# Properties

 Label 287.2.h.a 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + 2 \zeta_{10}^{3} q^{4} -\zeta_{10}^{3} q^{5} + \zeta_{10} q^{7} + 2 q^{9} +O(q^{10})$$ $$q + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + 2 \zeta_{10}^{3} q^{4} -\zeta_{10}^{3} q^{5} + \zeta_{10} q^{7} + 2 q^{9} + ( -\zeta_{10} - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{11} + ( 4 - 4 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{12} + ( -5 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{13} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{15} -4 \zeta_{10} q^{16} + ( -5 \zeta_{10} + 2 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{17} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{19} + 2 \zeta_{10} q^{20} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{21} + ( 2 - 5 \zeta_{10} + 5 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} + 4 \zeta_{10} q^{25} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{27} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{28} + ( -3 + 3 \zeta_{10} + 2 \zeta_{10}^{3} ) q^{29} + ( -3 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{31} + ( 5 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{33} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{35} + 4 \zeta_{10}^{3} q^{36} + ( 4 - 4 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{37} + ( 9 - \zeta_{10} + \zeta_{10}^{2} - 9 \zeta_{10}^{3} ) q^{39} + ( -6 + 7 \zeta_{10} - 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{41} + ( -5 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{43} + ( 6 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{44} -2 \zeta_{10}^{3} q^{45} + ( 4 - 3 \zeta_{10} + 3 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{47} + ( 8 - 4 \zeta_{10} + 8 \zeta_{10}^{2} ) q^{48} + \zeta_{10}^{2} q^{49} + ( \zeta_{10} + 12 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{51} + ( -4 \zeta_{10} - 6 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{52} + ( -4 + 4 \zeta_{10} ) q^{53} + ( -3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{55} + ( 6 + 7 \zeta_{10} + 6 \zeta_{10}^{2} ) q^{57} + ( -10 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{59} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{60} + ( -3 + \zeta_{10} - 3 \zeta_{10}^{2} ) q^{61} + 2 \zeta_{10} q^{63} + ( 8 - 8 \zeta_{10} + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{64} + ( 2 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{65} + 8 \zeta_{10}^{3} q^{67} + ( 6 + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{68} + ( 4 - 11 \zeta_{10} + 11 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69} + ( -10 \zeta_{10} + 2 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{71} + ( 9 + 7 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{73} + ( -8 + 4 \zeta_{10} - 8 \zeta_{10}^{2} ) q^{75} + ( 10 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{76} + ( 1 - \zeta_{10} - 3 \zeta_{10}^{3} ) q^{77} + ( -3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{79} + ( -4 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{80} -11 q^{81} + 7 q^{83} + ( 2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{84} + ( -3 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{85} + ( 1 - \zeta_{10} - 8 \zeta_{10}^{3} ) q^{87} + ( -11 + 4 \zeta_{10} - 11 \zeta_{10}^{2} ) q^{89} + ( -5 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{91} + ( -6 \zeta_{10} + 10 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{92} + ( 5 \zeta_{10} + 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{93} + ( -5 + \zeta_{10} - \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{95} + ( -4 + 4 \zeta_{10} + 11 \zeta_{10}^{3} ) q^{97} + ( -2 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - q^{5} + q^{7} + 8q^{9} + O(q^{10})$$ $$4q + 2q^{4} - q^{5} + q^{7} + 8q^{9} + 10q^{12} - 9q^{13} - 5q^{15} - 4q^{16} - 12q^{17} - 13q^{19} + 2q^{20} - 5q^{21} - 4q^{23} + 4q^{25} - 2q^{28} - 7q^{29} - 5q^{31} + 10q^{33} + q^{35} + 4q^{36} + 17q^{37} + 25q^{39} - 11q^{41} - q^{43} + 20q^{44} - 2q^{45} + 6q^{47} + 20q^{48} - q^{49} - 10q^{51} - 2q^{52} - 12q^{53} - 10q^{55} + 25q^{57} - 16q^{59} - 10q^{60} - 8q^{61} + 2q^{63} + 8q^{64} + q^{65} + 8q^{67} + 4q^{68} - 10q^{69} - 22q^{71} + 22q^{73} - 20q^{75} + 26q^{76} - 16q^{79} - 4q^{80} - 44q^{81} + 28q^{83} + 10q^{84} - 2q^{85} - 5q^{87} - 29q^{89} - 16q^{91} - 22q^{92} + 5q^{93} - 13q^{95} - q^{97} + O(q^{100})$$

## 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 2.23607 1.61803 + 1.17557i −0.809017 0.587785i 0 −0.309017 0.951057i 0 2.00000 0
78.1 0 −2.23607 −0.618034 + 1.90211i 0.309017 0.951057i 0 0.809017 + 0.587785i 0 2.00000 0
92.1 0 −2.23607 −0.618034 1.90211i 0.309017 + 0.951057i 0 0.809017 0.587785i 0 2.00000 0
141.1 0 2.23607 1.61803 1.17557i −0.809017 + 0.587785i 0 −0.309017 + 0.951057i 0 2.00000 0
 $$n$$: 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 $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$.