# Properties

 Label 4851.2.a.j Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ 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] = [4851,2,Mod(1,4851)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4851 = 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4851.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: $$38.7354300205$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - q^{5}+O(q^{10})$$ q - 2 * q^4 - q^5 $$q - 2 q^{4} - q^{5} + q^{11} + 4 q^{13} + 4 q^{16} + 2 q^{17} + 6 q^{19} + 2 q^{20} + 5 q^{23} - 4 q^{25} - 10 q^{29} - q^{31} - 5 q^{37} - 2 q^{41} - 8 q^{43} - 2 q^{44} + 8 q^{47} - 8 q^{52} + 6 q^{53} - q^{55} + 3 q^{59} + 2 q^{61} - 8 q^{64} - 4 q^{65} - 3 q^{67} - 4 q^{68} - q^{71} - 10 q^{73} - 12 q^{76} + 6 q^{79} - 4 q^{80} + 12 q^{83} - 2 q^{85} - 15 q^{89} - 10 q^{92} - 6 q^{95} + 5 q^{97}+O(q^{100})$$ q - 2 * q^4 - q^5 + q^11 + 4 * q^13 + 4 * q^16 + 2 * q^17 + 6 * q^19 + 2 * q^20 + 5 * q^23 - 4 * q^25 - 10 * q^29 - q^31 - 5 * q^37 - 2 * q^41 - 8 * q^43 - 2 * q^44 + 8 * q^47 - 8 * q^52 + 6 * q^53 - q^55 + 3 * q^59 + 2 * q^61 - 8 * q^64 - 4 * q^65 - 3 * q^67 - 4 * q^68 - q^71 - 10 * q^73 - 12 * q^76 + 6 * q^79 - 4 * q^80 + 12 * q^83 - 2 * q^85 - 15 * q^89 - 10 * q^92 - 6 * q^95 + 5 * q^97

## 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 0 −2.00000 −1.00000 0 0 0 0 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
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 4851.2.a.j 1
3.b odd 2 1 539.2.a.c 1
7.b odd 2 1 693.2.a.c 1
12.b even 2 1 8624.2.a.a 1
21.c even 2 1 77.2.a.a 1
21.g even 6 2 539.2.e.f 2
21.h odd 6 2 539.2.e.c 2
33.d even 2 1 5929.2.a.f 1
77.b even 2 1 7623.2.a.j 1
84.h odd 2 1 1232.2.a.l 1
105.g even 2 1 1925.2.a.h 1
105.k odd 4 2 1925.2.b.e 2
168.e odd 2 1 4928.2.a.a 1
168.i even 2 1 4928.2.a.bj 1
231.h odd 2 1 847.2.a.b 1
231.r odd 10 4 847.2.f.h 4
231.u even 10 4 847.2.f.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 21.c even 2 1
539.2.a.c 1 3.b odd 2 1
539.2.e.c 2 21.h odd 6 2
539.2.e.f 2 21.g even 6 2
693.2.a.c 1 7.b odd 2 1
847.2.a.b 1 231.h odd 2 1
847.2.f.h 4 231.r odd 10 4
847.2.f.i 4 231.u even 10 4
1232.2.a.l 1 84.h odd 2 1
1925.2.a.h 1 105.g even 2 1
1925.2.b.e 2 105.k odd 4 2
4851.2.a.j 1 1.a even 1 1 trivial
4928.2.a.a 1 168.e odd 2 1
4928.2.a.bj 1 168.i even 2 1
5929.2.a.f 1 33.d even 2 1
7623.2.a.j 1 77.b even 2 1
8624.2.a.a 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4851))$$:

 $$T_{2}$$ T2 $$T_{5} + 1$$ T5 + 1 $$T_{13} - 4$$ T13 - 4 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T - 4$$
$17$ $$T - 2$$
$19$ $$T - 6$$
$23$ $$T - 5$$
$29$ $$T + 10$$
$31$ $$T + 1$$
$37$ $$T + 5$$
$41$ $$T + 2$$
$43$ $$T + 8$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T - 3$$
$61$ $$T - 2$$
$67$ $$T + 3$$
$71$ $$T + 1$$
$73$ $$T + 10$$
$79$ $$T - 6$$
$83$ $$T - 12$$
$89$ $$T + 15$$
$97$ $$T - 5$$