# Properties

 Label 2816.2.c.k Level $2816$ Weight $2$ Character orbit 2816.c Analytic conductor $22.486$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$22.4858732092$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 3 i q^{5} + 2 q^{7} + 2 q^{9} +O(q^{10})$$ q + i * q^3 - 3*i * q^5 + 2 * q^7 + 2 * q^9 $$q + i q^{3} - 3 i q^{5} + 2 q^{7} + 2 q^{9} + i q^{11} + 4 i q^{13} + 3 q^{15} + 6 q^{17} + 8 i q^{19} + 2 i q^{21} - 3 q^{23} - 4 q^{25} + 5 i q^{27} - 5 q^{31} - q^{33} - 6 i q^{35} - i q^{37} - 4 q^{39} + 10 i q^{43} - 6 i q^{45} - 3 q^{49} + 6 i q^{51} - 6 i q^{53} + 3 q^{55} - 8 q^{57} - 3 i q^{59} + 4 i q^{61} + 4 q^{63} + 12 q^{65} - i q^{67} - 3 i q^{69} + 15 q^{71} + 4 q^{73} - 4 i q^{75} + 2 i q^{77} - 2 q^{79} + q^{81} + 6 i q^{83} - 18 i q^{85} + 9 q^{89} + 8 i q^{91} - 5 i q^{93} + 24 q^{95} - 7 q^{97} + 2 i q^{99} +O(q^{100})$$ q + i * q^3 - 3*i * q^5 + 2 * q^7 + 2 * q^9 + i * q^11 + 4*i * q^13 + 3 * q^15 + 6 * q^17 + 8*i * q^19 + 2*i * q^21 - 3 * q^23 - 4 * q^25 + 5*i * q^27 - 5 * q^31 - q^33 - 6*i * q^35 - i * q^37 - 4 * q^39 + 10*i * q^43 - 6*i * q^45 - 3 * q^49 + 6*i * q^51 - 6*i * q^53 + 3 * q^55 - 8 * q^57 - 3*i * q^59 + 4*i * q^61 + 4 * q^63 + 12 * q^65 - i * q^67 - 3*i * q^69 + 15 * q^71 + 4 * q^73 - 4*i * q^75 + 2*i * q^77 - 2 * q^79 + q^81 + 6*i * q^83 - 18*i * q^85 + 9 * q^89 + 8*i * q^91 - 5*i * q^93 + 24 * q^95 - 7 * q^97 + 2*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^7 + 4 * q^9 $$2 q + 4 q^{7} + 4 q^{9} + 6 q^{15} + 12 q^{17} - 6 q^{23} - 8 q^{25} - 10 q^{31} - 2 q^{33} - 8 q^{39} - 6 q^{49} + 6 q^{55} - 16 q^{57} + 8 q^{63} + 24 q^{65} + 30 q^{71} + 8 q^{73} - 4 q^{79} + 2 q^{81} + 18 q^{89} + 48 q^{95} - 14 q^{97}+O(q^{100})$$ 2 * q + 4 * q^7 + 4 * q^9 + 6 * q^15 + 12 * q^17 - 6 * q^23 - 8 * q^25 - 10 * q^31 - 2 * q^33 - 8 * q^39 - 6 * q^49 + 6 * q^55 - 16 * q^57 + 8 * q^63 + 24 * q^65 + 30 * q^71 + 8 * q^73 - 4 * q^79 + 2 * q^81 + 18 * q^89 + 48 * 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.

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 1.00000i 0 3.00000i 0 2.00000 0 2.00000 0
1409.2 0 1.00000i 0 3.00000i 0 2.00000 0 2.00000 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.k 2
4.b odd 2 1 2816.2.c.e 2
8.b even 2 1 inner 2816.2.c.k 2
8.d odd 2 1 2816.2.c.e 2
16.e even 4 1 176.2.a.a 1
16.e even 4 1 704.2.a.i 1
16.f odd 4 1 44.2.a.a 1
16.f odd 4 1 704.2.a.f 1
48.i odd 4 1 1584.2.a.p 1
48.i odd 4 1 6336.2.a.i 1
48.k even 4 1 396.2.a.c 1
48.k even 4 1 6336.2.a.j 1
80.i odd 4 1 4400.2.b.k 2
80.j even 4 1 1100.2.b.c 2
80.k odd 4 1 1100.2.a.b 1
80.q even 4 1 4400.2.a.v 1
80.s even 4 1 1100.2.b.c 2
80.t odd 4 1 4400.2.b.k 2
112.j even 4 1 2156.2.a.a 1
112.l odd 4 1 8624.2.a.w 1
112.u odd 12 2 2156.2.i.b 2
112.v even 12 2 2156.2.i.c 2
144.u even 12 2 3564.2.i.a 2
144.v odd 12 2 3564.2.i.j 2
176.i even 4 1 484.2.a.a 1
176.i even 4 1 7744.2.a.m 1
176.l odd 4 1 1936.2.a.c 1
176.l odd 4 1 7744.2.a.bc 1
176.v odd 20 4 484.2.e.a 4
176.x even 20 4 484.2.e.b 4
208.o odd 4 1 7436.2.a.d 1
240.t even 4 1 9900.2.a.h 1
240.z odd 4 1 9900.2.c.g 2
240.bd odd 4 1 9900.2.c.g 2
528.s odd 4 1 4356.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 16.f odd 4 1
176.2.a.a 1 16.e even 4 1
396.2.a.c 1 48.k even 4 1
484.2.a.a 1 176.i even 4 1
484.2.e.a 4 176.v odd 20 4
484.2.e.b 4 176.x even 20 4
704.2.a.f 1 16.f odd 4 1
704.2.a.i 1 16.e even 4 1
1100.2.a.b 1 80.k odd 4 1
1100.2.b.c 2 80.j even 4 1
1100.2.b.c 2 80.s even 4 1
1584.2.a.p 1 48.i odd 4 1
1936.2.a.c 1 176.l odd 4 1
2156.2.a.a 1 112.j even 4 1
2156.2.i.b 2 112.u odd 12 2
2156.2.i.c 2 112.v even 12 2
2816.2.c.e 2 4.b odd 2 1
2816.2.c.e 2 8.d odd 2 1
2816.2.c.k 2 1.a even 1 1 trivial
2816.2.c.k 2 8.b even 2 1 inner
3564.2.i.a 2 144.u even 12 2
3564.2.i.j 2 144.v odd 12 2
4356.2.a.j 1 528.s odd 4 1
4400.2.a.v 1 80.q even 4 1
4400.2.b.k 2 80.i odd 4 1
4400.2.b.k 2 80.t odd 4 1
6336.2.a.i 1 48.i odd 4 1
6336.2.a.j 1 48.k even 4 1
7436.2.a.d 1 208.o odd 4 1
7744.2.a.m 1 176.i even 4 1
7744.2.a.bc 1 176.l odd 4 1
8624.2.a.w 1 112.l odd 4 1
9900.2.a.h 1 240.t even 4 1
9900.2.c.g 2 240.z odd 4 1
9900.2.c.g 2 240.bd odd 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 9$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 1$$
$13$ $$T^{2} + 16$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 64$$
$23$ $$(T + 3)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 9$$
$61$ $$T^{2} + 16$$
$67$ $$T^{2} + 1$$
$71$ $$(T - 15)^{2}$$
$73$ $$(T - 4)^{2}$$
$79$ $$(T + 2)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 9)^{2}$$
$97$ $$(T + 7)^{2}$$