# Properties

 Label 2016.2.a.o Level $2016$ Weight $2$ Character orbit 2016.a Self dual yes Analytic conductor $16.098$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta q^{5} - q^{7} +O(q^{10})$$ $$q -2 \beta q^{5} - q^{7} -4 \beta q^{11} + ( 4 - 2 \beta ) q^{13} + ( 2 - 4 \beta ) q^{17} + ( -2 + 2 \beta ) q^{19} + 4 q^{23} + ( -1 + 4 \beta ) q^{25} + ( -2 + 4 \beta ) q^{29} + ( -4 + 4 \beta ) q^{31} + 2 \beta q^{35} + ( 2 - 4 \beta ) q^{37} + ( 2 + 4 \beta ) q^{41} -4 \beta q^{43} + ( 4 + 4 \beta ) q^{47} + q^{49} + 10 q^{53} + ( 8 + 8 \beta ) q^{55} + ( -6 - 2 \beta ) q^{59} + ( 8 + 2 \beta ) q^{61} + ( 4 - 4 \beta ) q^{65} -4 q^{67} + ( 8 - 8 \beta ) q^{71} + ( 10 - 8 \beta ) q^{73} + 4 \beta q^{77} -8 \beta q^{79} + ( -6 - 2 \beta ) q^{83} + ( 8 + 4 \beta ) q^{85} + 6 q^{89} + ( -4 + 2 \beta ) q^{91} -4 q^{95} + ( 6 + 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 2q^{7} + O(q^{10})$$ $$2q - 2q^{5} - 2q^{7} - 4q^{11} + 6q^{13} - 2q^{19} + 8q^{23} + 2q^{25} - 4q^{31} + 2q^{35} + 8q^{41} - 4q^{43} + 12q^{47} + 2q^{49} + 20q^{53} + 24q^{55} - 14q^{59} + 18q^{61} + 4q^{65} - 8q^{67} + 8q^{71} + 12q^{73} + 4q^{77} - 8q^{79} - 14q^{83} + 20q^{85} + 12q^{89} - 6q^{91} - 8q^{95} + 16q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 0 0 −3.23607 0 −1.00000 0 0 0
1.2 0 0 0 1.23607 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.o 2
3.b odd 2 1 224.2.a.d yes 2
4.b odd 2 1 2016.2.a.r 2
8.b even 2 1 4032.2.a.bv 2
8.d odd 2 1 4032.2.a.bw 2
12.b even 2 1 224.2.a.c 2
15.d odd 2 1 5600.2.a.z 2
21.c even 2 1 1568.2.a.k 2
21.g even 6 2 1568.2.i.w 4
21.h odd 6 2 1568.2.i.m 4
24.f even 2 1 448.2.a.j 2
24.h odd 2 1 448.2.a.i 2
48.i odd 4 2 1792.2.b.m 4
48.k even 4 2 1792.2.b.k 4
60.h even 2 1 5600.2.a.bk 2
84.h odd 2 1 1568.2.a.v 2
84.j odd 6 2 1568.2.i.n 4
84.n even 6 2 1568.2.i.v 4
168.e odd 2 1 3136.2.a.bf 2
168.i even 2 1 3136.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 12.b even 2 1
224.2.a.d yes 2 3.b odd 2 1
448.2.a.i 2 24.h odd 2 1
448.2.a.j 2 24.f even 2 1
1568.2.a.k 2 21.c even 2 1
1568.2.a.v 2 84.h odd 2 1
1568.2.i.m 4 21.h odd 6 2
1568.2.i.n 4 84.j odd 6 2
1568.2.i.v 4 84.n even 6 2
1568.2.i.w 4 21.g even 6 2
1792.2.b.k 4 48.k even 4 2
1792.2.b.m 4 48.i odd 4 2
2016.2.a.o 2 1.a even 1 1 trivial
2016.2.a.r 2 4.b odd 2 1
3136.2.a.bf 2 168.e odd 2 1
3136.2.a.by 2 168.i even 2 1
4032.2.a.bv 2 8.b even 2 1
4032.2.a.bw 2 8.d odd 2 1
5600.2.a.z 2 15.d odd 2 1
5600.2.a.bk 2 60.h even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} - 4$$ $$T_{11}^{2} + 4 T_{11} - 16$$ $$T_{13}^{2} - 6 T_{13} + 4$$ $$T_{17}^{2} - 20$$ $$T_{19}^{2} + 2 T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-16 + 4 T + T^{2}$$
$13$ $$4 - 6 T + T^{2}$$
$17$ $$-20 + T^{2}$$
$19$ $$-4 + 2 T + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$-20 + T^{2}$$
$41$ $$-4 - 8 T + T^{2}$$
$43$ $$-16 + 4 T + T^{2}$$
$47$ $$16 - 12 T + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$44 + 14 T + T^{2}$$
$61$ $$76 - 18 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-64 - 8 T + T^{2}$$
$73$ $$-44 - 12 T + T^{2}$$
$79$ $$-64 + 8 T + T^{2}$$
$83$ $$44 + 14 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$44 - 16 T + T^{2}$$