# Properties

 Label 8016.2.a.n Level $8016$ Weight $2$ Character orbit 8016.a Self dual yes Analytic conductor $64.008$ Analytic rank $0$ 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: $$0$$ 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 4008) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 2 - \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 2 - \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{7} + q^{9} + ( 1 + \beta_{1} - \beta_{2} ) q^{11} + 2 q^{13} + ( 2 - \beta_{2} ) q^{15} + 3 \beta_{2} q^{17} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 2 - 2 \beta_{1} ) q^{21} + 8 q^{23} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{25} + q^{27} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 1 + \beta_{1} - \beta_{2} ) q^{33} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -4 + 2 \beta_{1} ) q^{37} + 2 q^{39} + ( 6 - 6 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{43} + ( 2 - \beta_{2} ) q^{45} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{47} + ( 5 - 4 \beta_{1} + 4 \beta_{2} ) q^{49} + 3 \beta_{2} q^{51} + ( -4 + 4 \beta_{1} + 3 \beta_{2} ) q^{53} + ( 6 - 4 \beta_{2} ) q^{55} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} ) q^{63} + ( 4 - 2 \beta_{2} ) q^{65} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{67} + 8 q^{69} + ( 4 + 4 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 8 - 2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{75} + ( -4 - 4 \beta_{2} ) q^{77} + ( -4 + 2 \beta_{1} - 5 \beta_{2} ) q^{79} + q^{81} + ( 4 - 8 \beta_{1} - 6 \beta_{2} ) q^{83} + ( -9 + 3 \beta_{1} + 9 \beta_{2} ) q^{85} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -6 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 4 - 4 \beta_{1} ) q^{91} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -8 + 4 \beta_{1} + 8 \beta_{2} ) q^{95} + ( 2 - 6 \beta_{2} ) q^{97} + ( 1 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + 6q^{5} + 4q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} + 6q^{5} + 4q^{7} + 3q^{9} + 4q^{11} + 6q^{13} + 6q^{15} - 4q^{19} + 4q^{21} + 24q^{23} + 5q^{25} + 3q^{27} - 14q^{29} + 4q^{31} + 4q^{33} + 4q^{35} - 10q^{37} + 6q^{39} + 12q^{41} + 8q^{43} + 6q^{45} + 11q^{49} - 8q^{53} + 18q^{55} - 4q^{57} - 4q^{59} + 2q^{61} + 4q^{63} + 12q^{65} + 26q^{67} + 24q^{69} + 16q^{71} + 22q^{73} + 5q^{75} - 12q^{77} - 10q^{79} + 3q^{81} + 4q^{83} - 24q^{85} - 14q^{87} - 6q^{89} + 8q^{91} + 4q^{93} - 20q^{95} + 6q^{97} + 4q^{99} + O(q^{100})$$

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
 −1.48119 2.17009 0.311108
0 1.00000 0 0.324869 0 4.96239 0 1.00000 0
1.2 0 1.00000 0 1.46081 0 −2.34017 0 1.00000 0
1.3 0 1.00000 0 4.21432 0 1.37778 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.n 3
4.b odd 2 1 4008.2.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.e 3 4.b odd 2 1
8016.2.a.n 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{3} - 6 T_{5}^{2} + 8 T_{5} - 2$$ $$T_{7}^{3} - 4 T_{7}^{2} - 8 T_{7} + 16$$ $$T_{11}^{3} - 4 T_{11}^{2} - 4 T_{11} + 20$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-2 + 8 T - 6 T^{2} + T^{3}$$
$7$ $$16 - 8 T - 4 T^{2} + T^{3}$$
$11$ $$20 - 4 T - 4 T^{2} + T^{3}$$
$13$ $$( -2 + T )^{3}$$
$17$ $$54 - 36 T + T^{3}$$
$19$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$23$ $$( -8 + T )^{3}$$
$29$ $$-152 + 28 T + 14 T^{2} + T^{3}$$
$31$ $$32 - 16 T - 4 T^{2} + T^{3}$$
$37$ $$-8 + 20 T + 10 T^{2} + T^{3}$$
$41$ $$918 - 72 T - 12 T^{2} + T^{3}$$
$43$ $$130 - 16 T - 8 T^{2} + T^{3}$$
$47$ $$268 - 84 T + T^{3}$$
$53$ $$-290 - 44 T + 8 T^{2} + T^{3}$$
$59$ $$32 - 32 T + 4 T^{2} + T^{3}$$
$61$ $$-4 - 132 T - 2 T^{2} + T^{3}$$
$67$ $$-554 + 212 T - 26 T^{2} + T^{3}$$
$71$ $$1040 - 64 T - 16 T^{2} + T^{3}$$
$73$ $$-104 + 100 T - 22 T^{2} + T^{3}$$
$79$ $$-278 - 100 T + 10 T^{2} + T^{3}$$
$83$ $$1424 - 256 T - 4 T^{2} + T^{3}$$
$89$ $$-920 - 148 T + 6 T^{2} + T^{3}$$
$97$ $$-152 - 132 T - 6 T^{2} + T^{3}$$