Properties

 Label 583.1.n.a Level $583$ Weight $1$ Character orbit 583.n Analytic conductor $0.291$ Analytic rank $0$ Dimension $12$ Projective image $D_{26}$ CM discriminant -11 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$583 = 11 \cdot 53$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 583.n (of order $$26$$, degree $$12$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.290954902365$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{26})$$ Defining polynomial: $$x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{26}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{26} - \cdots)$$

$q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{26}^{6} - \zeta_{26}^{11} ) q^{3} + \zeta_{26} q^{4} + ( -\zeta_{26}^{2} + \zeta_{26}^{6} ) q^{5} + ( -\zeta_{26}^{4} - \zeta_{26}^{9} + \zeta_{26}^{12} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{26}^{6} - \zeta_{26}^{11} ) q^{3} + \zeta_{26} q^{4} + ( -\zeta_{26}^{2} + \zeta_{26}^{6} ) q^{5} + ( -\zeta_{26}^{4} - \zeta_{26}^{9} + \zeta_{26}^{12} ) q^{9} -\zeta_{26}^{3} q^{11} + ( -\zeta_{26}^{7} - \zeta_{26}^{12} ) q^{12} + ( -1 + \zeta_{26}^{4} + \zeta_{26}^{8} - \zeta_{26}^{12} ) q^{15} + \zeta_{26}^{2} q^{16} + ( -\zeta_{26}^{3} + \zeta_{26}^{7} ) q^{20} + ( \zeta_{26}^{3} + \zeta_{26}^{10} ) q^{23} + ( \zeta_{26}^{4} - \zeta_{26}^{8} + \zeta_{26}^{12} ) q^{25} + ( -\zeta_{26}^{2} + \zeta_{26}^{5} - \zeta_{26}^{7} + \zeta_{26}^{10} ) q^{27} + ( \zeta_{26}^{9} - \zeta_{26}^{11} ) q^{31} + ( -\zeta_{26} + \zeta_{26}^{9} ) q^{33} + ( -1 - \zeta_{26}^{5} - \zeta_{26}^{10} ) q^{36} + ( -\zeta_{26}^{6} + \zeta_{26}^{11} ) q^{37} -\zeta_{26}^{4} q^{44} + ( \zeta_{26} + \zeta_{26}^{2} - \zeta_{26}^{5} + \zeta_{26}^{6} - \zeta_{26}^{10} + \zeta_{26}^{11} ) q^{45} + ( \zeta_{26}^{7} + \zeta_{26}^{11} ) q^{47} + ( 1 - \zeta_{26}^{8} ) q^{48} -\zeta_{26} q^{49} -\zeta_{26}^{5} q^{53} + ( \zeta_{26}^{5} - \zeta_{26}^{9} ) q^{55} + ( -\zeta_{26}^{8} - \zeta_{26}^{10} ) q^{59} + ( 1 - \zeta_{26} + \zeta_{26}^{5} + \zeta_{26}^{9} ) q^{60} + \zeta_{26}^{3} q^{64} + ( \zeta_{26}^{7} + \zeta_{26}^{8} ) q^{67} + ( \zeta_{26} + \zeta_{26}^{3} + \zeta_{26}^{8} - \zeta_{26}^{9} ) q^{69} + ( \zeta_{26}^{3} + \zeta_{26}^{6} ) q^{71} + ( -\zeta_{26} + \zeta_{26}^{2} + \zeta_{26}^{5} - \zeta_{26}^{6} ) q^{75} + ( -\zeta_{26}^{4} + \zeta_{26}^{8} ) q^{80} + ( -1 + \zeta_{26}^{3} - \zeta_{26}^{5} + \zeta_{26}^{8} - \zeta_{26}^{11} ) q^{81} + ( -\zeta_{26}^{2} - \zeta_{26}^{8} ) q^{89} + ( \zeta_{26}^{4} + \zeta_{26}^{11} ) q^{92} + ( \zeta_{26}^{2} - \zeta_{26}^{4} + \zeta_{26}^{7} - \zeta_{26}^{9} ) q^{93} + ( -\zeta_{26} - \zeta_{26}^{7} ) q^{97} + ( \zeta_{26}^{2} + \zeta_{26}^{7} + \zeta_{26}^{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + q^{4} - q^{9} + O(q^{10})$$ $$12q + q^{4} - q^{9} - q^{11} - 13q^{15} - q^{16} - q^{25} - 12q^{36} + 2q^{37} + q^{44} + 2q^{47} + 13q^{48} - q^{49} - q^{53} + 2q^{59} + 13q^{60} + q^{64} - 14q^{81} + 2q^{89} - 2q^{97} - q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/583\mathbb{Z}\right)^\times$$.

 $$n$$ $$266$$ $$320$$ $$\chi(n)$$ $$-1$$ $$\zeta_{26}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.885456 + 0.464723i 0.354605 − 0.935016i −0.120537 − 0.992709i −0.120537 + 0.992709i 0.748511 − 0.663123i 0.970942 − 0.239316i −0.568065 − 0.822984i 0.748511 + 0.663123i 0.970942 + 0.239316i −0.885456 − 0.464723i −0.568065 + 0.822984i 0.354605 + 0.935016i
0 1.53901 + 1.06230i −0.885456 + 0.464723i −1.53901 + 0.583668i 0 0 0 0.885456 + 2.33476i 0
131.1 0 −1.31658 + 1.48611i 0.354605 0.935016i 1.31658 0.159861i 0 0 0 −0.354605 2.92043i 0
197.1 0 −0.222431 0.902438i −0.120537 0.992709i 0.222431 + 0.423807i 0 0 0 0.120537 0.0632625i 0
219.1 0 −0.222431 + 0.902438i −0.120537 + 0.992709i 0.222431 0.423807i 0 0 0 0.120537 + 0.0632625i 0
241.1 0 0.475142 + 0.0576926i 0.748511 0.663123i −0.475142 + 1.92773i 0 0 0 −0.748511 0.184491i 0
252.1 0 0.764919 + 1.45743i 0.970942 0.239316i −0.764919 0.527986i 0 0 0 −0.970942 + 1.40665i 0
274.1 0 −1.24006 0.470293i −0.568065 0.822984i 1.24006 1.39974i 0 0 0 0.568065 + 0.503261i 0
329.1 0 0.475142 0.0576926i 0.748511 + 0.663123i −0.475142 1.92773i 0 0 0 −0.748511 + 0.184491i 0
428.1 0 0.764919 1.45743i 0.970942 + 0.239316i −0.764919 + 0.527986i 0 0 0 −0.970942 1.40665i 0
461.1 0 1.53901 1.06230i −0.885456 0.464723i −1.53901 0.583668i 0 0 0 0.885456 2.33476i 0
483.1 0 −1.24006 + 0.470293i −0.568065 + 0.822984i 1.24006 + 1.39974i 0 0 0 0.568065 0.503261i 0
494.1 0 −1.31658 1.48611i 0.354605 + 0.935016i 1.31658 + 0.159861i 0 0 0 −0.354605 + 2.92043i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 494.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
53.e even 26 1 inner
583.n odd 26 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 583.1.n.a 12
11.b odd 2 1 CM 583.1.n.a 12
53.e even 26 1 inner 583.1.n.a 12
583.n odd 26 1 inner 583.1.n.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
583.1.n.a 12 1.a even 1 1 trivial
583.1.n.a 12 11.b odd 2 1 CM
583.1.n.a 12 53.e even 26 1 inner
583.1.n.a 12 583.n odd 26 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(583, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$13 - 39 T + 13 T^{2} + 39 T^{4} + 39 T^{5} + 13 T^{8} + T^{12}$$
$5$ $$13 - 13 T + 26 T^{2} + 52 T^{3} - 65 T^{5} + 13 T^{7} + T^{12}$$
$7$ $$T^{12}$$
$11$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$
$13$ $$T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$13 + 91 T^{2} + 182 T^{4} + 156 T^{6} + 65 T^{8} + 13 T^{10} + T^{12}$$
$29$ $$T^{12}$$
$31$ $$13 - 65 T + 156 T^{2} - 182 T^{3} + 91 T^{4} - 13 T^{5} + T^{12}$$
$37$ $$1 - 7 T + 23 T^{2} - 18 T^{3} + 9 T^{4} + 15 T^{5} + 12 T^{6} - 6 T^{7} + 3 T^{8} - 8 T^{9} + 4 T^{10} - 2 T^{11} + T^{12}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$1 - 7 T + 10 T^{2} + 8 T^{3} + 22 T^{4} - 11 T^{5} + 38 T^{6} - 19 T^{7} + 16 T^{8} - 8 T^{9} + 4 T^{10} - 2 T^{11} + T^{12}$$
$53$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12}$$
$59$ $$1 + 6 T + 10 T^{2} - 5 T^{3} + 35 T^{4} - 24 T^{5} + 12 T^{6} + 20 T^{7} - 10 T^{8} + 5 T^{9} + 4 T^{10} - 2 T^{11} + T^{12}$$
$61$ $$T^{12}$$
$67$ $$13 - 13 T + 26 T^{2} + 52 T^{3} - 65 T^{5} + 13 T^{7} + T^{12}$$
$71$ $$13 + 65 T + 156 T^{2} + 182 T^{3} + 91 T^{4} + 13 T^{5} + T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$1 - 7 T + 23 T^{2} - 18 T^{3} + 9 T^{4} + 15 T^{5} + 12 T^{6} - 6 T^{7} + 3 T^{8} - 8 T^{9} + 4 T^{10} - 2 T^{11} + T^{12}$$
$97$ $$1 + 7 T + 36 T^{2} + 96 T^{3} + 139 T^{4} + 115 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12}$$