Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,4,Mod(20,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.20");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.c (of order \(2\), degree \(1\), 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: | \(1.23904011012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-6}, \sqrt{-17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 46x^{2} + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 46x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 35\nu ) / 22 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 57\nu ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 11\nu^{2} + 35\nu + 253 ) / 44 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{3} + 2\beta _1 - 23 \)
|
| \(\nu^{3}\) | \(=\) |
\( -35\beta_{2} - 57\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 20.1 |
|
− | 4.12311i | −5.04975 | − | 1.22474i | −9.00000 | 10.0995 | −5.04975 | + | 20.8207i | 7.00000 | − | 17.1464i | 4.12311i | 24.0000 | + | 12.3693i | − | 41.6413i | ||||||||||||||||||||
| 20.2 | − | 4.12311i | 5.04975 | + | 1.22474i | −9.00000 | −10.0995 | 5.04975 | − | 20.8207i | 7.00000 | + | 17.1464i | 4.12311i | 24.0000 | + | 12.3693i | 41.6413i | ||||||||||||||||||||||
| 20.3 | 4.12311i | −5.04975 | + | 1.22474i | −9.00000 | 10.0995 | −5.04975 | − | 20.8207i | 7.00000 | + | 17.1464i | − | 4.12311i | 24.0000 | − | 12.3693i | 41.6413i | ||||||||||||||||||||||
| 20.4 | 4.12311i | 5.04975 | − | 1.22474i | −9.00000 | −10.0995 | 5.04975 | + | 20.8207i | 7.00000 | − | 17.1464i | − | 4.12311i | 24.0000 | − | 12.3693i | − | 41.6413i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.4.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 21.4.c.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 336.4.k.b | 4 | ||
| 7.b | odd | 2 | 1 | inner | 21.4.c.b | ✓ | 4 |
| 7.c | even | 3 | 2 | 147.4.g.c | 8 | ||
| 7.d | odd | 6 | 2 | 147.4.g.c | 8 | ||
| 12.b | even | 2 | 1 | 336.4.k.b | 4 | ||
| 21.c | even | 2 | 1 | inner | 21.4.c.b | ✓ | 4 |
| 21.g | even | 6 | 2 | 147.4.g.c | 8 | ||
| 21.h | odd | 6 | 2 | 147.4.g.c | 8 | ||
| 28.d | even | 2 | 1 | 336.4.k.b | 4 | ||
| 84.h | odd | 2 | 1 | 336.4.k.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 21.4.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 21.4.c.b | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 21.4.c.b | ✓ | 4 | 21.c | even | 2 | 1 | inner |
| 147.4.g.c | 8 | 7.c | even | 3 | 2 | ||
| 147.4.g.c | 8 | 7.d | odd | 6 | 2 | ||
| 147.4.g.c | 8 | 21.g | even | 6 | 2 | ||
| 147.4.g.c | 8 | 21.h | odd | 6 | 2 | ||
| 336.4.k.b | 4 | 4.b | odd | 2 | 1 | ||
| 336.4.k.b | 4 | 12.b | even | 2 | 1 | ||
| 336.4.k.b | 4 | 28.d | even | 2 | 1 | ||
| 336.4.k.b | 4 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 17 \)
acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 17)^{2} \)
|
| $3$ |
\( T^{4} - 48T^{2} + 729 \)
|
| $5$ |
\( (T^{2} - 102)^{2} \)
|
| $7$ |
\( (T^{2} - 14 T + 343)^{2} \)
|
| $11$ |
\( (T^{2} + 1088)^{2} \)
|
| $13$ |
\( (T^{2} + 3174)^{2} \)
|
| $17$ |
\( (T^{2} - 3672)^{2} \)
|
| $19$ |
\( (T^{2} + 1350)^{2} \)
|
| $23$ |
\( (T^{2} + 8228)^{2} \)
|
| $29$ |
\( (T^{2} + 3332)^{2} \)
|
| $31$ |
\( (T^{2} + 64896)^{2} \)
|
| $37$ |
\( (T - 230)^{4} \)
|
| $41$ |
\( (T^{2} - 19992)^{2} \)
|
| $43$ |
\( (T - 44)^{4} \)
|
| $47$ |
\( (T^{2} - 117912)^{2} \)
|
| $53$ |
\( (T^{2} + 42500)^{2} \)
|
| $59$ |
\( (T^{2} - 17238)^{2} \)
|
| $61$ |
\( (T^{2} + 5046)^{2} \)
|
| $67$ |
\( (T + 64)^{4} \)
|
| $71$ |
\( (T^{2} + 213248)^{2} \)
|
| $73$ |
\( (T^{2} + 7776)^{2} \)
|
| $79$ |
\( (T + 442)^{4} \)
|
| $83$ |
\( (T^{2} - 244902)^{2} \)
|
| $89$ |
\( (T^{2} - 235008)^{2} \)
|
| $97$ |
\( (T^{2} + 1193496)^{2} \)
|
| show more | |
| show less |