# Properties

 Label 579.1.s.a Level $579$ Weight $1$ Character orbit 579.s Analytic conductor $0.289$ Analytic rank $0$ Dimension $16$ Projective image $D_{32}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$579 = 3 \cdot 193$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 579.s (of order $$32$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.288958642315$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{32})$$ Defining polynomial: $$x^{16} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{32}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{32} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{32}^{10} q^{3} -\zeta_{32}^{2} q^{4} + ( \zeta_{32} + \zeta_{32}^{7} ) q^{7} -\zeta_{32}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{32}^{10} q^{3} -\zeta_{32}^{2} q^{4} + ( \zeta_{32} + \zeta_{32}^{7} ) q^{7} -\zeta_{32}^{4} q^{9} -\zeta_{32}^{12} q^{12} + ( -1 + \zeta_{32}^{5} ) q^{13} + \zeta_{32}^{4} q^{16} + ( \zeta_{32}^{12} + \zeta_{32}^{13} ) q^{19} + ( -\zeta_{32} + \zeta_{32}^{11} ) q^{21} + \zeta_{32}^{9} q^{25} -\zeta_{32}^{14} q^{27} + ( -\zeta_{32}^{3} - \zeta_{32}^{9} ) q^{28} + ( \zeta_{32}^{3} - \zeta_{32}^{15} ) q^{31} + \zeta_{32}^{6} q^{36} + ( -\zeta_{32}^{5} - \zeta_{32}^{8} ) q^{37} + ( -\zeta_{32}^{10} + \zeta_{32}^{15} ) q^{39} + ( -\zeta_{32}^{11} - \zeta_{32}^{13} ) q^{43} + \zeta_{32}^{14} q^{48} + ( \zeta_{32}^{2} + \zeta_{32}^{8} + \zeta_{32}^{14} ) q^{49} + ( \zeta_{32}^{2} - \zeta_{32}^{7} ) q^{52} + ( -\zeta_{32}^{6} - \zeta_{32}^{7} ) q^{57} + ( -\zeta_{32}^{8} + \zeta_{32}^{15} ) q^{61} + ( -\zeta_{32}^{5} - \zeta_{32}^{11} ) q^{63} -\zeta_{32}^{6} q^{64} + ( \zeta_{32}^{8} - \zeta_{32}^{10} ) q^{67} + ( -\zeta_{32}^{6} - \zeta_{32}^{13} ) q^{73} -\zeta_{32}^{3} q^{75} + ( -\zeta_{32}^{14} - \zeta_{32}^{15} ) q^{76} + ( \zeta_{32}^{6} - \zeta_{32}^{9} ) q^{79} + \zeta_{32}^{8} q^{81} + ( \zeta_{32}^{3} - \zeta_{32}^{13} ) q^{84} + ( -\zeta_{32} + \zeta_{32}^{6} - \zeta_{32}^{7} + \zeta_{32}^{12} ) q^{91} + ( \zeta_{32}^{9} + \zeta_{32}^{13} ) q^{93} + ( \zeta_{32}^{3} + \zeta_{32}^{11} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{13} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/579\mathbb{Z}\right)^\times$$.

 $$n$$ $$194$$ $$391$$ $$\chi(n)$$ $$-1$$ $$\zeta_{32}^{9}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 −0.195090 + 0.980785i 0.980785 − 0.195090i 0.831470 + 0.555570i −0.831470 − 0.555570i −0.980785 + 0.195090i 0.195090 − 0.980785i −0.555570 + 0.831470i −0.555570 − 0.831470i −0.980785 − 0.195090i −0.831470 + 0.555570i −0.195090 − 0.980785i 0.195090 + 0.980785i 0.831470 − 0.555570i 0.980785 + 0.195090i 0.555570 + 0.831470i 0.555570 − 0.831470i
0 0.382683 0.923880i 0.923880 + 0.382683i 0 0 0.785695 + 0.785695i 0 −0.707107 0.707107i 0
14.1 0 −0.382683 0.923880i −0.923880 + 0.382683i 0 0 1.17588 1.17588i 0 −0.707107 + 0.707107i 0
23.1 0 0.923880 0.382683i −0.382683 0.923880i 0 0 0.275899 0.275899i 0 0.707107 0.707107i 0
170.1 0 0.923880 0.382683i −0.382683 0.923880i 0 0 −0.275899 + 0.275899i 0 0.707107 0.707107i 0
179.1 0 −0.382683 0.923880i −0.923880 + 0.382683i 0 0 −1.17588 + 1.17588i 0 −0.707107 + 0.707107i 0
185.1 0 0.382683 0.923880i 0.923880 + 0.382683i 0 0 −0.785695 0.785695i 0 −0.707107 0.707107i 0
260.1 0 −0.923880 + 0.382683i 0.382683 + 0.923880i 0 0 −1.38704 + 1.38704i 0 0.707107 0.707107i 0
314.1 0 −0.923880 0.382683i 0.382683 0.923880i 0 0 −1.38704 1.38704i 0 0.707107 + 0.707107i 0
317.1 0 −0.382683 + 0.923880i −0.923880 0.382683i 0 0 −1.17588 1.17588i 0 −0.707107 0.707107i 0
344.1 0 0.923880 + 0.382683i −0.382683 + 0.923880i 0 0 −0.275899 0.275899i 0 0.707107 + 0.707107i 0
362.1 0 0.382683 + 0.923880i 0.923880 0.382683i 0 0 0.785695 0.785695i 0 −0.707107 + 0.707107i 0
410.1 0 0.382683 + 0.923880i 0.923880 0.382683i 0 0 −0.785695 + 0.785695i 0 −0.707107 + 0.707107i 0
428.1 0 0.923880 + 0.382683i −0.382683 + 0.923880i 0 0 0.275899 + 0.275899i 0 0.707107 + 0.707107i 0
455.1 0 −0.382683 + 0.923880i −0.923880 0.382683i 0 0 1.17588 + 1.17588i 0 −0.707107 0.707107i 0
458.1 0 −0.923880 0.382683i 0.382683 0.923880i 0 0 1.38704 + 1.38704i 0 0.707107 + 0.707107i 0
512.1 0 −0.923880 + 0.382683i 0.382683 + 0.923880i 0 0 1.38704 1.38704i 0 0.707107 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 512.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
193.j even 32 1 inner
579.s odd 32 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 579.1.s.a 16
3.b odd 2 1 CM 579.1.s.a 16
193.j even 32 1 inner 579.1.s.a 16
579.s odd 32 1 inner 579.1.s.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
579.1.s.a 16 1.a even 1 1 trivial
579.1.s.a 16 3.b odd 2 1 CM
579.1.s.a 16 193.j even 32 1 inner
579.1.s.a 16 579.s odd 32 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(579, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 1 + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16}$$
$11$ $$T^{16}$$
$13$ $$2 + 16 T + 120 T^{2} + 560 T^{3} + 1820 T^{4} + 4368 T^{5} + 8008 T^{6} + 11440 T^{7} + 12870 T^{8} + 11440 T^{9} + 8008 T^{10} + 4368 T^{11} + 1820 T^{12} + 560 T^{13} + 120 T^{14} + 16 T^{15} + T^{16}$$
$17$ $$T^{16}$$
$19$ $$2 - 16 T + 72 T^{2} - 80 T^{3} + 4 T^{4} + 56 T^{6} + 160 T^{7} + 6 T^{8} - 16 T^{11} + 4 T^{12} + T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$16 + 64 T^{4} + 128 T^{8} - 16 T^{12} + T^{16}$$
$37$ $$2 + 16 T + 8 T^{2} - 112 T^{3} + 28 T^{4} + 112 T^{5} + 56 T^{6} - 16 T^{7} + 70 T^{8} + 56 T^{10} + 28 T^{12} + 8 T^{14} + T^{16}$$
$41$ $$T^{16}$$
$43$ $$4 + 176 T^{4} + 148 T^{8} + 24 T^{12} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$T^{16}$$
$61$ $$2 - 16 T + 8 T^{2} + 112 T^{3} + 28 T^{4} - 112 T^{5} + 56 T^{6} + 16 T^{7} + 70 T^{8} + 56 T^{10} + 28 T^{12} + 8 T^{14} + T^{16}$$
$67$ $$( 2 - 8 T + 4 T^{2} + 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$71$ $$T^{16}$$
$73$ $$2 + 16 T + 40 T^{2} + 140 T^{4} - 48 T^{5} + 192 T^{7} + 2 T^{8} + 88 T^{10} + 16 T^{13} + T^{16}$$
$79$ $$2 - 16 T + 88 T^{2} - 192 T^{3} + 140 T^{4} - 16 T^{5} + 2 T^{8} + 48 T^{9} + 40 T^{10} + T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$256 + T^{16}$$