# Properties

 Label 9216.2.a.bf Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $1$ Dimension $4$ CM discriminant -3 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 2304) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 2 \beta_1) q^{7}+O(q^{10})$$ q + (-b2 - 2*b1) * q^7 $$q + ( - \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{2} + \beta_1) q^{13} - \beta_{3} q^{19} - 5 q^{25} + ( - 3 \beta_{2} + 2 \beta_1) q^{31} + (2 \beta_{2} + 5 \beta_1) q^{37} + 3 \beta_{3} q^{43} + (4 \beta_{3} + 7) q^{49} + ( - 2 \beta_{2} + 7 \beta_1) q^{61} - 16 q^{67} - 4 \beta_{3} q^{73} + (5 \beta_{2} + 2 \beta_1) q^{79} + ( - 5 \beta_{3} - 16) q^{91} - 4 \beta_{3} q^{97}+O(q^{100})$$ q + (-b2 - 2*b1) * q^7 + (2*b2 + b1) * q^13 - b3 * q^19 - 5 * q^25 + (-3*b2 + 2*b1) * q^31 + (2*b2 + 5*b1) * q^37 + 3*b3 * q^43 + (4*b3 + 7) * q^49 + (-2*b2 + 7*b1) * q^61 - 16 * q^67 - 4*b3 * q^73 + (5*b2 + 2*b1) * q^79 + (-5*b3 - 16) * q^91 - 4*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 20 q^{25} + 28 q^{49} - 64 q^{67} - 64 q^{91}+O(q^{100})$$ 4 * q - 20 * q^25 + 28 * q^49 - 64 * q^67 - 64 * 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 $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 5\nu$$ -v^3 + 5*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 4$$ 2*v^2 - 4
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 4 ) / 2$$ (b3 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{2} + 5\beta_1 ) / 2$$ (3*b2 + 5*b1) / 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
 1.93185 −0.517638 0.517638 −1.93185
0 0 0 0 0 −5.27792 0 0 0
1.2 0 0 0 0 0 −0.378937 0 0 0
1.3 0 0 0 0 0 0.378937 0 0 0
1.4 0 0 0 0 0 5.27792 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 CM by $$\Q(\sqrt{-3})$$
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.bf 4
3.b odd 2 1 CM 9216.2.a.bf 4
4.b odd 2 1 9216.2.a.bg 4
8.b even 2 1 9216.2.a.bg 4
8.d odd 2 1 inner 9216.2.a.bf 4
12.b even 2 1 9216.2.a.bg 4
24.f even 2 1 inner 9216.2.a.bf 4
24.h odd 2 1 9216.2.a.bg 4
32.g even 8 2 2304.2.k.g 8
32.g even 8 2 2304.2.k.j yes 8
32.h odd 8 2 2304.2.k.g 8
32.h odd 8 2 2304.2.k.j yes 8
96.o even 8 2 2304.2.k.g 8
96.o even 8 2 2304.2.k.j yes 8
96.p odd 8 2 2304.2.k.g 8
96.p odd 8 2 2304.2.k.j yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2304.2.k.g 8 32.g even 8 2
2304.2.k.g 8 32.h odd 8 2
2304.2.k.g 8 96.o even 8 2
2304.2.k.g 8 96.p odd 8 2
2304.2.k.j yes 8 32.g even 8 2
2304.2.k.j yes 8 32.h odd 8 2
2304.2.k.j yes 8 96.o even 8 2
2304.2.k.j yes 8 96.p odd 8 2
9216.2.a.bf 4 1.a even 1 1 trivial
9216.2.a.bf 4 3.b odd 2 1 CM
9216.2.a.bf 4 8.d odd 2 1 inner
9216.2.a.bf 4 24.f even 2 1 inner
9216.2.a.bg 4 4.b odd 2 1
9216.2.a.bg 4 8.b even 2 1
9216.2.a.bg 4 12.b even 2 1
9216.2.a.bg 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}$$ T5 $$T_{7}^{4} - 28T_{7}^{2} + 4$$ T7^4 - 28*T7^2 + 4 $$T_{11}$$ T11 $$T_{13}^{4} - 52T_{13}^{2} + 484$$ T13^4 - 52*T13^2 + 484 $$T_{17}$$ T17 $$T_{19}^{2} - 12$$ T19^2 - 12 $$T_{67} + 16$$ T67 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 28T^{2} + 4$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 52T^{2} + 484$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 12)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} - 124T^{2} + 2116$$
$37$ $$T^{4} - 148T^{2} + 676$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 108)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} - 244T^{2} + 5476$$
$67$ $$(T + 16)^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - 192)^{2}$$
$79$ $$T^{4} - 316 T^{2} + 20164$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} - 192)^{2}$$