# Properties

 Label 3200.1.m.e Level $3200$ Weight $1$ Character orbit 3200.m Analytic conductor $1.597$ Analytic rank $0$ Dimension $8$ Projective image $D_{6}$ CM discriminant -8 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.6400000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{3} q^{3}+O(q^{10})$$ q + z^3 * q^3 $$q + \zeta_{24}^{3} q^{3} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{11} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{17} + ( - \zeta_{24}^{10} + \zeta_{24}^{2}) q^{19} - \zeta_{24}^{9} q^{27} + ( - \zeta_{24}^{11} - \zeta_{24}^{7}) q^{33} - q^{41} + \zeta_{24}^{3} q^{43} - \zeta_{24}^{6} q^{49} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{51} + (\zeta_{24}^{5} + \zeta_{24}) q^{57} + \zeta_{24}^{9} q^{67} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{73} + q^{81} + \zeta_{24}^{3} q^{83} + \zeta_{24}^{6} q^{89} +O(q^{100})$$ q + z^3 * q^3 + (-z^8 - z^4) * q^11 + (-z^5 - z) * q^17 + (-z^10 + z^2) * q^19 - z^9 * q^27 + (-z^11 - z^7) * q^33 - q^41 + z^3 * q^43 - z^6 * q^49 + (-z^8 - z^4) * q^51 + (z^5 + z) * q^57 + z^9 * q^67 + (z^11 + z^7) * q^73 + q^81 + z^3 * q^83 + z^6 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 8 q^{41} + 8 q^{81}+O(q^{100})$$ 8 * q - 8 * q^41 + 8 * q^81

## Character values

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

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{24}^{6}$$

## 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
 0.258819 − 0.965926i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.258819 − 0.965926i
0 −0.707107 + 0.707107i 0 0 0 0 0 0 0
193.2 0 −0.707107 + 0.707107i 0 0 0 0 0 0 0
193.3 0 0.707107 0.707107i 0 0 0 0 0 0 0
193.4 0 0.707107 0.707107i 0 0 0 0 0 0 0
1857.1 0 −0.707107 0.707107i 0 0 0 0 0 0 0
1857.2 0 −0.707107 0.707107i 0 0 0 0 0 0 0
1857.3 0 0.707107 + 0.707107i 0 0 0 0 0 0 0
1857.4 0 0.707107 + 0.707107i 0 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1857.4 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 $$\Q(\sqrt{-2})$$
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.d odd 2 1 inner
20.e even 4 2 inner
40.e odd 2 1 inner
40.f even 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.e 8
4.b odd 2 1 inner 3200.1.m.e 8
5.b even 2 1 inner 3200.1.m.e 8
5.c odd 4 2 inner 3200.1.m.e 8
8.b even 2 1 inner 3200.1.m.e 8
8.d odd 2 1 CM 3200.1.m.e 8
20.d odd 2 1 inner 3200.1.m.e 8
20.e even 4 2 inner 3200.1.m.e 8
40.e odd 2 1 inner 3200.1.m.e 8
40.f even 2 1 inner 3200.1.m.e 8
40.i odd 4 2 inner 3200.1.m.e 8
40.k even 4 2 inner 3200.1.m.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.m.e 8 1.a even 1 1 trivial
3200.1.m.e 8 4.b odd 2 1 inner
3200.1.m.e 8 5.b even 2 1 inner
3200.1.m.e 8 5.c odd 4 2 inner
3200.1.m.e 8 8.b even 2 1 inner
3200.1.m.e 8 8.d odd 2 1 CM
3200.1.m.e 8 20.d odd 2 1 inner
3200.1.m.e 8 20.e even 4 2 inner
3200.1.m.e 8 40.e odd 2 1 inner
3200.1.m.e 8 40.f even 2 1 inner
3200.1.m.e 8 40.i odd 4 2 inner
3200.1.m.e 8 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 $$S_{1}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{4} + 1$$ T3^4 + 1 $$T_{7}$$ T7 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 3)^{4}$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 9)^{2}$$
$19$ $$(T^{2} - 3)^{4}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T + 1)^{8}$$
$43$ $$(T^{4} + 16)^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$(T^{4} + 1)^{2}$$
$71$ $$T^{8}$$
$73$ $$(T^{4} + 9)^{2}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} + 1)^{2}$$
$89$ $$(T^{2} + 1)^{4}$$
$97$ $$T^{8}$$