Properties

Label 2016.1.f.a
Level 2016
Weight 1
Character orbit 2016.f
Analytic conductor 1.006
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM disc. -84
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.1008.2

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} \) \( + \zeta_{8}^{2} q^{7} \) \(+O(q^{10})\) \( q\) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} \) \( + \zeta_{8}^{2} q^{7} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} \) \( -2 \zeta_{8}^{2} q^{19} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{23} \) \(- q^{25}\) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{35} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} \) \(- q^{49}\) \( + 2 \zeta_{8}^{2} q^{55} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{71} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{77} \) \( -2 q^{85} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{89} \) \( + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{95} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 8q^{85} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 0 0 1.41421i 0 1.00000i 0 0 0
1441.2 0 0 0 1.41421i 0 1.00000i 0 0 0
1441.3 0 0 0 1.41421i 0 1.00000i 0 0 0
1441.4 0 0 0 1.41421i 0 1.00000i 0 0 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
84.h Odd 1 CM by \(\Q(\sqrt{-21}) \) yes
3.b Odd 1 yes
4.b Odd 1 yes
7.b Odd 1 yes
12.b Even 1 yes
21.c Even 1 yes
28.d Even 1 yes

Hecke kernels

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