# Properties

 Label 2016.2.a.k Level $2016$ Weight $2$ Character orbit 2016.a Self dual yes Analytic conductor $16.098$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.0978410475$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - q^{7} + O(q^{10})$$ $$q + 2 q^{5} - q^{7} + 4 q^{11} - 6 q^{13} + 2 q^{17} + 4 q^{19} + 4 q^{23} - q^{25} + 2 q^{29} + 8 q^{31} - 2 q^{35} - 10 q^{37} + 2 q^{41} + 8 q^{43} + q^{49} + 10 q^{53} + 8 q^{55} + 12 q^{59} + 10 q^{61} - 12 q^{65} - 8 q^{67} - 12 q^{71} + 2 q^{73} - 4 q^{77} - 12 q^{83} + 4 q^{85} - 6 q^{89} + 6 q^{91} + 8 q^{95} + 2 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 2.00000 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.a.k 1
3.b odd 2 1 672.2.a.f yes 1
4.b odd 2 1 2016.2.a.l 1
8.b even 2 1 4032.2.a.f 1
8.d odd 2 1 4032.2.a.n 1
12.b even 2 1 672.2.a.b 1
21.c even 2 1 4704.2.a.m 1
24.f even 2 1 1344.2.a.r 1
24.h odd 2 1 1344.2.a.h 1
48.i odd 4 2 5376.2.c.s 2
48.k even 4 2 5376.2.c.m 2
84.h odd 2 1 4704.2.a.be 1
168.e odd 2 1 9408.2.a.g 1
168.i even 2 1 9408.2.a.cb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.b 1 12.b even 2 1
672.2.a.f yes 1 3.b odd 2 1
1344.2.a.h 1 24.h odd 2 1
1344.2.a.r 1 24.f even 2 1
2016.2.a.k 1 1.a even 1 1 trivial
2016.2.a.l 1 4.b odd 2 1
4032.2.a.f 1 8.b even 2 1
4032.2.a.n 1 8.d odd 2 1
4704.2.a.m 1 21.c even 2 1
4704.2.a.be 1 84.h odd 2 1
5376.2.c.m 2 48.k even 4 2
5376.2.c.s 2 48.i odd 4 2
9408.2.a.g 1 168.e odd 2 1
9408.2.a.cb 1 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2016))$$:

 $$T_{5} - 2$$ $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{17} - 2$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-2 + T$$
$7$ $$1 + T$$
$11$ $$-4 + T$$
$13$ $$6 + T$$
$17$ $$-2 + T$$
$19$ $$-4 + T$$
$23$ $$-4 + T$$
$29$ $$-2 + T$$
$31$ $$-8 + T$$
$37$ $$10 + T$$
$41$ $$-2 + T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$-10 + T$$
$59$ $$-12 + T$$
$61$ $$-10 + T$$
$67$ $$8 + T$$
$71$ $$12 + T$$
$73$ $$-2 + T$$
$79$ $$T$$
$83$ $$12 + T$$
$89$ $$6 + T$$
$97$ $$-2 + T$$