# Properties

 Label 3920.1.v.a Level $3920$ Weight $1$ Character orbit 3920.v Analytic conductor $1.956$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3920.v (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.2.3294172000000.3

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{5} q^{5} -\zeta_{16}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{5} q^{5} -\zeta_{16}^{4} q^{9} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{13} + ( -\zeta_{16}^{5} + \zeta_{16}^{7} ) q^{17} -\zeta_{16}^{2} q^{25} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{29} + 2 \zeta_{16}^{6} q^{37} + ( \zeta_{16}^{3} + \zeta_{16}^{5} ) q^{41} + \zeta_{16} q^{45} + ( -1 - \zeta_{16}^{4} ) q^{53} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{61} + ( -\zeta_{16}^{2} - \zeta_{16}^{4} ) q^{65} + ( -\zeta_{16} + \zeta_{16}^{3} ) q^{73} - q^{81} + ( \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{85} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{89} + ( -\zeta_{16} - \zeta_{16}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 8q^{53} - 8q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1471$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\zeta_{16}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
783.1
 −0.382683 + 0.923880i 0.923880 + 0.382683i −0.923880 − 0.382683i 0.382683 − 0.923880i −0.382683 − 0.923880i 0.923880 − 0.382683i −0.923880 + 0.382683i 0.382683 + 0.923880i
0 0 0 −0.923880 0.382683i 0 0 0 1.00000i 0
783.2 0 0 0 −0.382683 + 0.923880i 0 0 0 1.00000i 0
783.3 0 0 0 0.382683 0.923880i 0 0 0 1.00000i 0
783.4 0 0 0 0.923880 + 0.382683i 0 0 0 1.00000i 0
1567.1 0 0 0 −0.923880 + 0.382683i 0 0 0 1.00000i 0
1567.2 0 0 0 −0.382683 0.923880i 0 0 0 1.00000i 0
1567.3 0 0 0 0.382683 + 0.923880i 0 0 0 1.00000i 0
1567.4 0 0 0 0.923880 0.382683i 0 0 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
7.b odd 2 1 inner
20.e even 4 1 inner
28.d even 2 1 inner
35.f even 4 1 inner
140.j odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.v.a 8
4.b odd 2 1 CM 3920.1.v.a 8
5.c odd 4 1 inner 3920.1.v.a 8
7.b odd 2 1 inner 3920.1.v.a 8
7.c even 3 2 3920.1.cx.a 16
7.d odd 6 2 3920.1.cx.a 16
20.e even 4 1 inner 3920.1.v.a 8
28.d even 2 1 inner 3920.1.v.a 8
28.f even 6 2 3920.1.cx.a 16
28.g odd 6 2 3920.1.cx.a 16
35.f even 4 1 inner 3920.1.v.a 8
35.k even 12 2 3920.1.cx.a 16
35.l odd 12 2 3920.1.cx.a 16
140.j odd 4 1 inner 3920.1.v.a 8
140.w even 12 2 3920.1.cx.a 16
140.x odd 12 2 3920.1.cx.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.v.a 8 1.a even 1 1 trivial
3920.1.v.a 8 4.b odd 2 1 CM
3920.1.v.a 8 5.c odd 4 1 inner
3920.1.v.a 8 7.b odd 2 1 inner
3920.1.v.a 8 20.e even 4 1 inner
3920.1.v.a 8 28.d even 2 1 inner
3920.1.v.a 8 35.f even 4 1 inner
3920.1.v.a 8 140.j odd 4 1 inner
3920.1.cx.a 16 7.c even 3 2
3920.1.cx.a 16 7.d odd 6 2
3920.1.cx.a 16 28.f even 6 2
3920.1.cx.a 16 28.g odd 6 2
3920.1.cx.a 16 35.k even 12 2
3920.1.cx.a 16 35.l odd 12 2
3920.1.cx.a 16 140.w even 12 2
3920.1.cx.a 16 140.x odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3920, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$1 + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$4 + 12 T^{4} + T^{8}$$
$17$ $$4 + 12 T^{4} + T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$( 2 + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( 16 + T^{4} )^{2}$$
$41$ $$( 2 + 4 T^{2} + T^{4} )^{2}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$( 2 + 2 T + T^{2} )^{4}$$
$59$ $$T^{8}$$
$61$ $$( 2 + 4 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$4 + 12 T^{4} + T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$( 2 - 4 T^{2} + T^{4} )^{2}$$
$97$ $$4 + 12 T^{4} + T^{8}$$