# Properties

 Label 2016.2.s.e 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 2) q^{7}+O(q^{10})$$ q + (-z - 2) * q^7 $$q + ( - \zeta_{6} - 2) q^{7} + (2 \zeta_{6} - 2) q^{11} + q^{13} + (2 \zeta_{6} - 2) q^{17} - 5 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + 8 q^{29} + ( - 3 \zeta_{6} + 3) q^{31} + 9 \zeta_{6} q^{37} - 2 q^{41} + q^{43} + 8 \zeta_{6} q^{47} + (5 \zeta_{6} + 3) q^{49} + ( - 6 \zeta_{6} + 6) q^{53} + ( - 6 \zeta_{6} + 6) q^{59} + 2 \zeta_{6} q^{61} + ( - 5 \zeta_{6} + 5) q^{67} - 4 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + ( - 4 \zeta_{6} + 6) q^{77} + 5 \zeta_{6} q^{79} + 12 \zeta_{6} q^{89} + ( - \zeta_{6} - 2) q^{91} + 18 q^{97} +O(q^{100})$$ q + (-z - 2) * q^7 + (2*z - 2) * q^11 + q^13 + (2*z - 2) * q^17 - 5*z * q^19 + 6*z * q^23 + (-5*z + 5) * q^25 + 8 * q^29 + (-3*z + 3) * q^31 + 9*z * q^37 - 2 * q^41 + q^43 + 8*z * q^47 + (5*z + 3) * q^49 + (-6*z + 6) * q^53 + (-6*z + 6) * q^59 + 2*z * q^61 + (-5*z + 5) * q^67 - 4 * q^71 + (-11*z + 11) * q^73 + (-4*z + 6) * q^77 + 5*z * q^79 + 12*z * q^89 + (-z - 2) * q^91 + 18 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 5 q^{7}+O(q^{10})$$ 2 * q - 5 * q^7 $$2 q - 5 q^{7} - 2 q^{11} + 2 q^{13} - 2 q^{17} - 5 q^{19} + 6 q^{23} + 5 q^{25} + 16 q^{29} + 3 q^{31} + 9 q^{37} - 4 q^{41} + 2 q^{43} + 8 q^{47} + 11 q^{49} + 6 q^{53} + 6 q^{59} + 2 q^{61} + 5 q^{67} - 8 q^{71} + 11 q^{73} + 8 q^{77} + 5 q^{79} + 12 q^{89} - 5 q^{91} + 36 q^{97}+O(q^{100})$$ 2 * q - 5 * q^7 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 5 * q^19 + 6 * q^23 + 5 * q^25 + 16 * q^29 + 3 * q^31 + 9 * q^37 - 4 * q^41 + 2 * q^43 + 8 * q^47 + 11 * q^49 + 6 * q^53 + 6 * q^59 + 2 * q^61 + 5 * q^67 - 8 * q^71 + 11 * q^73 + 8 * q^77 + 5 * q^79 + 12 * q^89 - 5 * q^91 + 36 * q^97

## 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 −2.50000 + 0.866025i 0 0 0
865.1 0 0 0 0 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.e 2
3.b odd 2 1 672.2.q.h yes 2
4.b odd 2 1 2016.2.s.h 2
7.c even 3 1 inner 2016.2.s.e 2
12.b even 2 1 672.2.q.d 2
21.g even 6 1 4704.2.a.y 1
21.h odd 6 1 672.2.q.h yes 2
21.h odd 6 1 4704.2.a.g 1
24.f even 2 1 1344.2.q.q 2
24.h odd 2 1 1344.2.q.e 2
28.g odd 6 1 2016.2.s.h 2
84.j odd 6 1 4704.2.a.j 1
84.n even 6 1 672.2.q.d 2
84.n even 6 1 4704.2.a.ba 1
168.s odd 6 1 1344.2.q.e 2
168.s odd 6 1 9408.2.a.cn 1
168.v even 6 1 1344.2.q.q 2
168.v even 6 1 9408.2.a.t 1
168.ba even 6 1 9408.2.a.x 1
168.be odd 6 1 9408.2.a.ci 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.q.d 2 12.b even 2 1
672.2.q.d 2 84.n even 6 1
672.2.q.h yes 2 3.b odd 2 1
672.2.q.h yes 2 21.h odd 6 1
1344.2.q.e 2 24.h odd 2 1
1344.2.q.e 2 168.s odd 6 1
1344.2.q.q 2 24.f even 2 1
1344.2.q.q 2 168.v even 6 1
2016.2.s.e 2 1.a even 1 1 trivial
2016.2.s.e 2 7.c even 3 1 inner
2016.2.s.h 2 4.b odd 2 1
2016.2.s.h 2 28.g odd 6 1
4704.2.a.g 1 21.h odd 6 1
4704.2.a.j 1 84.j odd 6 1
4704.2.a.y 1 21.g even 6 1
4704.2.a.ba 1 84.n even 6 1
9408.2.a.t 1 168.v even 6 1
9408.2.a.x 1 168.ba even 6 1
9408.2.a.ci 1 168.be odd 6 1
9408.2.a.cn 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 $$S_{2}^{\mathrm{new}}(2016, [\chi])$$:

 $$T_{5}$$ T5 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2} - 3T + 9$$
$37$ $$T^{2} - 9T + 81$$
$41$ $$(T + 2)^{2}$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} - 5T + 25$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 12T + 144$$
$97$ $$(T - 18)^{2}$$