# Properties

 Label 3200.1.g.a Level $3200$ Weight $1$ Character orbit 3200.g Self dual yes Analytic conductor $1.597$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -8, 8 Inner twists $4$

# 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: yes Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\zeta_{8})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.12800.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{9} + O(q^{10})$$ $$q - q^{9} + 2 q^{17} + 2 q^{41} + q^{49} + 2 q^{73} + q^{81} - 2 q^{89} + 2 q^{97} + O(q^{100})$$

## 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
0 0 0 0 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.g.a 1
4.b odd 2 1 CM 3200.1.g.a 1
5.b even 2 1 128.1.d.a 1
5.c odd 4 2 3200.1.e.a 2
8.b even 2 1 RM 3200.1.g.a 1
8.d odd 2 1 CM 3200.1.g.a 1
15.d odd 2 1 1152.1.b.a 1
20.d odd 2 1 128.1.d.a 1
20.e even 4 2 3200.1.e.a 2
40.e odd 2 1 128.1.d.a 1
40.f even 2 1 128.1.d.a 1
40.i odd 4 2 3200.1.e.a 2
40.k even 4 2 3200.1.e.a 2
60.h even 2 1 1152.1.b.a 1
80.k odd 4 2 256.1.c.a 1
80.q even 4 2 256.1.c.a 1
120.i odd 2 1 1152.1.b.a 1
120.m even 2 1 1152.1.b.a 1
160.y odd 8 4 1024.1.f.b 2
160.z even 8 4 1024.1.f.b 2
240.t even 4 2 2304.1.g.b 1
240.bm odd 4 2 2304.1.g.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 5.b even 2 1
128.1.d.a 1 20.d odd 2 1
128.1.d.a 1 40.e odd 2 1
128.1.d.a 1 40.f even 2 1
256.1.c.a 1 80.k odd 4 2
256.1.c.a 1 80.q even 4 2
1024.1.f.b 2 160.y odd 8 4
1024.1.f.b 2 160.z even 8 4
1152.1.b.a 1 15.d odd 2 1
1152.1.b.a 1 60.h even 2 1
1152.1.b.a 1 120.i odd 2 1
1152.1.b.a 1 120.m even 2 1
2304.1.g.b 1 240.t even 4 2
2304.1.g.b 1 240.bm odd 4 2
3200.1.e.a 2 5.c odd 4 2
3200.1.e.a 2 20.e even 4 2
3200.1.e.a 2 40.i odd 4 2
3200.1.e.a 2 40.k even 4 2
3200.1.g.a 1 1.a even 1 1 trivial
3200.1.g.a 1 4.b odd 2 1 CM
3200.1.g.a 1 8.b even 2 1 RM
3200.1.g.a 1 8.d odd 2 1 CM

## Hecke kernels

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

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

## Hecke characteristic polynomials

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