# Properties

 Label 4560.2.a.bs Level $4560$ Weight $2$ Character orbit 4560.a Self dual yes Analytic conductor $36.412$ 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] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.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: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1772.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 12x + 8$$ x^3 - x^2 - 12*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - \beta_{2} q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^5 - b2 * q^7 + q^9 $$q - q^{3} + q^{5} - \beta_{2} q^{7} + q^{9} + ( - \beta_{2} - 2) q^{11} + ( - \beta_1 + 1) q^{13} - q^{15} + 4 q^{17} - q^{19} + \beta_{2} q^{21} + (\beta_{2} - \beta_1 + 1) q^{23} + q^{25} - q^{27} - \beta_{2} q^{29} + 2 q^{31} + (\beta_{2} + 2) q^{33} - \beta_{2} q^{35} + ( - 2 \beta_{2} + \beta_1 - 1) q^{37} + (\beta_1 - 1) q^{39} + ( - 2 \beta_{2} + \beta_1 + 1) q^{41} + ( - \beta_1 - 1) q^{43} + q^{45} + ( - \beta_{2} + \beta_1 - 1) q^{47} + ( - \beta_{2} - \beta_1 + 2) q^{49} - 4 q^{51} + ( - \beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_{2} - 2) q^{55} + q^{57} + (3 \beta_{2} - \beta_1 - 1) q^{59} + (\beta_{2} + \beta_1 + 5) q^{61} - \beta_{2} q^{63} + ( - \beta_1 + 1) q^{65} - 4 q^{67} + ( - \beta_{2} + \beta_1 - 1) q^{69} + (2 \beta_1 + 2) q^{71} + 6 q^{73} - q^{75} + (\beta_{2} - \beta_1 + 9) q^{77} + (3 \beta_{2} + \beta_1 + 1) q^{79} + q^{81} + (2 \beta_{2} + 2) q^{83} + 4 q^{85} + \beta_{2} q^{87} + (2 \beta_{2} - \beta_1 + 3) q^{89} + ( - 5 \beta_{2} + \beta_1 + 3) q^{91} - 2 q^{93} - q^{95} + ( - 2 \beta_{2} + \beta_1 + 11) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100})$$ q - q^3 + q^5 - b2 * q^7 + q^9 + (-b2 - 2) * q^11 + (-b1 + 1) * q^13 - q^15 + 4 * q^17 - q^19 + b2 * q^21 + (b2 - b1 + 1) * q^23 + q^25 - q^27 - b2 * q^29 + 2 * q^31 + (b2 + 2) * q^33 - b2 * q^35 + (-2*b2 + b1 - 1) * q^37 + (b1 - 1) * q^39 + (-2*b2 + b1 + 1) * q^41 + (-b1 - 1) * q^43 + q^45 + (-b2 + b1 - 1) * q^47 + (-b2 - b1 + 2) * q^49 - 4 * q^51 + (-b2 + b1 - 3) * q^53 + (-b2 - 2) * q^55 + q^57 + (3*b2 - b1 - 1) * q^59 + (b2 + b1 + 5) * q^61 - b2 * q^63 + (-b1 + 1) * q^65 - 4 * q^67 + (-b2 + b1 - 1) * q^69 + (2*b1 + 2) * q^71 + 6 * q^73 - q^75 + (b2 - b1 + 9) * q^77 + (3*b2 + b1 + 1) * q^79 + q^81 + (2*b2 + 2) * q^83 + 4 * q^85 + b2 * q^87 + (2*b2 - b1 + 3) * q^89 + (-5*b2 + b1 + 3) * q^91 - 2 * q^93 - q^95 + (-2*b2 + b1 + 11) * q^97 + (-b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 6 q^{11} + 4 q^{13} - 3 q^{15} + 12 q^{17} - 3 q^{19} + 4 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{31} + 6 q^{33} - 4 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} + 3 q^{45} - 4 q^{47} + 7 q^{49} - 12 q^{51} - 10 q^{53} - 6 q^{55} + 3 q^{57} - 2 q^{59} + 14 q^{61} + 4 q^{65} - 12 q^{67} - 4 q^{69} + 4 q^{71} + 18 q^{73} - 3 q^{75} + 28 q^{77} + 2 q^{79} + 3 q^{81} + 6 q^{83} + 12 q^{85} + 10 q^{89} + 8 q^{91} - 6 q^{93} - 3 q^{95} + 32 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 6 * q^11 + 4 * q^13 - 3 * q^15 + 12 * q^17 - 3 * q^19 + 4 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^31 + 6 * q^33 - 4 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 + 3 * q^45 - 4 * q^47 + 7 * q^49 - 12 * q^51 - 10 * q^53 - 6 * q^55 + 3 * q^57 - 2 * q^59 + 14 * q^61 + 4 * q^65 - 12 * q^67 - 4 * q^69 + 4 * q^71 + 18 * q^73 - 3 * q^75 + 28 * q^77 + 2 * q^79 + 3 * q^81 + 6 * q^83 + 12 * q^85 + 10 * q^89 + 8 * q^91 - 6 * q^93 - 3 * q^95 + 32 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 12x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} + 3\nu - 10 ) / 2$$ (v^2 + 3*v - 10) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{2} - \nu - 8 ) / 2$$ (v^2 - v - 8) / 2
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 2$$ (-b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} + \beta _1 + 17 ) / 2$$ (3*b2 + b1 + 17) / 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
 −3.32803 3.67370 0.654334
0 −1.00000 0 1.00000 0 −3.20191 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −0.911179 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.11309 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## 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 4560.2.a.bs 3
4.b odd 2 1 2280.2.a.u 3
12.b even 2 1 6840.2.a.bh 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.a.u 3 4.b odd 2 1
4560.2.a.bs 3 1.a even 1 1 trivial
6840.2.a.bh 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4560))$$:

 $$T_{7}^{3} - 14T_{7} - 12$$ T7^3 - 14*T7 - 12 $$T_{11}^{3} + 6T_{11}^{2} - 2T_{11} - 32$$ T11^3 + 6*T11^2 - 2*T11 - 32 $$T_{13}^{3} - 4T_{13}^{2} - 38T_{13} + 164$$ T13^3 - 4*T13^2 - 38*T13 + 164 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 14T - 12$$
$11$ $$T^{3} + 6 T^{2} - 2 T - 32$$
$13$ $$T^{3} - 4 T^{2} - 38 T + 164$$
$17$ $$(T - 4)^{3}$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 4 T^{2} - 44 T + 32$$
$29$ $$T^{3} - 14T - 12$$
$31$ $$(T - 2)^{3}$$
$37$ $$T^{3} + 4 T^{2} - 78 T + 180$$
$41$ $$T^{3} - 2 T^{2} - 82 T + 344$$
$43$ $$T^{3} + 2 T^{2} - 42 T + 80$$
$47$ $$T^{3} + 4 T^{2} - 44 T - 32$$
$53$ $$T^{3} + 10 T^{2} - 16 T - 96$$
$59$ $$T^{3} + 2 T^{2} - 144 T - 688$$
$61$ $$T^{3} - 14T^{2} + 144$$
$67$ $$(T + 4)^{3}$$
$71$ $$T^{3} - 4 T^{2} - 168 T - 640$$
$73$ $$(T - 6)^{3}$$
$79$ $$T^{3} - 2 T^{2} - 192 T + 1024$$
$83$ $$T^{3} - 6 T^{2} - 44 T + 200$$
$89$ $$T^{3} - 10 T^{2} - 50 T - 48$$
$97$ $$T^{3} - 32 T^{2} + 258 T - 36$$