# Properties

 Label 2016.2.s.l 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + 3*z * q^5 + (-3*z + 1) * q^7 $$q + 3 \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 1) q^{7} + ( - 5 \zeta_{6} + 5) q^{11} + 2 q^{13} + ( - 2 \zeta_{6} + 2) q^{17} - 6 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - q^{29} + (\zeta_{6} - 1) q^{31} + ( - 6 \zeta_{6} + 9) q^{35} - 10 \zeta_{6} q^{37} - 4 q^{41} - 4 q^{43} - 8 \zeta_{6} q^{47} + (3 \zeta_{6} - 8) q^{49} + ( - 5 \zeta_{6} + 5) q^{53} + 15 q^{55} + ( - 13 \zeta_{6} + 13) q^{59} + 8 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + (14 \zeta_{6} - 14) q^{67} + 12 q^{71} + ( - 6 \zeta_{6} + 6) q^{73} + ( - 5 \zeta_{6} - 10) q^{77} + 11 \zeta_{6} q^{79} + 7 q^{83} + 6 q^{85} - 6 \zeta_{6} q^{89} + ( - 6 \zeta_{6} + 2) q^{91} + ( - 18 \zeta_{6} + 18) q^{95} + 19 q^{97} +O(q^{100})$$ q + 3*z * q^5 + (-3*z + 1) * q^7 + (-5*z + 5) * q^11 + 2 * q^13 + (-2*z + 2) * q^17 - 6*z * q^19 + 2*z * q^23 + (4*z - 4) * q^25 - q^29 + (z - 1) * q^31 + (-6*z + 9) * q^35 - 10*z * q^37 - 4 * q^41 - 4 * q^43 - 8*z * q^47 + (3*z - 8) * q^49 + (-5*z + 5) * q^53 + 15 * q^55 + (-13*z + 13) * q^59 + 8*z * q^61 + 6*z * q^65 + (14*z - 14) * q^67 + 12 * q^71 + (-6*z + 6) * q^73 + (-5*z - 10) * q^77 + 11*z * q^79 + 7 * q^83 + 6 * q^85 - 6*z * q^89 + (-6*z + 2) * q^91 + (-18*z + 18) * q^95 + 19 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5} - q^{7}+O(q^{10})$$ 2 * q + 3 * q^5 - q^7 $$2 q + 3 q^{5} - q^{7} + 5 q^{11} + 4 q^{13} + 2 q^{17} - 6 q^{19} + 2 q^{23} - 4 q^{25} - 2 q^{29} - q^{31} + 12 q^{35} - 10 q^{37} - 8 q^{41} - 8 q^{43} - 8 q^{47} - 13 q^{49} + 5 q^{53} + 30 q^{55} + 13 q^{59} + 8 q^{61} + 6 q^{65} - 14 q^{67} + 24 q^{71} + 6 q^{73} - 25 q^{77} + 11 q^{79} + 14 q^{83} + 12 q^{85} - 6 q^{89} - 2 q^{91} + 18 q^{95} + 38 q^{97}+O(q^{100})$$ 2 * q + 3 * q^5 - q^7 + 5 * q^11 + 4 * q^13 + 2 * q^17 - 6 * q^19 + 2 * q^23 - 4 * q^25 - 2 * q^29 - q^31 + 12 * q^35 - 10 * q^37 - 8 * q^41 - 8 * q^43 - 8 * q^47 - 13 * q^49 + 5 * q^53 + 30 * q^55 + 13 * q^59 + 8 * q^61 + 6 * q^65 - 14 * q^67 + 24 * q^71 + 6 * q^73 - 25 * q^77 + 11 * q^79 + 14 * q^83 + 12 * q^85 - 6 * q^89 - 2 * q^91 + 18 * q^95 + 38 * 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 1.50000 2.59808i 0 −0.500000 + 2.59808i 0 0 0
865.1 0 0 0 1.50000 + 2.59808i 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.l yes 2
3.b odd 2 1 2016.2.s.c 2
4.b odd 2 1 2016.2.s.m yes 2
7.c even 3 1 inner 2016.2.s.l yes 2
12.b even 2 1 2016.2.s.d yes 2
21.h odd 6 1 2016.2.s.c 2
28.g odd 6 1 2016.2.s.m yes 2
84.n even 6 1 2016.2.s.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.s.c 2 3.b odd 2 1
2016.2.s.c 2 21.h odd 6 1
2016.2.s.d yes 2 12.b even 2 1
2016.2.s.d yes 2 84.n even 6 1
2016.2.s.l yes 2 1.a even 1 1 trivial
2016.2.s.l yes 2 7.c even 3 1 inner
2016.2.s.m yes 2 4.b odd 2 1
2016.2.s.m yes 2 28.g odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} - 5T_{11} + 25$$ T11^2 - 5*T11 + 25 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} + 6T + 36$$
$23$ $$T^{2} - 2T + 4$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + T + 1$$
$37$ $$T^{2} + 10T + 100$$
$41$ $$(T + 4)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} - 5T + 25$$
$59$ $$T^{2} - 13T + 169$$
$61$ $$T^{2} - 8T + 64$$
$67$ $$T^{2} + 14T + 196$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$(T - 7)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T - 19)^{2}$$