↑

Properties

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

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] = [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)
 
\(f(q)\) \(=\) \( q - 256 q^{2} + 6561 q^{3} + 65536 q^{4} - 72186 q^{5} - 1679616 q^{6} - 8640184 q^{7} - 16777216 q^{8} + 43046721 q^{9} + 18479616 q^{10} + 1159304460 q^{11} + 429981696 q^{12} + 2801062862 q^{13} + 2211887104 q^{14}+ \cdots + 49\!\cdots\!60 q^{99}+O(q^{100}) \) 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 −72186.0 −1.67962e6 −8.64018e6 −1.67772e7 4.30467e7 1.84796e7
\(n\): 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.b ✓ 1
3.b odd 2 1 18.18.a.d 1
4.b odd 2 1 48.18.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.18.a.b ✓ 1 1.a even 1 1 trivial
18.18.a.d 1 3.b odd 2 1
48.18.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 72186 \) 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
$5$ \( T + 72186 \) Copy content Toggle raw display
$7$ \( T + 8640184 \) Copy content Toggle raw display
$11$ \( T - 1159304460 \) Copy content Toggle raw display
$13$ \( T - 2801062862 \) Copy content Toggle raw display
$17$ \( T - 32979662226 \) Copy content Toggle raw display
$19$ \( T - 5778498836 \) Copy content Toggle raw display
$23$ \( T - 169116994200 \) Copy content Toggle raw display
$29$ \( T - 3631735478814 \) Copy content Toggle raw display
$31$ \( T - 6880978560608 \) Copy content Toggle raw display
$37$ \( T + 35464500749338 \) Copy content Toggle raw display
$41$ \( T + 8923766734806 \) Copy content Toggle raw display
$43$ \( T + 129966457018324 \) Copy content Toggle raw display
$47$ \( T - 129499777218480 \) Copy content Toggle raw display
$53$ \( T - 218262107088054 \) Copy content Toggle raw display
$59$ \( T + 1783401246652740 \) Copy content Toggle raw display
$61$ \( T - 1469145893932670 \) Copy content Toggle raw display
$67$ \( T - 5051560974054596 \) Copy content Toggle raw display
$71$ \( T + 793480696785720 \) Copy content Toggle raw display
$73$ \( T - 6343500933237962 \) Copy content Toggle raw display
$79$ \( T + 8292883305185392 \) Copy content Toggle raw display
$83$ \( T + 24\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T + 15\!\cdots\!22 \) Copy content Toggle raw display
$97$ \( T - 79\!\cdots\!22 \) Copy content Toggle raw display
show more
show less