Properties

Label 2015.1.bq.c
Level 2015
Weight 1
Character orbit 2015.bq
Analytic conductor 1.006
Analytic rank 0
Dimension 4
Projective image \(A_{4}\)
CM/RM No
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.4060225.2

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{12} q^{2} \) \( + \zeta_{12}^{4} q^{3} \) \( + \zeta_{12}^{3} q^{5} \) \( + \zeta_{12}^{5} q^{6} \) \( + \zeta_{12}^{5} q^{7} \) \( -\zeta_{12}^{3} q^{8} \) \(+O(q^{10})\) \( q\) \( + \zeta_{12} q^{2} \) \( + \zeta_{12}^{4} q^{3} \) \( + \zeta_{12}^{3} q^{5} \) \( + \zeta_{12}^{5} q^{6} \) \( + \zeta_{12}^{5} q^{7} \) \( -\zeta_{12}^{3} q^{8} \) \( + \zeta_{12}^{4} q^{10} \) \( -\zeta_{12} q^{11} \) \(- q^{13}\) \(- q^{14}\) \( -\zeta_{12} q^{15} \) \( -\zeta_{12}^{4} q^{16} \) \( + \zeta_{12}^{2} q^{17} \) \( + \zeta_{12}^{2} q^{19} \) \( -\zeta_{12}^{3} q^{21} \) \( -\zeta_{12}^{2} q^{22} \) \( + \zeta_{12}^{4} q^{23} \) \( + \zeta_{12} q^{24} \) \(- q^{25}\) \( -\zeta_{12} q^{26} \) \(- q^{27}\) \( -\zeta_{12} q^{29} \) \( -\zeta_{12}^{2} q^{30} \) \(- q^{31}\) \( -\zeta_{12}^{5} q^{33} \) \( + \zeta_{12}^{3} q^{34} \) \( -\zeta_{12}^{2} q^{35} \) \( -\zeta_{12}^{4} q^{37} \) \( + \zeta_{12}^{3} q^{38} \) \( -\zeta_{12}^{4} q^{39} \) \(+ q^{40}\) \( -\zeta_{12}^{4} q^{41} \) \( -\zeta_{12}^{4} q^{42} \) \( + \zeta_{12}^{2} q^{43} \) \( + \zeta_{12}^{5} q^{46} \) \( + \zeta_{12}^{2} q^{48} \) \( -\zeta_{12} q^{50} \) \(- q^{51}\) \( -\zeta_{12} q^{54} \) \( -\zeta_{12}^{4} q^{55} \) \( + \zeta_{12}^{2} q^{56} \) \(- q^{57}\) \( -\zeta_{12}^{2} q^{58} \) \( + \zeta_{12}^{2} q^{59} \) \( -\zeta_{12}^{5} q^{61} \) \( -\zeta_{12} q^{62} \) \(- q^{64}\) \( -\zeta_{12}^{3} q^{65} \) \(+ q^{66}\) \( -\zeta_{12} q^{67} \) \( -\zeta_{12}^{2} q^{69} \) \( -\zeta_{12}^{3} q^{70} \) \( + \zeta_{12}^{2} q^{71} \) \( -\zeta_{12}^{5} q^{74} \) \( -\zeta_{12}^{4} q^{75} \) \(+ q^{77}\) \( -\zeta_{12}^{5} q^{78} \) \( + 2 \zeta_{12}^{3} q^{79} \) \( + \zeta_{12} q^{80} \) \( -\zeta_{12}^{4} q^{81} \) \( -\zeta_{12}^{5} q^{82} \) \( + 2 q^{83} \) \( + \zeta_{12}^{5} q^{85} \) \( + \zeta_{12}^{3} q^{86} \) \( -\zeta_{12}^{5} q^{87} \) \( + \zeta_{12}^{4} q^{88} \) \( + \zeta_{12} q^{89} \) \( -\zeta_{12}^{5} q^{91} \) \( -\zeta_{12}^{4} q^{93} \) \( + \zeta_{12}^{5} q^{95} \) \( -\zeta_{12}^{5} q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut -\mathstrut 2q^{23} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 2q^{30} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut -\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 4q^{40} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut +\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut +\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut -\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut -\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 4q^{66} \) \(\mathstrut -\mathstrut 2q^{69} \) \(\mathstrut +\mathstrut 2q^{71} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 4q^{77} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 8q^{83} \) \(\mathstrut -\mathstrut 2q^{88} \) \(\mathstrut +\mathstrut 2q^{93} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−0.866025 0.500000i −0.500000 + 0.866025i 0 1.00000i 0.866025 0.500000i 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i
464.2 0.866025 + 0.500000i −0.500000 + 0.866025i 0 1.00000i −0.866025 + 0.500000i −0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i
1394.1 −0.866025 + 0.500000i −0.500000 0.866025i 0 1.00000i 0.866025 + 0.500000i 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i
1394.2 0.866025 0.500000i −0.500000 0.866025i 0 1.00000i −0.866025 0.500000i −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i
\(n\): 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
13.c Even 1 yes
155.c 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 \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

\(T_{2}^{4} \) \(\mathstrut -\mathstrut T_{2}^{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{3}^{2} \) \(\mathstrut +\mathstrut T_{3} \) \(\mathstrut +\mathstrut 1 \)