# Properties

 Label 287.2.a.f Level 287 Weight 2 Character orbit 287.a Self dual Yes Analytic conductor 2.292 Analytic rank 0 Dimension 6 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: $$6$$ Coefficient field: 6.6.185257757.1 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( \beta_{3} - \beta_{4} ) q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{6} - q^{7} + ( 1 - \beta_{1} - \beta_{2} ) q^{8} + ( 2 + \beta_{4} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{3} + ( 1 + \beta_{2} + \beta_{3} ) q^{4} + ( \beta_{3} - \beta_{4} ) q^{5} + ( -\beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{6} - q^{7} + ( 1 - \beta_{1} - \beta_{2} ) q^{8} + ( 2 + \beta_{4} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{10} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{12} + ( 1 + 2 \beta_{1} + \beta_{5} ) q^{13} + \beta_{1} q^{14} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + ( 1 + \beta_{4} + \beta_{5} ) q^{16} + ( 1 + \beta_{3} ) q^{17} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{18} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{19} + ( 3 - \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{20} + ( 1 - \beta_{2} ) q^{21} + ( 3 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{22} + ( 3 + \beta_{2} ) q^{23} + ( -5 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{24} + ( 4 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} + ( -6 - 3 \beta_{2} - \beta_{3} ) q^{26} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{27} + ( -1 - \beta_{2} - \beta_{3} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{29} + ( 6 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{30} + ( -4 - \beta_{3} - \beta_{4} ) q^{31} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{32} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{34} + ( -\beta_{3} + \beta_{4} ) q^{35} + ( 3 + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{36} + ( 4 + \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{37} + ( -3 - 2 \beta_{1} - 4 \beta_{3} - \beta_{5} ) q^{38} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{39} + ( 2 + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{40} + q^{41} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{42} + ( 3 - \beta_{3} - 2 \beta_{5} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{44} + ( -5 - 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{45} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{46} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{47} + ( -3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{48} + q^{49} + ( -8 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{50} + ( -3 + \beta_{1} - 2 \beta_{3} ) q^{51} + ( -3 + 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{52} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{53} + ( -5 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{54} + ( 2 - 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} ) q^{56} + ( 6 - 4 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{57} + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{58} + ( -2 + 3 \beta_{3} + \beta_{4} ) q^{59} + ( -15 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{60} + ( -1 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{61} + ( -1 + 5 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{62} + ( -2 - \beta_{4} ) q^{63} + ( -6 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{64} + ( -4 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{65} + ( -7 - 3 \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{66} + ( 4 + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{67} + ( 5 - \beta_{1} + 3 \beta_{3} + \beta_{5} ) q^{68} + ( 1 + 4 \beta_{2} + \beta_{4} ) q^{69} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{70} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{71} + ( 3 - 4 \beta_{1} - 5 \beta_{2} - \beta_{5} ) q^{72} + ( 9 - 3 \beta_{1} + \beta_{2} + \beta_{5} ) q^{73} + ( -3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{74} + ( -10 + \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 5 \beta_{5} ) q^{75} + ( 2 + 4 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{76} + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} ) q^{77} + ( -4 - \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{78} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{79} + ( -3 - 5 \beta_{1} - \beta_{2} + \beta_{5} ) q^{80} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{81} -\beta_{1} q^{82} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{83} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{84} + ( 4 + 3 \beta_{3} - \beta_{4} ) q^{85} + ( -1 - 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{86} + ( 7 - \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} ) q^{87} + ( 1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{88} + ( -6 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{89} + ( 12 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} ) q^{90} + ( -1 - 2 \beta_{1} - \beta_{5} ) q^{91} + ( 5 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{92} + ( 7 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{93} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{94} + ( 1 + 5 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{95} + ( -1 - 4 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{96} + ( 3 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{97} -\beta_{1} q^{98} + ( -2 + 6 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - q^{2} - 4q^{3} + 9q^{4} - q^{5} - 4q^{6} - 6q^{7} + 3q^{8} + 14q^{9} + O(q^{10})$$ $$6q - q^{2} - 4q^{3} + 9q^{4} - q^{5} - 4q^{6} - 6q^{7} + 3q^{8} + 14q^{9} + 10q^{10} + 6q^{11} + 9q^{12} + 7q^{13} + q^{14} - 13q^{15} + 7q^{16} + 7q^{17} - 5q^{18} + 2q^{19} + 11q^{20} + 4q^{21} + 15q^{22} + 20q^{23} - 36q^{24} + 29q^{25} - 43q^{26} + 2q^{27} - 9q^{28} - 9q^{29} + 13q^{30} - 27q^{31} - 10q^{32} + 17q^{33} + 6q^{34} + q^{35} + 29q^{36} + 19q^{37} - 23q^{38} + q^{39} + 23q^{40} + 6q^{41} + 4q^{42} + 19q^{43} + 21q^{44} - 35q^{45} - 8q^{46} - 19q^{47} - 9q^{48} + 6q^{49} - 58q^{50} - 19q^{51} + 5q^{53} - 37q^{54} + 3q^{55} - 3q^{56} + 37q^{57} + 13q^{58} - 7q^{59} - 110q^{60} - 12q^{61} - 14q^{63} - 37q^{64} - 13q^{65} - 54q^{66} + 27q^{67} + 31q^{68} + 16q^{69} - 10q^{70} - 6q^{71} + 5q^{72} + 52q^{73} - 14q^{74} - 46q^{75} + 13q^{76} - 6q^{77} - 45q^{78} - 26q^{80} - 22q^{81} - q^{82} + 12q^{83} - 9q^{84} + 25q^{85} - 10q^{86} + 42q^{87} - 2q^{88} - 38q^{89} + 93q^{90} - 7q^{91} + 45q^{92} + 33q^{93} - 8q^{94} + q^{95} - 12q^{96} + 8q^{97} - q^{98} + q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 10 x^{4} + 10 x^{3} + 23 x^{2} - 24 x + 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 5 \nu - 4$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 10 \nu^{3} + 2 \nu^{2} + 24 \nu - 10$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + \nu^{4} + 10 \nu^{3} - 8 \nu^{2} - 24 \nu + 13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{4} - 2 \beta_{3} + 8 \beta_{2} + 26 \beta_{1} - 6$$

## 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
 2.47904 2.05073 0.644787 0.306800 −2.01956 −2.46179
−2.47904 2.84004 4.14562 −3.76023 −7.04056 −1.00000 −5.31907 5.06582 9.32175
1.2 −2.05073 −1.62935 2.20548 4.18004 3.34135 −1.00000 −0.421375 −0.345215 −8.57212
1.3 −0.644787 −2.95586 −1.58425 −4.36552 1.90590 −1.00000 2.31108 5.73713 2.81483
1.4 −0.306800 −1.50512 −1.90587 0.333855 0.461772 −1.00000 1.19832 −0.734606 −0.102427
1.5 2.01956 1.86075 2.07864 −0.244521 3.75791 −1.00000 0.158810 0.462403 −0.493825
1.6 2.46179 −2.61045 4.06039 2.85638 −6.42638 −1.00000 5.07224 3.81447 7.03179
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(287))$$:

 $$T_{2}^{6} + T_{2}^{5} - 10 T_{2}^{4} - 10 T_{2}^{3} + 23 T_{2}^{2} + 24 T_{2} + 5$$ $$T_{3}^{6} + 4 T_{3}^{5} - 8 T_{3}^{4} - 46 T_{3}^{3} - 13 T_{3}^{2} + 111 T_{3} + 100$$