Properties

Label 2016.1.l.b
Level 2016
Weight 1
Character orbit 2016.l
Analytic conductor 1.006
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM disc. -7, -24, 168
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.l (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-6}, \sqrt{-7})\)
Artin image size \(16\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.1792336896.7

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(- q^{7}\) \(+O(q^{10})\) \( q\) \(- q^{7}\) \( -2 i q^{11} \) \(- q^{25}\) \( -2 i q^{29} \) \(+ q^{49}\) \( -2 i q^{53} \) \( + 2 i q^{77} \) \( + 2 q^{79} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
1.00000i
1.00000i
0 0 0 0 0 −1.00000 0 0 0
433.2 0 0 0 0 0 −1.00000 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
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
168.i Even 1 RM by \(\Q(\sqrt{42}) \) yes
3.b Odd 1 yes
8.b Even 1 yes
21.c Even 1 yes
56.h Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{11}^{2} \) \(\mathstrut +\mathstrut 4 \) acting on \(S_{1}^{\mathrm{new}}(2016, [\chi])\).