Properties

Label 126.10.a.h
Level $126$
Weight $10$
Character orbit 126.a
Self dual yes
Analytic conductor $64.895$
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] = [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: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 256 q^{4} + 2290 q^{5} - 2401 q^{7} + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + 256 q^{4} + 2290 q^{5} - 2401 q^{7} + 4096 q^{8} + 36640 q^{10} - 64468 q^{11} - 174402 q^{13} - 38416 q^{14} + 65536 q^{16} - 441322 q^{17} - 506332 q^{19} + 586240 q^{20} - 1031488 q^{22} - 1104880 q^{23} + 3290975 q^{25} - 2790432 q^{26} - 614656 q^{28} - 6115454 q^{29} + 2827296 q^{31} + 1048576 q^{32} - 7061152 q^{34} - 5498290 q^{35} + 9341222 q^{37} - 8101312 q^{38} + 9379840 q^{40} + 12641454 q^{41} + 30847772 q^{43} - 16503808 q^{44} - 17678080 q^{46} + 4249824 q^{47} + 5764801 q^{49} + 52655600 q^{50} - 44646912 q^{52} - 87962982 q^{53} - 147631720 q^{55} - 9834496 q^{56} - 97847264 q^{58} + 5995348 q^{59} - 672930 q^{61} + 45236736 q^{62} + 16777216 q^{64} - 399380580 q^{65} - 140689148 q^{67} - 112978432 q^{68} - 87972640 q^{70} + 322386224 q^{71} + 281507290 q^{73} + 149459552 q^{74} - 129620992 q^{76} + 154787668 q^{77} - 466481136 q^{79} + 150077440 q^{80} + 202263264 q^{82} - 495155764 q^{83} - 1010627380 q^{85} + 493564352 q^{86} - 264060928 q^{88} + 524640510 q^{89} + 418739202 q^{91} - 282849280 q^{92} + 67997184 q^{94} - 1159500280 q^{95} + 1666701490 q^{97} + 92236816 q^{98}+O(q^{100}) \) 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 2290.00 0 −2401.00 4096.00 0 36640.0
\(n\): 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.h 1
3.b odd 2 1 42.10.a.c 1
12.b even 2 1 336.10.a.a 1
21.c even 2 1 294.10.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.10.a.c 1 3.b odd 2 1
126.10.a.h 1 1.a even 1 1 trivial
294.10.a.d 1 21.c even 2 1
336.10.a.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2290 \) 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
$5$ \( T - 2290 \) Copy content Toggle raw display
$7$ \( T + 2401 \) Copy content Toggle raw display
$11$ \( T + 64468 \) Copy content Toggle raw display
$13$ \( T + 174402 \) Copy content Toggle raw display
$17$ \( T + 441322 \) Copy content Toggle raw display
$19$ \( T + 506332 \) Copy content Toggle raw display
$23$ \( T + 1104880 \) Copy content Toggle raw display
$29$ \( T + 6115454 \) Copy content Toggle raw display
$31$ \( T - 2827296 \) Copy content Toggle raw display
$37$ \( T - 9341222 \) Copy content Toggle raw display
$41$ \( T - 12641454 \) Copy content Toggle raw display
$43$ \( T - 30847772 \) Copy content Toggle raw display
$47$ \( T - 4249824 \) Copy content Toggle raw display
$53$ \( T + 87962982 \) Copy content Toggle raw display
$59$ \( T - 5995348 \) Copy content Toggle raw display
$61$ \( T + 672930 \) Copy content Toggle raw display
$67$ \( T + 140689148 \) Copy content Toggle raw display
$71$ \( T - 322386224 \) Copy content Toggle raw display
$73$ \( T - 281507290 \) Copy content Toggle raw display
$79$ \( T + 466481136 \) Copy content Toggle raw display
$83$ \( T + 495155764 \) Copy content Toggle raw display
$89$ \( T - 524640510 \) Copy content Toggle raw display
$97$ \( T - 1666701490 \) Copy content Toggle raw display
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