# 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

# Related objects

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