Properties

Label 583.1.o.a
Level $583$
Weight $1$
Character orbit 583.o
Analytic conductor $0.291$
Analytic rank $0$
Dimension $12$
Projective image $D_{13}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [583,1,Mod(10,583)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(583, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 24]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("583.10");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 583 = 11 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 583.o (of order \(26\), degree \(12\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.290954902365\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
comment: defining polynomial
 
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: \(D_{13}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\)

$q$-expansion

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

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

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

Character values

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

\(n\) \(266\) \(320\)
\(\chi(n)\) \(-1\) \(\zeta_{26}^{8}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
0.748511 + 0.663123i
0.970942 0.239316i
−0.120537 0.992709i
0.748511 0.663123i
0.354605 + 0.935016i
−0.885456 0.464723i
−0.120537 + 0.992709i
0.354605 0.935016i
−0.885456 + 0.464723i
−0.568065 + 0.822984i
−0.568065 0.822984i
0.970942 + 0.239316i
0 −0.402877 + 0.583668i 0.885456 0.464723i −0.402877 1.06230i 0 0 0 0.176246 + 0.464723i 0
142.1 0 −0.180446 + 0.159861i −0.354605 0.935016i −0.180446 + 1.48611i 0 0 0 −0.113532 + 0.935016i 0
153.1 0 0.530851 + 1.39974i 0.568065 0.822984i 0.530851 0.470293i 0 0 0 −0.928957 + 0.822984i 0
175.1 0 −0.402877 0.583668i 0.885456 + 0.464723i −0.402877 + 1.06230i 0 0 0 0.176246 0.464723i 0
208.1 0 1.00599 + 0.527986i −0.970942 0.239316i 1.00599 + 1.45743i 0 0 0 0.165188 + 0.239316i 0
307.1 0 −0.234068 1.92773i −0.748511 0.663123i −0.234068 + 0.0576926i 0 0 0 −2.69039 + 0.663123i 0
362.1 0 0.530851 1.39974i 0.568065 + 0.822984i 0.530851 + 0.470293i 0 0 0 −0.928957 0.822984i 0
384.1 0 1.00599 0.527986i −0.970942 + 0.239316i 1.00599 1.45743i 0 0 0 0.165188 0.239316i 0
395.1 0 −0.234068 + 1.92773i −0.748511 + 0.663123i −0.234068 0.0576926i 0 0 0 −2.69039 0.663123i 0
417.1 0 −1.71945 0.423807i 0.120537 0.992709i −1.71945 0.902438i 0 0 0 1.89145 + 0.992709i 0
439.1 0 −1.71945 + 0.423807i 0.120537 + 0.992709i −1.71945 + 0.902438i 0 0 0 1.89145 0.992709i 0
505.1 0 −0.180446 0.159861i −0.354605 + 0.935016i −0.180446 1.48611i 0 0 0 −0.113532 0.935016i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.1
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.d even 13 1 inner
583.o 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.o.a 12
11.b odd 2 1 CM 583.1.o.a 12
53.d even 13 1 inner 583.1.o.a 12
583.o odd 26 1 inner 583.1.o.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
583.1.o.a 12 1.a even 1 1 trivial
583.1.o.a 12 11.b odd 2 1 CM
583.1.o.a 12 53.d even 13 1 inner
583.1.o.a 12 583.o 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
$3$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + T^{10} + T^{9} + T^{8} + T^{7} + \cdots + 1 \) 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
$23$ \( (T^{6} + T^{5} - 5 T^{4} - 4 T^{3} + 6 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} + T^{10} + T^{9} + T^{8} + T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} + 2 T^{11} + 4 T^{10} - 5 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) 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
$89$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} + 2 T^{11} + 4 T^{10} + 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
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