# Properties

 Label 2015.1.bq.a Level 2015 Weight 1 Character orbit 2015.bq Analytic conductor 1.006 Analytic rank 0 Dimension 2 Projective image $$D_{3}$$ CM discriminant -155 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2015 = 5 \cdot 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2015.bq (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00561600046$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{3}$$ Projective field Galois closure of 3.1.26195.1 Artin image $C_3\times S_3$ Artin field Galois closure of 6.0.629334875.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{3} + \zeta_{6}^{2} q^{4} + q^{5} + 3 \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q -2 \zeta_{6} q^{3} + \zeta_{6}^{2} q^{4} + q^{5} + 3 \zeta_{6}^{2} q^{9} + 2 q^{12} + \zeta_{6}^{2} q^{13} -2 \zeta_{6} q^{15} -\zeta_{6} q^{16} -\zeta_{6}^{2} q^{17} -\zeta_{6}^{2} q^{19} + \zeta_{6}^{2} q^{20} + \zeta_{6} q^{23} + q^{25} + 4 q^{27} + q^{31} -3 \zeta_{6} q^{36} + \zeta_{6} q^{37} + 2 q^{39} + \zeta_{6} q^{41} -\zeta_{6}^{2} q^{43} + 3 \zeta_{6}^{2} q^{45} + 2 \zeta_{6}^{2} q^{48} -\zeta_{6} q^{49} -2 q^{51} -\zeta_{6} q^{52} - q^{53} -2 q^{57} -\zeta_{6}^{2} q^{59} + 2 q^{60} + q^{64} + \zeta_{6}^{2} q^{65} + \zeta_{6} q^{68} -2 \zeta_{6}^{2} q^{69} -\zeta_{6}^{2} q^{71} + 2 q^{73} -2 \zeta_{6} q^{75} + \zeta_{6} q^{76} -\zeta_{6} q^{80} -5 \zeta_{6} q^{81} - q^{83} -\zeta_{6}^{2} q^{85} - q^{92} -2 \zeta_{6} q^{93} -\zeta_{6}^{2} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - q^{4} + 2q^{5} - 3q^{9} + O(q^{10})$$ $$2q - 2q^{3} - q^{4} + 2q^{5} - 3q^{9} + 4q^{12} - q^{13} - 2q^{15} - q^{16} + q^{17} + q^{19} - q^{20} + q^{23} + 2q^{25} + 8q^{27} + 2q^{31} - 3q^{36} + q^{37} + 4q^{39} + q^{41} + q^{43} - 3q^{45} - 2q^{48} - q^{49} - 4q^{51} - q^{52} - 2q^{53} - 4q^{57} + q^{59} + 4q^{60} + 2q^{64} - q^{65} + q^{68} + 2q^{69} + q^{71} + 4q^{73} - 2q^{75} + q^{76} - q^{80} - 5q^{81} - 2q^{83} + q^{85} - 2q^{92} - 2q^{93} + q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2015\mathbb{Z}\right)^\times$$.

 $$n$$ $$716$$ $$807$$ $$1861$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
464.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 + 1.73205i −0.500000 0.866025i 1.00000 0 0 0 −1.50000 2.59808i 0
1394.1 0 −1.00000 1.73205i −0.500000 + 0.866025i 1.00000 0 0 0 −1.50000 + 2.59808i 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
155.c odd 2 1 CM by $$\Q(\sqrt{-155})$$
13.c even 3 1 inner
2015.bq odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2015.1.bq.a 2
5.b even 2 1 2015.1.bq.b yes 2
13.c even 3 1 inner 2015.1.bq.a 2
31.b odd 2 1 2015.1.bq.b yes 2
65.n even 6 1 2015.1.bq.b yes 2
155.c odd 2 1 CM 2015.1.bq.a 2
403.p odd 6 1 2015.1.bq.b yes 2
2015.bq odd 6 1 inner 2015.1.bq.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bq.a 2 1.a even 1 1 trivial
2015.1.bq.a 2 13.c even 3 1 inner
2015.1.bq.a 2 155.c odd 2 1 CM
2015.1.bq.a 2 2015.bq odd 6 1 inner
2015.1.bq.b yes 2 5.b even 2 1
2015.1.bq.b yes 2 31.b odd 2 1
2015.1.bq.b yes 2 65.n even 6 1
2015.1.bq.b yes 2 403.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2015, [\chi])$$:

 $$T_{2}$$ $$T_{3}^{2} + 2 T_{3} + 4$$

## Hecke characteristic polynomials

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