# Properties

 Label 3200.1.e.a Level $3200$ Weight $1$ Character orbit 3200.e Analytic conductor $1.597$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -8, 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: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{8})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.4096000000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{9} +O(q^{10})$$ $$q + q^{9} -2 i q^{17} + 2 q^{41} - q^{49} + 2 i q^{73} + q^{81} + 2 q^{89} -2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{9} + O(q^{10})$$ $$2q + 2q^{9} + 4q^{41} - 2q^{49} + 2q^{81} + 4q^{89} + O(q^{100})$$

## 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
 1.00000i − 1.00000i
0 0 0 0 0 0 0 1.00000 0
1599.2 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 $$\Q(\sqrt{-2})$$
5.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.a 2
4.b odd 2 1 CM 3200.1.e.a 2
5.b even 2 1 inner 3200.1.e.a 2
5.c odd 4 1 128.1.d.a 1
5.c odd 4 1 3200.1.g.a 1
8.b even 2 1 RM 3200.1.e.a 2
8.d odd 2 1 CM 3200.1.e.a 2
15.e even 4 1 1152.1.b.a 1
20.d odd 2 1 inner 3200.1.e.a 2
20.e even 4 1 128.1.d.a 1
20.e even 4 1 3200.1.g.a 1
40.e odd 2 1 inner 3200.1.e.a 2
40.f even 2 1 inner 3200.1.e.a 2
40.i odd 4 1 128.1.d.a 1
40.i odd 4 1 3200.1.g.a 1
40.k even 4 1 128.1.d.a 1
40.k even 4 1 3200.1.g.a 1
60.l odd 4 1 1152.1.b.a 1
80.i odd 4 1 256.1.c.a 1
80.j even 4 1 256.1.c.a 1
80.s even 4 1 256.1.c.a 1
80.t odd 4 1 256.1.c.a 1
120.q odd 4 1 1152.1.b.a 1
120.w even 4 1 1152.1.b.a 1
160.u even 8 2 1024.1.f.b 2
160.v odd 8 2 1024.1.f.b 2
160.ba even 8 2 1024.1.f.b 2
160.bb odd 8 2 1024.1.f.b 2
240.z odd 4 1 2304.1.g.b 1
240.bb even 4 1 2304.1.g.b 1
240.bd odd 4 1 2304.1.g.b 1
240.bf even 4 1 2304.1.g.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 5.c odd 4 1
128.1.d.a 1 20.e even 4 1
128.1.d.a 1 40.i odd 4 1
128.1.d.a 1 40.k even 4 1
256.1.c.a 1 80.i odd 4 1
256.1.c.a 1 80.j even 4 1
256.1.c.a 1 80.s even 4 1
256.1.c.a 1 80.t odd 4 1
1024.1.f.b 2 160.u even 8 2
1024.1.f.b 2 160.v odd 8 2
1024.1.f.b 2 160.ba even 8 2
1024.1.f.b 2 160.bb odd 8 2
1152.1.b.a 1 15.e even 4 1
1152.1.b.a 1 60.l odd 4 1
1152.1.b.a 1 120.q odd 4 1
1152.1.b.a 1 120.w even 4 1
2304.1.g.b 1 240.z odd 4 1
2304.1.g.b 1 240.bb even 4 1
2304.1.g.b 1 240.bd odd 4 1
2304.1.g.b 1 240.bf even 4 1
3200.1.e.a 2 1.a even 1 1 trivial
3200.1.e.a 2 4.b odd 2 1 CM
3200.1.e.a 2 5.b even 2 1 inner
3200.1.e.a 2 8.b even 2 1 RM
3200.1.e.a 2 8.d odd 2 1 CM
3200.1.e.a 2 20.d odd 2 1 inner
3200.1.e.a 2 40.e odd 2 1 inner
3200.1.e.a 2 40.f even 2 1 inner
3200.1.g.a 1 5.c odd 4 1
3200.1.g.a 1 20.e even 4 1
3200.1.g.a 1 40.i odd 4 1
3200.1.g.a 1 40.k even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

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