# Properties

 Label 287.2.h.b Level 287 Weight 2 Character orbit 287.h Analytic conductor 2.292 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 287.h (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/287\mathbb{Z}\right)^\times$$.

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 + 0.951057i
0.809017 2.48990i 2.61803 −3.92705 2.85317i −0.500000 0.363271i 2.11803 6.51864i 0.309017 + 0.951057i −6.04508 + 4.39201i 3.85410 −1.30902 + 0.951057i
78.1 −0.309017 + 0.224514i 0.381966 −0.572949 + 1.76336i −0.500000 + 1.53884i −0.118034 + 0.0857567i −0.809017 0.587785i −0.454915 1.40008i −2.85410 −0.190983 0.587785i
92.1 −0.309017 0.224514i 0.381966 −0.572949 1.76336i −0.500000 1.53884i −0.118034 0.0857567i −0.809017 + 0.587785i −0.454915 + 1.40008i −2.85410 −0.190983 + 0.587785i
141.1 0.809017 + 2.48990i 2.61803 −3.92705 + 2.85317i −0.500000 + 0.363271i 2.11803 + 6.51864i 0.309017 0.951057i −6.04508 4.39201i 3.85410 −1.30902 0.951057i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
41.d Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 6 T_{2}^{2} + 4 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$.