Properties

Label 584.1.br.a
Level $584$
Weight $1$
Character orbit 584.br
Analytic conductor $0.291$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [584,1,Mod(19,584)] 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("584.19"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(584, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([18, 18, 31])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 584 = 2^{3} \cdot 73 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 584.br (of order \(36\), 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: \(0.291453967378\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{36}^{2} q^{2} + ( - \zeta_{36}^{7} + \zeta_{36}^{5}) q^{3} + \zeta_{36}^{4} q^{4} + ( - \zeta_{36}^{9} + \zeta_{36}^{7}) q^{6} + \zeta_{36}^{6} q^{8} + (\zeta_{36}^{14} + \cdots + \zeta_{36}^{10}) q^{9}+ \cdots + ( - \zeta_{36}^{16} - \zeta_{36}^{13} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{8} + 6 q^{9} - 6 q^{11} - 12 q^{17} - 6 q^{18} + 6 q^{33} - 12 q^{36} - 6 q^{64} - 6 q^{66} + 12 q^{72} - 6 q^{75} + 18 q^{76} - 6 q^{81} + 6 q^{83} + 6 q^{88} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/584\mathbb{Z}\right)^\times\).

\(n\) \(293\) \(297\) \(439\)
\(\chi(n)\) \(-1\) \(-\zeta_{36}^{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.984808 + 0.173648i
0.342020 + 0.939693i
−0.642788 0.766044i
−0.984808 0.173648i
−0.342020 + 0.939693i
0.342020 0.939693i
0.984808 + 0.173648i
0.642788 + 0.766044i
−0.342020 0.939693i
0.984808 0.173648i
0.642788 0.766044i
−0.642788 + 0.766044i
0.939693 0.342020i −0.300767 0.173648i 0.766044 0.642788i 0 −0.342020 0.0603074i 0 0.500000 0.866025i −0.439693 0.761570i 0
35.1 −0.766044 + 0.642788i 1.62760 0.939693i 0.173648 0.984808i 0 −0.642788 + 1.76604i 0 0.500000 + 0.866025i 1.26604 2.19285i 0
67.1 −0.173648 + 0.984808i 1.32683 + 0.766044i −0.939693 0.342020i 0 −0.984808 + 1.17365i 0 0.500000 0.866025i 0.673648 + 1.16679i 0
123.1 0.939693 + 0.342020i −0.300767 + 0.173648i 0.766044 + 0.642788i 0 −0.342020 + 0.0603074i 0 0.500000 + 0.866025i −0.439693 + 0.761570i 0
171.1 −0.766044 0.642788i −1.62760 0.939693i 0.173648 + 0.984808i 0 0.642788 + 1.76604i 0 0.500000 0.866025i 1.26604 + 2.19285i 0
267.1 −0.766044 0.642788i 1.62760 + 0.939693i 0.173648 + 0.984808i 0 −0.642788 1.76604i 0 0.500000 0.866025i 1.26604 + 2.19285i 0
315.1 0.939693 + 0.342020i 0.300767 0.173648i 0.766044 + 0.642788i 0 0.342020 0.0603074i 0 0.500000 + 0.866025i −0.439693 + 0.761570i 0
371.1 −0.173648 + 0.984808i −1.32683 0.766044i −0.939693 0.342020i 0 0.984808 1.17365i 0 0.500000 0.866025i 0.673648 + 1.16679i 0
403.1 −0.766044 + 0.642788i −1.62760 + 0.939693i 0.173648 0.984808i 0 0.642788 1.76604i 0 0.500000 + 0.866025i 1.26604 2.19285i 0
419.1 0.939693 0.342020i 0.300767 + 0.173648i 0.766044 0.642788i 0 0.342020 + 0.0603074i 0 0.500000 0.866025i −0.439693 0.761570i 0
499.1 −0.173648 0.984808i −1.32683 + 0.766044i −0.939693 + 0.342020i 0 0.984808 + 1.17365i 0 0.500000 + 0.866025i 0.673648 1.16679i 0
523.1 −0.173648 0.984808i 1.32683 0.766044i −0.939693 + 0.342020i 0 −0.984808 1.17365i 0 0.500000 + 0.866025i 0.673648 1.16679i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
73.k even 36 1 inner
584.br odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 584.1.br.a 12
4.b odd 2 1 2336.1.ed.a 12
8.b even 2 1 2336.1.ed.a 12
8.d odd 2 1 CM 584.1.br.a 12
73.k even 36 1 inner 584.1.br.a 12
292.u odd 36 1 2336.1.ed.a 12
584.bo even 36 1 2336.1.ed.a 12
584.br odd 36 1 inner 584.1.br.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
584.1.br.a 12 1.a even 1 1 trivial
584.1.br.a 12 8.d odd 2 1 CM
584.1.br.a 12 73.k even 36 1 inner
584.1.br.a 12 584.br odd 36 1 inner
2336.1.ed.a 12 4.b odd 2 1
2336.1.ed.a 12 8.b even 2 1
2336.1.ed.a 12 292.u odd 36 1
2336.1.ed.a 12 584.bo even 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(584, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} + 5 T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$19$ \( (T^{6} + 9 T^{3} + 27)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{12} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
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