↑

Properties

Label 42.12.a.f
Level $42$
Weight $12$
Character orbit 42.a
Self dual yes
Analytic conductor $32.270$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [42,12,Mod(1,42)] 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("42.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,32,243,1024,-11880] 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: yes
Analytic conductor: \(32.2704135835\)
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)
 
\(f(q)\) \(=\) \( q + 32 q^{2} + 243 q^{3} + 1024 q^{4} - 11880 q^{5} + 7776 q^{6} + 16807 q^{7} + 32768 q^{8} + 59049 q^{9} - 380160 q^{10} + 727110 q^{11} + 248832 q^{12} - 1895734 q^{13} + 537824 q^{14} - 2886840 q^{15}+ \cdots + 42935118390 q^{99}+O(q^{100}) \) 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
32.0000 243.000 1024.00 −11880.0 7776.00 16807.0 32768.0 59049.0 −380160.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 42.12.a.f ✓ 1
3.b odd 2 1 126.12.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.12.a.f ✓ 1 1.a even 1 1 trivial
126.12.a.c 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 11880 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(42))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 32 \) Copy content Toggle raw display
$3$ \( T - 243 \) Copy content Toggle raw display
$5$ \( T + 11880 \) Copy content Toggle raw display
$7$ \( T - 16807 \) Copy content Toggle raw display
$11$ \( T - 727110 \) Copy content Toggle raw display
$13$ \( T + 1895734 \) Copy content Toggle raw display
$17$ \( T + 10233912 \) Copy content Toggle raw display
$19$ \( T + 12792796 \) Copy content Toggle raw display
$23$ \( T + 27412290 \) Copy content Toggle raw display
$29$ \( T + 108082914 \) Copy content Toggle raw display
$31$ \( T + 202243384 \) Copy content Toggle raw display
$37$ \( T - 412454954 \) Copy content Toggle raw display
$41$ \( T + 245108604 \) Copy content Toggle raw display
$43$ \( T - 509839844 \) Copy content Toggle raw display
$47$ \( T - 699876996 \) Copy content Toggle raw display
$53$ \( T + 4836690462 \) Copy content Toggle raw display
$59$ \( T - 2462248272 \) Copy content Toggle raw display
$61$ \( T - 9054716702 \) Copy content Toggle raw display
$67$ \( T + 2923148584 \) Copy content Toggle raw display
$71$ \( T + 11268869310 \) Copy content Toggle raw display
$73$ \( T - 2809231382 \) Copy content Toggle raw display
$79$ \( T - 25573574516 \) Copy content Toggle raw display
$83$ \( T + 38718767364 \) Copy content Toggle raw display
$89$ \( T - 90783707844 \) Copy content Toggle raw display
$97$ \( T - 130559880038 \) Copy content Toggle raw display
show more
show less