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 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.bq (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.21271518775.2

$q$-expansion

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

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

Character values

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

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

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 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.c even 3 1 inner
31.b odd 2 1 inner
65.n even 6 1 inner
403.p odd 6 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.d 4
5.b even 2 1 inner 2015.1.bq.d 4
13.c even 3 1 inner 2015.1.bq.d 4
31.b odd 2 1 inner 2015.1.bq.d 4
65.n even 6 1 inner 2015.1.bq.d 4
155.c odd 2 1 CM 2015.1.bq.d 4
403.p odd 6 1 inner 2015.1.bq.d 4
2015.bq odd 6 1 inner 2015.1.bq.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bq.d 4 1.a even 1 1 trivial
2015.1.bq.d 4 5.b even 2 1 inner
2015.1.bq.d 4 13.c even 3 1 inner
2015.1.bq.d 4 31.b odd 2 1 inner
2015.1.bq.d 4 65.n even 6 1 inner
2015.1.bq.d 4 155.c odd 2 1 CM
2015.1.bq.d 4 403.p odd 6 1 inner
2015.1.bq.d 4 2015.bq odd 6 1 inner

Hecke kernels

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

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