# Properties

 Label 287.2.a.d Level 287 Weight 2 Character orbit 287.a Self dual Yes Analytic conductor 2.292 Analytic rank 0 Dimension 3 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: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ 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 root $$\beta$$ of the polynomial $$x^{3} - x^{2} - 2 x + 1$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.24698 0.445042 1.80194
−0.246980 3.24698 −1.93900 0.890084 −0.801938 −1.00000 0.972853 7.54288 −0.219833
1.2 1.44504 1.55496 0.0881460 3.60388 2.24698 −1.00000 −2.76271 −0.582105 5.20775
1.3 2.80194 0.198062 5.85086 −2.49396 0.554958 −1.00000 10.7899 −2.96077 −6.98792
 $$n$$: 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(287))$$:

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