# Properties

 Label 9216.2.a.h 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 1536) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 2 \beta q^{7} +O(q^{10})$$ q + b * q^5 - 2*b * q^7 $$q + \beta q^{5} - 2 \beta q^{7} + 3 \beta q^{13} - 4 q^{17} - 8 q^{19} - 4 \beta q^{23} - 3 q^{25} - \beta q^{29} + 2 \beta q^{31} - 4 q^{35} + 3 \beta q^{37} + 4 q^{41} - 8 \beta q^{47} + q^{49} + 5 \beta q^{53} + 12 q^{59} + \beta q^{61} + 6 q^{65} + 4 q^{67} + 8 \beta q^{71} + 14 q^{73} + 6 \beta q^{79} + 16 q^{83} - 4 \beta q^{85} - 6 q^{89} - 12 q^{91} - 8 \beta q^{95} + 16 q^{97} +O(q^{100})$$ q + b * q^5 - 2*b * q^7 + 3*b * q^13 - 4 * q^17 - 8 * q^19 - 4*b * q^23 - 3 * q^25 - b * q^29 + 2*b * q^31 - 4 * q^35 + 3*b * q^37 + 4 * q^41 - 8*b * q^47 + q^49 + 5*b * q^53 + 12 * q^59 + b * q^61 + 6 * q^65 + 4 * q^67 + 8*b * q^71 + 14 * q^73 + 6*b * q^79 + 16 * q^83 - 4*b * q^85 - 6 * q^89 - 12 * q^91 - 8*b * q^95 + 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{17} - 16 q^{19} - 6 q^{25} - 8 q^{35} + 8 q^{41} + 2 q^{49} + 24 q^{59} + 12 q^{65} + 8 q^{67} + 28 q^{73} + 32 q^{83} - 12 q^{89} - 24 q^{91} + 32 q^{97}+O(q^{100})$$ 2 * q - 8 * q^17 - 16 * q^19 - 6 * q^25 - 8 * q^35 + 8 * q^41 + 2 * q^49 + 24 * q^59 + 12 * q^65 + 8 * q^67 + 28 * q^73 + 32 * q^83 - 12 * q^89 - 24 * q^91 + 32 * 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 2.82843 0 0 0
1.2 0 0 0 1.41421 0 −2.82843 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.h 2
3.b odd 2 1 3072.2.a.b 2
4.b odd 2 1 9216.2.a.i 2
8.b even 2 1 9216.2.a.i 2
8.d odd 2 1 inner 9216.2.a.h 2
12.b even 2 1 3072.2.a.h 2
24.f even 2 1 3072.2.a.b 2
24.h odd 2 1 3072.2.a.h 2
32.g even 8 2 4608.2.k.y 4
32.g even 8 2 4608.2.k.bb 4
32.h odd 8 2 4608.2.k.y 4
32.h odd 8 2 4608.2.k.bb 4
48.i odd 4 2 3072.2.d.c 4
48.k even 4 2 3072.2.d.c 4
96.o even 8 2 1536.2.j.b 4
96.o even 8 2 1536.2.j.c yes 4
96.p odd 8 2 1536.2.j.b 4
96.p odd 8 2 1536.2.j.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.b 4 96.o even 8 2
1536.2.j.b 4 96.p odd 8 2
1536.2.j.c yes 4 96.o even 8 2
1536.2.j.c yes 4 96.p odd 8 2
3072.2.a.b 2 3.b odd 2 1
3072.2.a.b 2 24.f even 2 1
3072.2.a.h 2 12.b even 2 1
3072.2.a.h 2 24.h odd 2 1
3072.2.d.c 4 48.i odd 4 2
3072.2.d.c 4 48.k even 4 2
4608.2.k.y 4 32.g even 8 2
4608.2.k.y 4 32.h odd 8 2
4608.2.k.bb 4 32.g even 8 2
4608.2.k.bb 4 32.h odd 8 2
9216.2.a.h 2 1.a even 1 1 trivial
9216.2.a.h 2 8.d odd 2 1 inner
9216.2.a.i 2 4.b odd 2 1
9216.2.a.i 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9216))$$:

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}$$ T11 $$T_{13}^{2} - 18$$ T13^2 - 18 $$T_{17} + 4$$ T17 + 4 $$T_{19} + 8$$ T19 + 8 $$T_{67} - 4$$ T67 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 18$$
$17$ $$(T + 4)^{2}$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} - 32$$
$29$ $$T^{2} - 2$$
$31$ $$T^{2} - 8$$
$37$ $$T^{2} - 18$$
$41$ $$(T - 4)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 128$$
$53$ $$T^{2} - 50$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} - 2$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} - 128$$
$73$ $$(T - 14)^{2}$$
$79$ $$T^{2} - 72$$
$83$ $$(T - 16)^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T - 16)^{2}$$