# Properties

 Label 2015.1.bf.a Level 2015 Weight 1 Character orbit 2015.bf Analytic conductor 1.006 Analytic rank 0 Dimension 8 Projective image $$D_{12}$$ CM discriminant -155 Inner twists 8

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.00561600046$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{12}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{3} + \zeta_{24}^{4} q^{4} -\zeta_{24}^{6} q^{5} -\zeta_{24}^{4} q^{9} +O(q^{10})$$ $$q + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{3} + \zeta_{24}^{4} q^{4} -\zeta_{24}^{6} q^{5} -\zeta_{24}^{4} q^{9} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{12} -\zeta_{24}^{7} q^{13} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{15} + \zeta_{24}^{8} q^{16} + ( \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{17} -\zeta_{24}^{10} q^{19} -\zeta_{24}^{10} q^{20} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{23} - q^{25} + \zeta_{24}^{6} q^{31} -\zeta_{24}^{8} q^{36} + ( \zeta_{24}^{7} - \zeta_{24}^{9} ) q^{37} + ( 1 + \zeta_{24}^{6} ) q^{39} + ( 1 + \zeta_{24}^{4} ) q^{41} + ( -\zeta_{24}^{9} + \zeta_{24}^{11} ) q^{43} + \zeta_{24}^{10} q^{45} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{48} -\zeta_{24}^{8} q^{49} + ( -\zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{8} + \zeta_{24}^{10} ) q^{51} -\zeta_{24}^{11} q^{52} + ( \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{53} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{57} + ( -1 + \zeta_{24}^{8} ) q^{59} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{60} - q^{64} -\zeta_{24} q^{65} + ( \zeta_{24}^{7} + \zeta_{24}^{9} ) q^{68} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{6} - \zeta_{24}^{8} ) q^{69} + \zeta_{24}^{10} q^{71} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{73} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{75} + \zeta_{24}^{2} q^{76} + \zeta_{24}^{2} q^{80} -\zeta_{24}^{8} q^{81} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{83} + ( -\zeta_{24}^{9} - \zeta_{24}^{11} ) q^{85} + ( -\zeta_{24} + \zeta_{24}^{11} ) q^{92} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{93} -\zeta_{24}^{4} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} - 4q^{9} + O(q^{10})$$ $$8q + 4q^{4} - 4q^{9} - 4q^{16} - 8q^{25} + 4q^{36} + 8q^{39} + 12q^{41} + 4q^{49} - 8q^{51} - 12q^{59} - 8q^{64} - 4q^{69} + 4q^{81} - 4q^{95} + O(q^{100})$$

## Character values

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

 $$n$$ $$716$$ $$807$$ $$1861$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
309.1
 −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.258819 − 0.965926i
0 −0.707107 1.22474i 0.500000 0.866025i 1.00000i 0 0 0 −0.500000 + 0.866025i 0
309.2 0 −0.707107 1.22474i 0.500000 0.866025i 1.00000i 0 0 0 −0.500000 + 0.866025i 0
309.3 0 0.707107 + 1.22474i 0.500000 0.866025i 1.00000i 0 0 0 −0.500000 + 0.866025i 0
309.4 0 0.707107 + 1.22474i 0.500000 0.866025i 1.00000i 0 0 0 −0.500000 + 0.866025i 0
1239.1 0 −0.707107 + 1.22474i 0.500000 + 0.866025i 1.00000i 0 0 0 −0.500000 0.866025i 0
1239.2 0 −0.707107 + 1.22474i 0.500000 + 0.866025i 1.00000i 0 0 0 −0.500000 0.866025i 0
1239.3 0 0.707107 1.22474i 0.500000 + 0.866025i 1.00000i 0 0 0 −0.500000 0.866025i 0
1239.4 0 0.707107 1.22474i 0.500000 + 0.866025i 1.00000i 0 0 0 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1239.4 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 $$\Q(\sqrt{-155})$$
5.b even 2 1 inner
13.e even 6 1 inner
31.b odd 2 1 inner
65.l even 6 1 inner
403.t odd 6 1 inner
2015.bf 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.bf.a 8
5.b even 2 1 inner 2015.1.bf.a 8
13.e even 6 1 inner 2015.1.bf.a 8
31.b odd 2 1 inner 2015.1.bf.a 8
65.l even 6 1 inner 2015.1.bf.a 8
155.c odd 2 1 CM 2015.1.bf.a 8
403.t odd 6 1 inner 2015.1.bf.a 8
2015.bf odd 6 1 inner 2015.1.bf.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bf.a 8 1.a even 1 1 trivial
2015.1.bf.a 8 5.b even 2 1 inner
2015.1.bf.a 8 13.e even 6 1 inner
2015.1.bf.a 8 31.b odd 2 1 inner
2015.1.bf.a 8 65.l even 6 1 inner
2015.1.bf.a 8 155.c odd 2 1 CM
2015.1.bf.a 8 403.t odd 6 1 inner
2015.1.bf.a 8 2015.bf odd 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2015, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$( 1 - T^{4} + T^{8} )^{2}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$( 1 - T^{2} + T^{4} )^{4}$$
$11$ $$( 1 - T^{2} + T^{4} )^{4}$$
$13$ $$1 - T^{4} + T^{8}$$
$17$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$19$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$23$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$29$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$31$ $$( 1 + T^{2} )^{4}$$
$37$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$41$ $$( 1 - T )^{8}( 1 - T + T^{2} )^{4}$$
$43$ $$( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} )$$
$47$ $$( 1 + T^{2} )^{8}$$
$53$ $$( 1 - T^{4} + T^{8} )^{2}$$
$59$ $$( 1 + T )^{8}( 1 + T + T^{2} )^{4}$$
$61$ $$( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4}$$
$67$ $$( 1 - T^{2} + T^{4} )^{4}$$
$71$ $$( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2}$$
$73$ $$( 1 + T^{4} )^{4}$$
$79$ $$( 1 - T )^{8}( 1 + T )^{8}$$
$83$ $$( 1 - T^{4} + T^{8} )^{2}$$
$89$ $$( 1 - T^{2} + T^{4} )^{4}$$
$97$ $$( 1 - T^{2} + T^{4} )^{4}$$
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