# Properties

 Label 4992.2.a.s Level $4992$ Weight $2$ Character orbit 4992.a Self dual yes Analytic conductor $39.861$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4992,2,Mod(1,4992)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4992, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4992.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4992 = 2^{7} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4992.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: $$39.8613206890$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_{2} + 1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b2 + 1) * q^5 + (-b2 + 1) * q^7 + q^9 $$q - q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_{2} + 1) q^{7} + q^{9} + ( - \beta_{2} + \beta_1 + 2) q^{11} + q^{13} + (\beta_{2} - 1) q^{15} + (2 \beta_1 - 4) q^{17} + (\beta_{2} + 1) q^{19} + (\beta_{2} - 1) q^{21} + ( - \beta_{2} + 3 \beta_1 + 2) q^{23} + ( - 3 \beta_{2} - \beta_1 + 3) q^{25} - q^{27} + (\beta_{2} + 3 \beta_1 - 4) q^{29} + (\beta_{2} - 2 \beta_1 + 1) q^{31} + (\beta_{2} - \beta_1 - 2) q^{33} + ( - 3 \beta_{2} - \beta_1 + 8) q^{35} + (2 \beta_{2} + 2 \beta_1 + 2) q^{37} - q^{39} + ( - 2 \beta_{2} - \beta_1 - 3) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{43} + ( - \beta_{2} + 1) q^{45} - 2 q^{47} + ( - 3 \beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_1 + 4) q^{51} + ( - \beta_{2} - 3 \beta_1) q^{53} + ( - 4 \beta_{2} - 2 \beta_1 + 10) q^{55} + ( - \beta_{2} - 1) q^{57} + ( - \beta_{2} + \beta_1 + 2) q^{59} + ( - \beta_{2} - 5 \beta_1 + 4) q^{61} + ( - \beta_{2} + 1) q^{63} + ( - \beta_{2} + 1) q^{65} + (\beta_{2} + 6 \beta_1 - 5) q^{67} + (\beta_{2} - 3 \beta_1 - 2) q^{69} + (2 \beta_{2} - 2 \beta_1 + 2) q^{71} + (2 \beta_{2} + 2 \beta_1 - 10) q^{73} + (3 \beta_{2} + \beta_1 - 3) q^{75} + ( - 4 \beta_{2} - 2 \beta_1 + 10) q^{77} + (2 \beta_{2} + 4 \beta_1 - 4) q^{79} + q^{81} + (\beta_{2} + \beta_1 + 12) q^{83} + (4 \beta_{2} - 2 \beta_1 - 2) q^{85} + ( - \beta_{2} - 3 \beta_1 + 4) q^{87} + (3 \beta_1 - 9) q^{89} + ( - \beta_{2} + 1) q^{91} + ( - \beta_{2} + 2 \beta_1 - 1) q^{93} + (\beta_{2} + \beta_1 - 6) q^{95} + (4 \beta_{2} + 2 \beta_1 - 4) q^{97} + ( - \beta_{2} + \beta_1 + 2) q^{99}+O(q^{100})$$ q - q^3 + (-b2 + 1) * q^5 + (-b2 + 1) * q^7 + q^9 + (-b2 + b1 + 2) * q^11 + q^13 + (b2 - 1) * q^15 + (2*b1 - 4) * q^17 + (b2 + 1) * q^19 + (b2 - 1) * q^21 + (-b2 + 3*b1 + 2) * q^23 + (-3*b2 - b1 + 3) * q^25 - q^27 + (b2 + 3*b1 - 4) * q^29 + (b2 - 2*b1 + 1) * q^31 + (b2 - b1 - 2) * q^33 + (-3*b2 - b1 + 8) * q^35 + (2*b2 + 2*b1 + 2) * q^37 - q^39 + (-2*b2 - b1 - 3) * q^41 + (-2*b2 - 4*b1 + 6) * q^43 + (-b2 + 1) * q^45 - 2 * q^47 + (-3*b2 - b1 + 1) * q^49 + (-2*b1 + 4) * q^51 + (-b2 - 3*b1) * q^53 + (-4*b2 - 2*b1 + 10) * q^55 + (-b2 - 1) * q^57 + (-b2 + b1 + 2) * q^59 + (-b2 - 5*b1 + 4) * q^61 + (-b2 + 1) * q^63 + (-b2 + 1) * q^65 + (b2 + 6*b1 - 5) * q^67 + (b2 - 3*b1 - 2) * q^69 + (2*b2 - 2*b1 + 2) * q^71 + (2*b2 + 2*b1 - 10) * q^73 + (3*b2 + b1 - 3) * q^75 + (-4*b2 - 2*b1 + 10) * q^77 + (2*b2 + 4*b1 - 4) * q^79 + q^81 + (b2 + b1 + 12) * q^83 + (4*b2 - 2*b1 - 2) * q^85 + (-b2 - 3*b1 + 4) * q^87 + (3*b1 - 9) * q^89 + (-b2 + 1) * q^91 + (-b2 + 2*b1 - 1) * q^93 + (b2 + b1 - 6) * q^95 + (4*b2 + 2*b1 - 4) * q^97 + (-b2 + b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 $$3 q - 3 q^{3} + 2 q^{5} + 2 q^{7} + 3 q^{9} + 6 q^{11} + 3 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} - 2 q^{21} + 8 q^{23} + 5 q^{25} - 3 q^{27} - 8 q^{29} + 2 q^{31} - 6 q^{33} + 20 q^{35} + 10 q^{37} - 3 q^{39} - 12 q^{41} + 12 q^{43} + 2 q^{45} - 6 q^{47} - q^{49} + 10 q^{51} - 4 q^{53} + 24 q^{55} - 4 q^{57} + 6 q^{59} + 6 q^{61} + 2 q^{63} + 2 q^{65} - 8 q^{67} - 8 q^{69} + 6 q^{71} - 26 q^{73} - 5 q^{75} + 24 q^{77} - 6 q^{79} + 3 q^{81} + 38 q^{83} - 4 q^{85} + 8 q^{87} - 24 q^{89} + 2 q^{91} - 2 q^{93} - 16 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 2 * q^5 + 2 * q^7 + 3 * q^9 + 6 * q^11 + 3 * q^13 - 2 * q^15 - 10 * q^17 + 4 * q^19 - 2 * q^21 + 8 * q^23 + 5 * q^25 - 3 * q^27 - 8 * q^29 + 2 * q^31 - 6 * q^33 + 20 * q^35 + 10 * q^37 - 3 * q^39 - 12 * q^41 + 12 * q^43 + 2 * q^45 - 6 * q^47 - q^49 + 10 * q^51 - 4 * q^53 + 24 * q^55 - 4 * q^57 + 6 * q^59 + 6 * q^61 + 2 * q^63 + 2 * q^65 - 8 * q^67 - 8 * q^69 + 6 * q^71 - 26 * q^73 - 5 * q^75 + 24 * q^77 - 6 * q^79 + 3 * q^81 + 38 * q^83 - 4 * q^85 + 8 * q^87 - 24 * q^89 + 2 * q^91 - 2 * q^93 - 16 * q^95 - 6 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2

## 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
 0.311108 2.17009 −1.48119
0 −1.00000 0 −1.52543 0 −1.52543 0 1.00000 0
1.2 0 −1.00000 0 −0.630898 0 −0.630898 0 1.00000 0
1.3 0 −1.00000 0 4.15633 0 4.15633 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$13$$ $$-1$$

## 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 4992.2.a.s yes 3
4.b odd 2 1 4992.2.a.z yes 3
8.b even 2 1 4992.2.a.w yes 3
8.d odd 2 1 4992.2.a.n 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4992.2.a.n 3 8.d odd 2 1
4992.2.a.s yes 3 1.a even 1 1 trivial
4992.2.a.w yes 3 8.b even 2 1
4992.2.a.z yes 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4992))$$:

 $$T_{5}^{3} - 2T_{5}^{2} - 8T_{5} - 4$$ T5^3 - 2*T5^2 - 8*T5 - 4 $$T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4$$ T7^3 - 2*T7^2 - 8*T7 - 4 $$T_{11}^{3} - 6T_{11}^{2} - 4T_{11} + 40$$ T11^3 - 6*T11^2 - 4*T11 + 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} - 2 T^{2} - 8 T - 4$$
$7$ $$T^{3} - 2 T^{2} - 8 T - 4$$
$11$ $$T^{3} - 6 T^{2} - 4 T + 40$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} + 10 T^{2} + 12 T - 40$$
$19$ $$T^{3} - 4 T^{2} - 4 T + 20$$
$23$ $$T^{3} - 8 T^{2} - 40 T + 304$$
$29$ $$T^{3} + 8 T^{2} - 32 T - 272$$
$31$ $$T^{3} - 2 T^{2} - 32 T - 52$$
$37$ $$T^{3} - 10 T^{2} - 20 T + 136$$
$41$ $$T^{3} + 12 T^{2} + 8 T - 172$$
$43$ $$T^{3} - 12 T^{2} - 64 T + 800$$
$47$ $$(T + 2)^{3}$$
$53$ $$T^{3} + 4 T^{2} - 48 T + 80$$
$59$ $$T^{3} - 6 T^{2} - 4 T + 40$$
$61$ $$T^{3} - 6 T^{2} - 124 T + 760$$
$67$ $$T^{3} + 8 T^{2} - 172 T - 1252$$
$71$ $$T^{3} - 6 T^{2} - 52 T - 8$$
$73$ $$T^{3} + 26 T^{2} + 172 T + 184$$
$79$ $$T^{3} + 6 T^{2} - 100 T - 632$$
$83$ $$T^{3} - 38 T^{2} + 468 T - 1864$$
$89$ $$T^{3} + 24 T^{2} + 144 T + 108$$
$97$ $$T^{3} + 6 T^{2} - 148 T + 296$$