Properties

Label 2016.1.ce.a
Level $2016$
Weight $1$
Character orbit 2016.ce
Analytic conductor $1.006$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -24
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.ce (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.9680832.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{5} + \zeta_{12}^{2} q^{7} +O(q^{10})\) \( q + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{5} + \zeta_{12}^{2} q^{7} + \zeta_{12}^{5} q^{11} + ( -1 - \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{25} -\zeta_{12}^{3} q^{29} + ( -1 + \zeta_{12}^{4} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{35} + \zeta_{12}^{4} q^{49} + \zeta_{12}^{5} q^{53} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{55} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} -\zeta_{12} q^{77} -\zeta_{12}^{4} q^{79} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{83} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{7} - 4q^{25} - 6q^{31} - 2q^{49} + 2q^{79} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 −0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 0 0
145.2 0 0 0 0.866025 1.50000i 0 0.500000 + 0.866025i 0 0 0
1585.1 0 0 0 −0.866025 1.50000i 0 0.500000 0.866025i 0 0 0
1585.2 0 0 0 0.866025 + 1.50000i 0 0.500000 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.1.ce.a 4
3.b odd 2 1 inner 2016.1.ce.a 4
4.b odd 2 1 504.1.bw.a 4
7.d odd 6 1 inner 2016.1.ce.a 4
8.b even 2 1 inner 2016.1.ce.a 4
8.d odd 2 1 504.1.bw.a 4
12.b even 2 1 504.1.bw.a 4
21.g even 6 1 inner 2016.1.ce.a 4
24.f even 2 1 504.1.bw.a 4
24.h odd 2 1 CM 2016.1.ce.a 4
28.d even 2 1 3528.1.bw.c 4
28.f even 6 1 504.1.bw.a 4
28.f even 6 1 3528.1.l.a 4
28.g odd 6 1 3528.1.l.a 4
28.g odd 6 1 3528.1.bw.c 4
56.e even 2 1 3528.1.bw.c 4
56.j odd 6 1 inner 2016.1.ce.a 4
56.k odd 6 1 3528.1.l.a 4
56.k odd 6 1 3528.1.bw.c 4
56.m even 6 1 504.1.bw.a 4
56.m even 6 1 3528.1.l.a 4
84.h odd 2 1 3528.1.bw.c 4
84.j odd 6 1 504.1.bw.a 4
84.j odd 6 1 3528.1.l.a 4
84.n even 6 1 3528.1.l.a 4
84.n even 6 1 3528.1.bw.c 4
168.e odd 2 1 3528.1.bw.c 4
168.v even 6 1 3528.1.l.a 4
168.v even 6 1 3528.1.bw.c 4
168.ba even 6 1 inner 2016.1.ce.a 4
168.be odd 6 1 504.1.bw.a 4
168.be odd 6 1 3528.1.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.bw.a 4 4.b odd 2 1
504.1.bw.a 4 8.d odd 2 1
504.1.bw.a 4 12.b even 2 1
504.1.bw.a 4 24.f even 2 1
504.1.bw.a 4 28.f even 6 1
504.1.bw.a 4 56.m even 6 1
504.1.bw.a 4 84.j odd 6 1
504.1.bw.a 4 168.be odd 6 1
2016.1.ce.a 4 1.a even 1 1 trivial
2016.1.ce.a 4 3.b odd 2 1 inner
2016.1.ce.a 4 7.d odd 6 1 inner
2016.1.ce.a 4 8.b even 2 1 inner
2016.1.ce.a 4 21.g even 6 1 inner
2016.1.ce.a 4 24.h odd 2 1 CM
2016.1.ce.a 4 56.j odd 6 1 inner
2016.1.ce.a 4 168.ba even 6 1 inner
3528.1.l.a 4 28.f even 6 1
3528.1.l.a 4 28.g odd 6 1
3528.1.l.a 4 56.k odd 6 1
3528.1.l.a 4 56.m even 6 1
3528.1.l.a 4 84.j odd 6 1
3528.1.l.a 4 84.n even 6 1
3528.1.l.a 4 168.v even 6 1
3528.1.l.a 4 168.be odd 6 1
3528.1.bw.c 4 28.d even 2 1
3528.1.bw.c 4 28.g odd 6 1
3528.1.bw.c 4 56.e even 2 1
3528.1.bw.c 4 56.k odd 6 1
3528.1.bw.c 4 84.h odd 2 1
3528.1.bw.c 4 84.n even 6 1
3528.1.bw.c 4 168.e odd 2 1
3528.1.bw.c 4 168.v even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2016, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 9 + 3 T^{2} + T^{4} \)
$7$ \( ( 1 - T + T^{2} )^{2} \)
$11$ \( 1 - T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( ( 3 + 3 T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 1 - T^{2} + T^{4} \)
$59$ \( 9 + 3 T^{2} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 1 - T + T^{2} )^{2} \)
$83$ \( ( -3 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 3 + T^{2} )^{2} \)
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