# Properties

 Label 3200.1.m.d Level $3200$ Weight $1$ Character orbit 3200.m Analytic conductor $1.597$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -8, -20, 40 Inner twists $16$

# 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{-5})$$ Artin image: $\OD_{16}:C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{3} q^{3} - 3 \zeta_{8}^{2} q^{9} +O(q^{10})$$ q - z^3 * q^3 - 3*z^2 * q^9 $$q - \zeta_{8}^{3} q^{3} - 3 \zeta_{8}^{2} q^{9} + 2 \zeta_{8} q^{27} + q^{41} + \zeta_{8}^{3} q^{43} + \zeta_{8}^{2} q^{49} - \zeta_{8} q^{67} + 3 q^{81} - \zeta_{8}^{3} q^{83} + \zeta_{8}^{2} q^{89} +O(q^{100})$$ q - z^3 * q^3 - 3*z^2 * q^9 + 2*z * q^27 + q^41 + z^3 * q^43 + z^2 * q^49 - z * q^67 + 3 * q^81 - z^3 * q^83 + z^2 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{41} - 20 q^{81}+O(q^{100})$$ 4 * q + 8 * q^41 - 20 * q^81

## Character values

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

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −1.41421 + 1.41421i 0 0 0 0 0 3.00000i 0
193.2 0 1.41421 1.41421i 0 0 0 0 0 3.00000i 0
1857.1 0 −1.41421 1.41421i 0 0 0 0 0 3.00000i 0
1857.2 0 1.41421 + 1.41421i 0 0 0 0 0 3.00000i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
40.f even 2 1 RM by $$\Q(\sqrt{10})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
8.b even 2 1 inner
20.e even 4 2 inner
40.e odd 2 1 inner
40.i odd 4 2 inner
40.k even 4 2 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} + 16$$ T3^4 + 16 $$T_{7}$$ T7 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 16$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T - 2)^{4}$$
$43$ $$T^{4} + 16$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4} + 16$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 16$$
$89$ $$(T^{2} + 4)^{2}$$
$97$ $$T^{4}$$