Properties

Label 2015.1.bq.d
Level 2015
Weight 1
Character orbit 2015.bq
Analytic conductor 1.006
Analytic rank 0
Dimension 4
Projective image \(D_{6}\)
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.bq (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.3430890125.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( -\zeta_{12}^{2} q^{4} \) \(- q^{5}\) \( + \zeta_{12}^{2} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{12}^{2} q^{4} \) \(- q^{5}\) \( + \zeta_{12}^{2} q^{9} \) \( + \zeta_{12}^{5} q^{13} \) \( + \zeta_{12}^{4} q^{16} \) \( + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{17} \) \( -\zeta_{12}^{2} q^{19} \) \( + \zeta_{12}^{2} q^{20} \) \( + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{23} \) \(+ q^{25}\) \(- q^{31}\) \( -\zeta_{12}^{4} q^{36} \) \( + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{37} \) \( -\zeta_{12}^{4} q^{41} \) \( + ( \zeta_{12} + \zeta_{12}^{3} ) q^{43} \) \( -\zeta_{12}^{2} q^{45} \) \( + \zeta_{12}^{4} q^{49} \) \( + \zeta_{12} q^{52} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{53} \) \( + \zeta_{12}^{2} q^{59} \) \(+ q^{64}\) \( -\zeta_{12}^{5} q^{65} \) \( + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{68} \) \( -\zeta_{12}^{2} q^{71} \) \( + \zeta_{12}^{4} q^{76} \) \( -\zeta_{12}^{4} q^{80} \) \( + \zeta_{12}^{4} q^{81} \) \( + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{83} \) \( + ( \zeta_{12} + \zeta_{12}^{3} ) q^{85} \) \( + ( \zeta_{12} - \zeta_{12}^{5} ) q^{92} \) \( + \zeta_{12}^{2} q^{95} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut -\mathstrut 2q^{71} \) \(\mathstrut -\mathstrut 2q^{76} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 −0.500000 0.866025i −1.00000 0 0 0 0.500000 + 0.866025i 0
464.2 0 0 −0.500000 0.866025i −1.00000 0 0 0 0.500000 + 0.866025i 0
1394.1 0 0 −0.500000 + 0.866025i −1.00000 0 0 0 0.500000 0.866025i 0
1394.2 0 0 −0.500000 + 0.866025i −1.00000 0 0 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
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.c Even 1 yes
31.b Odd 1 yes
65.n Even 1 yes
403.p Odd 1 yes
2015.bq Odd 1 yes

Hecke kernels

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

\(T_{2} \)
\(T_{3} \)