# Properties

 Label 8016.2.a.x Level 8016 Weight 2 Character orbit 8016.a Self dual Yes Analytic conductor 64.008 Analytic rank 1 Dimension 8 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} - 8 x^{6} + 28 x^{5} + 9 x^{4} - 64 x^{3} + 17 x^{2} + 23 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{7} + 38 \nu^{5} + 2 \nu^{4} - 147 \nu^{3} - 11 \nu^{2} + 168 \nu + 15$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 15 \nu^{5} + 3 \nu^{4} - 70 \nu^{3} - 20 \nu^{2} + 98 \nu + 19$$$$)/7$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} - 7 \nu^{6} - 16 \nu^{5} + 64 \nu^{4} + 21 \nu^{3} - 135 \nu^{2} + 21 \nu + 25$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 7 \nu^{6} + 24 \nu^{5} - 61 \nu^{4} - 28 \nu^{3} + 122 \nu^{2} - 42 \nu - 27$$$$)/7$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 18 \nu^{4} - 25 \nu^{3} + 37 \nu^{2} + 8 \nu - 5$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} - 7 \nu^{6} - 39 \nu^{5} + 58 \nu^{4} + 98 \nu^{3} - 95 \nu^{2} - 56 \nu - 13$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{5} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{4} - \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$6 \beta_{7} + 7 \beta_{5} + \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - \beta_{1} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 9 \beta_{6} + 8 \beta_{4} + 2 \beta_{3} - 9 \beta_{2} + 28 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$37 \beta_{7} + \beta_{6} + 45 \beta_{5} + 8 \beta_{4} + 57 \beta_{3} - 21 \beta_{2} - 8 \beta_{1} + 84$$ $$\nu^{7}$$ $$=$$ $$13 \beta_{7} + 65 \beta_{6} + \beta_{5} + 53 \beta_{4} + 27 \beta_{3} - 71 \beta_{2} + 165 \beta_{1} + 4$$

## 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
 −0.459587 2.63734 1.60389 −2.45154 −0.0452510 −1.71120 2.18779 1.23856
0 −1.00000 0 −2.63865 0 0.604857 0 1.00000 0
1.2 0 −1.00000 0 −1.29727 0 2.24503 0 1.00000 0
1.3 0 −1.00000 0 −0.122001 0 −3.47013 0 1.00000 0
1.4 0 −1.00000 0 1.48532 0 5.10210 0 1.00000 0
1.5 0 −1.00000 0 1.52778 0 −0.428515 0 1.00000 0
1.6 0 −1.00000 0 2.45529 0 −2.43610 0 1.00000 0
1.7 0 −1.00000 0 2.52133 0 0.307143 0 1.00000 0
1.8 0 −1.00000 0 3.06821 0 2.07562 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{5}^{8} - \cdots$$ $$T_{7}^{8} - \cdots$$ $$T_{11}^{8} + \cdots$$ $$T_{13}^{8} - 37 T_{13}^{6} - 52 T_{13}^{5} + 320 T_{13}^{4} + 973 T_{13}^{3} + 994 T_{13}^{2} + 412 T_{13} + 56$$