# Properties

 Label 2016.1.f.a Level 2016 Weight 1 Character orbit 2016.f Analytic conductor 1.006 Analytic rank 0 Dimension 4 Projective image $$D_{4}$$ CM disc. -84 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 2016.f (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Projective image $$D_{4}$$ Projective field Galois closure of 4.0.1008.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5}$$ $$+ \zeta_{8}^{2} q^{7}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5}$$ $$+ \zeta_{8}^{2} q^{7}$$ $$+ ( -\zeta_{8} + \zeta_{8}^{3} ) q^{11}$$ $$+ ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17}$$ $$-2 \zeta_{8}^{2} q^{19}$$ $$+ ( \zeta_{8} - \zeta_{8}^{3} ) q^{23}$$ $$- q^{25}$$ $$+ ( \zeta_{8} - \zeta_{8}^{3} ) q^{35}$$ $$+ ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41}$$ $$- q^{49}$$ $$+ 2 \zeta_{8}^{2} q^{55}$$ $$+ ( -\zeta_{8} + \zeta_{8}^{3} ) q^{71}$$ $$+ ( -\zeta_{8} - \zeta_{8}^{3} ) q^{77}$$ $$-2 q^{85}$$ $$+ ( \zeta_{8} + \zeta_{8}^{3} ) q^{89}$$ $$+ ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{95}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 4q^{25}$$ $$\mathstrut -\mathstrut 4q^{49}$$ $$\mathstrut -\mathstrut 8q^{85}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1441.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0 0 0 1.41421i 0 1.00000i 0 0 0
1441.2 0 0 0 1.41421i 0 1.00000i 0 0 0
1441.3 0 0 0 1.41421i 0 1.00000i 0 0 0
1441.4 0 0 0 1.41421i 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
84.h Odd 1 CM by $$\Q(\sqrt{-21})$$ yes
3.b Odd 1 yes
4.b Odd 1 yes
7.b Odd 1 yes
12.b Even 1 yes
21.c Even 1 yes
28.d Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{1}^{\mathrm{new}}(2016, [\chi])$$.