38 * q^37 + 360 * q^39 - 89*b * q^41 + 34 * q^43 - 322*b * q^45 - 370*b * q^47 + 10 * q^51 - 74 * q^53 - 196*b * q^55 + 10 * q^57 - 307*b * q^59 + 10*b * q^61 - 1008 * q^65 - 684 * q^67 - 700*b * q^69 - 588 * q^71 - 191*b * q^73 + 1335*b * q^75 - 1220 * q^79 - 821 * q^81 - 299*b * q^83 - 28 * q^85 - 1430*b * q^87 + 437*b * q^89 + 660 * q^93 - 28 * q^95 + 1049*b * q^97 + 322 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 46 q^{9}+O(q^{10})$$ 2 * q + 46 * q^9 $$2 q + 46 q^{9} + 28 q^{11} - 280 q^{15} - 280 q^{23} + 534 q^{25} - 572 q^{29} - 76 q^{37} + 720 q^{39} + 68 q^{43} + 20 q^{51} - 148 q^{53} + 20 q^{57} - 2016 q^{65} - 1368 q^{67} - 1176 q^{71} - 2440 q^{79} - 1642 q^{81} - 56 q^{85} + 1320 q^{93} - 56 q^{95} + 644 q^{99}+O(q^{100})$$ 2 * q + 46 * q^9 + 28 * q^11 - 280 * q^15 - 280 * q^23 + 534 * q^25 - 572 * q^29 - 76 * q^37 + 720 * q^39 + 68 * q^43 + 20 * q^51 - 148 * q^53 + 20 * q^57 - 2016 * q^65 - 1368 * q^67 - 1176 * q^71 - 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 1320 * q^93 - 56 * q^95 + 644 * 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
 −1.41421 1.41421
0 −7.07107 0 19.7990 0 0 0 23.0000 0
1.2 0 7.07107 0 −19.7990 0 0 0 23.0000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.y 2
4.b odd 2 1 98.4.a.g 2
7.b odd 2 1 inner 784.4.a.y 2
12.b even 2 1 882.4.a.bg 2
20.d odd 2 1 2450.4.a.bx 2
28.d even 2 1 98.4.a.g 2
28.f even 6 2 98.4.c.h 4
28.g odd 6 2 98.4.c.h 4
84.h odd 2 1 882.4.a.bg 2
84.j odd 6 2 882.4.g.ba 4
84.n even 6 2 882.4.g.ba 4
140.c even 2 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 4.b odd 2 1
98.4.a.g 2 28.d even 2 1
98.4.c.h 4 28.f even 6 2
98.4.c.h 4 28.g odd 6 2
784.4.a.y 2 1.a even 1 1 trivial
784.4.a.y 2 7.b odd 2 1 inner
882.4.a.bg 2 12.b even 2 1
882.4.a.bg 2 84.h odd 2 1
882.4.g.ba 4 84.j odd 6 2
882.4.g.ba 4 84.n even 6 2
2450.4.a.bx 2 20.d odd 2 1
2450.4.a.bx 2 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 50$$ T3^2 - 50 $$T_{5}^{2} - 392$$ T5^2 - 392

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 50$$
$5$ $$T^{2} - 392$$
$7$ $$T^{2}$$
$11$ $$(T - 14)^{2}$$
$13$ $$T^{2} - 2592$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} - 2$$
$23$ $$(T + 140)^{2}$$
$29$ $$(T + 286)^{2}$$
$31$ $$T^{2} - 8712$$
$37$ $$(T + 38)^{2}$$
$41$ $$T^{2} - 15842$$
$43$ $$(T - 34)^{2}$$
$47$ $$T^{2} - 273800$$
$53$ $$(T + 74)^{2}$$
$59$ $$T^{2} - 188498$$
$61$ $$T^{2} - 200$$
$67$ $$(T + 684)^{2}$$
$71$ $$(T + 588)^{2}$$
$73$ $$T^{2} - 72962$$
$79$ $$(T + 1220)^{2}$$
$83$ $$T^{2} - 178802$$
$89$ $$T^{2} - 381938$$
$97$ $$T^{2} - 2200802$$