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Properties

Label 336.6.a.a
Level $336$
Weight $6$
Character orbit 336.a
Self dual yes
Analytic conductor $53.889$
Analytic rank $1$
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] = [336,6,Mod(1,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-9,0,-106,0,49] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.8889634572\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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)
 
\(f(q)\) \(=\) \( q - 9 q^{3} - 106 q^{5} + 49 q^{7} + 81 q^{9} - 92 q^{11} + 670 q^{13} + 954 q^{15} - 222 q^{17} + 908 q^{19} - 441 q^{21} + 1176 q^{23} + 8111 q^{25} - 729 q^{27} + 1118 q^{29} - 3696 q^{31} + 828 q^{33}+ \cdots - 7452 q^{99}+O(q^{100}) \) 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 −9.00000 0 −106.000 0 49.0000 0 81.0000 0
\(n\): 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 336.6.a.a 1
3.b odd 2 1 1008.6.a.bc 1
4.b odd 2 1 21.6.a.d ✓ 1
12.b even 2 1 63.6.a.a 1
20.d odd 2 1 525.6.a.a 1
20.e even 4 2 525.6.d.a 2
28.d even 2 1 147.6.a.g 1
28.f even 6 2 147.6.e.b 2
28.g odd 6 2 147.6.e.a 2
84.h odd 2 1 441.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.d ✓ 1 4.b odd 2 1
63.6.a.a 1 12.b even 2 1
147.6.a.g 1 28.d even 2 1
147.6.e.a 2 28.g odd 6 2
147.6.e.b 2 28.f even 6 2
336.6.a.a 1 1.a even 1 1 trivial
441.6.a.b 1 84.h odd 2 1
525.6.a.a 1 20.d odd 2 1
525.6.d.a 2 20.e even 4 2
1008.6.a.bc 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 106 \) Copy content Toggle raw display
\( T_{11} + 92 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T + 106 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T + 92 \) Copy content Toggle raw display
$13$ \( T - 670 \) Copy content Toggle raw display
$17$ \( T + 222 \) Copy content Toggle raw display
$19$ \( T - 908 \) Copy content Toggle raw display
$23$ \( T - 1176 \) Copy content Toggle raw display
$29$ \( T - 1118 \) Copy content Toggle raw display
$31$ \( T + 3696 \) Copy content Toggle raw display
$37$ \( T - 4182 \) Copy content Toggle raw display
$41$ \( T + 6662 \) Copy content Toggle raw display
$43$ \( T - 3700 \) Copy content Toggle raw display
$47$ \( T - 7056 \) Copy content Toggle raw display
$53$ \( T + 37578 \) Copy content Toggle raw display
$59$ \( T + 32700 \) Copy content Toggle raw display
$61$ \( T + 10802 \) Copy content Toggle raw display
$67$ \( T + 64996 \) Copy content Toggle raw display
$71$ \( T - 61320 \) Copy content Toggle raw display
$73$ \( T - 38922 \) Copy content Toggle raw display
$79$ \( T - 88096 \) Copy content Toggle raw display
$83$ \( T + 71892 \) Copy content Toggle raw display
$89$ \( T - 111818 \) Copy content Toggle raw display
$97$ \( T + 150846 \) Copy content Toggle raw display
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