# Properties

 Label 3200.1.g.e Level $3200$ Weight $1$ Character orbit 3200.g Analytic conductor $1.597$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3200.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 640) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.1600.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + q^{9} + 2 \zeta_{8}^{2} q^{21} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{23} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{43} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{47} - q^{49} -2 \zeta_{8}^{2} q^{61} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{63} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{67} + 2 \zeta_{8}^{2} q^{69} - q^{81} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{83} -2 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{9} - 4 q^{49} - 4 q^{81} - 8 q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2751.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −1.41421 0 0 0 1.41421i 0 1.00000 0
2751.2 0 −1.41421 0 0 0 1.41421i 0 1.00000 0
2751.3 0 1.41421 0 0 0 1.41421i 0 1.00000 0
2751.4 0 1.41421 0 0 0 1.41421i 0 1.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.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.g.e 4
4.b odd 2 1 inner 3200.1.g.e 4
5.b even 2 1 inner 3200.1.g.e 4
5.c odd 4 2 640.1.e.c 4
8.b even 2 1 inner 3200.1.g.e 4
8.d odd 2 1 inner 3200.1.g.e 4
20.d odd 2 1 CM 3200.1.g.e 4
20.e even 4 2 640.1.e.c 4
40.e odd 2 1 inner 3200.1.g.e 4
40.f even 2 1 inner 3200.1.g.e 4
40.i odd 4 2 640.1.e.c 4
40.k even 4 2 640.1.e.c 4
80.i odd 4 1 1280.1.h.a 2
80.i odd 4 1 1280.1.h.c 2
80.j even 4 1 1280.1.h.a 2
80.j even 4 1 1280.1.h.c 2
80.s even 4 1 1280.1.h.a 2
80.s even 4 1 1280.1.h.c 2
80.t odd 4 1 1280.1.h.a 2
80.t odd 4 1 1280.1.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.1.e.c 4 5.c odd 4 2
640.1.e.c 4 20.e even 4 2
640.1.e.c 4 40.i odd 4 2
640.1.e.c 4 40.k even 4 2
1280.1.h.a 2 80.i odd 4 1
1280.1.h.a 2 80.j even 4 1
1280.1.h.a 2 80.s even 4 1
1280.1.h.a 2 80.t odd 4 1
1280.1.h.c 2 80.i odd 4 1
1280.1.h.c 2 80.j even 4 1
1280.1.h.c 2 80.s even 4 1
1280.1.h.c 2 80.t odd 4 1
3200.1.g.e 4 1.a even 1 1 trivial
3200.1.g.e 4 4.b odd 2 1 inner
3200.1.g.e 4 5.b even 2 1 inner
3200.1.g.e 4 8.b even 2 1 inner
3200.1.g.e 4 8.d odd 2 1 inner
3200.1.g.e 4 20.d odd 2 1 CM
3200.1.g.e 4 40.e odd 2 1 inner
3200.1.g.e 4 40.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{2} - 2$$ $$T_{13}$$ $$T_{17}$$

## Hecke characteristic polynomials

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