Properties

Label 2016.1.e.a
Level 2016
Weight 1
Character orbit 2016.e
Analytic conductor 1.006
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM discriminant -7
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.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.84672.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q + \zeta_{8}^{2} q^{7} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} + q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + 2 \zeta_{8}^{2} q^{37} - q^{49} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{53} + 2 q^{67} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{71} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{77} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{25} - 4q^{49} + 8q^{67} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1007.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 1.00000i 0 0 0
1007.2 0 0 0 0 0 1.00000i 0 0 0
1007.3 0 0 0 0 0 1.00000i 0 0 0
1007.4 0 0 0 0 0 1.00000i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
24.f even 2 1 inner
56.e even 2 1 inner
168.e odd 2 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + T^{4} )^{2} \)
$13$ \( ( 1 + T^{2} )^{4} \)
$17$ \( ( 1 + T^{2} )^{4} \)
$19$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( 1 + T^{4} )^{2} \)
$31$ \( ( 1 + T^{2} )^{4} \)
$37$ \( ( 1 + T^{2} )^{4} \)
$41$ \( ( 1 + T^{2} )^{4} \)
$43$ \( ( 1 + T^{2} )^{4} \)
$47$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$53$ \( ( 1 + T^{4} )^{2} \)
$59$ \( ( 1 + T^{2} )^{4} \)
$61$ \( ( 1 + T^{2} )^{4} \)
$67$ \( ( 1 - T )^{8} \)
$71$ \( ( 1 + T^{4} )^{2} \)
$73$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$79$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$83$ \( ( 1 + T^{2} )^{4} \)
$89$ \( ( 1 + T^{2} )^{4} \)
$97$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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