# Properties

 Label 8016.2.a.s Level 8016 Weight 2 Character orbit 8016.a Self dual yes Analytic conductor 64.008 Analytic rank 1 Dimension 5 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: $$5$$ Coefficient field: 5.5.284897.1 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -\beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + q^{9} + ( \beta_{3} + \beta_{4} ) q^{11} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{13} + ( \beta_{3} - \beta_{4} ) q^{15} + ( -3 + 2 \beta_{3} ) q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} + ( -1 - \beta_{2} + \beta_{3} ) q^{21} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( -1 + 3 \beta_{1} + \beta_{3} ) q^{25} - q^{27} + ( 4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{31} + ( -\beta_{3} - \beta_{4} ) q^{33} + ( 2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{35} + ( -1 - 3 \beta_{2} ) q^{37} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{39} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{41} + ( 4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{43} + ( -\beta_{3} + \beta_{4} ) q^{45} + ( 1 - \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{47} + ( -2 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{49} + ( 3 - 2 \beta_{3} ) q^{51} + ( -2 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{53} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{55} + ( -\beta_{1} + \beta_{2} ) q^{57} + ( -2 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{59} + ( -4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( 1 + \beta_{2} - \beta_{3} ) q^{63} + ( -3 - \beta_{1} + 3 \beta_{3} - 4 \beta_{4} ) q^{65} + ( 4 - \beta_{1} + \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{67} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} ) q^{71} + ( -5 + 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{73} + ( 1 - 3 \beta_{1} - \beta_{3} ) q^{75} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{77} + ( 2 + \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( 2 + 2 \beta_{1} + 5 \beta_{2} - \beta_{3} + \beta_{4} ) q^{83} + ( -4 - 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{85} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{87} + ( -8 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{89} + ( -1 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{91} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{93} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{95} + ( -6 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{97} + ( \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 5q^{3} + q^{5} + 4q^{7} + 5q^{9} + O(q^{10})$$ $$5q - 5q^{3} + q^{5} + 4q^{7} + 5q^{9} + 3q^{11} - 14q^{13} - q^{15} - 13q^{17} + 2q^{19} - 4q^{21} + 5q^{23} + 2q^{25} - 5q^{27} + 13q^{29} - 2q^{31} - 3q^{33} + 12q^{35} - 5q^{37} + 14q^{39} - 20q^{41} + 20q^{43} + q^{45} - q^{47} - 9q^{49} + 13q^{51} - 3q^{53} + 3q^{55} - 2q^{57} + q^{59} - 34q^{61} + 4q^{63} - 22q^{65} + 16q^{67} - 5q^{69} + 5q^{71} - 12q^{73} - 2q^{75} - 8q^{77} + 20q^{79} + 5q^{81} + 15q^{83} - 27q^{85} - 13q^{87} - 48q^{89} - 7q^{91} + 2q^{93} + 5q^{95} - 21q^{97} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 6 x^{3} + 5 x^{2} + 10 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 3 \nu^{3} - 3 \nu^{2} + 8 \nu + 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 9 \nu + 6$$ $$\beta_{4}$$ $$=$$ $$2 \nu^{4} - 5 \nu^{3} - 9 \nu^{2} + 14 \nu + 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - 3 \beta_{3} + \beta_{2} + 5 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{4} - 12 \beta_{3} + 7 \beta_{2} + 10 \beta_{1} + 18$$

## 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
 1.80442 −0.759623 −1.65065 −0.435916 3.04177
0 −1.00000 0 −3.40664 0 −0.548479 0 1.00000 0
1.2 0 −1.00000 0 −0.473647 0 −0.663349 0 1.00000 0
1.3 0 −1.00000 0 −0.457884 0 2.37531 0 1.00000 0
1.4 0 −1.00000 0 2.07208 0 −1.37406 0 1.00000 0
1.5 0 −1.00000 0 3.26608 0 4.21058 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.s 5
4.b odd 2 1 4008.2.a.f 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.f 5 4.b odd 2 1
8016.2.a.s 5 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$167$$ $$1$$

## Hecke kernels

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

 $$T_{5}^{5} - T_{5}^{4} - 13 T_{5}^{3} + 12 T_{5}^{2} + 19 T_{5} + 5$$ $$T_{7}^{5} - 4 T_{7}^{4} - 5 T_{7}^{3} + 13 T_{7}^{2} + 17 T_{7} + 5$$ $$T_{11}^{5} - 3 T_{11}^{4} - 19 T_{11}^{3} + 30 T_{11}^{2} + 47 T_{11} - 55$$ $$T_{13}^{5} + 14 T_{13}^{4} + 61 T_{13}^{3} + 62 T_{13}^{2} - 132 T_{13} - 215$$