# Properties

 Label 2816.2.c.i Level $2816$ Weight $2$ Character orbit 2816.c Analytic conductor $22.486$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2816 = 2^{8} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2816.c (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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - 3 i q^{5} + 2 q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 - 3*i * q^5 + 2 * q^7 - 6 * q^9 $$q + 3 i q^{3} - 3 i q^{5} + 2 q^{7} - 6 q^{9} - i q^{11} + 9 q^{15} - 6 q^{17} - 4 i q^{19} + 6 i q^{21} - q^{23} - 4 q^{25} - 9 i q^{27} + 8 i q^{29} - 7 q^{31} + 3 q^{33} - 6 i q^{35} - i q^{37} - 4 q^{41} + 6 i q^{43} + 18 i q^{45} - 8 q^{47} - 3 q^{49} - 18 i q^{51} + 2 i q^{53} - 3 q^{55} + 12 q^{57} - i q^{59} - 4 i q^{61} - 12 q^{63} + 5 i q^{67} - 3 i q^{69} - 3 q^{71} - 16 q^{73} - 12 i q^{75} - 2 i q^{77} + 2 q^{79} + 9 q^{81} + 2 i q^{83} + 18 i q^{85} - 24 q^{87} - 15 q^{89} - 21 i q^{93} - 12 q^{95} - 7 q^{97} + 6 i q^{99} +O(q^{100})$$ q + 3*i * q^3 - 3*i * q^5 + 2 * q^7 - 6 * q^9 - i * q^11 + 9 * q^15 - 6 * q^17 - 4*i * q^19 + 6*i * q^21 - q^23 - 4 * q^25 - 9*i * q^27 + 8*i * q^29 - 7 * q^31 + 3 * q^33 - 6*i * q^35 - i * q^37 - 4 * q^41 + 6*i * q^43 + 18*i * q^45 - 8 * q^47 - 3 * q^49 - 18*i * q^51 + 2*i * q^53 - 3 * q^55 + 12 * q^57 - i * q^59 - 4*i * q^61 - 12 * q^63 + 5*i * q^67 - 3*i * q^69 - 3 * q^71 - 16 * q^73 - 12*i * q^75 - 2*i * q^77 + 2 * q^79 + 9 * q^81 + 2*i * q^83 + 18*i * q^85 - 24 * q^87 - 15 * q^89 - 21*i * q^93 - 12 * q^95 - 7 * q^97 + 6*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7} - 12 q^{9}+O(q^{10})$$ 2 * q + 4 * q^7 - 12 * q^9 $$2 q + 4 q^{7} - 12 q^{9} + 18 q^{15} - 12 q^{17} - 2 q^{23} - 8 q^{25} - 14 q^{31} + 6 q^{33} - 8 q^{41} - 16 q^{47} - 6 q^{49} - 6 q^{55} + 24 q^{57} - 24 q^{63} - 6 q^{71} - 32 q^{73} + 4 q^{79} + 18 q^{81} - 48 q^{87} - 30 q^{89} - 24 q^{95} - 14 q^{97}+O(q^{100})$$ 2 * q + 4 * q^7 - 12 * q^9 + 18 * q^15 - 12 * q^17 - 2 * q^23 - 8 * q^25 - 14 * q^31 + 6 * q^33 - 8 * q^41 - 16 * q^47 - 6 * q^49 - 6 * q^55 + 24 * q^57 - 24 * q^63 - 6 * q^71 - 32 * q^73 + 4 * q^79 + 18 * q^81 - 48 * q^87 - 30 * q^89 - 24 * q^95 - 14 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1409.1
 − 1.00000i 1.00000i
0 3.00000i 0 3.00000i 0 2.00000 0 −6.00000 0
1409.2 0 3.00000i 0 3.00000i 0 2.00000 0 −6.00000 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
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 2816.2.c.i 2
4.b odd 2 1 2816.2.c.d 2
8.b even 2 1 inner 2816.2.c.i 2
8.d odd 2 1 2816.2.c.d 2
16.e even 4 1 88.2.a.a 1
16.e even 4 1 704.2.a.l 1
16.f odd 4 1 176.2.a.c 1
16.f odd 4 1 704.2.a.b 1
48.i odd 4 1 792.2.a.g 1
48.i odd 4 1 6336.2.a.h 1
48.k even 4 1 1584.2.a.q 1
48.k even 4 1 6336.2.a.k 1
80.i odd 4 1 2200.2.b.a 2
80.j even 4 1 4400.2.b.b 2
80.k odd 4 1 4400.2.a.a 1
80.q even 4 1 2200.2.a.k 1
80.s even 4 1 4400.2.b.b 2
80.t odd 4 1 2200.2.b.a 2
112.j even 4 1 8624.2.a.c 1
112.l odd 4 1 4312.2.a.l 1
176.i even 4 1 1936.2.a.l 1
176.i even 4 1 7744.2.a.b 1
176.l odd 4 1 968.2.a.a 1
176.l odd 4 1 7744.2.a.bk 1
176.u odd 20 4 968.2.i.i 4
176.w even 20 4 968.2.i.j 4
528.x even 4 1 8712.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 16.e even 4 1
176.2.a.c 1 16.f odd 4 1
704.2.a.b 1 16.f odd 4 1
704.2.a.l 1 16.e even 4 1
792.2.a.g 1 48.i odd 4 1
968.2.a.a 1 176.l odd 4 1
968.2.i.i 4 176.u odd 20 4
968.2.i.j 4 176.w even 20 4
1584.2.a.q 1 48.k even 4 1
1936.2.a.l 1 176.i even 4 1
2200.2.a.k 1 80.q even 4 1
2200.2.b.a 2 80.i odd 4 1
2200.2.b.a 2 80.t odd 4 1
2816.2.c.d 2 4.b odd 2 1
2816.2.c.d 2 8.d odd 2 1
2816.2.c.i 2 1.a even 1 1 trivial
2816.2.c.i 2 8.b even 2 1 inner
4312.2.a.l 1 112.l odd 4 1
4400.2.a.a 1 80.k odd 4 1
4400.2.b.b 2 80.j even 4 1
4400.2.b.b 2 80.s even 4 1
6336.2.a.h 1 48.i odd 4 1
6336.2.a.k 1 48.k even 4 1
7744.2.a.b 1 176.i even 4 1
7744.2.a.bk 1 176.l odd 4 1
8624.2.a.c 1 112.j even 4 1
8712.2.a.x 1 528.x even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{5}^{2} + 9$$ T5^2 + 9 $$T_{7} - 2$$ T7 - 2 $$T_{23} + 1$$ T23 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2} + 9$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 1$$
$13$ $$T^{2}$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 64$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 4)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2} + 1$$
$61$ $$T^{2} + 16$$
$67$ $$T^{2} + 25$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T + 16)^{2}$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} + 4$$
$89$ $$(T + 15)^{2}$$
$97$ $$(T + 7)^{2}$$