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Properties

Label 558.2.e.a
Level $558$
Weight $2$
Character orbit 558.e
Analytic conductor $4.456$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [558,2,Mod(253,558)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(558, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("558.253"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 558.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.45565243279\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 186)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} - q^{8} + 2 \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} + 4 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{14} + q^{16} + ( - 8 \zeta_{6} + 8) q^{19} + \cdots - 6 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - q^{7} - 2 q^{8} + 2 q^{10} + 3 q^{11} + 4 q^{13} + q^{14} + 2 q^{16} + 8 q^{19} - 2 q^{20} - 3 q^{22} + 4 q^{23} + q^{25} - 4 q^{26} - q^{28} + 18 q^{29} - 7 q^{31}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/558\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(497\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
253.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−1.00000 0 1.00000 −1.00000 − 1.73205i 0 −0.500000 + 0.866025i −1.00000 0 1.00000 + 1.73205i
397.1 −1.00000 0 1.00000 −1.00000 + 1.73205i 0 −0.500000 − 0.866025i −1.00000 0 1.00000 − 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 558.2.e.a 2
3.b odd 2 1 186.2.e.b ✓ 2
12.b even 2 1 1488.2.q.e 2
31.c even 3 1 inner 558.2.e.a 2
93.g even 6 1 5766.2.a.g 1
93.h odd 6 1 186.2.e.b ✓ 2
93.h odd 6 1 5766.2.a.i 1
372.p even 6 1 1488.2.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
186.2.e.b ✓ 2 3.b odd 2 1
186.2.e.b ✓ 2 93.h odd 6 1
558.2.e.a 2 1.a even 1 1 trivial
558.2.e.a 2 31.c even 3 1 inner
1488.2.q.e 2 12.b even 2 1
1488.2.q.e 2 372.p even 6 1
5766.2.a.g 1 93.g even 6 1
5766.2.a.i 1 93.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(558, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 31 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$83$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$89$ \( (T - 14)^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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