# Properties

 Label 2016.1.dp.a Level $2016$ Weight $1$ Character orbit 2016.dp Analytic conductor $1.006$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2016.dp (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.59663538192384.15

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{7} - \zeta_{8}^{3} q^{8} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 - z^3 * q^7 - z^3 * q^8 $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{7} - \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{3} - 1) q^{11} - q^{14} - q^{16} + (\zeta_{8} - 1) q^{22} + ( - \zeta_{8}^{2} - 1) q^{23} - \zeta_{8}^{3} q^{25} + \zeta_{8} q^{28} + ( - \zeta_{8} + 1) q^{29} + \zeta_{8} q^{32} + (\zeta_{8}^{2} - \zeta_{8}) q^{37} + (\zeta_{8}^{3} + 1) q^{43} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{44} + (\zeta_{8}^{3} + \zeta_{8}) q^{46} - \zeta_{8}^{2} q^{49} - q^{50} + (\zeta_{8}^{2} + \zeta_{8}) q^{53} - \zeta_{8}^{2} q^{56} + (\zeta_{8}^{2} - \zeta_{8}) q^{58} - \zeta_{8}^{2} q^{64} + ( - \zeta_{8} + 1) q^{67} - \zeta_{8}^{3} q^{71} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{74} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{77} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{79} + ( - \zeta_{8} + 1) q^{86} + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{88} + ( - \zeta_{8}^{2} + 1) q^{92} + \zeta_{8}^{3} q^{98} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 - z^3 * q^7 - z^3 * q^8 + (-z^3 - 1) * q^11 - q^14 - q^16 + (z - 1) * q^22 + (-z^2 - 1) * q^23 - z^3 * q^25 + z * q^28 + (-z + 1) * q^29 + z * q^32 + (z^2 - z) * q^37 + (z^3 + 1) * q^43 + (-z^2 + z) * q^44 + (z^3 + z) * q^46 - z^2 * q^49 - q^50 + (z^2 + z) * q^53 - z^2 * q^56 + (z^2 - z) * q^58 - z^2 * q^64 + (-z + 1) * q^67 - z^3 * q^71 + (-z^3 + z^2) * q^74 + (z^3 - z^2) * q^77 + (-z^3 - z) * q^79 + (-z + 1) * q^86 + (z^3 - z^2) * q^88 + (-z^2 + 1) * q^92 + z^3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{11} - 4 q^{14} - 4 q^{16} - 4 q^{22} - 4 q^{23} + 4 q^{29} + 4 q^{43} - 4 q^{50} + 4 q^{67} + 4 q^{86} + 4 q^{92}+O(q^{100})$$ 4 * q - 4 * q^11 - 4 * q^14 - 4 * q^16 - 4 * q^22 - 4 * q^23 + 4 * q^29 + 4 * q^43 - 4 * q^50 + 4 * q^67 + 4 * q^86 + 4 * q^92

## Character values

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

 $$n$$ $$127$$ $$577$$ $$1765$$ $$1793$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{8}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
685.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
1189.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1693.1 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 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 $$\Q(\sqrt{-7})$$
32.g even 8 1 inner
224.v odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.1.dp.a 4
3.b odd 2 1 2016.1.dp.c yes 4
7.b odd 2 1 CM 2016.1.dp.a 4
21.c even 2 1 2016.1.dp.c yes 4
32.g even 8 1 inner 2016.1.dp.a 4
96.p odd 8 1 2016.1.dp.c yes 4
224.v odd 8 1 inner 2016.1.dp.a 4
672.bo even 8 1 2016.1.dp.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.1.dp.a 4 1.a even 1 1 trivial
2016.1.dp.a 4 7.b odd 2 1 CM
2016.1.dp.a 4 32.g even 8 1 inner
2016.1.dp.a 4 224.v odd 8 1 inner
2016.1.dp.c yes 4 3.b odd 2 1
2016.1.dp.c yes 4 21.c even 2 1
2016.1.dp.c yes 4 96.p odd 8 1
2016.1.dp.c yes 4 672.bo even 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 4T_{11}^{3} + 6T_{11}^{2} + 4T_{11} + 2$$ acting on $$S_{1}^{\mathrm{new}}(2016, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 1$$
$11$ $$T^{4} + 4 T^{3} + 6 T^{2} + 4 T + 2$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 2 T + 2)^{2}$$
$29$ $$T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 2 T^{2} + 4 T + 2$$
$41$ $$T^{4}$$
$43$ $$T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 2 T^{2} + 4 T + 2$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2$$
$71$ $$T^{4} + 16$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 2)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$