# Properties

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

# 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: 4.4.4352.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 6x^{2} - 4x + 2$$ x^4 - 6*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{5} + ( - \beta_{2} - 1) q^{7}+O(q^{10})$$ q + (b3 + 1) * q^5 + (-b2 - 1) * q^7 $$q + (\beta_{3} + 1) q^{5} + ( - \beta_{2} - 1) q^{7} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{11} + (\beta_{3} + \beta_{2} - 2) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{19} - 2 \beta_1 q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + ( - \beta_{3} - 2 \beta_1 + 3) q^{29} + (\beta_{2} - 3) q^{31} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{35} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{37} + ( - \beta_{3} + \beta_{2} - 3 \beta_1) q^{41} + ( - 3 \beta_{3} - \beta_{2} - \beta_1) q^{43} + 2 \beta_1 q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + (\beta_{3} + 2 \beta_1 + 5) q^{53} + ( - 4 \beta_1 - 4) q^{55} + 4 \beta_1 q^{59} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{61} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 + 2) q^{65} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{67} + ( - 2 \beta_{2} - 2) q^{71} + (2 \beta_{2} + 4 \beta_1 - 2) q^{73} + (2 \beta_{2} - 2 \beta_1 + 6) q^{77} + (4 \beta_{3} + \beta_{2} + 4 \beta_1 - 3) q^{79} + ( - 3 \beta_{3} + \beta_{2} - 5 \beta_1) q^{83} + (2 \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 2) q^{89} + ( - \beta_{3} + \beta_{2} + \beta_1 - 4) q^{91} + (2 \beta_{2} + 2 \beta_1 - 6) q^{95} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{97}+O(q^{100})$$ q + (b3 + 1) * q^5 + (-b2 - 1) * q^7 + (-b3 - b2 + b1) * q^11 + (b3 + b2 - 2) * q^13 + (-b3 + b2 + b1) * q^17 + (-b3 + b2 + b1) * q^19 - 2*b1 * q^23 + (2*b3 + 2*b1 + 1) * q^25 + (-b3 - 2*b1 + 3) * q^29 + (b2 - 3) * q^31 + (-b3 - b2 - 3*b1) * q^35 + (b3 - b2 - 2*b1 - 4) * q^37 + (-b3 + b2 - 3*b1) * q^41 + (-3*b3 - b2 - b1) * q^43 + 2*b1 * q^47 + (2*b2 - 4*b1 + 1) * q^49 + (b3 + 2*b1 + 5) * q^53 + (-4*b1 - 4) * q^55 + 4*b1 * q^59 + (-b3 + b2 - 2*b1 - 4) * q^61 + (-b3 + b2 + 5*b1 + 2) * q^65 + (-2*b3 - 2*b2 - 2*b1 - 4) * q^67 + (-2*b2 - 2) * q^71 + (2*b2 + 4*b1 - 2) * q^73 + (2*b2 - 2*b1 + 6) * q^77 + (4*b3 + b2 + 4*b1 - 3) * q^79 + (-3*b3 + b2 - 5*b1) * q^83 + (2*b2 + 2*b1 - 6) * q^85 + (2*b3 + 2*b2 - 6*b1 + 2) * q^89 + (-b3 + b2 + b1 - 4) * q^91 + (2*b2 + 2*b1 - 6) * q^95 + (-2*b3 - 2*b2 - 6*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q + 4 * q^5 - 4 * q^7 $$4 q + 4 q^{5} - 4 q^{7} - 8 q^{13} + 4 q^{25} + 12 q^{29} - 12 q^{31} - 16 q^{37} + 4 q^{49} + 20 q^{53} - 16 q^{55} - 16 q^{61} + 8 q^{65} - 16 q^{67} - 8 q^{71} - 8 q^{73} + 24 q^{77} - 12 q^{79} - 24 q^{85} + 8 q^{89} - 16 q^{91} - 24 q^{95}+O(q^{100})$$ 4 * q + 4 * q^5 - 4 * q^7 - 8 * q^13 + 4 * q^25 + 12 * q^29 - 12 * q^31 - 16 * q^37 + 4 * q^49 + 20 * q^53 - 16 * q^55 - 16 * q^61 + 8 * q^65 - 16 * q^67 - 8 * q^71 - 8 * q^73 + 24 * q^77 - 12 * q^79 - 24 * q^85 + 8 * q^89 - 16 * q^91 - 24 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu$$ v^3 - v^2 - 4*v $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 3$$ -v^2 + 2*v + 3 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2\nu^{2} + 4\nu - 3$$ -v^3 + 2*v^2 + 4*v - 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b3 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 3$$ b3 + b1 + 3 $$\nu^{3}$$ $$=$$ $$3\beta_{3} + 2\beta_{2} + 4\beta _1 + 3$$ 3*b3 + 2*b2 + 4*b1 + 3

## 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
 −1.27133 0.334904 −1.74912 2.68554
0 0 0 −1.79793 0 0.158942 0 0 0
1.2 0 0 0 −0.473626 0 −4.55765 0 0 0
1.3 0 0 0 2.47363 0 2.55765 0 0 0
1.4 0 0 0 3.79793 0 −2.15894 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

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 9216.2.a.bn 4
3.b odd 2 1 3072.2.a.o 4
4.b odd 2 1 9216.2.a.bo 4
8.b even 2 1 9216.2.a.x 4
8.d odd 2 1 9216.2.a.y 4
12.b even 2 1 3072.2.a.i 4
24.f even 2 1 3072.2.a.t 4
24.h odd 2 1 3072.2.a.n 4
32.g even 8 2 576.2.k.b 8
32.g even 8 2 1152.2.k.f 8
32.h odd 8 2 144.2.k.b 8
32.h odd 8 2 1152.2.k.c 8
48.i odd 4 2 3072.2.d.i 8
48.k even 4 2 3072.2.d.f 8
96.o even 8 2 48.2.j.a 8
96.o even 8 2 384.2.j.b 8
96.p odd 8 2 192.2.j.a 8
96.p odd 8 2 384.2.j.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 96.o even 8 2
144.2.k.b 8 32.h odd 8 2
192.2.j.a 8 96.p odd 8 2
384.2.j.a 8 96.p odd 8 2
384.2.j.b 8 96.o even 8 2
576.2.k.b 8 32.g even 8 2
1152.2.k.c 8 32.h odd 8 2
1152.2.k.f 8 32.g even 8 2
3072.2.a.i 4 12.b even 2 1
3072.2.a.n 4 24.h odd 2 1
3072.2.a.o 4 3.b odd 2 1
3072.2.a.t 4 24.f even 2 1
3072.2.d.f 8 48.k even 4 2
3072.2.d.i 8 48.i odd 4 2
9216.2.a.x 4 8.b even 2 1
9216.2.a.y 4 8.d odd 2 1
9216.2.a.bn 4 1.a even 1 1 trivial
9216.2.a.bo 4 4.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} - 4T_{5}^{3} - 4T_{5}^{2} + 16T_{5} + 8$$ T5^4 - 4*T5^3 - 4*T5^2 + 16*T5 + 8 $$T_{7}^{4} + 4T_{7}^{3} - 8T_{7}^{2} - 24T_{7} + 4$$ T7^4 + 4*T7^3 - 8*T7^2 - 24*T7 + 4 $$T_{11}^{4} - 24T_{11}^{2} - 32T_{11} + 32$$ T11^4 - 24*T11^2 - 32*T11 + 32 $$T_{13}^{4} + 8T_{13}^{3} + 4T_{13}^{2} - 48T_{13} + 4$$ T13^4 + 8*T13^3 + 4*T13^2 - 48*T13 + 4 $$T_{17}^{4} - 32T_{17}^{2} + 64T_{17} + 16$$ T17^4 - 32*T17^2 + 64*T17 + 16 $$T_{19}^{4} - 32T_{19}^{2} + 64T_{19} + 16$$ T19^4 - 32*T19^2 + 64*T19 + 16 $$T_{67}^{4} + 16T_{67}^{3} - 256T_{67} + 256$$ T67^4 + 16*T67^3 - 256*T67 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 4 T^{3} - 4 T^{2} + 16 T + 8$$
$7$ $$T^{4} + 4 T^{3} - 8 T^{2} - 24 T + 4$$
$11$ $$T^{4} - 24 T^{2} - 32 T + 32$$
$13$ $$T^{4} + 8 T^{3} + 4 T^{2} - 48 T + 4$$
$17$ $$T^{4} - 32 T^{2} + 64 T + 16$$
$19$ $$T^{4} - 32 T^{2} + 64 T + 16$$
$23$ $$(T^{2} - 8)^{2}$$
$29$ $$T^{4} - 12 T^{3} + 28 T^{2} + \cdots - 248$$
$31$ $$T^{4} + 12 T^{3} + 40 T^{2} + 24 T - 28$$
$37$ $$T^{4} + 16 T^{3} + 52 T^{2} + \cdots - 1052$$
$41$ $$T^{4} - 64 T^{2} - 192 T - 112$$
$43$ $$T^{4} - 96 T^{2} + 256 T - 112$$
$47$ $$(T^{2} - 8)^{2}$$
$53$ $$T^{4} - 20 T^{3} + 124 T^{2} + \cdots + 136$$
$59$ $$(T^{2} - 32)^{2}$$
$61$ $$T^{4} + 16 T^{3} + 52 T^{2} + \cdots - 1052$$
$67$ $$T^{4} + 16 T^{3} - 256 T + 256$$
$71$ $$T^{4} + 8 T^{3} - 32 T^{2} - 192 T + 64$$
$73$ $$T^{4} + 8 T^{3} - 96 T^{2} + 64 T + 64$$
$79$ $$T^{4} + 12 T^{3} - 168 T^{2} + \cdots - 10108$$
$83$ $$T^{4} - 216 T^{2} - 160 T + 32$$
$89$ $$T^{4} - 8 T^{3} - 200 T^{2} + \cdots - 1904$$
$97$ $$T^{4} - 224 T^{2} + 768 T + 512$$