# Properties

 Label 9216.2.a.bb Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 256) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7}+O(q^{10})$$ q + b3 * q^5 + (-b3 - b2) * q^7 $$q + \beta_{3} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_1 - 3) q^{11} + (\beta_{3} - 2 \beta_{2}) q^{13} + 2 \beta_1 q^{17} + (\beta_1 + 3) q^{19} + ( - \beta_{3} + 3 \beta_{2}) q^{23} + q^{25} - \beta_{3} q^{29} + 4 \beta_{2} q^{31} + ( - 2 \beta_1 - 6) q^{35} + (\beta_{3} + 2 \beta_{2}) q^{37} - 4 \beta_1 q^{41} + ( - 3 \beta_1 + 3) q^{43} - 4 \beta_{3} q^{47} + (4 \beta_1 + 1) q^{49} + (\beta_{3} + 6 \beta_{2}) q^{53} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{55} + (\beta_1 - 9) q^{59} + ( - \beta_{3} - 2 \beta_{2}) q^{61} + ( - 4 \beta_1 + 6) q^{65} + (3 \beta_1 + 1) q^{67} + (\beta_{3} - 3 \beta_{2}) q^{71} + (2 \beta_1 - 6) q^{73} + (4 \beta_{3} + 6 \beta_{2}) q^{77} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{79} + (3 \beta_1 - 3) q^{83} + 6 \beta_{2} q^{85} + ( - 2 \beta_1 + 6) q^{89} + (2 \beta_1 - 2) q^{91} + (3 \beta_{3} + 3 \beta_{2}) q^{95} + 2 \beta_1 q^{97}+O(q^{100})$$ q + b3 * q^5 + (-b3 - b2) * q^7 + (-b1 - 3) * q^11 + (b3 - 2*b2) * q^13 + 2*b1 * q^17 + (b1 + 3) * q^19 + (-b3 + 3*b2) * q^23 + q^25 - b3 * q^29 + 4*b2 * q^31 + (-2*b1 - 6) * q^35 + (b3 + 2*b2) * q^37 - 4*b1 * q^41 + (-3*b1 + 3) * q^43 - 4*b3 * q^47 + (4*b1 + 1) * q^49 + (b3 + 6*b2) * q^53 + (-3*b3 - 3*b2) * q^55 + (b1 - 9) * q^59 + (-b3 - 2*b2) * q^61 + (-4*b1 + 6) * q^65 + (3*b1 + 1) * q^67 + (b3 - 3*b2) * q^71 + (2*b1 - 6) * q^73 + (4*b3 + 6*b2) * q^77 + (-4*b3 + 4*b2) * q^79 + (3*b1 - 3) * q^83 + 6*b2 * q^85 + (-2*b1 + 6) * q^89 + (2*b1 - 2) * q^91 + (3*b3 + 3*b2) * q^95 + 2*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{11} + 12 q^{19} + 4 q^{25} - 24 q^{35} + 12 q^{43} + 4 q^{49} - 36 q^{59} + 24 q^{65} + 4 q^{67} - 24 q^{73} - 12 q^{83} + 24 q^{89} - 8 q^{91}+O(q^{100})$$ 4 * q - 12 * q^11 + 12 * q^19 + 4 * q^25 - 24 * q^35 + 12 * q^43 + 4 * q^49 - 36 * q^59 + 24 * q^65 + 4 * q^67 - 24 * q^73 - 12 * q^83 + 24 * q^89 - 8 * q^91

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 5\nu$$ -v^3 + 5*v
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} + 5\beta_{2} ) / 2$$ (3*b3 + 5*b2) / 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
 −0.517638 −1.93185 1.93185 0.517638
0 0 0 −2.44949 0 1.03528 0 0 0
1.2 0 0 0 −2.44949 0 3.86370 0 0 0
1.3 0 0 0 2.44949 0 −3.86370 0 0 0
1.4 0 0 0 2.44949 0 −1.03528 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
8.d 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.bb 4
3.b odd 2 1 1024.2.a.j 4
4.b odd 2 1 9216.2.a.bk 4
8.b even 2 1 9216.2.a.bk 4
8.d odd 2 1 inner 9216.2.a.bb 4
12.b even 2 1 1024.2.a.g 4
24.f even 2 1 1024.2.a.j 4
24.h odd 2 1 1024.2.a.g 4
32.g even 8 2 2304.2.k.f 8
32.g even 8 2 2304.2.k.k 8
32.h odd 8 2 2304.2.k.f 8
32.h odd 8 2 2304.2.k.k 8
48.i odd 4 2 1024.2.b.h 8
48.k even 4 2 1024.2.b.h 8
96.o even 8 2 256.2.e.a 8
96.o even 8 2 256.2.e.b yes 8
96.p odd 8 2 256.2.e.a 8
96.p odd 8 2 256.2.e.b yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.e.a 8 96.o even 8 2
256.2.e.a 8 96.p odd 8 2
256.2.e.b yes 8 96.o even 8 2
256.2.e.b yes 8 96.p odd 8 2
1024.2.a.g 4 12.b even 2 1
1024.2.a.g 4 24.h odd 2 1
1024.2.a.j 4 3.b odd 2 1
1024.2.a.j 4 24.f even 2 1
1024.2.b.h 8 48.i odd 4 2
1024.2.b.h 8 48.k even 4 2
2304.2.k.f 8 32.g even 8 2
2304.2.k.f 8 32.h odd 8 2
2304.2.k.k 8 32.g even 8 2
2304.2.k.k 8 32.h odd 8 2
9216.2.a.bb 4 1.a even 1 1 trivial
9216.2.a.bb 4 8.d odd 2 1 inner
9216.2.a.bk 4 4.b odd 2 1
9216.2.a.bk 4 8.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 6$$ T5^2 - 6 $$T_{7}^{4} - 16T_{7}^{2} + 16$$ T7^4 - 16*T7^2 + 16 $$T_{11}^{2} + 6T_{11} + 6$$ T11^2 + 6*T11 + 6 $$T_{13}^{4} - 28T_{13}^{2} + 4$$ T13^4 - 28*T13^2 + 4 $$T_{17}^{2} - 12$$ T17^2 - 12 $$T_{19}^{2} - 6T_{19} + 6$$ T19^2 - 6*T19 + 6 $$T_{67}^{2} - 2T_{67} - 26$$ T67^2 - 2*T67 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 6)^{2}$$
$7$ $$T^{4} - 16T^{2} + 16$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$T^{4} - 28T^{2} + 4$$
$17$ $$(T^{2} - 12)^{2}$$
$19$ $$(T^{2} - 6 T + 6)^{2}$$
$23$ $$T^{4} - 48T^{2} + 144$$
$29$ $$(T^{2} - 6)^{2}$$
$31$ $$(T^{2} - 32)^{2}$$
$37$ $$T^{4} - 28T^{2} + 4$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$(T^{2} - 6 T - 18)^{2}$$
$47$ $$(T^{2} - 96)^{2}$$
$53$ $$T^{4} - 156T^{2} + 4356$$
$59$ $$(T^{2} + 18 T + 78)^{2}$$
$61$ $$T^{4} - 28T^{2} + 4$$
$67$ $$(T^{2} - 2 T - 26)^{2}$$
$71$ $$T^{4} - 48T^{2} + 144$$
$73$ $$(T^{2} + 12 T + 24)^{2}$$
$79$ $$T^{4} - 256T^{2} + 4096$$
$83$ $$(T^{2} + 6 T - 18)^{2}$$
$89$ $$(T^{2} - 12 T + 24)^{2}$$
$97$ $$(T^{2} - 12)^{2}$$