# Properties

 Label 2016.2.s.f 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 3 \zeta_{6} ) q^{7} + ( -2 + 2 \zeta_{6} ) q^{11} + 5 q^{13} + ( -2 + 2 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{19} -2 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} -8 q^{29} + ( -1 + \zeta_{6} ) q^{31} + 5 \zeta_{6} q^{37} -2 q^{41} -7 q^{43} + 8 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -2 + 2 \zeta_{6} ) q^{53} + ( -10 + 10 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( -11 + 11 \zeta_{6} ) q^{67} + 12 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + ( -2 - 4 \zeta_{6} ) q^{77} + 17 \zeta_{6} q^{79} -16 q^{83} + 12 \zeta_{6} q^{89} + ( -10 + 15 \zeta_{6} ) q^{91} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{7} + O(q^{10})$$ $$2q - q^{7} - 2q^{11} + 10q^{13} - 2q^{17} + 3q^{19} - 2q^{23} + 5q^{25} - 16q^{29} - q^{31} + 5q^{37} - 4q^{41} - 14q^{43} + 8q^{47} - 13q^{49} - 2q^{53} - 10q^{59} + 2q^{61} - 11q^{67} + 24q^{71} + 3q^{73} - 8q^{77} + 17q^{79} - 32q^{83} + 12q^{89} - 5q^{91} - 28q^{97} + O(q^{100})$$

## 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 0 0 −0.500000 2.59808i 0 0 0
865.1 0 0 0 0 0 −0.500000 + 2.59808i 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.f 2
3.b odd 2 1 672.2.q.i yes 2
4.b odd 2 1 2016.2.s.g 2
7.c even 3 1 inner 2016.2.s.f 2
12.b even 2 1 672.2.q.c 2
21.g even 6 1 4704.2.a.x 1
21.h odd 6 1 672.2.q.i yes 2
21.h odd 6 1 4704.2.a.h 1
24.f even 2 1 1344.2.q.p 2
24.h odd 2 1 1344.2.q.f 2
28.g odd 6 1 2016.2.s.g 2
84.j odd 6 1 4704.2.a.i 1
84.n even 6 1 672.2.q.c 2
84.n even 6 1 4704.2.a.bb 1
168.s odd 6 1 1344.2.q.f 2
168.s odd 6 1 9408.2.a.cm 1
168.v even 6 1 1344.2.q.p 2
168.v even 6 1 9408.2.a.s 1
168.ba even 6 1 9408.2.a.y 1
168.be odd 6 1 9408.2.a.cj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.c 2 12.b even 2 1
672.2.q.c 2 84.n even 6 1
672.2.q.i yes 2 3.b odd 2 1
672.2.q.i yes 2 21.h odd 6 1
1344.2.q.f 2 24.h odd 2 1
1344.2.q.f 2 168.s odd 6 1
1344.2.q.p 2 24.f even 2 1
1344.2.q.p 2 168.v even 6 1
2016.2.s.f 2 1.a even 1 1 trivial
2016.2.s.f 2 7.c even 3 1 inner
2016.2.s.g 2 4.b odd 2 1
2016.2.s.g 2 28.g odd 6 1
4704.2.a.h 1 21.h odd 6 1
4704.2.a.i 1 84.j odd 6 1
4704.2.a.x 1 21.g even 6 1
4704.2.a.bb 1 84.n even 6 1
9408.2.a.s 1 168.v even 6 1
9408.2.a.y 1 168.ba even 6 1
9408.2.a.cj 1 168.be odd 6 1
9408.2.a.cm 1 168.s odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$4 + 2 T + T^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$9 - 3 T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$1 + T + T^{2}$$
$37$ $$25 - 5 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$( 7 + T )^{2}$$
$47$ $$64 - 8 T + T^{2}$$
$53$ $$4 + 2 T + T^{2}$$
$59$ $$100 + 10 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$289 - 17 T + T^{2}$$
$83$ $$( 16 + T )^{2}$$
$89$ $$144 - 12 T + T^{2}$$
$97$ $$( 14 + T )^{2}$$