Properties

Label 2015.1.bq.b
Level 2015
Weight 1
Character orbit 2015.bq
Analytic conductor 1.006
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -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 \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00561600046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.26195.1

$q$-expansion

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

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

Character Values

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
155.c Odd 1 CM by \(\Q(\sqrt{-155}) \) yes
13.c Even 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 \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

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