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Properties

Label 21.8.a.a
Level $21$
Weight $8$
Character orbit 21.a
Self dual yes
Analytic conductor $6.560$
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] = [21,8,Mod(1,21)] 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("21.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.56008553517\)
Analytic rank: \(1\)
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

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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 27 q^{3} - 124 q^{4} - 278 q^{5} + 54 q^{6} - 343 q^{7} - 504 q^{8} + 729 q^{9} - 556 q^{10} - 4496 q^{11} - 3348 q^{12} - 7274 q^{13} - 686 q^{14} - 7506 q^{15} + 14864 q^{16} + 11382 q^{17}+ \cdots - 3277584 q^{99}+O(q^{100}) \) 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
2.00000 27.0000 −124.000 −278.000 54.0000 −343.000 −504.000 729.000 −556.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)

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 21.8.a.a ✓ 1
3.b odd 2 1 63.8.a.a 1
4.b odd 2 1 336.8.a.b 1
5.b even 2 1 525.8.a.b 1
7.b odd 2 1 147.8.a.a 1
7.c even 3 2 147.8.e.c 2
7.d odd 6 2 147.8.e.d 2
21.c even 2 1 441.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.a.a ✓ 1 1.a even 1 1 trivial
63.8.a.a 1 3.b odd 2 1
147.8.a.a 1 7.b odd 2 1
147.8.e.c 2 7.c even 3 2
147.8.e.d 2 7.d odd 6 2
336.8.a.b 1 4.b odd 2 1
441.8.a.b 1 21.c even 2 1
525.8.a.b 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T + 278 \) Copy content Toggle raw display
$7$ \( T + 343 \) Copy content Toggle raw display
$11$ \( T + 4496 \) Copy content Toggle raw display
$13$ \( T + 7274 \) Copy content Toggle raw display
$17$ \( T - 11382 \) Copy content Toggle raw display
$19$ \( T + 15884 \) Copy content Toggle raw display
$23$ \( T - 86100 \) Copy content Toggle raw display
$29$ \( T - 40702 \) Copy content Toggle raw display
$31$ \( T + 44760 \) Copy content Toggle raw display
$37$ \( T + 580962 \) Copy content Toggle raw display
$41$ \( T + 171658 \) Copy content Toggle raw display
$43$ \( T + 741148 \) Copy content Toggle raw display
$47$ \( T - 1071720 \) Copy content Toggle raw display
$53$ \( T + 1721778 \) Copy content Toggle raw display
$59$ \( T + 1557012 \) Copy content Toggle raw display
$61$ \( T - 2597998 \) Copy content Toggle raw display
$67$ \( T + 963548 \) Copy content Toggle raw display
$71$ \( T + 4063380 \) Copy content Toggle raw display
$73$ \( T + 5370222 \) Copy content Toggle raw display
$79$ \( T - 4094936 \) Copy content Toggle raw display
$83$ \( T + 1343124 \) Copy content Toggle raw display
$89$ \( T - 9081574 \) Copy content Toggle raw display
$97$ \( T - 6487914 \) Copy content Toggle raw display
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