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:

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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