# Properties

 Label 287.1.d.a Level 287 Weight 1 Character orbit 287.d Self dual yes Analytic conductor 0.143 Analytic rank 0 Dimension 3 Projective image $$D_{7}$$ CM discriminant -287 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$287 = 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 287.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.143231658626$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{7}$$ Projective field Galois closure of 7.1.23639903.1 Artin image $D_7$ Artin field Galois closure of 7.1.23639903.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a root $$\beta$$ of the polynomial $$x^{3} - x^{2} - 2 x + 1$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta - \beta^{2} ) q^{2} -\beta q^{3} + ( 1 - \beta ) q^{4} + ( -1 + \beta ) q^{6} + q^{7} + ( -1 + \beta ) q^{8} + ( -1 + \beta^{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta - \beta^{2} ) q^{2} -\beta q^{3} + ( 1 - \beta ) q^{4} + ( -1 + \beta ) q^{6} + q^{7} + ( -1 + \beta ) q^{8} + ( -1 + \beta^{2} ) q^{9} + ( -\beta + \beta^{2} ) q^{12} + ( 1 + \beta - \beta^{2} ) q^{13} + ( 1 + \beta - \beta^{2} ) q^{14} + ( -1 - \beta + \beta^{2} ) q^{16} + ( -2 + \beta^{2} ) q^{17} - q^{18} + ( -2 + \beta^{2} ) q^{19} -\beta q^{21} -\beta q^{23} + ( \beta - \beta^{2} ) q^{24} + q^{25} + ( 2 - \beta ) q^{26} + ( 1 - \beta^{2} ) q^{27} + ( 1 - \beta ) q^{28} - q^{32} + ( -2 - \beta + \beta^{2} ) q^{34} -\beta q^{36} + ( 1 + \beta - \beta^{2} ) q^{37} + ( -2 - \beta + \beta^{2} ) q^{38} + ( -1 + \beta ) q^{39} + q^{41} + ( -1 + \beta ) q^{42} + ( -2 + \beta^{2} ) q^{43} + ( -1 + \beta ) q^{46} + ( 1 + \beta - \beta^{2} ) q^{47} + ( 1 - \beta ) q^{48} + q^{49} + ( 1 + \beta - \beta^{2} ) q^{50} + ( 1 - \beta^{2} ) q^{51} + ( 2 \beta - \beta^{2} ) q^{52} + q^{54} + ( -1 + \beta ) q^{56} + ( 1 - \beta^{2} ) q^{57} + ( -1 + \beta^{2} ) q^{63} - q^{68} + \beta^{2} q^{69} + \beta q^{72} + ( 2 - \beta ) q^{74} -\beta q^{75} - q^{76} + ( -2 \beta + \beta^{2} ) q^{78} + \beta q^{81} + ( 1 + \beta - \beta^{2} ) q^{82} + ( -\beta + \beta^{2} ) q^{84} + ( -2 - \beta + \beta^{2} ) q^{86} + ( 1 + \beta - \beta^{2} ) q^{89} + ( 1 + \beta - \beta^{2} ) q^{91} + ( -\beta + \beta^{2} ) q^{92} + ( 2 - \beta ) q^{94} + \beta q^{96} -\beta q^{97} + ( 1 + \beta - \beta^{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} - q^{3} + 2q^{4} - 2q^{6} + 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$3q - q^{2} - q^{3} + 2q^{4} - 2q^{6} + 3q^{7} - 2q^{8} + 2q^{9} + 4q^{12} - q^{13} - q^{14} + q^{16} - q^{17} - 3q^{18} - q^{19} - q^{21} - q^{23} - 4q^{24} + 3q^{25} + 5q^{26} - 2q^{27} + 2q^{28} - 3q^{32} - 2q^{34} - q^{36} - q^{37} - 2q^{38} - 2q^{39} + 3q^{41} - 2q^{42} - q^{43} - 2q^{46} - q^{47} + 2q^{48} + 3q^{49} - q^{50} - 2q^{51} - 3q^{52} + 3q^{54} - 2q^{56} - 2q^{57} + 2q^{63} - 3q^{68} + 5q^{69} + q^{72} + 5q^{74} - q^{75} - 3q^{76} + 3q^{78} + q^{81} - q^{82} + 4q^{84} - 2q^{86} - q^{89} - q^{91} + 4q^{92} + 5q^{94} + q^{96} - q^{97} - q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$206$$ $$211$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
286.1
 −1.24698 1.80194 0.445042
−1.80194 1.24698 2.24698 0 −2.24698 1.00000 −2.24698 0.554958 0
286.2 −0.445042 −1.80194 −0.801938 0 0.801938 1.00000 0.801938 2.24698 0
286.3 1.24698 −0.445042 0.554958 0 −0.554958 1.00000 −0.554958 −0.801938 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
287.d odd 2 1 CM by $$\Q(\sqrt{-287})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.1.d.a 3
3.b odd 2 1 2583.1.f.b 3
7.b odd 2 1 287.1.d.b yes 3
7.c even 3 2 2009.1.i.b 6
7.d odd 6 2 2009.1.i.a 6
21.c even 2 1 2583.1.f.a 3
41.b even 2 1 287.1.d.b yes 3
123.b odd 2 1 2583.1.f.a 3
287.d odd 2 1 CM 287.1.d.a 3
287.i odd 6 2 2009.1.i.b 6
287.j even 6 2 2009.1.i.a 6
861.e even 2 1 2583.1.f.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.1.d.a 3 1.a even 1 1 trivial
287.1.d.a 3 287.d odd 2 1 CM
287.1.d.b yes 3 7.b odd 2 1
287.1.d.b yes 3 41.b even 2 1
2009.1.i.a 6 7.d odd 6 2
2009.1.i.a 6 287.j even 6 2
2009.1.i.b 6 7.c even 3 2
2009.1.i.b 6 287.i odd 6 2
2583.1.f.a 3 21.c even 2 1
2583.1.f.a 3 123.b odd 2 1
2583.1.f.b 3 3.b odd 2 1
2583.1.f.b 3 861.e even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$5$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$7$ $$( 1 - T )^{3}$$
$11$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$13$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$17$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$19$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$23$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$29$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$31$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$37$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$41$ $$( 1 - T )^{3}$$
$43$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$47$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$53$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$59$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$61$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$67$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$71$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$73$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$79$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$83$ $$( 1 - T )^{3}( 1 + T )^{3}$$
$89$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$97$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$