Properties

Label 126.6.a.e
Level $126$
Weight $6$
Character orbit 126.a
Self dual yes
Analytic conductor $20.208$
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] = [126,6,Mod(1,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("126.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 126.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: \(20.2083612964\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} + 54 q^{5} + 49 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} + 54 q^{5} + 49 q^{7} - 64 q^{8} - 216 q^{10} + 594 q^{11} + 26 q^{13} - 196 q^{14} + 256 q^{16} - 534 q^{17} - 3004 q^{19} + 864 q^{20} - 2376 q^{22} + 3510 q^{23} - 209 q^{25} - 104 q^{26} + 784 q^{28} + 4296 q^{29} + 8036 q^{31} - 1024 q^{32} + 2136 q^{34} + 2646 q^{35} - 502 q^{37} + 12016 q^{38} - 3456 q^{40} + 9870 q^{41} + 9068 q^{43} + 9504 q^{44} - 14040 q^{46} + 1140 q^{47} + 2401 q^{49} + 836 q^{50} + 416 q^{52} + 28356 q^{53} + 32076 q^{55} - 3136 q^{56} - 17184 q^{58} - 8196 q^{59} + 29822 q^{61} - 32144 q^{62} + 4096 q^{64} + 1404 q^{65} - 62884 q^{67} - 8544 q^{68} - 10584 q^{70} - 34398 q^{71} + 56990 q^{73} + 2008 q^{74} - 48064 q^{76} + 29106 q^{77} + 49496 q^{79} + 13824 q^{80} - 39480 q^{82} - 52512 q^{83} - 28836 q^{85} - 36272 q^{86} - 38016 q^{88} - 48282 q^{89} + 1274 q^{91} + 56160 q^{92} - 4560 q^{94} - 162216 q^{95} - 83938 q^{97} - 9604 q^{98}+O(q^{100}) \) 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
−4.00000 0 16.0000 54.0000 0 49.0000 −64.0000 0 −216.000
\(n\): 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 126.6.a.e 1
3.b odd 2 1 126.6.a.g yes 1
4.b odd 2 1 1008.6.a.w 1
7.b odd 2 1 882.6.a.b 1
12.b even 2 1 1008.6.a.f 1
21.c even 2 1 882.6.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.6.a.e 1 1.a even 1 1 trivial
126.6.a.g yes 1 3.b odd 2 1
882.6.a.b 1 7.b odd 2 1
882.6.a.w 1 21.c even 2 1
1008.6.a.f 1 12.b even 2 1
1008.6.a.w 1 4.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(126))\):

\( T_{5} - 54 \) Copy content Toggle raw display
\( T_{11} - 594 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 54 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 594 \) Copy content Toggle raw display
$13$ \( T - 26 \) Copy content Toggle raw display
$17$ \( T + 534 \) Copy content Toggle raw display
$19$ \( T + 3004 \) Copy content Toggle raw display
$23$ \( T - 3510 \) Copy content Toggle raw display
$29$ \( T - 4296 \) Copy content Toggle raw display
$31$ \( T - 8036 \) Copy content Toggle raw display
$37$ \( T + 502 \) Copy content Toggle raw display
$41$ \( T - 9870 \) Copy content Toggle raw display
$43$ \( T - 9068 \) Copy content Toggle raw display
$47$ \( T - 1140 \) Copy content Toggle raw display
$53$ \( T - 28356 \) Copy content Toggle raw display
$59$ \( T + 8196 \) Copy content Toggle raw display
$61$ \( T - 29822 \) Copy content Toggle raw display
$67$ \( T + 62884 \) Copy content Toggle raw display
$71$ \( T + 34398 \) Copy content Toggle raw display
$73$ \( T - 56990 \) Copy content Toggle raw display
$79$ \( T - 49496 \) Copy content Toggle raw display
$83$ \( T + 52512 \) Copy content Toggle raw display
$89$ \( T + 48282 \) Copy content Toggle raw display
$97$ \( T + 83938 \) Copy content Toggle raw display
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