Properties

Label 126.6.a.f
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: no (minimal twist has level 14)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} - 84 q^{5} + 49 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} - 84 q^{5} + 49 q^{7} + 64 q^{8} - 336 q^{10} + 336 q^{11} + 584 q^{13} + 196 q^{14} + 256 q^{16} + 1458 q^{17} + 470 q^{19} - 1344 q^{20} + 1344 q^{22} + 4200 q^{23} + 3931 q^{25} + 2336 q^{26} + 784 q^{28} - 4866 q^{29} - 7372 q^{31} + 1024 q^{32} + 5832 q^{34} - 4116 q^{35} + 14330 q^{37} + 1880 q^{38} - 5376 q^{40} - 6222 q^{41} + 3704 q^{43} + 5376 q^{44} + 16800 q^{46} + 1812 q^{47} + 2401 q^{49} + 15724 q^{50} + 9344 q^{52} + 37242 q^{53} - 28224 q^{55} + 3136 q^{56} - 19464 q^{58} - 34302 q^{59} + 24476 q^{61} - 29488 q^{62} + 4096 q^{64} - 49056 q^{65} - 17452 q^{67} + 23328 q^{68} - 16464 q^{70} - 28224 q^{71} + 3602 q^{73} + 57320 q^{74} + 7520 q^{76} + 16464 q^{77} + 42872 q^{79} - 21504 q^{80} - 24888 q^{82} + 35202 q^{83} - 122472 q^{85} + 14816 q^{86} + 21504 q^{88} - 26730 q^{89} + 28616 q^{91} + 67200 q^{92} + 7248 q^{94} - 39480 q^{95} - 16978 q^{97} + 9604 q^{98}+O(q^{100}) \) 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 −84.0000 0 49.0000 64.0000 0 −336.000
\(n\): 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.f 1
3.b odd 2 1 14.6.a.a 1
4.b odd 2 1 1008.6.a.b 1
7.b odd 2 1 882.6.a.x 1
12.b even 2 1 112.6.a.c 1
15.d odd 2 1 350.6.a.i 1
15.e even 4 2 350.6.c.d 2
21.c even 2 1 98.6.a.a 1
21.g even 6 2 98.6.c.d 2
21.h odd 6 2 98.6.c.c 2
24.f even 2 1 448.6.a.l 1
24.h odd 2 1 448.6.a.e 1
84.h odd 2 1 784.6.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 3.b odd 2 1
98.6.a.a 1 21.c even 2 1
98.6.c.c 2 21.h odd 6 2
98.6.c.d 2 21.g even 6 2
112.6.a.c 1 12.b even 2 1
126.6.a.f 1 1.a even 1 1 trivial
350.6.a.i 1 15.d odd 2 1
350.6.c.d 2 15.e even 4 2
448.6.a.e 1 24.h odd 2 1
448.6.a.l 1 24.f even 2 1
784.6.a.i 1 84.h odd 2 1
882.6.a.x 1 7.b odd 2 1
1008.6.a.b 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(126))\):

\( T_{5} + 84 \) Copy content Toggle raw display
\( T_{11} - 336 \) 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
$5$ \( T + 84 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 336 \) Copy content Toggle raw display
$13$ \( T - 584 \) Copy content Toggle raw display
$17$ \( T - 1458 \) Copy content Toggle raw display
$19$ \( T - 470 \) Copy content Toggle raw display
$23$ \( T - 4200 \) Copy content Toggle raw display
$29$ \( T + 4866 \) Copy content Toggle raw display
$31$ \( T + 7372 \) Copy content Toggle raw display
$37$ \( T - 14330 \) Copy content Toggle raw display
$41$ \( T + 6222 \) Copy content Toggle raw display
$43$ \( T - 3704 \) Copy content Toggle raw display
$47$ \( T - 1812 \) Copy content Toggle raw display
$53$ \( T - 37242 \) Copy content Toggle raw display
$59$ \( T + 34302 \) Copy content Toggle raw display
$61$ \( T - 24476 \) Copy content Toggle raw display
$67$ \( T + 17452 \) Copy content Toggle raw display
$71$ \( T + 28224 \) Copy content Toggle raw display
$73$ \( T - 3602 \) Copy content Toggle raw display
$79$ \( T - 42872 \) Copy content Toggle raw display
$83$ \( T - 35202 \) Copy content Toggle raw display
$89$ \( T + 26730 \) Copy content Toggle raw display
$97$ \( T + 16978 \) Copy content Toggle raw display
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