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 disc. -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 \(6\) and degree \(2\))

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: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
155.c Odd 1 CM by \(\Q(\sqrt{-155}) \) yes
5.b Even 1 yes
13.e Even 1 yes
31.b Odd 1 yes
65.l Even 1 yes
403.t Odd 1 yes
2015.bf Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(2015, [\chi])\).