# Properties

 Label 287.2.e.b 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: $$\Q(\sqrt{-3}, \sqrt{13})$$ Defining polynomial: $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu^{2} - 4 \nu - 3$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 7$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{3} - 7$$

## Character values

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

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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
 1.15139 − 1.99426i −0.651388 + 1.12824i 1.15139 + 1.99426i −0.651388 − 1.12824i
0 −1.15139 + 1.99426i 1.00000 1.73205i −0.651388 1.12824i 0 −0.500000 2.59808i 0 −1.15139 1.99426i 0
165.2 0 0.651388 1.12824i 1.00000 1.73205i 1.15139 + 1.99426i 0 −0.500000 2.59808i 0 0.651388 + 1.12824i 0
247.1 0 −1.15139 1.99426i 1.00000 + 1.73205i −0.651388 + 1.12824i 0 −0.500000 + 2.59808i 0 −1.15139 + 1.99426i 0
247.2 0 0.651388 + 1.12824i 1.00000 + 1.73205i 1.15139 1.99426i 0 −0.500000 + 2.59808i 0 0.651388 1.12824i 0
 $$n$$: 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.b 4
7.c even 3 1 inner 287.2.e.b 4
7.c even 3 1 2009.2.a.d 2
7.d odd 6 1 2009.2.a.c 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 3 T + 4 T^{2} + T^{3} + T^{4}$$
$5$ $$9 + 3 T + 4 T^{2} - T^{3} + T^{4}$$
$7$ $$( 7 + T + T^{2} )^{2}$$
$11$ $$9 - 15 T + 22 T^{2} - 5 T^{3} + T^{4}$$
$13$ $$( -3 + T + T^{2} )^{2}$$
$17$ $$729 - 81 T + 36 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$169 + 13 T^{2} + T^{4}$$
$23$ $$81 + 36 T + 25 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$( -9 - 4 T + T^{2} )^{2}$$
$31$ $$9 + 3 T + 4 T^{2} - T^{3} + T^{4}$$
$37$ $$2704 + 52 T^{2} + T^{4}$$
$41$ $$( 1 + T )^{4}$$
$43$ $$( 23 + 12 T + T^{2} )^{2}$$
$47$ $$( 81 - 9 T + T^{2} )^{2}$$
$53$ $$81 - 81 T + 90 T^{2} + 9 T^{3} + T^{4}$$
$59$ $$9 + 24 T + 61 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$2704 + 52 T^{2} + T^{4}$$
$67$ $$16 + 24 T + 40 T^{2} - 6 T^{3} + T^{4}$$
$71$ $$( -117 + T^{2} )^{2}$$
$73$ $$2809 + 795 T + 172 T^{2} + 15 T^{3} + T^{4}$$
$79$ $$39601 + 1194 T + 235 T^{2} - 6 T^{3} + T^{4}$$
$83$ $$( 51 - 16 T + T^{2} )^{2}$$
$89$ $$9 + 24 T + 61 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( 53 - 15 T + T^{2} )^{2}$$