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Properties

Label 336.6.a.j
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 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);
 
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")
 
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,-54,0,-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 42)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} - 54 q^{5} - 49 q^{7} + 81 q^{9} - 216 q^{11} + 998 q^{13} - 486 q^{15} + 1302 q^{17} - 884 q^{19} - 441 q^{21} + 2268 q^{23} - 209 q^{25} + 729 q^{27} - 1482 q^{29} - 8360 q^{31} - 1944 q^{33}+ \cdots - 17496 q^{99}+O(q^{100}) \) 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 −54.0000 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.j 1
3.b odd 2 1 1008.6.a.x 1
4.b odd 2 1 42.6.a.a ✓ 1
12.b even 2 1 126.6.a.k 1
20.d odd 2 1 1050.6.a.n 1
20.e even 4 2 1050.6.g.o 2
28.d even 2 1 294.6.a.h 1
28.f even 6 2 294.6.e.h 2
28.g odd 6 2 294.6.e.r 2
84.h odd 2 1 882.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a ✓ 1 4.b odd 2 1
126.6.a.k 1 12.b even 2 1
294.6.a.h 1 28.d even 2 1
294.6.e.h 2 28.f even 6 2
294.6.e.r 2 28.g odd 6 2
336.6.a.j 1 1.a even 1 1 trivial
882.6.a.o 1 84.h odd 2 1
1008.6.a.x 1 3.b odd 2 1
1050.6.a.n 1 20.d odd 2 1
1050.6.g.o 2 20.e even 4 2

Hecke kernels

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

\( T_{5} + 54 \) Copy content Toggle raw display
\( T_{11} + 216 \) 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
$5$ \( T + 54 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T + 216 \) Copy content Toggle raw display
$13$ \( T - 998 \) Copy content Toggle raw display
$17$ \( T - 1302 \) Copy content Toggle raw display
$19$ \( T + 884 \) Copy content Toggle raw display
$23$ \( T - 2268 \) Copy content Toggle raw display
$29$ \( T + 1482 \) Copy content Toggle raw display
$31$ \( T + 8360 \) Copy content Toggle raw display
$37$ \( T + 4714 \) Copy content Toggle raw display
$41$ \( T + 9786 \) Copy content Toggle raw display
$43$ \( T + 19436 \) Copy content Toggle raw display
$47$ \( T + 22200 \) Copy content Toggle raw display
$53$ \( T - 26790 \) Copy content Toggle raw display
$59$ \( T + 28092 \) Copy content Toggle raw display
$61$ \( T + 38866 \) Copy content Toggle raw display
$67$ \( T + 23948 \) Copy content Toggle raw display
$71$ \( T - 20628 \) Copy content Toggle raw display
$73$ \( T - 290 \) Copy content Toggle raw display
$79$ \( T - 99544 \) Copy content Toggle raw display
$83$ \( T + 19308 \) Copy content Toggle raw display
$89$ \( T - 36390 \) Copy content Toggle raw display
$97$ \( T + 79078 \) Copy content Toggle raw display
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