Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,6,Mod(5,21)]
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([3, 5]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.5");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.g (of order \(6\), 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 \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 17x^{2} + 289 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 17x^{2} + 289 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 17\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 17\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−7.14143 | − | 4.12311i | −15.5707 | + | 0.743545i | 18.0000 | + | 31.1769i | 3.57071 | − | 6.18466i | 114.263 | + | 58.8897i | 122.500 | + | 42.4352i | − | 32.9848i | 241.894 | − | 23.1550i | −51.0000 | + | 29.4449i | |||||||||||||
| 5.2 | 7.14143 | + | 4.12311i | −8.42929 | + | 13.1129i | 18.0000 | + | 31.1769i | −3.57071 | + | 6.18466i | −114.263 | + | 58.8897i | 122.500 | + | 42.4352i | 32.9848i | −100.894 | − | 221.064i | −51.0000 | + | 29.4449i | |||||||||||||||
| 17.1 | −7.14143 | + | 4.12311i | −15.5707 | − | 0.743545i | 18.0000 | − | 31.1769i | 3.57071 | + | 6.18466i | 114.263 | − | 58.8897i | 122.500 | − | 42.4352i | 32.9848i | 241.894 | + | 23.1550i | −51.0000 | − | 29.4449i | |||||||||||||||
| 17.2 | 7.14143 | − | 4.12311i | −8.42929 | − | 13.1129i | 18.0000 | − | 31.1769i | −3.57071 | − | 6.18466i | −114.263 | − | 58.8897i | 122.500 | − | 42.4352i | − | 32.9848i | −100.894 | + | 221.064i | −51.0000 | − | 29.4449i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 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 | 21.6.g.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 21.6.g.b | ✓ | 4 |
| 7.c | even | 3 | 1 | 147.6.c.b | 4 | ||
| 7.d | odd | 6 | 1 | inner | 21.6.g.b | ✓ | 4 |
| 7.d | odd | 6 | 1 | 147.6.c.b | 4 | ||
| 21.g | even | 6 | 1 | inner | 21.6.g.b | ✓ | 4 |
| 21.g | even | 6 | 1 | 147.6.c.b | 4 | ||
| 21.h | odd | 6 | 1 | 147.6.c.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.6.g.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 21.6.g.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 21.6.g.b | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 21.6.g.b | ✓ | 4 | 21.g | even | 6 | 1 | inner |
| 147.6.c.b | 4 | 7.c | even | 3 | 1 | ||
| 147.6.c.b | 4 | 7.d | odd | 6 | 1 | ||
| 147.6.c.b | 4 | 21.g | even | 6 | 1 | ||
| 147.6.c.b | 4 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 68T_{2}^{2} + 4624 \)
acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 68T^{2} + 4624 \)
|
| $3$ |
\( T^{4} + 48 T^{3} + \cdots + 59049 \)
|
| $5$ |
\( T^{4} + 51T^{2} + 2601 \)
|
| $7$ |
\( (T^{2} - 245 T + 16807)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 40794708529 \)
|
| $13$ |
\( (T^{2} + 273612)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 1975025486736 \)
|
| $19$ |
\( (T^{2} + 942 T + 295788)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 208843771239184 \)
|
| $29$ |
\( (T^{2} + 6898073)^{2} \)
|
| $31$ |
\( (T^{2} + 15021 T + 75210147)^{2} \)
|
| $37$ |
\( (T^{2} - 6106 T + 37283236)^{2} \)
|
| $41$ |
\( (T^{2} - 39984)^{2} \)
|
| $43$ |
\( (T + 4744)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 46\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 35\!\cdots\!25 \)
|
| $59$ |
\( T^{4} + \cdots + 71\!\cdots\!25 \)
|
| $61$ |
\( (T^{2} + 7704 T + 19783872)^{2} \)
|
| $67$ |
\( (T^{2} + 1382 T + 1909924)^{2} \)
|
| $71$ |
\( (T^{2} + 6928440848)^{2} \)
|
| $73$ |
\( (T^{2} - 56316 T + 1057163952)^{2} \)
|
| $79$ |
\( (T^{2} + 21089 T + 444745921)^{2} \)
|
| $83$ |
\( (T^{2} - 1342685211)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 14\!\cdots\!76 \)
|
| $97$ |
\( (T^{2} + 14477825883)^{2} \)
|
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