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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,128,-2187] 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: \(8.56161030600\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 128 q^{2} - 2187 q^{3} + 16384 q^{4} - 114810 q^{5} - 279936 q^{6} - 3034528 q^{7} + 2097152 q^{8} + 4782969 q^{9} - 14695680 q^{10} - 103451700 q^{11} - 35831808 q^{12} - 104365834 q^{13} - 388419584 q^{14}+ \cdots - 494806274097300 q^{99}+O(q^{100}) \) 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
128.000 −2187.00 16384.0 −114810. −279936. −3.03453e6 2.09715e6 4.78297e6 −1.46957e7
\(n\): 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 6.16.a.b ✓ 1
3.b odd 2 1 18.16.a.b 1
4.b odd 2 1 48.16.a.d 1
5.b even 2 1 150.16.a.f 1
5.c odd 4 2 150.16.c.a 2
12.b even 2 1 144.16.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b ✓ 1 1.a even 1 1 trivial
18.16.a.b 1 3.b odd 2 1
48.16.a.d 1 4.b odd 2 1
144.16.a.j 1 12.b even 2 1
150.16.a.f 1 5.b even 2 1
150.16.c.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 114810 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(6))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 128 \) Copy content Toggle raw display
$3$ \( T + 2187 \) Copy content Toggle raw display
$5$ \( T + 114810 \) Copy content Toggle raw display
$7$ \( T + 3034528 \) Copy content Toggle raw display
$11$ \( T + 103451700 \) Copy content Toggle raw display
$13$ \( T + 104365834 \) Copy content Toggle raw display
$17$ \( T - 997689762 \) Copy content Toggle raw display
$19$ \( T - 4934015444 \) Copy content Toggle raw display
$23$ \( T - 8324920200 \) Copy content Toggle raw display
$29$ \( T - 104128242846 \) Copy content Toggle raw display
$31$ \( T + 296696681512 \) Copy content Toggle raw display
$37$ \( T + 178337455666 \) Copy content Toggle raw display
$41$ \( T + 1790882416086 \) Copy content Toggle raw display
$43$ \( T + 2863459422772 \) Copy content Toggle raw display
$47$ \( T - 4332907521600 \) Copy content Toggle raw display
$53$ \( T - 9732317104422 \) Copy content Toggle raw display
$59$ \( T + 13514837176500 \) Copy content Toggle raw display
$61$ \( T - 5352663511190 \) Copy content Toggle raw display
$67$ \( T + 53233909720108 \) Copy content Toggle raw display
$71$ \( T + 20229661643400 \) Copy content Toggle raw display
$73$ \( T - 26264166466106 \) Copy content Toggle raw display
$79$ \( T + 339031361615128 \) Copy content Toggle raw display
$83$ \( T - 131684771045076 \) Copy content Toggle raw display
$89$ \( T + 39352148322678 \) Copy content Toggle raw display
$97$ \( T - 1128750908801474 \) Copy content Toggle raw display
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