# Properties

 Label 3200.1.e.b Level $3200$ Weight $1$ Character orbit 3200.e Analytic conductor $1.597$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3200.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.1280000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{3} + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{3} + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{9} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{11} -\zeta_{12}^{3} q^{17} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{19} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{27} + ( -\zeta_{12} - 2 \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{33} - q^{41} - q^{49} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{51} + ( -\zeta_{12} - 2 \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{57} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{67} + \zeta_{12}^{3} q^{73} + q^{81} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{83} - q^{89} + 2 \zeta_{12}^{3} q^{97} + ( -2 \zeta_{12} + 2 \zeta_{12}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{9} + O(q^{10})$$ $$4 q - 8 q^{9} - 4 q^{41} - 4 q^{49} + 4 q^{81} - 4 q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1599.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 1.73205i 0 0 0 0 0 −2.00000 0
1599.2 0 1.73205i 0 0 0 0 0 −2.00000 0
1599.3 0 1.73205i 0 0 0 0 0 −2.00000 0
1599.4 0 1.73205i 0 0 0 0 0 −2.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.e.b 4
4.b odd 2 1 inner 3200.1.e.b 4
5.b even 2 1 inner 3200.1.e.b 4
5.c odd 4 1 3200.1.g.c 2
5.c odd 4 1 3200.1.g.d yes 2
8.b even 2 1 inner 3200.1.e.b 4
8.d odd 2 1 CM 3200.1.e.b 4
20.d odd 2 1 inner 3200.1.e.b 4
20.e even 4 1 3200.1.g.c 2
20.e even 4 1 3200.1.g.d yes 2
40.e odd 2 1 inner 3200.1.e.b 4
40.f even 2 1 inner 3200.1.e.b 4
40.i odd 4 1 3200.1.g.c 2
40.i odd 4 1 3200.1.g.d yes 2
40.k even 4 1 3200.1.g.c 2
40.k even 4 1 3200.1.g.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.e.b 4 1.a even 1 1 trivial
3200.1.e.b 4 4.b odd 2 1 inner
3200.1.e.b 4 5.b even 2 1 inner
3200.1.e.b 4 8.b even 2 1 inner
3200.1.e.b 4 8.d odd 2 1 CM
3200.1.e.b 4 20.d odd 2 1 inner
3200.1.e.b 4 40.e odd 2 1 inner
3200.1.e.b 4 40.f even 2 1 inner
3200.1.g.c 2 5.c odd 4 1
3200.1.g.c 2 20.e even 4 1
3200.1.g.c 2 40.i odd 4 1
3200.1.g.c 2 40.k even 4 1
3200.1.g.d yes 2 5.c odd 4 1
3200.1.g.d yes 2 20.e even 4 1
3200.1.g.d yes 2 40.i odd 4 1
3200.1.g.d yes 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3$$ acting on $$S_{1}^{\mathrm{new}}(3200, [\chi])$$.

## Hecke characteristic polynomials

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