# Properties

 Label 9576.2.a.cb Level $9576$ Weight $2$ Character orbit 9576.a Self dual yes Analytic conductor $76.465$ 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] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9576.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: $$76.4647449756$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3192) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} - q^{7}+O(q^{10})$$ q + b1 * q^5 - q^7 $$q + \beta_1 q^{5} - q^{7} + (\beta_{2} + \beta_1 + 1) q^{11} + 4 q^{13} + \beta_1 q^{17} - q^{19} + (\beta_{2} + \beta_1 + 1) q^{23} + (\beta_{2} - \beta_1 + 2) q^{25} + ( - \beta_{2} - 1) q^{29} + ( - \beta_{2} - \beta_1 - 3) q^{31} - \beta_1 q^{35} + 2 q^{37} + 2 \beta_{2} q^{41} + (\beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} + 1) q^{47} + q^{49} + (\beta_{2} + 2 \beta_1 + 5) q^{53} + (2 \beta_{2} + 2 \beta_1 + 6) q^{55} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{59} + ( - 2 \beta_{2} + 2 \beta_1) q^{61} + 4 \beta_1 q^{65} + (\beta_{2} + \beta_1 + 1) q^{67} + ( - \beta_{2} + 2 \beta_1 - 3) q^{71} + ( - 2 \beta_{2} + 4) q^{73} + ( - \beta_{2} - \beta_1 - 1) q^{77} + (\beta_{2} - \beta_1 + 1) q^{79} + (\beta_{2} - 2 \beta_1 - 1) q^{83} + (\beta_{2} - \beta_1 + 7) q^{85} + ( - 2 \beta_{2} - 4) q^{89} - 4 q^{91} - \beta_1 q^{95} + ( - \beta_{2} + \beta_1 + 9) q^{97}+O(q^{100})$$ q + b1 * q^5 - q^7 + (b2 + b1 + 1) * q^11 + 4 * q^13 + b1 * q^17 - q^19 + (b2 + b1 + 1) * q^23 + (b2 - b1 + 2) * q^25 + (-b2 - 1) * q^29 + (-b2 - b1 - 3) * q^31 - b1 * q^35 + 2 * q^37 + 2*b2 * q^41 + (b2 + b1 - 1) * q^43 + (-b2 + 1) * q^47 + q^49 + (b2 + 2*b1 + 5) * q^53 + (2*b2 + 2*b1 + 6) * q^55 + (-2*b2 + 2*b1 - 2) * q^59 + (-2*b2 + 2*b1) * q^61 + 4*b1 * q^65 + (b2 + b1 + 1) * q^67 + (-b2 + 2*b1 - 3) * q^71 + (-2*b2 + 4) * q^73 + (-b2 - b1 - 1) * q^77 + (b2 - b1 + 1) * q^79 + (b2 - 2*b1 - 1) * q^83 + (b2 - b1 + 7) * q^85 + (-2*b2 - 4) * q^89 - 4 * q^91 - b1 * q^95 + (-b2 + b1 + 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{7}+O(q^{10})$$ 3 * q - 3 * q^7 $$3 q - 3 q^{7} + 2 q^{11} + 12 q^{13} - 3 q^{19} + 2 q^{23} + 5 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{37} - 2 q^{41} - 4 q^{43} + 4 q^{47} + 3 q^{49} + 14 q^{53} + 16 q^{55} - 4 q^{59} + 2 q^{61} + 2 q^{67} - 8 q^{71} + 14 q^{73} - 2 q^{77} + 2 q^{79} - 4 q^{83} + 20 q^{85} - 10 q^{89} - 12 q^{91} + 28 q^{97}+O(q^{100})$$ 3 * q - 3 * q^7 + 2 * q^11 + 12 * q^13 - 3 * q^19 + 2 * q^23 + 5 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^37 - 2 * q^41 - 4 * q^43 + 4 * q^47 + 3 * q^49 + 14 * q^53 + 16 * q^55 - 4 * q^59 + 2 * q^61 + 2 * q^67 - 8 * q^71 + 14 * q^73 - 2 * q^77 + 2 * q^79 - 4 * q^83 + 20 * q^85 - 10 * q^89 - 12 * q^91 + 28 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 3\nu + 3$$ -v^2 + 3*v + 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 3\beta _1 + 9 ) / 2$$ (b2 + 3*b1 + 9) / 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
 −0.363328 −1.76156 3.12489
0 0 0 −3.50466 0 −1.00000 0 0 0
1.2 0 0 0 0.864641 0 −1.00000 0 0 0
1.3 0 0 0 2.64002 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$19$$ $$1$$

## 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 9576.2.a.cb 3
3.b odd 2 1 3192.2.a.u 3
12.b even 2 1 6384.2.a.bw 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.u 3 3.b odd 2 1
6384.2.a.bw 3 12.b even 2 1
9576.2.a.cb 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9576))$$:

 $$T_{5}^{3} - 10T_{5} + 8$$ T5^3 - 10*T5 + 8 $$T_{11}^{3} - 2T_{11}^{2} - 24T_{11} - 16$$ T11^3 - 2*T11^2 - 24*T11 - 16 $$T_{13} - 4$$ T13 - 4 $$T_{17}^{3} - 10T_{17} + 8$$ T17^3 - 10*T17 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 10T + 8$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} - 2 T^{2} - 24 T - 16$$
$13$ $$(T - 4)^{3}$$
$17$ $$T^{3} - 10T + 8$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 2 T^{2} - 24 T - 16$$
$29$ $$T^{3} + 2 T^{2} - 18 T - 44$$
$31$ $$T^{3} + 8 T^{2} - 4 T - 16$$
$37$ $$(T - 2)^{3}$$
$41$ $$T^{3} + 2 T^{2} - 76 T + 200$$
$43$ $$T^{3} + 4 T^{2} - 20 T - 64$$
$47$ $$T^{3} - 4 T^{2} - 14 T - 8$$
$53$ $$T^{3} - 14 T^{2} + 14 T + 4$$
$59$ $$T^{3} + 4 T^{2} - 128 T - 256$$
$61$ $$T^{3} - 2 T^{2} - 132 T + 8$$
$67$ $$T^{3} - 2 T^{2} - 24 T - 16$$
$71$ $$T^{3} + 8 T^{2} - 46 T - 16$$
$73$ $$T^{3} - 14 T^{2} - 12 T + 8$$
$79$ $$T^{3} - 2 T^{2} - 32 T + 32$$
$83$ $$T^{3} + 4 T^{2} - 62 T - 232$$
$89$ $$T^{3} + 10 T^{2} - 44 T - 472$$
$97$ $$T^{3} - 28 T^{2} + 228 T - 512$$