Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,6,Mod(4,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.4");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.e (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(3.36806021607\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-83})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 20x^{2} - 21x + 441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 20x^{2} - 21x + 441 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 20\nu^{2} - 20\nu - 441 ) / 420 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 41\nu ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 20\nu - 41 ) / 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 61\beta _1 + 62 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 40\beta_{3} - 20\beta_{2} + 20\beta _1 + 103 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−3.19493 | − | 5.53379i | 4.50000 | − | 7.79423i | −4.41520 | + | 7.64735i | −19.3645 | − | 33.5404i | −57.5088 | −87.5000 | + | 95.6596i | −148.051 | −40.5000 | − | 70.1481i | −123.737 | + | 214.318i | ||||||||||||||||
| 4.2 | 4.69493 | + | 8.13186i | 4.50000 | − | 7.79423i | −28.0848 | + | 48.6443i | 35.8645 | + | 62.1192i | 84.5088 | −87.5000 | − | 95.6596i | −226.949 | −40.5000 | − | 70.1481i | −336.763 | + | 583.291i | |||||||||||||||||
| 16.1 | −3.19493 | + | 5.53379i | 4.50000 | + | 7.79423i | −4.41520 | − | 7.64735i | −19.3645 | + | 33.5404i | −57.5088 | −87.5000 | − | 95.6596i | −148.051 | −40.5000 | + | 70.1481i | −123.737 | − | 214.318i | |||||||||||||||||
| 16.2 | 4.69493 | − | 8.13186i | 4.50000 | + | 7.79423i | −28.0848 | − | 48.6443i | 35.8645 | − | 62.1192i | 84.5088 | −87.5000 | + | 95.6596i | −226.949 | −40.5000 | + | 70.1481i | −336.763 | − | 583.291i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.6.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 63.6.e.c | 4 | ||
| 4.b | odd | 2 | 1 | 336.6.q.e | 4 | ||
| 7.b | odd | 2 | 1 | 147.6.e.l | 4 | ||
| 7.c | even | 3 | 1 | inner | 21.6.e.b | ✓ | 4 |
| 7.c | even | 3 | 1 | 147.6.a.i | 2 | ||
| 7.d | odd | 6 | 1 | 147.6.a.k | 2 | ||
| 7.d | odd | 6 | 1 | 147.6.e.l | 4 | ||
| 21.g | even | 6 | 1 | 441.6.a.s | 2 | ||
| 21.h | odd | 6 | 1 | 63.6.e.c | 4 | ||
| 21.h | odd | 6 | 1 | 441.6.a.t | 2 | ||
| 28.g | odd | 6 | 1 | 336.6.q.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.6.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 21.6.e.b | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 63.6.e.c | 4 | 3.b | odd | 2 | 1 | ||
| 63.6.e.c | 4 | 21.h | odd | 6 | 1 | ||
| 147.6.a.i | 2 | 7.c | even | 3 | 1 | ||
| 147.6.a.k | 2 | 7.d | odd | 6 | 1 | ||
| 147.6.e.l | 4 | 7.b | odd | 2 | 1 | ||
| 147.6.e.l | 4 | 7.d | odd | 6 | 1 | ||
| 336.6.q.e | 4 | 4.b | odd | 2 | 1 | ||
| 336.6.q.e | 4 | 28.g | odd | 6 | 1 | ||
| 441.6.a.s | 2 | 21.g | even | 6 | 1 | ||
| 441.6.a.t | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} + 69T_{2}^{2} + 180T_{2} + 3600 \)
acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 3600 \)
|
| $3$ |
\( (T^{2} - 9 T + 81)^{2} \)
|
| $5$ |
\( T^{4} - 33 T^{3} + \cdots + 7717284 \)
|
| $7$ |
\( (T^{2} + 175 T + 16807)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 104412996900 \)
|
| $13$ |
\( (T^{2} - 925 T + 208864)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 1788317798400 \)
|
| $19$ |
\( T^{4} + \cdots + 1663584040000 \)
|
| $23$ |
\( T^{4} + \cdots + 27756881510400 \)
|
| $29$ |
\( (T^{2} + 2217 T + 1102716)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 10\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots + 20\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + 12858 T - 3221280)^{2} \)
|
| $43$ |
\( (T^{2} + 2771 T - 257902490)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 28\!\cdots\!44 \)
|
| $61$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} - 102096 T + 2483190108)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 22\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots + 79\!\cdots\!69 \)
|
| $83$ |
\( (T^{2} - 61179 T + 711231498)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 45\!\cdots\!16 \)
|
| $97$ |
\( (T^{2} + 9791 T - 40418570)^{2} \)
|
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