# Properties

 Label 287.2.a.b Level 287 Weight 2 Character orbit 287.a Self dual Yes Analytic conductor 2.292 Analytic rank 1 Dimension 2 CM No Inner twists 1

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## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 287.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$2.29170653801$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{3} + ( -1 + \beta ) q^{4} + ( 1 - \beta ) q^{5} - q^{6} - q^{7} + ( -1 + 2 \beta ) q^{8} + ( -1 - \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{2} + ( -1 + \beta ) q^{3} + ( -1 + \beta ) q^{4} + ( 1 - \beta ) q^{5} - q^{6} - q^{7} + ( -1 + 2 \beta ) q^{8} + ( -1 - \beta ) q^{9} + q^{10} - q^{11} + ( 2 - \beta ) q^{12} + ( -5 + 2 \beta ) q^{13} + \beta q^{14} + ( -2 + \beta ) q^{15} -3 \beta q^{16} + ( -3 + 2 \beta ) q^{17} + ( 1 + 2 \beta ) q^{18} + ( 1 - 3 \beta ) q^{19} + ( -2 + \beta ) q^{20} + ( 1 - \beta ) q^{21} + \beta q^{22} + ( -3 + \beta ) q^{23} + ( 3 - \beta ) q^{24} + ( -3 - \beta ) q^{25} + ( -2 + 3 \beta ) q^{26} + ( 3 - 4 \beta ) q^{27} + ( 1 - \beta ) q^{28} + ( -4 + 3 \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( 5 - 5 \beta ) q^{31} + ( 5 - \beta ) q^{32} + ( 1 - \beta ) q^{33} + ( -2 + \beta ) q^{34} + ( -1 + \beta ) q^{35} -\beta q^{36} + ( -4 - 2 \beta ) q^{37} + ( 3 + 2 \beta ) q^{38} + ( 7 - 5 \beta ) q^{39} + ( -3 + \beta ) q^{40} - q^{41} + q^{42} - q^{43} + ( 1 - \beta ) q^{44} + \beta q^{45} + ( -1 + 2 \beta ) q^{46} + ( 6 - 6 \beta ) q^{47} -3 q^{48} + q^{49} + ( 1 + 4 \beta ) q^{50} + ( 5 - 3 \beta ) q^{51} + ( 7 - 5 \beta ) q^{52} + ( 3 + \beta ) q^{53} + ( 4 + \beta ) q^{54} + ( -1 + \beta ) q^{55} + ( 1 - 2 \beta ) q^{56} + ( -4 + \beta ) q^{57} + ( -3 + \beta ) q^{58} + ( 9 + \beta ) q^{59} + ( 3 - 2 \beta ) q^{60} + ( 7 - 4 \beta ) q^{61} + 5 q^{62} + ( 1 + \beta ) q^{63} + ( 1 + 2 \beta ) q^{64} + ( -7 + 5 \beta ) q^{65} + q^{66} + ( -9 + 5 \beta ) q^{67} + ( 5 - 3 \beta ) q^{68} + ( 4 - 3 \beta ) q^{69} - q^{70} + ( -1 + 8 \beta ) q^{71} + ( -1 - 3 \beta ) q^{72} + ( -5 + 8 \beta ) q^{73} + ( 2 + 6 \beta ) q^{74} + ( 2 - 3 \beta ) q^{75} + ( -4 + \beta ) q^{76} + q^{77} + ( 5 - 2 \beta ) q^{78} + ( -8 + 6 \beta ) q^{79} + 3 q^{80} + ( -4 + 6 \beta ) q^{81} + \beta q^{82} + ( -3 + 4 \beta ) q^{83} + ( -2 + \beta ) q^{84} + ( -5 + 3 \beta ) q^{85} + \beta q^{86} + ( 7 - 4 \beta ) q^{87} + ( 1 - 2 \beta ) q^{88} + ( -5 + 7 \beta ) q^{89} + ( -1 - \beta ) q^{90} + ( 5 - 2 \beta ) q^{91} + ( 4 - 3 \beta ) q^{92} + ( -10 + 5 \beta ) q^{93} + 6 q^{94} + ( 4 - \beta ) q^{95} + ( -6 + 5 \beta ) q^{96} + ( -3 - 9 \beta ) q^{97} -\beta q^{98} + ( 1 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} + q^{5} - 2q^{6} - 2q^{7} - 3q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} + q^{5} - 2q^{6} - 2q^{7} - 3q^{9} + 2q^{10} - 2q^{11} + 3q^{12} - 8q^{13} + q^{14} - 3q^{15} - 3q^{16} - 4q^{17} + 4q^{18} - q^{19} - 3q^{20} + q^{21} + q^{22} - 5q^{23} + 5q^{24} - 7q^{25} - q^{26} + 2q^{27} + q^{28} - 5q^{29} - q^{30} + 5q^{31} + 9q^{32} + q^{33} - 3q^{34} - q^{35} - q^{36} - 10q^{37} + 8q^{38} + 9q^{39} - 5q^{40} - 2q^{41} + 2q^{42} - 2q^{43} + q^{44} + q^{45} + 6q^{47} - 6q^{48} + 2q^{49} + 6q^{50} + 7q^{51} + 9q^{52} + 7q^{53} + 9q^{54} - q^{55} - 7q^{57} - 5q^{58} + 19q^{59} + 4q^{60} + 10q^{61} + 10q^{62} + 3q^{63} + 4q^{64} - 9q^{65} + 2q^{66} - 13q^{67} + 7q^{68} + 5q^{69} - 2q^{70} + 6q^{71} - 5q^{72} - 2q^{73} + 10q^{74} + q^{75} - 7q^{76} + 2q^{77} + 8q^{78} - 10q^{79} + 6q^{80} - 2q^{81} + q^{82} - 2q^{83} - 3q^{84} - 7q^{85} + q^{86} + 10q^{87} - 3q^{89} - 3q^{90} + 8q^{91} + 5q^{92} - 15q^{93} + 12q^{94} + 7q^{95} - 7q^{96} - 15q^{97} - q^{98} + 3q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
−1.61803 0.618034 0.618034 −0.618034 −1.00000 −1.00000 2.23607 −2.61803 1.00000
1.2 0.618034 −1.61803 −1.61803 1.61803 −1.00000 −1.00000 −2.23607 −0.381966 1.00000
 $$n$$: 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
$$7$$ $$1$$
$$41$$ $$1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(287))$$:

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