Properties

Label 2015.1.bq.a
Level $2015$
Weight $1$
Character orbit 2015.bq
Analytic conductor $1.006$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -155
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.26195.1
Artin image $C_3\times S_3$
Artin field Galois closure of 6.0.629334875.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{3} + \zeta_{6}^{2} q^{4} + q^{5} + 3 \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -2 \zeta_{6} q^{3} + \zeta_{6}^{2} q^{4} + q^{5} + 3 \zeta_{6}^{2} q^{9} + 2 q^{12} + \zeta_{6}^{2} q^{13} -2 \zeta_{6} q^{15} -\zeta_{6} q^{16} -\zeta_{6}^{2} q^{17} -\zeta_{6}^{2} q^{19} + \zeta_{6}^{2} q^{20} + \zeta_{6} q^{23} + q^{25} + 4 q^{27} + q^{31} -3 \zeta_{6} q^{36} + \zeta_{6} q^{37} + 2 q^{39} + \zeta_{6} q^{41} -\zeta_{6}^{2} q^{43} + 3 \zeta_{6}^{2} q^{45} + 2 \zeta_{6}^{2} q^{48} -\zeta_{6} q^{49} -2 q^{51} -\zeta_{6} q^{52} - q^{53} -2 q^{57} -\zeta_{6}^{2} q^{59} + 2 q^{60} + q^{64} + \zeta_{6}^{2} q^{65} + \zeta_{6} q^{68} -2 \zeta_{6}^{2} q^{69} -\zeta_{6}^{2} q^{71} + 2 q^{73} -2 \zeta_{6} q^{75} + \zeta_{6} q^{76} -\zeta_{6} q^{80} -5 \zeta_{6} q^{81} - q^{83} -\zeta_{6}^{2} q^{85} - q^{92} -2 \zeta_{6} q^{93} -\zeta_{6}^{2} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - q^{4} + 2q^{5} - 3q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - q^{4} + 2q^{5} - 3q^{9} + 4q^{12} - q^{13} - 2q^{15} - q^{16} + q^{17} + q^{19} - q^{20} + q^{23} + 2q^{25} + 8q^{27} + 2q^{31} - 3q^{36} + q^{37} + 4q^{39} + q^{41} + q^{43} - 3q^{45} - 2q^{48} - q^{49} - 4q^{51} - q^{52} - 2q^{53} - 4q^{57} + q^{59} + 4q^{60} + 2q^{64} - q^{65} + q^{68} + 2q^{69} + q^{71} + 4q^{73} - 2q^{75} + q^{76} - q^{80} - 5q^{81} - 2q^{83} + q^{85} - 2q^{92} - 2q^{93} + q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(716\) \(807\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 + 1.73205i −0.500000 0.866025i 1.00000 0 0 0 −1.50000 2.59808i 0
1394.1 0 −1.00000 1.73205i −0.500000 + 0.866025i 1.00000 0 0 0 −1.50000 + 2.59808i 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 \(\Q(\sqrt{-155}) \)
13.c even 3 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.a 2
5.b even 2 1 2015.1.bq.b yes 2
13.c even 3 1 inner 2015.1.bq.a 2
31.b odd 2 1 2015.1.bq.b yes 2
65.n even 6 1 2015.1.bq.b yes 2
155.c odd 2 1 CM 2015.1.bq.a 2
403.p odd 6 1 2015.1.bq.b yes 2
2015.bq odd 6 1 inner 2015.1.bq.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bq.a 2 1.a even 1 1 trivial
2015.1.bq.a 2 13.c even 3 1 inner
2015.1.bq.a 2 155.c odd 2 1 CM
2015.1.bq.a 2 2015.bq odd 6 1 inner
2015.1.bq.b yes 2 5.b even 2 1
2015.1.bq.b yes 2 31.b odd 2 1
2015.1.bq.b yes 2 65.n even 6 1
2015.1.bq.b yes 2 403.p odd 6 1

Hecke kernels

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

\( T_{2} \)
\( T_{3}^{2} + 2 T_{3} + 4 \)