Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,6,Mod(7,18)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.c (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: | \(2.88690875663\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
| 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{3} + 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\nu^{3} + 12\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
2.00000 | − | 3.46410i | −14.6969 | − | 5.19615i | −8.00000 | − | 13.8564i | −28.1969 | − | 48.8385i | −47.3939 | + | 40.5194i | 11.1515 | − | 19.3150i | −64.0000 | 189.000 | + | 152.735i | −225.576 | ||||||||||||||||
| 7.2 | 2.00000 | − | 3.46410i | 14.6969 | − | 5.19615i | −8.00000 | − | 13.8564i | 1.19694 | + | 2.07316i | 11.3939 | − | 61.3040i | 25.8485 | − | 44.7709i | −64.0000 | 189.000 | − | 152.735i | 9.57551 | |||||||||||||||||
| 13.1 | 2.00000 | + | 3.46410i | −14.6969 | + | 5.19615i | −8.00000 | + | 13.8564i | −28.1969 | + | 48.8385i | −47.3939 | − | 40.5194i | 11.1515 | + | 19.3150i | −64.0000 | 189.000 | − | 152.735i | −225.576 | |||||||||||||||||
| 13.2 | 2.00000 | + | 3.46410i | 14.6969 | + | 5.19615i | −8.00000 | + | 13.8564i | 1.19694 | − | 2.07316i | 11.3939 | + | 61.3040i | 25.8485 | + | 44.7709i | −64.0000 | 189.000 | + | 152.735i | 9.57551 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 18.6.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 54.6.c.a | 4 | ||
| 4.b | odd | 2 | 1 | 144.6.i.a | 4 | ||
| 9.c | even | 3 | 1 | inner | 18.6.c.a | ✓ | 4 |
| 9.c | even | 3 | 1 | 162.6.a.e | 2 | ||
| 9.d | odd | 6 | 1 | 54.6.c.a | 4 | ||
| 9.d | odd | 6 | 1 | 162.6.a.f | 2 | ||
| 12.b | even | 2 | 1 | 432.6.i.a | 4 | ||
| 36.f | odd | 6 | 1 | 144.6.i.a | 4 | ||
| 36.h | even | 6 | 1 | 432.6.i.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.6.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 18.6.c.a | ✓ | 4 | 9.c | even | 3 | 1 | inner |
| 54.6.c.a | 4 | 3.b | odd | 2 | 1 | ||
| 54.6.c.a | 4 | 9.d | odd | 6 | 1 | ||
| 144.6.i.a | 4 | 4.b | odd | 2 | 1 | ||
| 144.6.i.a | 4 | 36.f | odd | 6 | 1 | ||
| 162.6.a.e | 2 | 9.c | even | 3 | 1 | ||
| 162.6.a.f | 2 | 9.d | odd | 6 | 1 | ||
| 432.6.i.a | 4 | 12.b | even | 2 | 1 | ||
| 432.6.i.a | 4 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 54T_{5}^{3} + 3051T_{5}^{2} - 7290T_{5} + 18225 \)
acting on \(S_{6}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( T^{4} - 378 T^{2} + 59049 \)
|
| $5$ |
\( T^{4} + 54 T^{3} + \cdots + 18225 \)
|
| $7$ |
\( T^{4} - 74 T^{3} + \cdots + 1329409 \)
|
| $11$ |
\( T^{4} + \cdots + 158294966769 \)
|
| $13$ |
\( T^{4} + \cdots + 48140309281 \)
|
| $17$ |
\( (T^{2} - 492 T - 1848060)^{2} \)
|
| $19$ |
\( (T^{2} + 1640 T - 2648816)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 56736462234321 \)
|
| $29$ |
\( T^{4} + \cdots + 220517585649 \)
|
| $31$ |
\( T^{4} + \cdots + 12\!\cdots\!61 \)
|
| $37$ |
\( (T^{2} + 4796 T - 141621212)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 34\!\cdots\!49 \)
|
| $43$ |
\( T^{4} + \cdots + 64\!\cdots\!89 \)
|
| $47$ |
\( T^{4} + \cdots + 20\!\cdots\!69 \)
|
| $53$ |
\( (T^{2} + 16332 T + 59839812)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 99\!\cdots\!09 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!01 \)
|
| $67$ |
\( T^{4} + \cdots + 70\!\cdots\!25 \)
|
| $71$ |
\( (T^{2} + 73848 T + 665096112)^{2} \)
|
| $73$ |
\( (T^{2} + 51188 T + 491562436)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 27\!\cdots\!25 \)
|
| $83$ |
\( T^{4} + \cdots + 42\!\cdots\!89 \)
|
| $89$ |
\( (T^{2} + 9300 T - 7978887036)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 82\!\cdots\!25 \)
|
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