# Properties

 Label 3200.2.f.s Level $3200$ Weight $2$ Character orbit 3200.f Analytic conductor $25.552$ Analytic rank $0$ Dimension $8$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{10}\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{9} + ( -3 + \zeta_{24} - \zeta_{24}^{3} + 6 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{11} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{17} + ( 1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{19} + ( -5 \zeta_{24} + 6 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{27} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 11 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{33} + ( 3 - 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{41} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{43} + 7 q^{49} + ( 7 - 9 \zeta_{24} + 9 \zeta_{24}^{3} - 14 \zeta_{24}^{4} + 9 \zeta_{24}^{5} ) q^{51} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{57} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{59} + ( -3 \zeta_{24} - 14 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 7 \zeta_{24}^{6} ) q^{67} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{73} + ( 13 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{81} + ( -\zeta_{24} - 18 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + 9 \zeta_{24}^{6} ) q^{83} + ( -9 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{89} -10 \zeta_{24}^{6} q^{97} + ( -10 + 20 \zeta_{24} - 20 \zeta_{24}^{3} + 20 \zeta_{24}^{4} - 20 \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{9} + O(q^{10})$$ $$8q + 16q^{9} + 24q^{41} + 56q^{49} + 104q^{81} - 72q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i −0.965926 − 0.258819i
0 −3.14626 0 0 0 0 0 6.89898 0
449.2 0 −3.14626 0 0 0 0 0 6.89898 0
449.3 0 −0.317837 0 0 0 0 0 −2.89898 0
449.4 0 −0.317837 0 0 0 0 0 −2.89898 0
449.5 0 0.317837 0 0 0 0 0 −2.89898 0
449.6 0 0.317837 0 0 0 0 0 −2.89898 0
449.7 0 3.14626 0 0 0 0 0 6.89898 0
449.8 0 3.14626 0 0 0 0 0 6.89898 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 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
8.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 3200.2.f.s 8
4.b odd 2 1 inner 3200.2.f.s 8
5.b even 2 1 inner 3200.2.f.s 8
5.c odd 4 1 3200.2.d.n 4
5.c odd 4 1 3200.2.d.q yes 4
8.b even 2 1 inner 3200.2.f.s 8
8.d odd 2 1 CM 3200.2.f.s 8
20.d odd 2 1 inner 3200.2.f.s 8
20.e even 4 1 3200.2.d.n 4
20.e even 4 1 3200.2.d.q yes 4
40.e odd 2 1 inner 3200.2.f.s 8
40.f even 2 1 inner 3200.2.f.s 8
40.i odd 4 1 3200.2.d.n 4
40.i odd 4 1 3200.2.d.q yes 4
40.k even 4 1 3200.2.d.n 4
40.k even 4 1 3200.2.d.q yes 4
80.i odd 4 1 6400.2.a.cq 4
80.i odd 4 1 6400.2.a.cr 4
80.j even 4 1 6400.2.a.cq 4
80.j even 4 1 6400.2.a.cr 4
80.s even 4 1 6400.2.a.cq 4
80.s even 4 1 6400.2.a.cr 4
80.t odd 4 1 6400.2.a.cq 4
80.t odd 4 1 6400.2.a.cr 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.n 4 5.c odd 4 1
3200.2.d.n 4 20.e even 4 1
3200.2.d.n 4 40.i odd 4 1
3200.2.d.n 4 40.k even 4 1
3200.2.d.q yes 4 5.c odd 4 1
3200.2.d.q yes 4 20.e even 4 1
3200.2.d.q yes 4 40.i odd 4 1
3200.2.d.q yes 4 40.k even 4 1
3200.2.f.s 8 1.a even 1 1 trivial
3200.2.f.s 8 4.b odd 2 1 inner
3200.2.f.s 8 5.b even 2 1 inner
3200.2.f.s 8 8.b even 2 1 inner
3200.2.f.s 8 8.d odd 2 1 CM
3200.2.f.s 8 20.d odd 2 1 inner
3200.2.f.s 8 40.e odd 2 1 inner
3200.2.f.s 8 40.f even 2 1 inner
6400.2.a.cq 4 80.i odd 4 1
6400.2.a.cq 4 80.j even 4 1
6400.2.a.cq 4 80.s even 4 1
6400.2.a.cq 4 80.t odd 4 1
6400.2.a.cr 4 80.i odd 4 1
6400.2.a.cr 4 80.j even 4 1
6400.2.a.cr 4 80.s even 4 1
6400.2.a.cr 4 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3}^{4} - 10 T_{3}^{2} + 1$$ $$T_{7}$$ $$T_{11}^{4} + 58 T_{11}^{2} + 625$$ $$T_{13}$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 - 10 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 625 + 58 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( 225 + 66 T^{2} + T^{4} )^{2}$$
$19$ $$( 225 + 42 T^{2} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$( -87 - 6 T + T^{2} )^{4}$$
$43$ $$( -72 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 200 + T^{2} )^{4}$$
$61$ $$T^{8}$$
$67$ $$( 16641 - 330 T^{2} + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 46225 + 434 T^{2} + T^{4} )^{2}$$
$79$ $$T^{8}$$
$83$ $$( 58081 - 490 T^{2} + T^{4} )^{2}$$
$89$ $$( 57 + 18 T + T^{2} )^{4}$$
$97$ $$( 100 + T^{2} )^{4}$$