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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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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [583,1,Mod(43,583)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("583.43"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(583, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([13, 11])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 583 = 11 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 583.n (of order \(26\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.290954902365\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{26}^{11} - \zeta_{26}^{6}) q^{3} + \zeta_{26} q^{4} + (\zeta_{26}^{6} - \zeta_{26}^{2}) q^{5} + (\zeta_{26}^{12} + \cdots - \zeta_{26}^{4}) q^{9} - \zeta_{26}^{3} q^{11} + ( - \zeta_{26}^{12} - \zeta_{26}^{7}) q^{12} + \cdots + (\zeta_{26}^{12} + \cdots + \zeta_{26}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{4} - q^{9} - q^{11} - 13 q^{15} - q^{16} - q^{25} - 12 q^{36} + 2 q^{37} + q^{44} + 2 q^{47} + 13 q^{48} - q^{49} - q^{53} + 2 q^{59} + 13 q^{60} + q^{64} - 14 q^{81} + 2 q^{89} - 2 q^{97}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
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 43.1
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

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 13 T^{8} + \cdots + 13 \) Copy content Toggle raw display
$5$ \( T^{12} + 13 T^{7} + \cdots + 13 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 13 T^{10} + \cdots + 13 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 13 T^{5} + \cdots + 13 \) Copy content Toggle raw display
$37$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + 13 T^{7} + \cdots + 13 \) Copy content Toggle raw display
$71$ \( T^{12} + 13 T^{5} + \cdots + 13 \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
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