# Properties

 Label 4851.2.a.r Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4}+O(q^{10})$$ q + 2 * q^2 + 2 * q^4 $$q + 2 q^{2} + 2 q^{4} + q^{11} - 5 q^{13} - 4 q^{16} - 6 q^{17} + 7 q^{19} + 2 q^{22} + 4 q^{23} - 5 q^{25} - 10 q^{26} + 2 q^{29} - 7 q^{31} - 8 q^{32} - 12 q^{34} + 7 q^{37} + 14 q^{38} - 4 q^{41} - 9 q^{43} + 2 q^{44} + 8 q^{46} - 6 q^{47} - 10 q^{50} - 10 q^{52} + 2 q^{53} + 4 q^{58} - 12 q^{59} - 2 q^{61} - 14 q^{62} - 8 q^{64} + 7 q^{67} - 12 q^{68} - 8 q^{71} - 5 q^{73} + 14 q^{74} + 14 q^{76} - 11 q^{79} - 8 q^{82} + 4 q^{83} - 18 q^{86} - 6 q^{89} + 8 q^{92} - 12 q^{94} + 2 q^{97}+O(q^{100})$$ q + 2 * q^2 + 2 * q^4 + q^11 - 5 * q^13 - 4 * q^16 - 6 * q^17 + 7 * q^19 + 2 * q^22 + 4 * q^23 - 5 * q^25 - 10 * q^26 + 2 * q^29 - 7 * q^31 - 8 * q^32 - 12 * q^34 + 7 * q^37 + 14 * q^38 - 4 * q^41 - 9 * q^43 + 2 * q^44 + 8 * q^46 - 6 * q^47 - 10 * q^50 - 10 * q^52 + 2 * q^53 + 4 * q^58 - 12 * q^59 - 2 * q^61 - 14 * q^62 - 8 * q^64 + 7 * q^67 - 12 * q^68 - 8 * q^71 - 5 * q^73 + 14 * q^74 + 14 * q^76 - 11 * q^79 - 8 * q^82 + 4 * q^83 - 18 * q^86 - 6 * q^89 + 8 * q^92 - 12 * q^94 + 2 * 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
2.00000 0 2.00000 0 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.r 1
3.b odd 2 1 1617.2.a.a 1
7.b odd 2 1 4851.2.a.s 1
7.c even 3 2 693.2.i.a 2
21.c even 2 1 1617.2.a.b 1
21.h odd 6 2 231.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.c 2 21.h odd 6 2
693.2.i.a 2 7.c even 3 2
1617.2.a.a 1 3.b odd 2 1
1617.2.a.b 1 21.c even 2 1
4851.2.a.r 1 1.a even 1 1 trivial
4851.2.a.s 1 7.b odd 2 1

## Hecke kernels

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

 $$T_{2} - 2$$ T2 - 2 $$T_{5}$$ T5 $$T_{13} + 5$$ T13 + 5 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T + 5$$
$17$ $$T + 6$$
$19$ $$T - 7$$
$23$ $$T - 4$$
$29$ $$T - 2$$
$31$ $$T + 7$$
$37$ $$T - 7$$
$41$ $$T + 4$$
$43$ $$T + 9$$
$47$ $$T + 6$$
$53$ $$T - 2$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T - 7$$
$71$ $$T + 8$$
$73$ $$T + 5$$
$79$ $$T + 11$$
$83$ $$T - 4$$
$89$ $$T + 6$$
$97$ $$T - 2$$