Properties

Label 2016.1.bi.a
Level 2016
Weight 1
Character orbit 2016.bi
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: \( N \) = \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2016.bi (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Projective image \(A_{4}\)
Projective field Galois closure of 4.0.254016.3

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{12}^{2} q^{3} \) \( -\zeta_{12}^{3} q^{5} \) \( + \zeta_{12} q^{7} \) \( + \zeta_{12}^{4} q^{9} \) \(+O(q^{10})\) \( q\) \( -\zeta_{12}^{2} q^{3} \) \( -\zeta_{12}^{3} q^{5} \) \( + \zeta_{12} q^{7} \) \( + \zeta_{12}^{4} q^{9} \) \(+ q^{11}\) \( + \zeta_{12} q^{13} \) \( + \zeta_{12}^{5} q^{15} \) \( -\zeta_{12}^{4} q^{17} \) \( + \zeta_{12}^{2} q^{19} \) \( -\zeta_{12}^{3} q^{21} \) \( + \zeta_{12}^{3} q^{23} \) \(+ q^{27}\) \( + \zeta_{12}^{5} q^{29} \) \( -\zeta_{12}^{2} q^{33} \) \( -\zeta_{12}^{4} q^{35} \) \( + \zeta_{12}^{5} q^{37} \) \( -\zeta_{12}^{3} q^{39} \) \( -\zeta_{12}^{4} q^{41} \) \( -\zeta_{12}^{2} q^{43} \) \( + \zeta_{12} q^{45} \) \( + \zeta_{12}^{2} q^{49} \) \(- q^{51}\) \( -\zeta_{12} q^{53} \) \( -\zeta_{12}^{3} q^{55} \) \( -\zeta_{12}^{4} q^{57} \) \( + \zeta_{12}^{5} q^{63} \) \( -\zeta_{12}^{4} q^{65} \) \( -\zeta_{12}^{5} q^{69} \) \( + \zeta_{12}^{4} q^{73} \) \( + \zeta_{12} q^{77} \) \( -2 \zeta_{12} q^{79} \) \( -\zeta_{12}^{2} q^{81} \) \( -\zeta_{12}^{2} q^{83} \) \( -\zeta_{12} q^{85} \) \( + \zeta_{12} q^{87} \) \( -\zeta_{12}^{2} q^{89} \) \( + \zeta_{12}^{2} q^{91} \) \( -\zeta_{12}^{5} q^{95} \) \( -\zeta_{12}^{2} q^{97} \) \( + \zeta_{12}^{4} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut -\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut -\mathstrut 4q^{51} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 2q^{89} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.500000 + 0.866025i 0 1.00000i 0 −0.866025 + 0.500000i 0 −0.500000 0.866025i 0
79.2 0 −0.500000 + 0.866025i 0 1.00000i 0 0.866025 0.500000i 0 −0.500000 0.866025i 0
1327.1 0 −0.500000 0.866025i 0 1.00000i 0 0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0
1327.2 0 −0.500000 0.866025i 0 1.00000i 0 −0.866025 0.500000i 0 −0.500000 + 0.866025i 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
8.d Odd 1 yes
63.g Even 1 yes
504.ba Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(2016, [\chi])\).