Properties

Label 336.6.a.o
Level $336$
Weight $6$
Character orbit 336.a
Self dual yes
Analytic conductor $53.889$
Analytic rank $0$
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: \(0\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 9 q^{3} + 44 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + 44 q^{5} + 49 q^{7} + 81 q^{9} + 470 q^{11} - 1158 q^{13} + 396 q^{15} + 1204 q^{17} + 2644 q^{19} + 441 q^{21} + 1190 q^{23} - 1189 q^{25} + 729 q^{27} + 3614 q^{29} - 5616 q^{31} + 4230 q^{33} + 2156 q^{35} - 6478 q^{37} - 10422 q^{39} + 2856 q^{41} + 13492 q^{43} + 3564 q^{45} + 18372 q^{47} + 2401 q^{49} + 10836 q^{51} - 4374 q^{53} + 20680 q^{55} + 23796 q^{57} - 30248 q^{59} + 19542 q^{61} + 3969 q^{63} - 50952 q^{65} - 54328 q^{67} + 10710 q^{69} + 10730 q^{71} + 35374 q^{73} - 10701 q^{75} + 23030 q^{77} + 49956 q^{79} + 6561 q^{81} + 26948 q^{83} + 52976 q^{85} + 32526 q^{87} + 100776 q^{89} - 56742 q^{91} - 50544 q^{93} + 116336 q^{95} + 77134 q^{97} + 38070 q^{99}+O(q^{100}) \) 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 44.0000 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.o 1
3.b odd 2 1 1008.6.a.g 1
4.b odd 2 1 42.6.a.b 1
12.b even 2 1 126.6.a.h 1
20.d odd 2 1 1050.6.a.o 1
20.e even 4 2 1050.6.g.l 2
28.d even 2 1 294.6.a.f 1
28.f even 6 2 294.6.e.k 2
28.g odd 6 2 294.6.e.o 2
84.h odd 2 1 882.6.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.b 1 4.b odd 2 1
126.6.a.h 1 12.b even 2 1
294.6.a.f 1 28.d even 2 1
294.6.e.k 2 28.f even 6 2
294.6.e.o 2 28.g odd 6 2
336.6.a.o 1 1.a even 1 1 trivial
882.6.a.v 1 84.h odd 2 1
1008.6.a.g 1 3.b odd 2 1
1050.6.a.o 1 20.d odd 2 1
1050.6.g.l 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):

\( T_{5} - 44 \) Copy content Toggle raw display
\( T_{11} - 470 \) 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
$5$ \( T - 44 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 470 \) Copy content Toggle raw display
$13$ \( T + 1158 \) Copy content Toggle raw display
$17$ \( T - 1204 \) Copy content Toggle raw display
$19$ \( T - 2644 \) Copy content Toggle raw display
$23$ \( T - 1190 \) Copy content Toggle raw display
$29$ \( T - 3614 \) Copy content Toggle raw display
$31$ \( T + 5616 \) Copy content Toggle raw display
$37$ \( T + 6478 \) Copy content Toggle raw display
$41$ \( T - 2856 \) Copy content Toggle raw display
$43$ \( T - 13492 \) Copy content Toggle raw display
$47$ \( T - 18372 \) Copy content Toggle raw display
$53$ \( T + 4374 \) Copy content Toggle raw display
$59$ \( T + 30248 \) Copy content Toggle raw display
$61$ \( T - 19542 \) Copy content Toggle raw display
$67$ \( T + 54328 \) Copy content Toggle raw display
$71$ \( T - 10730 \) Copy content Toggle raw display
$73$ \( T - 35374 \) Copy content Toggle raw display
$79$ \( T - 49956 \) Copy content Toggle raw display
$83$ \( T - 26948 \) Copy content Toggle raw display
$89$ \( T - 100776 \) Copy content Toggle raw display
$97$ \( T - 77134 \) Copy content Toggle raw display
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