# Properties

 Label 9216.2.a.bp Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 2$$ x^4 - 4*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 512) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 2) q^{5} + (\beta_{3} + \beta_1) q^{7}+O(q^{10})$$ q + (-b2 + 2) * q^5 + (b3 + b1) * q^7 $$q + ( - \beta_{2} + 2) q^{5} + (\beta_{3} + \beta_1) q^{7} + ( - \beta_{3} + 2 \beta_1) q^{11} + ( - \beta_{2} - 2) q^{13} - 2 \beta_{2} q^{17} + (\beta_{3} + 2 \beta_1) q^{19} + ( - \beta_{3} + 3 \beta_1) q^{23} + ( - 4 \beta_{2} + 1) q^{25} + (\beta_{2} + 6) q^{29} + ( - 2 \beta_{3} + 2 \beta_1) q^{31} + 2 \beta_{3} q^{35} + ( - 5 \beta_{2} - 2) q^{37} + 4 q^{41} + \beta_{3} q^{43} + (2 \beta_{3} - 2 \beta_1) q^{47} + (4 \beta_{2} + 1) q^{49} + ( - \beta_{2} + 6) q^{53} + ( - 5 \beta_{3} + 3 \beta_1) q^{55} + \beta_{3} q^{59} + (5 \beta_{2} - 6) q^{61} - 2 q^{65} + (3 \beta_{3} - 2 \beta_1) q^{67} + (5 \beta_{3} + \beta_1) q^{71} + ( - 6 \beta_{2} + 2) q^{73} + (8 \beta_{2} + 4) q^{77} + ( - 4 \beta_{3} - 4 \beta_1) q^{79} + (3 \beta_{3} + 4 \beta_1) q^{83} + ( - 4 \beta_{2} + 4) q^{85} + ( - 2 \beta_{2} - 2) q^{89} + ( - 2 \beta_{3} - 4 \beta_1) q^{91} + (\beta_{3} + \beta_1) q^{95} + ( - 2 \beta_{2} + 8) q^{97}+O(q^{100})$$ q + (-b2 + 2) * q^5 + (b3 + b1) * q^7 + (-b3 + 2*b1) * q^11 + (-b2 - 2) * q^13 - 2*b2 * q^17 + (b3 + 2*b1) * q^19 + (-b3 + 3*b1) * q^23 + (-4*b2 + 1) * q^25 + (b2 + 6) * q^29 + (-2*b3 + 2*b1) * q^31 + 2*b3 * q^35 + (-5*b2 - 2) * q^37 + 4 * q^41 + b3 * q^43 + (2*b3 - 2*b1) * q^47 + (4*b2 + 1) * q^49 + (-b2 + 6) * q^53 + (-5*b3 + 3*b1) * q^55 + b3 * q^59 + (5*b2 - 6) * q^61 - 2 * q^65 + (3*b3 - 2*b1) * q^67 + (5*b3 + b1) * q^71 + (-6*b2 + 2) * q^73 + (8*b2 + 4) * q^77 + (-4*b3 - 4*b1) * q^79 + (3*b3 + 4*b1) * q^83 + (-4*b2 + 4) * q^85 + (-2*b2 - 2) * q^89 + (-2*b3 - 4*b1) * q^91 + (b3 + b1) * q^95 + (-2*b2 + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{5}+O(q^{10})$$ 4 * q + 8 * q^5 $$4 q + 8 q^{5} - 8 q^{13} + 4 q^{25} + 24 q^{29} - 8 q^{37} + 16 q^{41} + 4 q^{49} + 24 q^{53} - 24 q^{61} - 8 q^{65} + 8 q^{73} + 16 q^{77} + 16 q^{85} - 8 q^{89} + 32 q^{97}+O(q^{100})$$ 4 * q + 8 * q^5 - 8 * q^13 + 4 * q^25 + 24 * q^29 - 8 * q^37 + 16 * q^41 + 4 * q^49 + 24 * q^53 - 24 * q^61 - 8 * q^65 + 8 * q^73 + 16 * q^77 + 16 * q^85 - 8 * q^89 + 32 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{16} + \zeta_{16}^{-1}$$:

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

## 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
 −1.84776 1.84776 −0.765367 0.765367
0 0 0 0.585786 0 −3.69552 0 0 0
1.2 0 0 0 0.585786 0 3.69552 0 0 0
1.3 0 0 0 3.41421 0 −1.53073 0 0 0
1.4 0 0 0 3.41421 0 1.53073 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
4.b odd 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.bp 4
3.b odd 2 1 1024.2.a.h 4
4.b odd 2 1 inner 9216.2.a.bp 4
8.b even 2 1 9216.2.a.w 4
8.d odd 2 1 9216.2.a.w 4
12.b even 2 1 1024.2.a.h 4
24.f even 2 1 1024.2.a.i 4
24.h odd 2 1 1024.2.a.i 4
32.g even 8 2 4608.2.k.bd 8
32.g even 8 2 4608.2.k.bi 8
32.h odd 8 2 4608.2.k.bd 8
32.h odd 8 2 4608.2.k.bi 8
48.i odd 4 2 1024.2.b.g 8
48.k even 4 2 1024.2.b.g 8
96.o even 8 2 512.2.e.i 8
96.o even 8 2 512.2.e.j yes 8
96.p odd 8 2 512.2.e.i 8
96.p odd 8 2 512.2.e.j yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.i 8 96.o even 8 2
512.2.e.i 8 96.p odd 8 2
512.2.e.j yes 8 96.o even 8 2
512.2.e.j yes 8 96.p odd 8 2
1024.2.a.h 4 3.b odd 2 1
1024.2.a.h 4 12.b even 2 1
1024.2.a.i 4 24.f even 2 1
1024.2.a.i 4 24.h odd 2 1
1024.2.b.g 8 48.i odd 4 2
1024.2.b.g 8 48.k even 4 2
4608.2.k.bd 8 32.g even 8 2
4608.2.k.bd 8 32.h odd 8 2
4608.2.k.bi 8 32.g even 8 2
4608.2.k.bi 8 32.h odd 8 2
9216.2.a.w 4 8.b even 2 1
9216.2.a.w 4 8.d odd 2 1
9216.2.a.bp 4 1.a even 1 1 trivial
9216.2.a.bp 4 4.b odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} - 4T_{5} + 2$$ T5^2 - 4*T5 + 2 $$T_{7}^{4} - 16T_{7}^{2} + 32$$ T7^4 - 16*T7^2 + 32 $$T_{11}^{4} - 40T_{11}^{2} + 392$$ T11^4 - 40*T11^2 + 392 $$T_{13}^{2} + 4T_{13} + 2$$ T13^2 + 4*T13 + 2 $$T_{17}^{2} - 8$$ T17^2 - 8 $$T_{19}^{4} - 40T_{19}^{2} + 8$$ T19^4 - 40*T19^2 + 8 $$T_{67}^{4} - 104T_{67}^{2} + 392$$ T67^4 - 104*T67^2 + 392

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 4 T + 2)^{2}$$
$7$ $$T^{4} - 16T^{2} + 32$$
$11$ $$T^{4} - 40T^{2} + 392$$
$13$ $$(T^{2} + 4 T + 2)^{2}$$
$17$ $$(T^{2} - 8)^{2}$$
$19$ $$T^{4} - 40T^{2} + 8$$
$23$ $$T^{4} - 80T^{2} + 1568$$
$29$ $$(T^{2} - 12 T + 34)^{2}$$
$31$ $$T^{4} - 64T^{2} + 512$$
$37$ $$(T^{2} + 4 T - 46)^{2}$$
$41$ $$(T - 4)^{4}$$
$43$ $$T^{4} - 8T^{2} + 8$$
$47$ $$T^{4} - 64T^{2} + 512$$
$53$ $$(T^{2} - 12 T + 34)^{2}$$
$59$ $$T^{4} - 8T^{2} + 8$$
$61$ $$(T^{2} + 12 T - 14)^{2}$$
$67$ $$T^{4} - 104T^{2} + 392$$
$71$ $$T^{4} - 208T^{2} + 9248$$
$73$ $$(T^{2} - 4 T - 68)^{2}$$
$79$ $$T^{4} - 256T^{2} + 8192$$
$83$ $$T^{4} - 200T^{2} + 2312$$
$89$ $$(T^{2} + 4 T - 4)^{2}$$
$97$ $$(T^{2} - 16 T + 56)^{2}$$