# Properties

 Label 9360.2.a.cy Level $9360$ Weight $2$ Character orbit 9360.a Self dual yes Analytic conductor $74.740$ 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] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9360.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: $$74.7399762919$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.940.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 7x - 4$$ x^3 - 7*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1560) 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.

 $$f(q)$$ $$=$$ $$q - q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q - q^5 - b2 * q^7 $$q - q^{5} - \beta_{2} q^{7} + (\beta_1 + 2) q^{11} - q^{13} + (\beta_{2} + 2 \beta_1 - 2) q^{17} + ( - \beta_{2} - \beta_1 - 2) q^{19} - \beta_{2} q^{23} + q^{25} + (\beta_{2} - \beta_1 - 4) q^{29} + 2 \beta_1 q^{31} + \beta_{2} q^{35} + (\beta_1 + 4) q^{37} - \beta_1 q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + ( - 2 \beta_1 - 4) q^{47} + ( - 2 \beta_{2} - \beta_1 + 7) q^{49} + ( - 2 \beta_{2} + \beta_1) q^{53} + ( - \beta_1 - 2) q^{55} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{59} + ( - 2 \beta_{2} + \beta_1) q^{61} + q^{65} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{67} + (\beta_1 - 6) q^{71} + (\beta_{2} - \beta_1 + 8) q^{73} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{77} + (\beta_1 - 6) q^{79} + ( - 2 \beta_{2} + 2 \beta_1) q^{83} + ( - \beta_{2} - 2 \beta_1 + 2) q^{85} + (\beta_1 - 4) q^{89} + \beta_{2} q^{91} + (\beta_{2} + \beta_1 + 2) q^{95} + (3 \beta_{2} + 2) q^{97}+O(q^{100})$$ q - q^5 - b2 * q^7 + (b1 + 2) * q^11 - q^13 + (b2 + 2*b1 - 2) * q^17 + (-b2 - b1 - 2) * q^19 - b2 * q^23 + q^25 + (b2 - b1 - 4) * q^29 + 2*b1 * q^31 + b2 * q^35 + (b1 + 4) * q^37 - b1 * q^41 + (2*b2 + 2*b1) * q^43 + (-2*b1 - 4) * q^47 + (-2*b2 - b1 + 7) * q^49 + (-2*b2 + b1) * q^53 + (-b1 - 2) * q^55 + (-2*b2 + 2*b1 + 4) * q^59 + (-2*b2 + b1) * q^61 + q^65 + (-2*b2 - 2*b1 + 4) * q^67 + (b1 - 6) * q^71 + (b2 - b1 + 8) * q^73 + (-2*b2 - 3*b1 + 2) * q^77 + (b1 - 6) * q^79 + (-2*b2 + 2*b1) * q^83 + (-b2 - 2*b1 + 2) * q^85 + (b1 - 4) * q^89 + b2 * q^91 + (b2 + b1 + 2) * q^95 + (3*b2 + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - q^{7}+O(q^{10})$$ 3 * q - 3 * q^5 - q^7 $$3 q - 3 q^{5} - q^{7} + 5 q^{11} - 3 q^{13} - 7 q^{17} - 6 q^{19} - q^{23} + 3 q^{25} - 10 q^{29} - 2 q^{31} + q^{35} + 11 q^{37} + q^{41} - 10 q^{47} + 20 q^{49} - 3 q^{53} - 5 q^{55} + 8 q^{59} - 3 q^{61} + 3 q^{65} + 12 q^{67} - 19 q^{71} + 26 q^{73} + 7 q^{77} - 19 q^{79} - 4 q^{83} + 7 q^{85} - 13 q^{89} + q^{91} + 6 q^{95} + 9 q^{97}+O(q^{100})$$ 3 * q - 3 * q^5 - q^7 + 5 * q^11 - 3 * q^13 - 7 * q^17 - 6 * q^19 - q^23 + 3 * q^25 - 10 * q^29 - 2 * q^31 + q^35 + 11 * q^37 + q^41 - 10 * q^47 + 20 * q^49 - 3 * q^53 - 5 * q^55 + 8 * q^59 - 3 * q^61 + 3 * q^65 + 12 * q^67 - 19 * q^71 + 26 * q^73 + 7 * q^77 - 19 * q^79 - 4 * q^83 + 7 * q^85 - 13 * q^89 + q^91 + 6 * q^95 + 9 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 7x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 5$$ -v^2 + 2*v + 5
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 5$$ b1 + 5

## 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
 −0.602705 2.89511 −2.29240
0 0 0 −1.00000 0 −3.43134 0 0 0
1.2 0 0 0 −1.00000 0 −2.40857 0 0 0
1.3 0 0 0 −1.00000 0 4.83991 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 9360.2.a.cy 3
3.b odd 2 1 3120.2.a.bi 3
4.b odd 2 1 4680.2.a.bh 3
12.b even 2 1 1560.2.a.q 3
60.h even 2 1 7800.2.a.bi 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.q 3 12.b even 2 1
3120.2.a.bi 3 3.b odd 2 1
4680.2.a.bh 3 4.b odd 2 1
7800.2.a.bi 3 60.h even 2 1
9360.2.a.cy 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9360))$$:

 $$T_{7}^{3} + T_{7}^{2} - 20T_{7} - 40$$ T7^3 + T7^2 - 20*T7 - 40 $$T_{11}^{3} - 5T_{11}^{2} - 8T_{11} + 32$$ T11^3 - 5*T11^2 - 8*T11 + 32 $$T_{17}^{3} + 7T_{17}^{2} - 52T_{17} - 356$$ T17^3 + 7*T17^2 - 52*T17 - 356 $$T_{19}^{3} + 6T_{19}^{2} - 16T_{19} - 16$$ T19^3 + 6*T19^2 - 16*T19 - 16 $$T_{31}^{3} + 2T_{31}^{2} - 64T_{31} + 32$$ T31^3 + 2*T31^2 - 64*T31 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + T^{2} - 20 T - 40$$
$11$ $$T^{3} - 5 T^{2} - 8 T + 32$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} + 7 T^{2} - 52 T - 356$$
$19$ $$T^{3} + 6 T^{2} - 16 T - 16$$
$23$ $$T^{3} + T^{2} - 20 T - 40$$
$29$ $$T^{3} + 10 T^{2} - 12 T - 184$$
$31$ $$T^{3} + 2 T^{2} - 64 T + 32$$
$37$ $$T^{3} - 11 T^{2} + 24 T + 20$$
$41$ $$T^{3} - T^{2} - 16T - 4$$
$43$ $$T^{3} - 112T - 256$$
$47$ $$T^{3} + 10 T^{2} - 32 T - 256$$
$53$ $$T^{3} + 3 T^{2} - 112 T - 164$$
$59$ $$T^{3} - 8 T^{2} - 160 T + 1024$$
$61$ $$T^{3} + 3 T^{2} - 112 T - 164$$
$67$ $$T^{3} - 12 T^{2} - 64 T + 640$$
$71$ $$T^{3} + 19 T^{2} + 104 T + 160$$
$73$ $$T^{3} - 26 T^{2} + 180 T - 328$$
$79$ $$T^{3} + 19 T^{2} + 104 T + 160$$
$83$ $$T^{3} + 4 T^{2} - 176 T + 320$$
$89$ $$T^{3} + 13 T^{2} + 40 T + 20$$
$97$ $$T^{3} - 9 T^{2} - 156 T + 1420$$