# Properties

 Label 2816.2.g.a Level $2816$ Weight $2$ Character orbit 2816.g Analytic conductor $22.486$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1407,2816)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2816.1407");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2816 = 2^{8} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2816.g (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$22.4858732092$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 176) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - 3 \beta_1 q^{5} + 8 q^{9}+O(q^{10})$$ q + b2 * q^3 - 3*b1 * q^5 + 8 * q^9 $$q + \beta_{2} q^{3} - 3 \beta_1 q^{5} + 8 q^{9} + \beta_{2} q^{11} - 3 \beta_{3} q^{15} + \beta_{3} q^{23} - 4 q^{25} + 5 \beta_{2} q^{27} + 3 \beta_{3} q^{31} + 11 q^{33} - 7 \beta_1 q^{37} - 24 \beta_1 q^{45} - 2 \beta_{3} q^{47} - 7 q^{49} - 6 \beta_1 q^{53} - 3 \beta_{3} q^{55} - \beta_{2} q^{59} - 3 \beta_{2} q^{67} + 11 \beta_1 q^{69} + 5 \beta_{3} q^{71} - 4 \beta_{2} q^{75} + 31 q^{81} + 9 q^{89} + 33 \beta_1 q^{93} - 17 q^{97} + 8 \beta_{2} q^{99}+O(q^{100})$$ q + b2 * q^3 - 3*b1 * q^5 + 8 * q^9 + b2 * q^11 - 3*b3 * q^15 + b3 * q^23 - 4 * q^25 + 5*b2 * q^27 + 3*b3 * q^31 + 11 * q^33 - 7*b1 * q^37 - 24*b1 * q^45 - 2*b3 * q^47 - 7 * q^49 - 6*b1 * q^53 - 3*b3 * q^55 - b2 * q^59 - 3*b2 * q^67 + 11*b1 * q^69 + 5*b3 * q^71 - 4*b2 * q^75 + 31 * q^81 + 9 * q^89 + 33*b1 * q^93 - 17 * q^97 + 8*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 32 q^{9}+O(q^{10})$$ 4 * q + 32 * q^9 $$4 q + 32 q^{9} - 16 q^{25} + 44 q^{33} - 28 q^{49} + 124 q^{81} + 36 q^{89} - 68 q^{97}+O(q^{100})$$ 4 * q + 32 * q^9 - 16 * q^25 + 44 * q^33 - 28 * q^49 + 124 * q^81 + 36 * q^89 - 68 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 2\nu ) / 3$$ (v^3 - 2*v) / 3 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 8\nu ) / 3$$ (-v^3 + 8*v) / 3 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 5$$ 2*v^2 - 5
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 5 ) / 2$$ (b3 + 5) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 4\beta_1$$ b2 + 4*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$1541$$ $$2047$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1407.1
 −1.65831 + 0.500000i −1.65831 − 0.500000i 1.65831 + 0.500000i 1.65831 − 0.500000i
0 −3.31662 0 3.00000i 0 0 0 8.00000 0
1407.2 0 −3.31662 0 3.00000i 0 0 0 8.00000 0
1407.3 0 3.31662 0 3.00000i 0 0 0 8.00000 0
1407.4 0 3.31662 0 3.00000i 0 0 0 8.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
44.c even 2 1 inner
88.b odd 2 1 inner
88.g even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2816.2.g.a 4
4.b odd 2 1 inner 2816.2.g.a 4
8.b even 2 1 inner 2816.2.g.a 4
8.d odd 2 1 inner 2816.2.g.a 4
11.b odd 2 1 CM 2816.2.g.a 4
16.e even 4 1 176.2.e.a 2
16.e even 4 1 704.2.e.a 2
16.f odd 4 1 176.2.e.a 2
16.f odd 4 1 704.2.e.a 2
44.c even 2 1 inner 2816.2.g.a 4
48.i odd 4 1 1584.2.o.a 2
48.k even 4 1 1584.2.o.a 2
88.b odd 2 1 inner 2816.2.g.a 4
88.g even 2 1 inner 2816.2.g.a 4
176.i even 4 1 176.2.e.a 2
176.i even 4 1 704.2.e.a 2
176.l odd 4 1 176.2.e.a 2
176.l odd 4 1 704.2.e.a 2
528.s odd 4 1 1584.2.o.a 2
528.x even 4 1 1584.2.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.e.a 2 16.e even 4 1
176.2.e.a 2 16.f odd 4 1
176.2.e.a 2 176.i even 4 1
176.2.e.a 2 176.l odd 4 1
704.2.e.a 2 16.e even 4 1
704.2.e.a 2 16.f odd 4 1
704.2.e.a 2 176.i even 4 1
704.2.e.a 2 176.l odd 4 1
1584.2.o.a 2 48.i odd 4 1
1584.2.o.a 2 48.k even 4 1
1584.2.o.a 2 528.s odd 4 1
1584.2.o.a 2 528.x even 4 1
2816.2.g.a 4 1.a even 1 1 trivial
2816.2.g.a 4 4.b odd 2 1 inner
2816.2.g.a 4 8.b even 2 1 inner
2816.2.g.a 4 8.d odd 2 1 inner
2816.2.g.a 4 11.b odd 2 1 CM
2816.2.g.a 4 44.c even 2 1 inner
2816.2.g.a 4 88.b odd 2 1 inner
2816.2.g.a 4 88.g even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2816, [\chi])$$:

 $$T_{3}^{2} - 11$$ T3^2 - 11 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 11)^{2}$$
$5$ $$(T^{2} + 9)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 11)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 11)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 99)^{2}$$
$37$ $$(T^{2} + 49)^{2}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 44)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 11)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} - 99)^{2}$$
$71$ $$(T^{2} + 275)^{2}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T - 9)^{4}$$
$97$ $$(T + 17)^{4}$$