# Properties

 Label 3600.1.dx.a Level 3600 Weight 1 Character orbit 3600.dx Analytic conductor 1.797 Analytic rank 0 Dimension 16 Projective image $$D_{20}$$ CM discriminant -4 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.dx (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: $$\Q(\zeta_{40})$$ Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{20}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{20} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{40}^{7} q^{5} +O(q^{10})$$ $$q -\zeta_{40}^{7} q^{5} + ( -\zeta_{40}^{4} + \zeta_{40}^{18} ) q^{13} + ( \zeta_{40} - \zeta_{40}^{13} ) q^{17} + \zeta_{40}^{14} q^{25} + ( -\zeta_{40}^{7} - \zeta_{40}^{9} ) q^{29} + ( \zeta_{40}^{10} + \zeta_{40}^{12} ) q^{37} + ( -\zeta_{40}^{3} - \zeta_{40}^{9} ) q^{41} -\zeta_{40}^{10} q^{49} + ( -\zeta_{40}^{11} + \zeta_{40}^{15} ) q^{53} + ( -\zeta_{40}^{2} - \zeta_{40}^{6} ) q^{61} + ( \zeta_{40}^{5} + \zeta_{40}^{11} ) q^{65} + ( -\zeta_{40}^{2} - \zeta_{40}^{16} ) q^{73} + ( -1 - \zeta_{40}^{8} ) q^{85} + ( -\zeta_{40}^{13} + \zeta_{40}^{15} ) q^{89} + ( \zeta_{40}^{12} - \zeta_{40}^{14} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 4q^{13} + 4q^{37} + 4q^{73} - 12q^{85} + 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$-\zeta_{40}^{18}$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
287.1
 −0.156434 + 0.987688i 0.156434 − 0.987688i 0.987688 + 0.156434i −0.987688 − 0.156434i −0.891007 + 0.453990i 0.891007 − 0.453990i 0.987688 − 0.156434i −0.987688 + 0.156434i 0.453990 − 0.891007i −0.453990 + 0.891007i −0.891007 − 0.453990i 0.891007 + 0.453990i −0.156434 − 0.987688i 0.156434 + 0.987688i 0.453990 + 0.891007i −0.453990 − 0.891007i
0 0 0 −0.891007 + 0.453990i 0 0 0 0 0
287.2 0 0 0 0.891007 0.453990i 0 0 0 0 0
863.1 0 0 0 −0.453990 0.891007i 0 0 0 0 0
863.2 0 0 0 0.453990 + 0.891007i 0 0 0 0 0
1583.1 0 0 0 −0.987688 + 0.156434i 0 0 0 0 0
1583.2 0 0 0 0.987688 0.156434i 0 0 0 0 0
1727.1 0 0 0 −0.453990 + 0.891007i 0 0 0 0 0
1727.2 0 0 0 0.453990 0.891007i 0 0 0 0 0
2303.1 0 0 0 −0.156434 + 0.987688i 0 0 0 0 0
2303.2 0 0 0 0.156434 0.987688i 0 0 0 0 0
2447.1 0 0 0 −0.987688 0.156434i 0 0 0 0 0
2447.2 0 0 0 0.987688 + 0.156434i 0 0 0 0 0
3023.1 0 0 0 −0.891007 0.453990i 0 0 0 0 0
3023.2 0 0 0 0.891007 + 0.453990i 0 0 0 0 0
3167.1 0 0 0 −0.156434 0.987688i 0 0 0 0 0
3167.2 0 0 0 0.156434 + 0.987688i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3167.2 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.b even 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
100.l even 20 1 inner
300.u odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.dx.a 16
3.b odd 2 1 inner 3600.1.dx.a 16
4.b odd 2 1 CM 3600.1.dx.a 16
12.b even 2 1 inner 3600.1.dx.a 16
25.f odd 20 1 inner 3600.1.dx.a 16
75.l even 20 1 inner 3600.1.dx.a 16
100.l even 20 1 inner 3600.1.dx.a 16
300.u odd 20 1 inner 3600.1.dx.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.dx.a 16 1.a even 1 1 trivial
3600.1.dx.a 16 3.b odd 2 1 inner
3600.1.dx.a 16 4.b odd 2 1 CM
3600.1.dx.a 16 12.b even 2 1 inner
3600.1.dx.a 16 25.f odd 20 1 inner
3600.1.dx.a 16 75.l even 20 1 inner
3600.1.dx.a 16 100.l even 20 1 inner
3600.1.dx.a 16 300.u odd 20 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T^{4} + T^{8} - T^{12} + T^{16}$$
$7$ $$( 1 + T^{4} )^{8}$$
$11$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$13$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$17$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$19$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$23$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$29$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$37$ $$( 1 + T^{2} )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{4}$$
$41$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$43$ $$( 1 + T^{4} )^{8}$$
$47$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$53$ $$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$
$59$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4}$$
$61$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$67$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$71$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$73$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$
$79$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4}$$
$83$ $$( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2}$$
$89$ $$( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} )$$
$97$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}$$