# Properties

 Label 4851.2.a.bg Level $4851$ Weight $2$ Character orbit 4851.a Self dual yes Analytic conductor $38.735$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1617) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + 3 \beta q^{4} + ( - 2 \beta + 1) q^{5} + (4 \beta + 1) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + 3*b * q^4 + (-2*b + 1) * q^5 + (4*b + 1) * q^8 $$q + (\beta + 1) q^{2} + 3 \beta q^{4} + ( - 2 \beta + 1) q^{5} + (4 \beta + 1) q^{8} + ( - 3 \beta - 1) q^{10} + q^{11} + ( - 4 \beta + 3) q^{13} + (3 \beta + 5) q^{16} + 6 q^{17} + (2 \beta + 1) q^{19} + ( - 3 \beta - 6) q^{20} + (\beta + 1) q^{22} + (4 \beta - 4) q^{23} + ( - 5 \beta - 1) q^{26} + (2 \beta + 7) q^{29} + ( - 4 \beta - 2) q^{31} + (3 \beta + 6) q^{32} + (6 \beta + 6) q^{34} + (4 \beta - 7) q^{37} + (5 \beta + 3) q^{38} + ( - 6 \beta - 7) q^{40} + 6 q^{41} + ( - 4 \beta + 6) q^{43} + 3 \beta q^{44} + 4 \beta q^{46} + (4 \beta + 5) q^{47} + ( - 3 \beta - 12) q^{52} + 8 \beta q^{53} + ( - 2 \beta + 1) q^{55} + (11 \beta + 9) q^{58} + ( - 8 \beta + 1) q^{59} + (8 \beta - 6) q^{61} + ( - 10 \beta - 6) q^{62} + (6 \beta - 1) q^{64} + ( - 2 \beta + 11) q^{65} + (6 \beta - 9) q^{67} + 18 \beta q^{68} + ( - 4 \beta + 2) q^{71} + q^{73} + (\beta - 3) q^{74} + (9 \beta + 6) q^{76} + ( - 4 \beta + 4) q^{79} + ( - 13 \beta - 1) q^{80} + (6 \beta + 6) q^{82} + (4 \beta + 6) q^{83} + ( - 12 \beta + 6) q^{85} + ( - 2 \beta + 2) q^{86} + (4 \beta + 1) q^{88} + ( - 12 \beta + 6) q^{89} + 12 q^{92} + (13 \beta + 9) q^{94} + ( - 4 \beta - 3) q^{95} + (8 \beta - 6) q^{97}+O(q^{100})$$ q + (b + 1) * q^2 + 3*b * q^4 + (-2*b + 1) * q^5 + (4*b + 1) * q^8 + (-3*b - 1) * q^10 + q^11 + (-4*b + 3) * q^13 + (3*b + 5) * q^16 + 6 * q^17 + (2*b + 1) * q^19 + (-3*b - 6) * q^20 + (b + 1) * q^22 + (4*b - 4) * q^23 + (-5*b - 1) * q^26 + (2*b + 7) * q^29 + (-4*b - 2) * q^31 + (3*b + 6) * q^32 + (6*b + 6) * q^34 + (4*b - 7) * q^37 + (5*b + 3) * q^38 + (-6*b - 7) * q^40 + 6 * q^41 + (-4*b + 6) * q^43 + 3*b * q^44 + 4*b * q^46 + (4*b + 5) * q^47 + (-3*b - 12) * q^52 + 8*b * q^53 + (-2*b + 1) * q^55 + (11*b + 9) * q^58 + (-8*b + 1) * q^59 + (8*b - 6) * q^61 + (-10*b - 6) * q^62 + (6*b - 1) * q^64 + (-2*b + 11) * q^65 + (6*b - 9) * q^67 + 18*b * q^68 + (-4*b + 2) * q^71 + q^73 + (b - 3) * q^74 + (9*b + 6) * q^76 + (-4*b + 4) * q^79 + (-13*b - 1) * q^80 + (6*b + 6) * q^82 + (4*b + 6) * q^83 + (-12*b + 6) * q^85 + (-2*b + 2) * q^86 + (4*b + 1) * q^88 + (-12*b + 6) * q^89 + 12 * q^92 + (13*b + 9) * q^94 + (-4*b - 3) * q^95 + (8*b - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 $$2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} - 5 q^{10} + 2 q^{11} + 2 q^{13} + 13 q^{16} + 12 q^{17} + 4 q^{19} - 15 q^{20} + 3 q^{22} - 4 q^{23} - 7 q^{26} + 16 q^{29} - 8 q^{31} + 15 q^{32} + 18 q^{34} - 10 q^{37} + 11 q^{38} - 20 q^{40} + 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} + 14 q^{47} - 27 q^{52} + 8 q^{53} + 29 q^{58} - 6 q^{59} - 4 q^{61} - 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} + 18 q^{68} + 2 q^{73} - 5 q^{74} + 21 q^{76} + 4 q^{79} - 15 q^{80} + 18 q^{82} + 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} + 31 q^{94} - 10 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 - 5 * q^10 + 2 * q^11 + 2 * q^13 + 13 * q^16 + 12 * q^17 + 4 * q^19 - 15 * q^20 + 3 * q^22 - 4 * q^23 - 7 * q^26 + 16 * q^29 - 8 * q^31 + 15 * q^32 + 18 * q^34 - 10 * q^37 + 11 * q^38 - 20 * q^40 + 12 * q^41 + 8 * q^43 + 3 * q^44 + 4 * q^46 + 14 * q^47 - 27 * q^52 + 8 * q^53 + 29 * q^58 - 6 * q^59 - 4 * q^61 - 22 * q^62 + 4 * q^64 + 20 * q^65 - 12 * q^67 + 18 * q^68 + 2 * q^73 - 5 * q^74 + 21 * q^76 + 4 * q^79 - 15 * q^80 + 18 * q^82 + 16 * q^83 + 2 * q^86 + 6 * q^88 + 24 * q^92 + 31 * q^94 - 10 * q^95 - 4 * 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
 −0.618034 1.61803
0.381966 0 −1.85410 2.23607 0 0 −1.47214 0 0.854102
1.2 2.61803 0 4.85410 −2.23607 0 0 7.47214 0 −5.85410
 $$n$$: 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.bg 2
3.b odd 2 1 1617.2.a.l yes 2
7.b odd 2 1 4851.2.a.bh 2
21.c even 2 1 1617.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.k 2 21.c even 2 1
1617.2.a.l yes 2 3.b odd 2 1
4851.2.a.bg 2 1.a even 1 1 trivial
4851.2.a.bh 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4851))$$:

 $$T_{2}^{2} - 3T_{2} + 1$$ T2^2 - 3*T2 + 1 $$T_{5}^{2} - 5$$ T5^2 - 5 $$T_{13}^{2} - 2T_{13} - 19$$ T13^2 - 2*T13 - 19 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 2T - 19$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} - 4T - 1$$
$23$ $$T^{2} + 4T - 16$$
$29$ $$T^{2} - 16T + 59$$
$31$ $$T^{2} + 8T - 4$$
$37$ $$T^{2} + 10T + 5$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} - 8T - 4$$
$47$ $$T^{2} - 14T + 29$$
$53$ $$T^{2} - 8T - 64$$
$59$ $$T^{2} + 6T - 71$$
$61$ $$T^{2} + 4T - 76$$
$67$ $$T^{2} + 12T - 9$$
$71$ $$T^{2} - 20$$
$73$ $$(T - 1)^{2}$$
$79$ $$T^{2} - 4T - 16$$
$83$ $$T^{2} - 16T + 44$$
$89$ $$T^{2} - 180$$
$97$ $$T^{2} + 4T - 76$$