# Properties

 Label 8016.2.a.t 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: 5.5.149169.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2004) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + q^{9} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{13} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{15} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{17} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( \beta_{1} + \beta_{4} ) q^{21} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{23} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{25} + q^{27} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{29} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{31} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{33} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{35} + ( -4 - \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{39} + ( -2 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -2 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{43} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{45} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -4 + \beta_{2} - \beta_{3} ) q^{49} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{51} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{53} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{55} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{57} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{59} + ( -4 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{61} + ( \beta_{1} + \beta_{4} ) q^{63} + ( -5 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{65} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{67} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{69} + ( -5 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} ) q^{71} + ( -4 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{73} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{75} + ( -1 - \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{77} + ( 6 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{79} + q^{81} + ( 1 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{83} + ( 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{85} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{87} + ( -7 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{89} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{91} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{93} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{95} + ( -4 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{97} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 5q^{3} - 3q^{5} + 2q^{7} + 5q^{9} + O(q^{10})$$ $$5q + 5q^{3} - 3q^{5} + 2q^{7} + 5q^{9} - 3q^{11} - 4q^{13} - 3q^{15} - 7q^{17} + 2q^{19} + 2q^{21} - 13q^{23} - 2q^{25} + 5q^{27} - 3q^{29} + 12q^{31} - 3q^{33} - 10q^{35} - 7q^{37} - 4q^{39} - 16q^{41} - 3q^{45} - q^{47} - 17q^{49} - 7q^{51} + 3q^{53} + 23q^{55} + 2q^{57} - q^{59} - 22q^{61} + 2q^{63} - 20q^{65} - 2q^{67} - 13q^{69} - 9q^{71} - 28q^{73} - 2q^{75} - 10q^{77} + 28q^{79} + 5q^{81} - 7q^{83} - 11q^{85} - 3q^{87} - 30q^{89} + 13q^{91} + 12q^{93} - 3q^{95} - 33q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 6 x^{3} - 3 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4 \nu^{2} + \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{4} + 6 \beta_{3} + \beta_{2} + 3 \beta_{1} + 14$$

## 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
 0.752046 −0.228573 −1.25464 2.54970 −1.81853
0 1.00000 0 −3.18647 0 1.57076 0 1.00000 0
1.2 0 1.00000 0 −2.71918 0 2.29630 0 1.00000 0
1.3 0 1.00000 0 −0.171230 0 −2.92708 0 1.00000 0
1.4 0 1.00000 0 0.951283 0 1.28171 0 1.00000 0
1.5 0 1.00000 0 2.12560 0 −0.221696 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.t 5
4.b odd 2 1 2004.2.a.a 5
12.b even 2 1 6012.2.a.e 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.a 5 4.b odd 2 1
6012.2.a.e 5 12.b even 2 1
8016.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{5} + 3 T_{5}^{4} - 7 T_{5}^{3} - 16 T_{5}^{2} + 15 T_{5} + 3$$ $$T_{7}^{5} - 2 T_{7}^{4} - 7 T_{7}^{3} + 19 T_{7}^{2} - 9 T_{7} - 3$$ $$T_{11}^{5} + 3 T_{11}^{4} - 7 T_{11}^{3} - 16 T_{11}^{2} + 15 T_{11} + 3$$ $$T_{13}^{5} + 4 T_{13}^{4} - 13 T_{13}^{3} - 78 T_{13}^{2} - 94 T_{13} - 11$$