Properties

Label 336.6.a.e
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 / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,6,Mod(1,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 84)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 9 q^{3} - 34 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} - 34 q^{5} + 49 q^{7} + 81 q^{9} + 332 q^{11} - 1026 q^{13} + 306 q^{15} + 922 q^{17} - 452 q^{19} - 441 q^{21} + 3776 q^{23} - 1969 q^{25} - 729 q^{27} + 1166 q^{29} + 9792 q^{31} - 2988 q^{33} - 1666 q^{35} + 2390 q^{37} + 9234 q^{39} - 7230 q^{41} - 4652 q^{43} - 2754 q^{45} - 24672 q^{47} + 2401 q^{49} - 8298 q^{51} + 1110 q^{53} - 11288 q^{55} + 4068 q^{57} - 46892 q^{59} - 9762 q^{61} + 3969 q^{63} + 34884 q^{65} + 26252 q^{67} - 33984 q^{69} - 65440 q^{71} - 5606 q^{73} + 17721 q^{75} + 16268 q^{77} + 9840 q^{79} + 6561 q^{81} - 61108 q^{83} - 31348 q^{85} - 10494 q^{87} - 62958 q^{89} - 50274 q^{91} - 88128 q^{93} + 15368 q^{95} - 37838 q^{97} + 26892 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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 −34.0000 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.e 1
3.b odd 2 1 1008.6.a.u 1
4.b odd 2 1 84.6.a.b 1
12.b even 2 1 252.6.a.c 1
28.d even 2 1 588.6.a.b 1
28.f even 6 2 588.6.i.e 2
28.g odd 6 2 588.6.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.b 1 4.b odd 2 1
252.6.a.c 1 12.b even 2 1
336.6.a.e 1 1.a even 1 1 trivial
588.6.a.b 1 28.d even 2 1
588.6.i.c 2 28.g odd 6 2
588.6.i.e 2 28.f even 6 2
1008.6.a.u 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} + 34 \) Copy content Toggle raw display
\( T_{11} - 332 \) 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
$5$ \( T + 34 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 332 \) Copy content Toggle raw display
$13$ \( T + 1026 \) Copy content Toggle raw display
$17$ \( T - 922 \) Copy content Toggle raw display
$19$ \( T + 452 \) Copy content Toggle raw display
$23$ \( T - 3776 \) Copy content Toggle raw display
$29$ \( T - 1166 \) Copy content Toggle raw display
$31$ \( T - 9792 \) Copy content Toggle raw display
$37$ \( T - 2390 \) Copy content Toggle raw display
$41$ \( T + 7230 \) Copy content Toggle raw display
$43$ \( T + 4652 \) Copy content Toggle raw display
$47$ \( T + 24672 \) Copy content Toggle raw display
$53$ \( T - 1110 \) Copy content Toggle raw display
$59$ \( T + 46892 \) Copy content Toggle raw display
$61$ \( T + 9762 \) Copy content Toggle raw display
$67$ \( T - 26252 \) Copy content Toggle raw display
$71$ \( T + 65440 \) Copy content Toggle raw display
$73$ \( T + 5606 \) Copy content Toggle raw display
$79$ \( T - 9840 \) Copy content Toggle raw display
$83$ \( T + 61108 \) Copy content Toggle raw display
$89$ \( T + 62958 \) Copy content Toggle raw display
$97$ \( T + 37838 \) Copy content Toggle raw display
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