# Properties

 Label 287.2.a.e Level 287 Weight 2 Character orbit 287.a Self dual Yes Analytic conductor 2.292 Analytic rank 0 Dimension 5 CM No Inner twists 1

# Related objects

## 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.633117.1 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

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

## 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
 2.45719 1.20098 0.460315 −1.08727 −2.03121
−2.45719 −1.45719 4.03778 −2.26685 3.58059 1.00000 −5.00722 −0.876597 5.57007
1.2 −1.20098 −0.200978 −0.557652 −3.21704 0.241370 1.00000 3.07168 −2.95961 3.86360
1.3 −0.460315 0.539685 −1.78811 4.10136 −0.248425 1.00000 1.74372 −2.70874 −1.88791
1.4 1.08727 2.08727 −0.817843 0.209668 2.26943 1.00000 −3.06376 1.35670 0.227965
1.5 2.03121 3.03121 2.12582 −3.82713 6.15703 1.00000 0.255573 6.18825 −7.77372
 $$n$$: 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.

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(287))$$:

 $$T_{2}^{5} + T_{2}^{4} - 6 T_{2}^{3} - 4 T_{2}^{2} + 6 T_{2} + 3$$ $$T_{3}^{5} - 4 T_{3}^{4} + 10 T_{3}^{2} - 3 T_{3} - 1$$