# Properties

 Label 9216.2.a.d Level $9216$ Weight $2$ Character orbit 9216.a Self dual yes Analytic conductor $73.590$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 16) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 2 q^{7}+O(q^{10})$$ q + b * q^5 - 2 * q^7 $$q + \beta q^{5} - 2 q^{7} + \beta q^{11} - \beta q^{13} - 2 q^{17} + 3 \beta q^{19} + 6 q^{23} - 3 q^{25} + 3 \beta q^{29} - 8 q^{31} - 2 \beta q^{35} + 3 \beta q^{37} - 5 \beta q^{43} + 8 q^{47} - 3 q^{49} - 5 \beta q^{53} + 2 q^{55} + 3 \beta q^{59} + 9 \beta q^{61} - 2 q^{65} + 5 \beta q^{67} + 10 q^{71} + 4 q^{73} - 2 \beta q^{77} + \beta q^{83} - 2 \beta q^{85} - 4 q^{89} + 2 \beta q^{91} + 6 q^{95} - 2 q^{97} +O(q^{100})$$ q + b * q^5 - 2 * q^7 + b * q^11 - b * q^13 - 2 * q^17 + 3*b * q^19 + 6 * q^23 - 3 * q^25 + 3*b * q^29 - 8 * q^31 - 2*b * q^35 + 3*b * q^37 - 5*b * q^43 + 8 * q^47 - 3 * q^49 - 5*b * q^53 + 2 * q^55 + 3*b * q^59 + 9*b * q^61 - 2 * q^65 + 5*b * q^67 + 10 * q^71 + 4 * q^73 - 2*b * q^77 + b * q^83 - 2*b * q^85 - 4 * q^89 + 2*b * q^91 + 6 * q^95 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^7 $$2 q - 4 q^{7} - 4 q^{17} + 12 q^{23} - 6 q^{25} - 16 q^{31} + 16 q^{47} - 6 q^{49} + 4 q^{55} - 4 q^{65} + 20 q^{71} + 8 q^{73} - 8 q^{89} + 12 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 - 4 * q^17 + 12 * q^23 - 6 * q^25 - 16 * q^31 + 16 * q^47 - 6 * q^49 + 4 * q^55 - 4 * q^65 + 20 * q^71 + 8 * q^73 - 8 * q^89 + 12 * q^95 - 4 * q^97

## 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
 −1.41421 1.41421
0 0 0 −1.41421 0 −2.00000 0 0 0
1.2 0 0 0 1.41421 0 −2.00000 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.b 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.d 2
3.b odd 2 1 1024.2.a.b 2
4.b odd 2 1 9216.2.a.s 2
8.b even 2 1 inner 9216.2.a.d 2
8.d odd 2 1 9216.2.a.s 2
12.b even 2 1 1024.2.a.e 2
24.f even 2 1 1024.2.a.e 2
24.h odd 2 1 1024.2.a.b 2
32.g even 8 2 144.2.k.a 2
32.g even 8 2 1152.2.k.b 2
32.h odd 8 2 576.2.k.a 2
32.h odd 8 2 1152.2.k.a 2
48.i odd 4 2 1024.2.b.e 2
48.k even 4 2 1024.2.b.b 2
96.o even 8 2 64.2.e.a 2
96.o even 8 2 128.2.e.a 2
96.p odd 8 2 16.2.e.a 2
96.p odd 8 2 128.2.e.b 2
480.bq odd 8 2 1600.2.q.b 2
480.br even 8 2 400.2.q.b 2
480.bs even 8 2 1600.2.l.a 2
480.bu odd 8 2 400.2.l.c 2
480.ca odd 8 2 1600.2.q.a 2
480.cb even 8 2 400.2.q.a 2
672.bo even 8 2 784.2.m.b 2
672.ce odd 24 4 784.2.x.f 4
672.cl even 24 4 784.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 96.p odd 8 2
64.2.e.a 2 96.o even 8 2
128.2.e.a 2 96.o even 8 2
128.2.e.b 2 96.p odd 8 2
144.2.k.a 2 32.g even 8 2
400.2.l.c 2 480.bu odd 8 2
400.2.q.a 2 480.cb even 8 2
400.2.q.b 2 480.br even 8 2
576.2.k.a 2 32.h odd 8 2
784.2.m.b 2 672.bo even 8 2
784.2.x.c 4 672.cl even 24 4
784.2.x.f 4 672.ce odd 24 4
1024.2.a.b 2 3.b odd 2 1
1024.2.a.b 2 24.h odd 2 1
1024.2.a.e 2 12.b even 2 1
1024.2.a.e 2 24.f even 2 1
1024.2.b.b 2 48.k even 4 2
1024.2.b.e 2 48.i odd 4 2
1152.2.k.a 2 32.h odd 8 2
1152.2.k.b 2 32.g even 8 2
1600.2.l.a 2 480.bs even 8 2
1600.2.q.a 2 480.ca odd 8 2
1600.2.q.b 2 480.bq odd 8 2
9216.2.a.d 2 1.a even 1 1 trivial
9216.2.a.d 2 8.b even 2 1 inner
9216.2.a.s 2 4.b odd 2 1
9216.2.a.s 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} - 2$$ T11^2 - 2 $$T_{13}^{2} - 2$$ T13^2 - 2 $$T_{17} + 2$$ T17 + 2 $$T_{19}^{2} - 18$$ T19^2 - 18 $$T_{67}^{2} - 50$$ T67^2 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} - 2$$
$13$ $$T^{2} - 2$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 18$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 18$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} - 18$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 50$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} - 50$$
$59$ $$T^{2} - 18$$
$61$ $$T^{2} - 162$$
$67$ $$T^{2} - 50$$
$71$ $$(T - 10)^{2}$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 2$$
$89$ $$(T + 4)^{2}$$
$97$ $$(T + 2)^{2}$$