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 \(6\), 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.4060225.2

$q$-expansion

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
155.c odd 2 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.c 4
5.b even 2 1 2015.1.bq.e yes 4
13.c even 3 1 inner 2015.1.bq.c 4
31.b odd 2 1 2015.1.bq.e yes 4
65.n even 6 1 2015.1.bq.e yes 4
155.c odd 2 1 inner 2015.1.bq.c 4
403.p odd 6 1 2015.1.bq.e yes 4
2015.bq odd 6 1 inner 2015.1.bq.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2015.1.bq.c 4 1.a even 1 1 trivial
2015.1.bq.c 4 13.c even 3 1 inner
2015.1.bq.c 4 155.c odd 2 1 inner
2015.1.bq.c 4 2015.bq odd 6 1 inner
2015.1.bq.e yes 4 5.b even 2 1
2015.1.bq.e yes 4 31.b odd 2 1
2015.1.bq.e yes 4 65.n even 6 1
2015.1.bq.e yes 4 403.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 \(S_{1}^{\mathrm{new}}(2015, [\chi])\):

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