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 $4$

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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:

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

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