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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [21,8,Mod(5,21)] 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("21.5"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.56008553517\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (27 \zeta_{6} + 27) q^{3} + (128 \zeta_{6} - 128) q^{4} + (249 \zeta_{6} - 1006) q^{7} + 2187 \zeta_{6} q^{9} + (3456 \zeta_{6} - 6912) q^{12} + (18146 \zeta_{6} - 9073) q^{13} - 16384 \zeta_{6} q^{16} + \cdots + ( - 15198608 \zeta_{6} + 7599304) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 81 q^{3} - 128 q^{4} - 1763 q^{7} + 2187 q^{9} - 10368 q^{12} - 16384 q^{16} + 100539 q^{19} - 81486 q^{21} + 78125 q^{25} + 65024 q^{28} - 26445 q^{31} - 559872 q^{36} + 335663 q^{37} - 734913 q^{39}+ \cdots - 714015 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 40.5000 − 23.3827i −64.0000 − 110.851i 0 0 −881.500 − 215.640i 0 1093.50 − 1894.00i 0
17.1 0 40.5000 + 23.3827i −64.0000 + 110.851i 0 0 −881.500 + 215.640i 0 1093.50 + 1894.00i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.8.g.a ✓ 2
3.b odd 2 1 CM 21.8.g.a ✓ 2
7.c even 3 1 147.8.c.a 2
7.d odd 6 1 inner 21.8.g.a ✓ 2
7.d odd 6 1 147.8.c.a 2
21.g even 6 1 inner 21.8.g.a ✓ 2
21.g even 6 1 147.8.c.a 2
21.h odd 6 1 147.8.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.8.g.a ✓ 2 1.a even 1 1 trivial
21.8.g.a ✓ 2 3.b odd 2 1 CM
21.8.g.a ✓ 2 7.d odd 6 1 inner
21.8.g.a ✓ 2 21.g even 6 1 inner
147.8.c.a 2 7.c even 3 1
147.8.c.a 2 7.d odd 6 1
147.8.c.a 2 21.g even 6 1
147.8.c.a 2 21.h odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 81T + 2187 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1763 T + 823543 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 246957987 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 3369363507 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 26445 T + 233112675 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 112669649569 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 409495)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 70885825968 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 19744932199729 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 42659897573907 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 20408863358689 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 173248263853248 \) Copy content Toggle raw display
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