# Properties

 Label 4840.2.a.t Level $4840$ Weight $2$ Character orbit 4840.a Self dual yes Analytic conductor $38.648$ Analytic rank $1$ 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] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x - 1$$ x^3 - x^2 - 5*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - q^{5} - \beta_1 q^{7} + ( - \beta_{2} + \beta_1 + 2) q^{9}+O(q^{10})$$ q + b2 * q^3 - q^5 - b1 * q^7 + (-b2 + b1 + 2) * q^9 $$q + \beta_{2} q^{3} - q^{5} - \beta_1 q^{7} + ( - \beta_{2} + \beta_1 + 2) q^{9} + (\beta_{2} - 3 \beta_1) q^{13} - \beta_{2} q^{15} + (2 \beta_1 - 2) q^{17} + ( - \beta_{2} + 3 \beta_1 + 2) q^{19} + ( - 2 \beta_1 - 1) q^{21} + (\beta_{2} + 3 \beta_1 - 2) q^{23} + q^{25} + (\beta_1 - 4) q^{27} + ( - 2 \beta_{2} - 4) q^{29} + ( - \beta_{2} - \beta_1) q^{31} + \beta_1 q^{35} + ( - 2 \beta_{2} + 2) q^{37} + ( - \beta_{2} - 5 \beta_1 + 2) q^{39} + ( - \beta_{2} - \beta_1 - 3) q^{41} + (\beta_1 - 10) q^{43} + (\beta_{2} - \beta_1 - 2) q^{45} + (3 \beta_{2} - 2 \beta_1) q^{47} + (\beta_{2} + \beta_1 - 4) q^{49} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{51} + ( - 3 \beta_{2} + \beta_1 + 4) q^{53} + (3 \beta_{2} + 5 \beta_1 - 2) q^{57} + ( - 3 \beta_{2} + 3 \beta_1 + 2) q^{59} + ( - \beta_{2} - 3 \beta_1 + 5) q^{61} + ( - \beta_{2} - \beta_1 - 2) q^{63} + ( - \beta_{2} + 3 \beta_1) q^{65} + (5 \beta_{2} - 4 \beta_1 + 2) q^{67} + ( - 3 \beta_{2} + 7 \beta_1 + 8) q^{69} + (\beta_{2} - \beta_1 - 2) q^{71} + ( - 6 \beta_{2} + 2 \beta_1) q^{73} + \beta_{2} q^{75} + ( - 4 \beta_{2} - 2 \beta_1) q^{79} + ( - \beta_{2} - \beta_1 - 5) q^{81} + ( - 3 \beta_{2} + 3 \beta_1 + 2) q^{83} + ( - 2 \beta_1 + 2) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 - 10) q^{87} + (2 \beta_1 - 5) q^{89} + (3 \beta_{2} + \beta_1 + 8) q^{91} + (\beta_{2} - 3 \beta_1 - 6) q^{93} + (\beta_{2} - 3 \beta_1 - 2) q^{95} + (\beta_{2} - 3 \beta_1 + 8) q^{97}+O(q^{100})$$ q + b2 * q^3 - q^5 - b1 * q^7 + (-b2 + b1 + 2) * q^9 + (b2 - 3*b1) * q^13 - b2 * q^15 + (2*b1 - 2) * q^17 + (-b2 + 3*b1 + 2) * q^19 + (-2*b1 - 1) * q^21 + (b2 + 3*b1 - 2) * q^23 + q^25 + (b1 - 4) * q^27 + (-2*b2 - 4) * q^29 + (-b2 - b1) * q^31 + b1 * q^35 + (-2*b2 + 2) * q^37 + (-b2 - 5*b1 + 2) * q^39 + (-b2 - b1 - 3) * q^41 + (b1 - 10) * q^43 + (b2 - b1 - 2) * q^45 + (3*b2 - 2*b1) * q^47 + (b2 + b1 - 4) * q^49 + (-2*b2 + 4*b1 + 2) * q^51 + (-3*b2 + b1 + 4) * q^53 + (3*b2 + 5*b1 - 2) * q^57 + (-3*b2 + 3*b1 + 2) * q^59 + (-b2 - 3*b1 + 5) * q^61 + (-b2 - b1 - 2) * q^63 + (-b2 + 3*b1) * q^65 + (5*b2 - 4*b1 + 2) * q^67 + (-3*b2 + 7*b1 + 8) * q^69 + (b2 - b1 - 2) * q^71 + (-6*b2 + 2*b1) * q^73 + b2 * q^75 + (-4*b2 - 2*b1) * q^79 + (-b2 - b1 - 5) * q^81 + (-3*b2 + 3*b1 + 2) * q^83 + (-2*b1 + 2) * q^85 + (-2*b2 - 2*b1 - 10) * q^87 + (2*b1 - 5) * q^89 + (3*b2 + b1 + 8) * q^91 + (b2 - 3*b1 - 6) * q^93 + (b2 - 3*b1 - 2) * q^95 + (b2 - 3*b1 + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} - q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 - q^7 + 6 * q^9 $$3 q + q^{3} - 3 q^{5} - q^{7} + 6 q^{9} - 2 q^{13} - q^{15} - 4 q^{17} + 8 q^{19} - 5 q^{21} - 2 q^{23} + 3 q^{25} - 11 q^{27} - 14 q^{29} - 2 q^{31} + q^{35} + 4 q^{37} - 11 q^{41} - 29 q^{43} - 6 q^{45} + q^{47} - 10 q^{49} + 8 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 11 q^{61} - 8 q^{63} + 2 q^{65} + 7 q^{67} + 28 q^{69} - 6 q^{71} - 4 q^{73} + q^{75} - 6 q^{79} - 17 q^{81} + 6 q^{83} + 4 q^{85} - 34 q^{87} - 13 q^{89} + 28 q^{91} - 20 q^{93} - 8 q^{95} + 22 q^{97}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 - q^7 + 6 * q^9 - 2 * q^13 - q^15 - 4 * q^17 + 8 * q^19 - 5 * q^21 - 2 * q^23 + 3 * q^25 - 11 * q^27 - 14 * q^29 - 2 * q^31 + q^35 + 4 * q^37 - 11 * q^41 - 29 * q^43 - 6 * q^45 + q^47 - 10 * q^49 + 8 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 11 * q^61 - 8 * q^63 + 2 * q^65 + 7 * q^67 + 28 * q^69 - 6 * q^71 - 4 * q^73 + q^75 - 6 * q^79 - 17 * q^81 + 6 * q^83 + 4 * q^85 - 34 * q^87 - 13 * q^89 + 28 * q^91 - 20 * q^93 - 8 * q^95 + 22 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 −0.210756 −1.65544 2.86620
0 −2.74483 0 −1.00000 0 0.210756 0 4.53407 0
1.2 0 1.39593 0 −1.00000 0 1.65544 0 −1.05137 0
1.3 0 2.34889 0 −1.00000 0 −2.86620 0 2.51730 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 4840.2.a.t 3
4.b odd 2 1 9680.2.a.cc 3
11.b odd 2 1 4840.2.a.u yes 3
44.c even 2 1 9680.2.a.ca 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4840.2.a.t 3 1.a even 1 1 trivial
4840.2.a.u yes 3 11.b odd 2 1
9680.2.a.ca 3 44.c even 2 1
9680.2.a.cc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4840))$$:

 $$T_{3}^{3} - T_{3}^{2} - 7T_{3} + 9$$ T3^3 - T3^2 - 7*T3 + 9 $$T_{7}^{3} + T_{7}^{2} - 5T_{7} + 1$$ T7^3 + T7^2 - 5*T7 + 1 $$T_{13}^{3} + 2T_{13}^{2} - 40T_{13} - 84$$ T13^3 + 2*T13^2 - 40*T13 - 84

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 7T + 9$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + T^{2} - 5T + 1$$
$11$ $$T^{3}$$
$13$ $$T^{3} + 2 T^{2} - 40 T - 84$$
$17$ $$T^{3} + 4 T^{2} - 16 T - 48$$
$19$ $$T^{3} - 8 T^{2} - 20 T + 148$$
$23$ $$T^{3} + 2 T^{2} - 68 T - 268$$
$29$ $$T^{3} + 14 T^{2} + 36 T - 88$$
$31$ $$T^{3} + 2 T^{2} - 16 T + 4$$
$37$ $$T^{3} - 4 T^{2} - 24 T - 16$$
$41$ $$T^{3} + 11 T^{2} + 23 T + 1$$
$43$ $$T^{3} + 29 T^{2} + 275 T + 849$$
$47$ $$T^{3} - T^{2} - 59 T + 77$$
$53$ $$T^{3} - 10 T^{2} - 24 T - 4$$
$59$ $$T^{3} - 6 T^{2} - 60 T + 244$$
$61$ $$T^{3} - 11 T^{2} - 29 T + 427$$
$67$ $$T^{3} - 7 T^{2} - 159 T + 387$$
$71$ $$T^{3} + 6 T^{2} + 4 T - 12$$
$73$ $$T^{3} + 4 T^{2} - 224 T - 1568$$
$79$ $$T^{3} + 6 T^{2} - 164 T - 392$$
$83$ $$T^{3} - 6 T^{2} - 60 T + 244$$
$89$ $$T^{3} + 13 T^{2} + 35 T - 33$$
$97$ $$T^{3} - 22 T^{2} + 120 T - 148$$