↑

Properties

Label 12.12.a.a
Level $12$
Weight $12$
Character orbit 12.a
Self dual yes
Analytic conductor $9.220$
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] = [12,12,Mod(1,12)] 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("12.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-243] 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: \(9.22011816672\)
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)
 
\(f(q)\) \(=\) \( q - 243 q^{3} + 9990 q^{5} - 86128 q^{7} + 59049 q^{9} - 806004 q^{11} - 960250 q^{13} - 2427570 q^{15} - 4306878 q^{17} + 401300 q^{19} + 20929104 q^{21} + 17751528 q^{23} + 50971975 q^{25} - 14348907 q^{27}+ \cdots - 47593730196 q^{99}+O(q^{100}) \) 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
0 −243.000 0 9990.00 0 −86128.0 0 59049.0 0
\(n\): e.g. 2-40 or 990-1000
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 12.12.a.a ✓ 1
3.b odd 2 1 36.12.a.a 1
4.b odd 2 1 48.12.a.i 1
8.b even 2 1 192.12.a.k 1
8.d odd 2 1 192.12.a.a 1
12.b even 2 1 144.12.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.12.a.a ✓ 1 1.a even 1 1 trivial
36.12.a.a 1 3.b odd 2 1
48.12.a.i 1 4.b odd 2 1
144.12.a.a 1 12.b even 2 1
192.12.a.a 1 8.d odd 2 1
192.12.a.k 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 9990 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(12))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 243 \) Copy content Toggle raw display
$5$ \( T - 9990 \) Copy content Toggle raw display
$7$ \( T + 86128 \) Copy content Toggle raw display
$11$ \( T + 806004 \) Copy content Toggle raw display
$13$ \( T + 960250 \) Copy content Toggle raw display
$17$ \( T + 4306878 \) Copy content Toggle raw display
$19$ \( T - 401300 \) Copy content Toggle raw display
$23$ \( T - 17751528 \) Copy content Toggle raw display
$29$ \( T + 84704994 \) Copy content Toggle raw display
$31$ \( T - 140930504 \) Copy content Toggle raw display
$37$ \( T + 413506594 \) Copy content Toggle raw display
$41$ \( T - 150094890 \) Copy content Toggle raw display
$43$ \( T - 706702028 \) Copy content Toggle raw display
$47$ \( T + 2475725472 \) Copy content Toggle raw display
$53$ \( T - 1600124166 \) Copy content Toggle raw display
$59$ \( T - 3945492396 \) Copy content Toggle raw display
$61$ \( T + 885973498 \) Copy content Toggle raw display
$67$ \( T + 4881597772 \) Copy content Toggle raw display
$71$ \( T - 12631469400 \) Copy content Toggle raw display
$73$ \( T - 1423335194 \) Copy content Toggle raw display
$79$ \( T - 667407512 \) Copy content Toggle raw display
$83$ \( T - 5716071828 \) Copy content Toggle raw display
$89$ \( T + 85738736790 \) Copy content Toggle raw display
$97$ \( T + 52302647806 \) Copy content Toggle raw display
show more
show less