# Properties

 Label 2016.2.s.b Level $2016$ Weight $2$ Character orbit 2016.s Analytic conductor $16.098$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2016.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.0978410475$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -4 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + ( -6 + 6 \zeta_{6} ) q^{11} + 5 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + ( -3 + 3 \zeta_{6} ) q^{31} + ( 4 - 12 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + 6 q^{41} -5 q^{43} + 4 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( -6 + 6 \zeta_{6} ) q^{53} + 24 q^{55} + ( 6 - 6 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} -20 \zeta_{6} q^{65} + ( 7 - 7 \zeta_{6} ) q^{67} + 16 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + ( -18 + 12 \zeta_{6} ) q^{77} + 11 \zeta_{6} q^{79} + 12 q^{83} -8 q^{85} + 4 \zeta_{6} q^{89} + ( 10 + 5 \zeta_{6} ) q^{91} + ( 4 - 4 \zeta_{6} ) q^{95} -6 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} + 5 q^{7} + O(q^{10})$$ $$2 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 10 q^{13} + 2 q^{17} + q^{19} + 6 q^{23} - 11 q^{25} - 3 q^{31} - 4 q^{35} - 3 q^{37} + 12 q^{41} - 10 q^{43} + 4 q^{47} + 11 q^{49} - 6 q^{53} + 48 q^{55} + 6 q^{59} + 2 q^{61} - 20 q^{65} + 7 q^{67} + 32 q^{71} + 3 q^{73} - 24 q^{77} + 11 q^{79} + 24 q^{83} - 16 q^{85} + 4 q^{89} + 25 q^{91} + 4 q^{95} - 12 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1765$$ $$1793$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −2.00000 + 3.46410i 0 2.50000 0.866025i 0 0 0
865.1 0 0 0 −2.00000 3.46410i 0 2.50000 + 0.866025i 0 0 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.s.b 2
3.b odd 2 1 672.2.q.e 2
4.b odd 2 1 2016.2.s.a 2
7.c even 3 1 inner 2016.2.s.b 2
12.b even 2 1 672.2.q.j yes 2
21.g even 6 1 4704.2.a.p 1
21.h odd 6 1 672.2.q.e 2
21.h odd 6 1 4704.2.a.r 1
24.f even 2 1 1344.2.q.a 2
24.h odd 2 1 1344.2.q.l 2
28.g odd 6 1 2016.2.s.a 2
84.j odd 6 1 4704.2.a.bh 1
84.n even 6 1 672.2.q.j yes 2
84.n even 6 1 4704.2.a.a 1
168.s odd 6 1 1344.2.q.l 2
168.s odd 6 1 9408.2.a.bp 1
168.v even 6 1 1344.2.q.a 2
168.v even 6 1 9408.2.a.dd 1
168.ba even 6 1 9408.2.a.bs 1
168.be odd 6 1 9408.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.e 2 3.b odd 2 1
672.2.q.e 2 21.h odd 6 1
672.2.q.j yes 2 12.b even 2 1
672.2.q.j yes 2 84.n even 6 1
1344.2.q.a 2 24.f even 2 1
1344.2.q.a 2 168.v even 6 1
1344.2.q.l 2 24.h odd 2 1
1344.2.q.l 2 168.s odd 6 1
2016.2.s.a 2 4.b odd 2 1
2016.2.s.a 2 28.g odd 6 1
2016.2.s.b 2 1.a even 1 1 trivial
2016.2.s.b 2 7.c even 3 1 inner
4704.2.a.a 1 84.n even 6 1
4704.2.a.p 1 21.g even 6 1
4704.2.a.r 1 21.h odd 6 1
4704.2.a.bh 1 84.j odd 6 1
9408.2.a.a 1 168.be odd 6 1
9408.2.a.bp 1 168.s odd 6 1
9408.2.a.bs 1 168.ba even 6 1
9408.2.a.dd 1 168.v even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2016, [\chi])$$:

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{13} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$36 + 6 T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$9 + 3 T + T^{2}$$
$37$ $$9 + 3 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$16 - 4 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$36 - 6 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$( -16 + T )^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$121 - 11 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$16 - 4 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$