Properties

Label 126.10.a.d
Level $126$
Weight $10$
Character orbit 126.a
Self dual yes
Analytic conductor $64.895$
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] = [126,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(64.8945153566\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 256 q^{4} - 1590 q^{5} + 2401 q^{7} + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + 256 q^{4} - 1590 q^{5} + 2401 q^{7} + 4096 q^{8} - 25440 q^{10} + 22668 q^{11} - 64186 q^{13} + 38416 q^{14} + 65536 q^{16} - 29946 q^{17} + 301484 q^{19} - 407040 q^{20} + 362688 q^{22} - 1237488 q^{23} + 574975 q^{25} - 1026976 q^{26} + 614656 q^{28} - 391806 q^{29} + 6802688 q^{31} + 1048576 q^{32} - 479136 q^{34} - 3817590 q^{35} + 19279766 q^{37} + 4823744 q^{38} - 6512640 q^{40} + 10487358 q^{41} + 38420876 q^{43} + 5803008 q^{44} - 19799808 q^{46} + 2928 q^{47} + 5764801 q^{49} + 9199600 q^{50} - 16431616 q^{52} + 68190666 q^{53} - 36042120 q^{55} + 9834496 q^{56} - 6268896 q^{58} - 79636692 q^{59} - 37203370 q^{61} + 108843008 q^{62} + 16777216 q^{64} + 102055740 q^{65} + 58234052 q^{67} - 7666176 q^{68} - 61081440 q^{70} + 49349376 q^{71} + 345747674 q^{73} + 308476256 q^{74} + 77179904 q^{76} + 54425868 q^{77} + 455981888 q^{79} - 104202240 q^{80} + 167797728 q^{82} - 446212188 q^{83} + 47614140 q^{85} + 614734016 q^{86} + 92848128 q^{88} + 571901790 q^{89} - 154110586 q^{91} - 316796928 q^{92} + 46848 q^{94} - 479359560 q^{95} + 244250210 q^{97} + 92236816 q^{98}+O(q^{100}) \) 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
16.0000 0 256.000 −1590.00 0 2401.00 4096.00 0 −25440.0
\(n\): 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 126.10.a.d 1
3.b odd 2 1 42.10.a.b 1
12.b even 2 1 336.10.a.g 1
21.c even 2 1 294.10.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.10.a.b 1 3.b odd 2 1
126.10.a.d 1 1.a even 1 1 trivial
294.10.a.e 1 21.c even 2 1
336.10.a.g 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1590 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(126))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1590 \) Copy content Toggle raw display
$7$ \( T - 2401 \) Copy content Toggle raw display
$11$ \( T - 22668 \) Copy content Toggle raw display
$13$ \( T + 64186 \) Copy content Toggle raw display
$17$ \( T + 29946 \) Copy content Toggle raw display
$19$ \( T - 301484 \) Copy content Toggle raw display
$23$ \( T + 1237488 \) Copy content Toggle raw display
$29$ \( T + 391806 \) Copy content Toggle raw display
$31$ \( T - 6802688 \) Copy content Toggle raw display
$37$ \( T - 19279766 \) Copy content Toggle raw display
$41$ \( T - 10487358 \) Copy content Toggle raw display
$43$ \( T - 38420876 \) Copy content Toggle raw display
$47$ \( T - 2928 \) Copy content Toggle raw display
$53$ \( T - 68190666 \) Copy content Toggle raw display
$59$ \( T + 79636692 \) Copy content Toggle raw display
$61$ \( T + 37203370 \) Copy content Toggle raw display
$67$ \( T - 58234052 \) Copy content Toggle raw display
$71$ \( T - 49349376 \) Copy content Toggle raw display
$73$ \( T - 345747674 \) Copy content Toggle raw display
$79$ \( T - 455981888 \) Copy content Toggle raw display
$83$ \( T + 446212188 \) Copy content Toggle raw display
$89$ \( T - 571901790 \) Copy content Toggle raw display
$97$ \( T - 244250210 \) Copy content Toggle raw display
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