# Properties

 Label 9216.2.a.be Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $1$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9216 = 2^{10} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9216.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: $$73.5901305028$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 2304) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} - \beta_1 q^{7}+O(q^{10})$$ q + b2 * q^5 - b1 * q^7 $$q + \beta_{2} q^{5} - \beta_1 q^{7} + \beta_{3} q^{11} - \beta_1 q^{13} - \beta_{3} q^{17} - 6 q^{19} - 2 \beta_{2} q^{23} + 7 q^{25} - \beta_{2} q^{29} + \beta_1 q^{31} - \beta_{3} q^{35} + 7 \beta_1 q^{37} - \beta_{3} q^{41} - 6 q^{43} - 2 \beta_{2} q^{47} - 5 q^{49} + \beta_{2} q^{53} + 12 \beta_1 q^{55} + 2 \beta_{3} q^{59} + 5 \beta_1 q^{61} - \beta_{3} q^{65} + 8 q^{67} - 4 \beta_{2} q^{71} - 12 q^{73} - 2 \beta_{2} q^{77} - 11 \beta_1 q^{79} - 3 \beta_{3} q^{83} - 12 \beta_1 q^{85} - 2 \beta_{3} q^{89} + 2 q^{91} - 6 \beta_{2} q^{95}+O(q^{100})$$ q + b2 * q^5 - b1 * q^7 + b3 * q^11 - b1 * q^13 - b3 * q^17 - 6 * q^19 - 2*b2 * q^23 + 7 * q^25 - b2 * q^29 + b1 * q^31 - b3 * q^35 + 7*b1 * q^37 - b3 * q^41 - 6 * q^43 - 2*b2 * q^47 - 5 * q^49 + b2 * q^53 + 12*b1 * q^55 + 2*b3 * q^59 + 5*b1 * q^61 - b3 * q^65 + 8 * q^67 - 4*b2 * q^71 - 12 * q^73 - 2*b2 * q^77 - 11*b1 * q^79 - 3*b3 * q^83 - 12*b1 * q^85 - 2*b3 * q^89 + 2 * q^91 - 6*b2 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 24 q^{19} + 28 q^{25} - 24 q^{43} - 20 q^{49} + 32 q^{67} - 48 q^{73} + 8 q^{91}+O(q^{100})$$ 4 * q - 24 * q^19 + 28 * q^25 - 24 * q^43 - 20 * q^49 + 32 * q^67 - 48 * q^73 + 8 * q^91

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{2}$$ $$=$$ $$2\nu^{2} - 4$$ 2*v^2 - 4 $$\beta_{3}$$ $$=$$ $$-2\nu^{3} + 10\nu$$ -2*v^3 + 10*v
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 4$$ (b3 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 4 ) / 2$$ (b2 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} + 10\beta_1 ) / 4$$ (3*b3 + 10*b1) / 4

## 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
 −0.517638 0.517638 1.93185 −1.93185
0 0 0 −3.46410 0 −1.41421 0 0 0
1.2 0 0 0 −3.46410 0 1.41421 0 0 0
1.3 0 0 0 3.46410 0 −1.41421 0 0 0
1.4 0 0 0 3.46410 0 1.41421 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
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.be 4
3.b odd 2 1 inner 9216.2.a.be 4
4.b odd 2 1 9216.2.a.bh 4
8.b even 2 1 9216.2.a.bh 4
8.d odd 2 1 inner 9216.2.a.be 4
12.b even 2 1 9216.2.a.bh 4
24.f even 2 1 inner 9216.2.a.be 4
24.h odd 2 1 9216.2.a.bh 4
32.g even 8 2 2304.2.k.h 8
32.g even 8 2 2304.2.k.i yes 8
32.h odd 8 2 2304.2.k.h 8
32.h odd 8 2 2304.2.k.i yes 8
96.o even 8 2 2304.2.k.h 8
96.o even 8 2 2304.2.k.i yes 8
96.p odd 8 2 2304.2.k.h 8
96.p odd 8 2 2304.2.k.i yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.k.h 8 32.g even 8 2
2304.2.k.h 8 32.h odd 8 2
2304.2.k.h 8 96.o even 8 2
2304.2.k.h 8 96.p odd 8 2
2304.2.k.i yes 8 32.g even 8 2
2304.2.k.i yes 8 32.h odd 8 2
2304.2.k.i yes 8 96.o even 8 2
2304.2.k.i yes 8 96.p odd 8 2
9216.2.a.be 4 1.a even 1 1 trivial
9216.2.a.be 4 3.b odd 2 1 inner
9216.2.a.be 4 8.d odd 2 1 inner
9216.2.a.be 4 24.f even 2 1 inner
9216.2.a.bh 4 4.b odd 2 1
9216.2.a.bh 4 8.b even 2 1
9216.2.a.bh 4 12.b even 2 1
9216.2.a.bh 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{2} - 12$$ T5^2 - 12 $$T_{7}^{2} - 2$$ T7^2 - 2 $$T_{11}^{2} - 24$$ T11^2 - 24 $$T_{13}^{2} - 2$$ T13^2 - 2 $$T_{17}^{2} - 24$$ T17^2 - 24 $$T_{19} + 6$$ T19 + 6 $$T_{67} - 8$$ T67 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 12)^{2}$$
$7$ $$(T^{2} - 2)^{2}$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$(T^{2} - 2)^{2}$$
$17$ $$(T^{2} - 24)^{2}$$
$19$ $$(T + 6)^{4}$$
$23$ $$(T^{2} - 48)^{2}$$
$29$ $$(T^{2} - 12)^{2}$$
$31$ $$(T^{2} - 2)^{2}$$
$37$ $$(T^{2} - 98)^{2}$$
$41$ $$(T^{2} - 24)^{2}$$
$43$ $$(T + 6)^{4}$$
$47$ $$(T^{2} - 48)^{2}$$
$53$ $$(T^{2} - 12)^{2}$$
$59$ $$(T^{2} - 96)^{2}$$
$61$ $$(T^{2} - 50)^{2}$$
$67$ $$(T - 8)^{4}$$
$71$ $$(T^{2} - 192)^{2}$$
$73$ $$(T + 12)^{4}$$
$79$ $$(T^{2} - 242)^{2}$$
$83$ $$(T^{2} - 216)^{2}$$
$89$ $$(T^{2} - 96)^{2}$$
$97$ $$T^{4}$$