# Properties

 Label 287.2.e.a Level $287$ Weight $2$ Character orbit 287.e 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.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

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

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−0.809017 1.40126i −0.500000 + 0.866025i −0.309017 + 0.535233i 0.500000 + 0.866025i 1.61803 −2.00000 + 1.73205i −2.23607 1.00000 + 1.73205i 0.809017 1.40126i
165.2 0.309017 + 0.535233i −0.500000 + 0.866025i 0.809017 1.40126i 0.500000 + 0.866025i −0.618034 −2.00000 + 1.73205i 2.23607 1.00000 + 1.73205i −0.309017 + 0.535233i
247.1 −0.809017 + 1.40126i −0.500000 0.866025i −0.309017 0.535233i 0.500000 0.866025i 1.61803 −2.00000 1.73205i −2.23607 1.00000 1.73205i 0.809017 + 1.40126i
247.2 0.309017 0.535233i −0.500000 0.866025i 0.809017 + 1.40126i 0.500000 0.866025i −0.618034 −2.00000 1.73205i 2.23607 1.00000 1.73205i −0.309017 0.535233i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.e.a 4
7.c even 3 1 inner 287.2.e.a 4
7.c even 3 1 2009.2.a.f 2
7.d odd 6 1 2009.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.a 4 1.a even 1 1 trivial
287.2.e.a 4 7.c even 3 1 inner
2009.2.a.e 2 7.d odd 6 1
2009.2.a.f 2 7.c even 3 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 2 T^{2} + T^{3} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( 7 + 4 T + T^{2} )^{2}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$( -20 + T^{2} )^{2}$$
$17$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$121 + 66 T + 47 T^{2} - 6 T^{3} + T^{4}$$
$23$ $$1681 + 164 T + 57 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$1 - 4 T + 17 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$361 - 38 T + 23 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$( 1 + T )^{4}$$
$43$ $$( -16 + 4 T + T^{2} )^{2}$$
$47$ $$361 - 38 T + 23 T^{2} + 2 T^{3} + T^{4}$$
$53$ $$3481 - 944 T + 197 T^{2} - 16 T^{3} + T^{4}$$
$59$ $$1681 - 164 T + 57 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$25 + 50 T + 95 T^{2} + 10 T^{3} + T^{4}$$
$67$ $$5041 + 426 T + 107 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( 80 + 20 T + T^{2} )^{2}$$
$73$ $$841 - 406 T + 167 T^{2} - 14 T^{3} + T^{4}$$
$79$ $$( 1 - T + T^{2} )^{2}$$
$83$ $$( 16 + 12 T + T^{2} )^{2}$$
$89$ $$19321 - 3336 T + 437 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$( -4 - 8 T + T^{2} )^{2}$$