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Properties

Label 6.18.a.c
Level $6$
Weight $18$
Character orbit 6.a
Self dual yes
Analytic conductor $10.993$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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] = [6,18,Mod(1,6)] 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("6.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,256,-6561] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.9933252407\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 
\(f(q)\) \(=\) \( q + 256 q^{2} - 6561 q^{3} + 65536 q^{4} - 199650 q^{5} - 1679616 q^{6} + 24959264 q^{7} + 16777216 q^{8} + 43046721 q^{9} - 51110400 q^{10} + 125556420 q^{11} - 429981696 q^{12} + 4227195518 q^{13} + 6389571584 q^{14}+ \cdots + 54\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
256.000 −6561.00 65536.0 −199650. −1.67962e6 2.49593e7 1.67772e7 4.30467e7 −5.11104e7
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.18.a.c ✓ 1
3.b odd 2 1 18.18.a.b 1
4.b odd 2 1 48.18.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.18.a.c ✓ 1 1.a even 1 1 trivial
18.18.a.b 1 3.b odd 2 1
48.18.a.d 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 199650 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(6))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 256 \) Copy content Toggle raw display
$3$ \( T + 6561 \) Copy content Toggle raw display
$5$ \( T + 199650 \) Copy content Toggle raw display
$7$ \( T - 24959264 \) Copy content Toggle raw display
$11$ \( T - 125556420 \) Copy content Toggle raw display
$13$ \( T - 4227195518 \) Copy content Toggle raw display
$17$ \( T - 35551782594 \) Copy content Toggle raw display
$19$ \( T + 64354589764 \) Copy content Toggle raw display
$23$ \( T + 245819296200 \) Copy content Toggle raw display
$29$ \( T + 2280393162906 \) Copy content Toggle raw display
$31$ \( T - 4349964811688 \) Copy content Toggle raw display
$37$ \( T - 20770411877318 \) Copy content Toggle raw display
$41$ \( T + 97624823830086 \) Copy content Toggle raw display
$43$ \( T - 76137596568644 \) Copy content Toggle raw display
$47$ \( T - 296069387010240 \) Copy content Toggle raw display
$53$ \( T + 213113313107874 \) Copy content Toggle raw display
$59$ \( T + 1776690045107580 \) Copy content Toggle raw display
$61$ \( T + 1424434275760450 \) Copy content Toggle raw display
$67$ \( T + 1599652965063556 \) Copy content Toggle raw display
$71$ \( T - 5439386569413960 \) Copy content Toggle raw display
$73$ \( T + 3725056002188662 \) Copy content Toggle raw display
$79$ \( T - 10\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T + 29\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T + 43\!\cdots\!02 \) Copy content Toggle raw display
$97$ \( T - 34\!\cdots\!78 \) Copy content Toggle raw display
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