# Properties

 Label 3200.1.m.b Level $3200$ Weight $1$ Character orbit 3200.m Analytic conductor $1.597$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3200.m (of order $$4$$, degree $$2$$, 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 640) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.32000.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.131072000.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{9} +O(q^{10})$$ $$q -i q^{9} + ( 1 + i ) q^{13} + ( -1 - i ) q^{17} + 2 q^{29} + ( 1 - i ) q^{37} -i q^{49} + ( -1 - i ) q^{53} + 2 i q^{61} + ( 1 - i ) q^{73} - q^{81} + ( 1 + i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 2 q^{13} - 2 q^{17} + 4 q^{29} + 2 q^{37} - 2 q^{53} + 2 q^{73} - 2 q^{81} + 2 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 0 0 0 0 0 0 1.00000i 0
1857.1 0 0 0 0 0 0 0 1.00000i 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 $$\Q(\sqrt{-1})$$
40.i odd 4 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.m.b 2
4.b odd 2 1 CM 3200.1.m.b 2
5.b even 2 1 640.1.m.a 2
5.c odd 4 1 640.1.m.b yes 2
5.c odd 4 1 3200.1.m.a 2
8.b even 2 1 3200.1.m.a 2
8.d odd 2 1 3200.1.m.a 2
20.d odd 2 1 640.1.m.a 2
20.e even 4 1 640.1.m.b yes 2
20.e even 4 1 3200.1.m.a 2
40.e odd 2 1 640.1.m.b yes 2
40.f even 2 1 640.1.m.b yes 2
40.i odd 4 1 640.1.m.a 2
40.i odd 4 1 inner 3200.1.m.b 2
40.k even 4 1 640.1.m.a 2
40.k even 4 1 inner 3200.1.m.b 2
80.i odd 4 1 1280.1.p.b 2
80.j even 4 1 1280.1.p.a 2
80.k odd 4 1 1280.1.p.a 2
80.k odd 4 1 1280.1.p.b 2
80.q even 4 1 1280.1.p.a 2
80.q even 4 1 1280.1.p.b 2
80.s even 4 1 1280.1.p.b 2
80.t odd 4 1 1280.1.p.a 2

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

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}$$ $$T_{13}^{2} - 2 T_{13} + 2$$

## Hecke characteristic polynomials

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