Properties

Label 336.6.a.n
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} + 6 q^{5} - 49 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + 6 q^{5} - 49 q^{7} + 81 q^{9} + 108 q^{11} - 346 q^{13} + 54 q^{15} - 1398 q^{17} + 1012 q^{19} - 441 q^{21} + 1536 q^{23} - 3089 q^{25} + 729 q^{27} - 3762 q^{29} + 736 q^{31} + 972 q^{33} - 294 q^{35} + 2054 q^{37} - 3114 q^{39} - 15534 q^{41} - 11036 q^{43} + 486 q^{45} - 4560 q^{47} + 2401 q^{49} - 12582 q^{51} - 7962 q^{53} + 648 q^{55} + 9108 q^{57} + 7020 q^{59} + 26870 q^{61} - 3969 q^{63} - 2076 q^{65} - 52148 q^{67} + 13824 q^{69} + 2544 q^{71} - 9766 q^{73} - 27801 q^{75} - 5292 q^{77} - 68672 q^{79} + 6561 q^{81} + 61668 q^{83} - 8388 q^{85} - 33858 q^{87} - 41454 q^{89} + 16954 q^{91} + 6624 q^{93} + 6072 q^{95} - 111262 q^{97} + 8748 q^{99}+O(q^{100}) \) 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 6.00000 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.n 1
3.b odd 2 1 1008.6.a.o 1
4.b odd 2 1 84.6.a.a 1
12.b even 2 1 252.6.a.b 1
28.d even 2 1 588.6.a.e 1
28.f even 6 2 588.6.i.b 2
28.g odd 6 2 588.6.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.a 1 4.b odd 2 1
252.6.a.b 1 12.b even 2 1
336.6.a.n 1 1.a even 1 1 trivial
588.6.a.e 1 28.d even 2 1
588.6.i.b 2 28.f even 6 2
588.6.i.f 2 28.g odd 6 2
1008.6.a.o 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} - 6 \) Copy content Toggle raw display
\( T_{11} - 108 \) 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
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 108 \) Copy content Toggle raw display
$13$ \( T + 346 \) Copy content Toggle raw display
$17$ \( T + 1398 \) Copy content Toggle raw display
$19$ \( T - 1012 \) Copy content Toggle raw display
$23$ \( T - 1536 \) Copy content Toggle raw display
$29$ \( T + 3762 \) Copy content Toggle raw display
$31$ \( T - 736 \) Copy content Toggle raw display
$37$ \( T - 2054 \) Copy content Toggle raw display
$41$ \( T + 15534 \) Copy content Toggle raw display
$43$ \( T + 11036 \) Copy content Toggle raw display
$47$ \( T + 4560 \) Copy content Toggle raw display
$53$ \( T + 7962 \) Copy content Toggle raw display
$59$ \( T - 7020 \) Copy content Toggle raw display
$61$ \( T - 26870 \) Copy content Toggle raw display
$67$ \( T + 52148 \) Copy content Toggle raw display
$71$ \( T - 2544 \) Copy content Toggle raw display
$73$ \( T + 9766 \) Copy content Toggle raw display
$79$ \( T + 68672 \) Copy content Toggle raw display
$83$ \( T - 61668 \) Copy content Toggle raw display
$89$ \( T + 41454 \) Copy content Toggle raw display
$97$ \( T + 111262 \) Copy content Toggle raw display
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