# Properties

 Label 8015.2.a.g Level 8015 Weight 2 Character orbit 8015.a Self dual Yes Analytic conductor 64.000 Analytic rank 1 Dimension 3 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8015 = 5 \cdot 7 \cdot 229$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8015.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$64.0000972201$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( -3 - \beta_{2} ) q^{6} - q^{7} + ( 1 + \beta_{2} ) q^{8} + ( 4 - 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( -3 - \beta_{2} ) q^{6} - q^{7} + ( 1 + \beta_{2} ) q^{8} + ( 4 - 2 \beta_{1} ) q^{9} -\beta_{1} q^{10} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} + ( 1 - 2 \beta_{1} - 2 \beta_{2} ) q^{12} + ( -2 + 3 \beta_{1} + 2 \beta_{2} ) q^{13} -\beta_{1} q^{14} + ( \beta_{1} - \beta_{2} ) q^{15} + ( -1 - 2 \beta_{2} ) q^{16} + ( -2 + 2 \beta_{1} ) q^{17} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{18} + ( -\beta_{1} - \beta_{2} ) q^{20} + ( \beta_{1} - \beta_{2} ) q^{21} + ( -3 - \beta_{1} - \beta_{2} ) q^{22} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{23} + ( 4 - 3 \beta_{1} ) q^{24} + q^{25} + ( 4 + 3 \beta_{1} + 3 \beta_{2} ) q^{26} + ( 6 - \beta_{1} + 3 \beta_{2} ) q^{27} + ( -\beta_{1} - \beta_{2} ) q^{28} + ( 3 + \beta_{1} - \beta_{2} ) q^{29} + ( 3 + \beta_{2} ) q^{30} + ( 1 + 3 \beta_{1} - 3 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( 7 - \beta_{1} - \beta_{2} ) q^{33} + ( 4 + 2 \beta_{2} ) q^{34} + q^{35} + ( -2 + 2 \beta_{2} ) q^{36} + ( -3 + 3 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -1 - 2 \beta_{1} - 7 \beta_{2} ) q^{39} + ( -1 - \beta_{2} ) q^{40} + ( -9 + \beta_{1} - \beta_{2} ) q^{41} + ( 3 + \beta_{2} ) q^{42} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 1 - 3 \beta_{1} - 3 \beta_{2} ) q^{44} + ( -4 + 2 \beta_{1} ) q^{45} + ( -5 + \beta_{1} - \beta_{2} ) q^{46} + ( -3 + \beta_{1} + \beta_{2} ) q^{47} + ( -8 + 5 \beta_{1} + \beta_{2} ) q^{48} + q^{49} + \beta_{1} q^{50} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{51} + ( 7 + 4 \beta_{1} - \beta_{2} ) q^{52} + ( 1 - \beta_{1} - \beta_{2} ) q^{53} + ( -5 + 8 \beta_{1} - \beta_{2} ) q^{54} + ( 1 + \beta_{1} - \beta_{2} ) q^{55} + ( -1 - \beta_{2} ) q^{56} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{58} + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{60} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{61} + ( 9 + \beta_{1} + 3 \beta_{2} ) q^{62} + ( -4 + 2 \beta_{1} ) q^{63} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{64} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -1 + 5 \beta_{1} - \beta_{2} ) q^{66} + ( -5 + 3 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 2 + 2 \beta_{1} ) q^{68} + ( 15 - 5 \beta_{1} - 3 \beta_{2} ) q^{69} + \beta_{1} q^{70} + ( 5 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 6 - 4 \beta_{1} + 4 \beta_{2} ) q^{72} + ( -4 - 3 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{74} + ( -\beta_{1} + \beta_{2} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} ) q^{77} + ( 3 - 10 \beta_{1} - 2 \beta_{2} ) q^{78} + ( 3 - 4 \beta_{1} - 3 \beta_{2} ) q^{79} + ( 1 + 2 \beta_{2} ) q^{80} + ( 3 - 6 \beta_{1} + 4 \beta_{2} ) q^{81} + ( 3 - 9 \beta_{1} + \beta_{2} ) q^{82} + ( -2 - \beta_{1} - \beta_{2} ) q^{83} + ( -1 + 2 \beta_{1} + 2 \beta_{2} ) q^{84} + ( 2 - 2 \beta_{1} ) q^{85} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( -7 - \beta_{1} + 3 \beta_{2} ) q^{87} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{88} + ( -6 \beta_{1} - 8 \beta_{2} ) q^{89} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{90} + ( 2 - 3 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 5 - 3 \beta_{1} - 5 \beta_{2} ) q^{92} + ( -21 + 5 \beta_{1} + \beta_{2} ) q^{93} + ( 1 - \beta_{1} + \beta_{2} ) q^{94} + ( 1 + 4 \beta_{1} + 5 \beta_{2} ) q^{96} + ( -8 - 4 \beta_{2} ) q^{97} + \beta_{1} q^{98} + ( 2 - 2 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} - q^{3} + q^{4} - 3q^{5} - 9q^{6} - 3q^{7} + 3q^{8} + 10q^{9} + O(q^{10})$$ $$3q + q^{2} - q^{3} + q^{4} - 3q^{5} - 9q^{6} - 3q^{7} + 3q^{8} + 10q^{9} - q^{10} - 4q^{11} + q^{12} - 3q^{13} - q^{14} + q^{15} - 3q^{16} - 4q^{17} - 10q^{18} - q^{20} + q^{21} - 10q^{22} - 4q^{23} + 9q^{24} + 3q^{25} + 15q^{26} + 17q^{27} - q^{28} + 10q^{29} + 9q^{30} + 6q^{31} - 3q^{32} + 20q^{33} + 12q^{34} + 3q^{35} - 6q^{36} - 6q^{37} - 5q^{39} - 3q^{40} - 26q^{41} + 9q^{42} + 16q^{43} - 10q^{45} - 14q^{46} - 8q^{47} - 19q^{48} + 3q^{49} + q^{50} - 16q^{51} + 25q^{52} + 2q^{53} - 7q^{54} + 4q^{55} - 3q^{56} + 12q^{58} - 15q^{59} - q^{60} + q^{61} + 28q^{62} - 10q^{63} - 11q^{64} + 3q^{65} + 2q^{66} - 12q^{67} + 8q^{68} + 40q^{69} + q^{70} + 18q^{71} + 14q^{72} - 15q^{73} + 12q^{74} - q^{75} + 4q^{77} - q^{78} + 5q^{79} + 3q^{80} + 3q^{81} - 7q^{83} - q^{84} + 4q^{85} - 4q^{86} - 22q^{87} + 6q^{88} - 6q^{89} + 10q^{90} + 3q^{91} + 12q^{92} - 58q^{93} + 2q^{94} + 7q^{96} - 24q^{97} + q^{98} + 4q^{99} + O(q^{100})$$

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 0.311108 2.17009
−1.48119 3.15633 0.193937 −1.00000 −4.67513 −1.00000 2.67513 6.96239 1.48119
1.2 0.311108 −2.52543 −1.90321 −1.00000 −0.785680 −1.00000 −1.21432 3.37778 −0.311108
1.3 2.17009 −1.63090 2.70928 −1.00000 −3.53919 −1.00000 1.53919 −0.340173 −2.17009
 $$n$$: e.g. 2-40 or 990-1000 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
$$5$$ $$1$$
$$7$$ $$1$$
$$229$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8015))$$:

 $$T_{2}^{3} - T_{2}^{2} - 3 T_{2} + 1$$ $$T_{3}^{3} + T_{3}^{2} - 9 T_{3} - 13$$