# Properties

 Label 8016.2.a.l Level $8016$ Weight $2$ Character orbit 8016.a Self dual yes Analytic conductor $64.008$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8016 = 2^{4} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.0080822603$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1002) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 + \beta_{2} ) q^{5} + ( 2 - \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{11} + ( -3 + \beta_{1} - \beta_{2} ) q^{13} + ( -1 + \beta_{2} ) q^{15} + ( -5 - \beta_{1} - \beta_{2} ) q^{17} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{19} + ( 2 - \beta_{1} + \beta_{2} ) q^{21} + ( 1 + \beta_{1} - \beta_{2} ) q^{23} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{25} + q^{27} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{29} + ( 5 - 6 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{33} + ( 2 - \beta_{1} ) q^{35} + ( -1 - 2 \beta_{1} - 3 \beta_{2} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} ) q^{39} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{43} + ( -1 + \beta_{2} ) q^{45} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -5 - \beta_{1} - \beta_{2} ) q^{51} + ( -6 + 3 \beta_{1} + 4 \beta_{2} ) q^{53} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{55} + ( 1 - \beta_{1} - 3 \beta_{2} ) q^{57} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{59} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} ) q^{63} + ( -1 + \beta_{1} - \beta_{2} ) q^{65} + ( -3 + 6 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} ) q^{69} + ( 1 + \beta_{1} - 3 \beta_{2} ) q^{71} + ( 1 + 3 \beta_{1} - 5 \beta_{2} ) q^{73} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{75} + ( -7 + 5 \beta_{1} - 3 \beta_{2} ) q^{77} + ( -10 - 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( -2 + 3 \beta_{1} ) q^{83} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{85} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{87} + ( -8 - 3 \beta_{1} - 5 \beta_{2} ) q^{89} + ( -13 + 7 \beta_{1} - 5 \beta_{2} ) q^{91} + ( 5 - 6 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -9 + 3 \beta_{1} + 7 \beta_{2} ) q^{95} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{97} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 3q^{5} + 5q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 3q^{5} + 5q^{7} + 3q^{9} - 8q^{13} - 3q^{15} - 16q^{17} + 2q^{19} + 5q^{21} + 4q^{23} - 4q^{25} + 3q^{27} + 9q^{31} + 5q^{35} - 5q^{37} - 8q^{39} - 14q^{41} + 2q^{43} - 3q^{45} + 3q^{47} + 6q^{49} - 16q^{51} - 15q^{53} + 2q^{55} + 2q^{57} + 5q^{59} - 4q^{61} + 5q^{63} - 2q^{65} - 3q^{67} + 4q^{69} + 4q^{71} + 6q^{73} - 4q^{75} - 16q^{77} - 32q^{79} + 3q^{81} - 3q^{83} + 10q^{85} - 27q^{89} - 32q^{91} + 9q^{93} - 24q^{95} + 5q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 0.311108 2.17009 −1.48119
0 1.00000 0 −3.21432 0 −0.525428 0 1.00000 0
1.2 0 1.00000 0 −0.460811 0 0.369102 0 1.00000 0
1.3 0 1.00000 0 0.675131 0 5.15633 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$167$$ $$1$$

## 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 8016.2.a.l 3
4.b odd 2 1 1002.2.a.h 3
12.b even 2 1 3006.2.a.o 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.h 3 4.b odd 2 1
3006.2.a.o 3 12.b even 2 1
8016.2.a.l 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{3} + 3 T_{5}^{2} - T_{5} - 1$$ $$T_{7}^{3} - 5 T_{7}^{2} - T_{7} + 1$$ $$T_{11}^{3} - 28 T_{11} - 52$$ $$T_{13}^{3} + 8 T_{13}^{2} + 12 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-1 - T + 3 T^{2} + T^{3}$$
$7$ $$1 - T - 5 T^{2} + T^{3}$$
$11$ $$-52 - 28 T + T^{3}$$
$13$ $$4 + 12 T + 8 T^{2} + T^{3}$$
$17$ $$124 + 80 T + 16 T^{2} + T^{3}$$
$19$ $$-52 - 32 T - 2 T^{2} + T^{3}$$
$23$ $$20 - 4 T - 4 T^{2} + T^{3}$$
$29$ $$-52 - 28 T + T^{3}$$
$31$ $$725 - 85 T - 9 T^{2} + T^{3}$$
$37$ $$-107 - 29 T + 5 T^{2} + T^{3}$$
$41$ $$-152 + 28 T + 14 T^{2} + T^{3}$$
$43$ $$20 - 44 T - 2 T^{2} + T^{3}$$
$47$ $$351 - 81 T - 3 T^{2} + T^{3}$$
$53$ $$-139 + 5 T + 15 T^{2} + T^{3}$$
$59$ $$-25 - 57 T - 5 T^{2} + T^{3}$$
$61$ $$-64 - 80 T + 4 T^{2} + T^{3}$$
$67$ $$-621 - 117 T + 3 T^{2} + T^{3}$$
$71$ $$68 - 40 T - 4 T^{2} + T^{3}$$
$73$ $$740 - 148 T - 6 T^{2} + T^{3}$$
$79$ $$992 + 320 T + 32 T^{2} + T^{3}$$
$83$ $$-31 - 27 T + 3 T^{2} + T^{3}$$
$89$ $$-439 + 143 T + 27 T^{2} + T^{3}$$
$97$ $$61 - 37 T - 5 T^{2} + T^{3}$$