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Properties

Label 126.8.a.e
Level $126$
Weight $8$
Character orbit 126.a
Self dual yes
Analytic conductor $39.361$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,8,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,8,0,64,-270] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.3605132110\)
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 8 q^{2} + 64 q^{4} - 270 q^{5} + 343 q^{7} + 512 q^{8} - 2160 q^{10} + 1404 q^{11} - 2362 q^{13} + 2744 q^{14} + 4096 q^{16} - 18354 q^{17} + 27164 q^{19} - 17280 q^{20} + 11232 q^{22} - 86184 q^{23}+ \cdots + 941192 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
8.00000 0 64.0000 −270.000 0 343.000 512.000 0 −2160.00
\(n\): 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 126.8.a.e 1
3.b odd 2 1 42.8.a.d ✓ 1
12.b even 2 1 336.8.a.e 1
21.c even 2 1 294.8.a.a 1
21.g even 6 2 294.8.e.q 2
21.h odd 6 2 294.8.e.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.8.a.d ✓ 1 3.b odd 2 1
126.8.a.e 1 1.a even 1 1 trivial
294.8.a.a 1 21.c even 2 1
294.8.e.h 2 21.h odd 6 2
294.8.e.q 2 21.g even 6 2
336.8.a.e 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 270 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(126))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 270 \) Copy content Toggle raw display
$7$ \( T - 343 \) Copy content Toggle raw display
$11$ \( T - 1404 \) Copy content Toggle raw display
$13$ \( T + 2362 \) Copy content Toggle raw display
$17$ \( T + 18354 \) Copy content Toggle raw display
$19$ \( T - 27164 \) Copy content Toggle raw display
$23$ \( T + 86184 \) Copy content Toggle raw display
$29$ \( T + 251862 \) Copy content Toggle raw display
$31$ \( T + 27040 \) Copy content Toggle raw display
$37$ \( T + 410290 \) Copy content Toggle raw display
$41$ \( T - 246246 \) Copy content Toggle raw display
$43$ \( T - 49508 \) Copy content Toggle raw display
$47$ \( T - 409200 \) Copy content Toggle raw display
$53$ \( T + 1486398 \) Copy content Toggle raw display
$59$ \( T + 427956 \) Copy content Toggle raw display
$61$ \( T + 2048554 \) Copy content Toggle raw display
$67$ \( T - 940748 \) Copy content Toggle raw display
$71$ \( T + 789048 \) Copy content Toggle raw display
$73$ \( T - 374330 \) Copy content Toggle raw display
$79$ \( T + 4260880 \) Copy content Toggle raw display
$83$ \( T - 8772132 \) Copy content Toggle raw display
$89$ \( T + 2703786 \) Copy content Toggle raw display
$97$ \( T + 13666078 \) Copy content Toggle raw display
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