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 discs -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 \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 504)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-6}, \sqrt{-7})\)
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.

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

Character values

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
168.i even 2 1 RM by \(\Q(\sqrt{42}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
56.h 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.l.b 2
3.b odd 2 1 inner 2016.1.l.b 2
4.b odd 2 1 504.1.l.b 2
7.b odd 2 1 CM 2016.1.l.b 2
8.b even 2 1 inner 2016.1.l.b 2
8.d odd 2 1 504.1.l.b 2
12.b even 2 1 504.1.l.b 2
21.c even 2 1 inner 2016.1.l.b 2
24.f even 2 1 504.1.l.b 2
24.h odd 2 1 CM 2016.1.l.b 2
28.d even 2 1 504.1.l.b 2
28.f even 6 2 3528.1.bw.b 4
28.g odd 6 2 3528.1.bw.b 4
56.e even 2 1 504.1.l.b 2
56.h odd 2 1 inner 2016.1.l.b 2
56.k odd 6 2 3528.1.bw.b 4
56.m even 6 2 3528.1.bw.b 4
84.h odd 2 1 504.1.l.b 2
84.j odd 6 2 3528.1.bw.b 4
84.n even 6 2 3528.1.bw.b 4
168.e odd 2 1 504.1.l.b 2
168.i even 2 1 RM 2016.1.l.b 2
168.v even 6 2 3528.1.bw.b 4
168.be odd 6 2 3528.1.bw.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.1.l.b 2 4.b odd 2 1
504.1.l.b 2 8.d odd 2 1
504.1.l.b 2 12.b even 2 1
504.1.l.b 2 24.f even 2 1
504.1.l.b 2 28.d even 2 1
504.1.l.b 2 56.e even 2 1
504.1.l.b 2 84.h odd 2 1
504.1.l.b 2 168.e odd 2 1
2016.1.l.b 2 1.a even 1 1 trivial
2016.1.l.b 2 3.b odd 2 1 inner
2016.1.l.b 2 7.b odd 2 1 CM
2016.1.l.b 2 8.b even 2 1 inner
2016.1.l.b 2 21.c even 2 1 inner
2016.1.l.b 2 24.h odd 2 1 CM
2016.1.l.b 2 56.h odd 2 1 inner
2016.1.l.b 2 168.i even 2 1 RM
3528.1.bw.b 4 28.f even 6 2
3528.1.bw.b 4 28.g odd 6 2
3528.1.bw.b 4 56.k odd 6 2
3528.1.bw.b 4 56.m even 6 2
3528.1.bw.b 4 84.j odd 6 2
3528.1.bw.b 4 84.n even 6 2
3528.1.bw.b 4 168.v even 6 2
3528.1.bw.b 4 168.be odd 6 2

Hecke kernels

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

Hecke characteristic polynomials

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