# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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