Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(1.06203438010\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-35})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} - 9x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{4} - x^{3} - 8x^{2} - 9x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 8\nu^{2} - 8\nu - 81 ) / 72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 17\nu ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 8\nu - 17 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 25\beta _1 + 26 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 16\beta_{3} - 8\beta_{2} + 8\beta _1 + 43 ) / 3 \)
|
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 |
|
1.00000 | − | 1.73205i | −1.81174 | − | 4.87007i | −2.00000 | − | 3.46410i | 9.93521 | + | 17.2083i | −10.2470 | − | 1.73205i | 2.93521 | − | 5.08394i | −8.00000 | −20.4352 | + | 17.6466i | 39.7409 | ||||||||||||||||
| 7.2 | 1.00000 | − | 1.73205i | 3.31174 | + | 4.00405i | −2.00000 | − | 3.46410i | −5.43521 | − | 9.41407i | 10.2470 | − | 1.73205i | −12.4352 | + | 21.5384i | −8.00000 | −5.06479 | + | 26.5207i | −21.7409 | |||||||||||||||||
| 13.1 | 1.00000 | + | 1.73205i | −1.81174 | + | 4.87007i | −2.00000 | + | 3.46410i | 9.93521 | − | 17.2083i | −10.2470 | + | 1.73205i | 2.93521 | + | 5.08394i | −8.00000 | −20.4352 | − | 17.6466i | 39.7409 | |||||||||||||||||
| 13.2 | 1.00000 | + | 1.73205i | 3.31174 | − | 4.00405i | −2.00000 | + | 3.46410i | −5.43521 | + | 9.41407i | 10.2470 | + | 1.73205i | −12.4352 | − | 21.5384i | −8.00000 | −5.06479 | − | 26.5207i | −21.7409 | |||||||||||||||||
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 | 18.4.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 54.4.c.b | 4 | ||
| 4.b | odd | 2 | 1 | 144.4.i.b | 4 | ||
| 9.c | even | 3 | 1 | inner | 18.4.c.b | ✓ | 4 |
| 9.c | even | 3 | 1 | 162.4.a.f | 2 | ||
| 9.d | odd | 6 | 1 | 54.4.c.b | 4 | ||
| 9.d | odd | 6 | 1 | 162.4.a.g | 2 | ||
| 12.b | even | 2 | 1 | 432.4.i.b | 4 | ||
| 36.f | odd | 6 | 1 | 144.4.i.b | 4 | ||
| 36.f | odd | 6 | 1 | 1296.4.a.l | 2 | ||
| 36.h | even | 6 | 1 | 432.4.i.b | 4 | ||
| 36.h | even | 6 | 1 | 1296.4.a.r | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.4.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 18.4.c.b | ✓ | 4 | 9.c | even | 3 | 1 | inner |
| 54.4.c.b | 4 | 3.b | odd | 2 | 1 | ||
| 54.4.c.b | 4 | 9.d | odd | 6 | 1 | ||
| 144.4.i.b | 4 | 4.b | odd | 2 | 1 | ||
| 144.4.i.b | 4 | 36.f | odd | 6 | 1 | ||
| 162.4.a.f | 2 | 9.c | even | 3 | 1 | ||
| 162.4.a.g | 2 | 9.d | odd | 6 | 1 | ||
| 432.4.i.b | 4 | 12.b | even | 2 | 1 | ||
| 432.4.i.b | 4 | 36.h | even | 6 | 1 | ||
| 1296.4.a.l | 2 | 36.f | odd | 6 | 1 | ||
| 1296.4.a.r | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 9T_{5}^{3} + 297T_{5}^{2} + 1944T_{5} + 46656 \)
acting on \(S_{4}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 729 \)
|
| $5$ |
\( T^{4} - 9 T^{3} + \cdots + 46656 \)
|
| $7$ |
\( T^{4} + 19 T^{3} + \cdots + 21316 \)
|
| $11$ |
\( T^{4} - 24 T^{3} + \cdots + 641601 \)
|
| $13$ |
\( T^{4} + 61 T^{3} + \cdots + 481636 \)
|
| $17$ |
\( (T^{2} - 3 T - 234)^{2} \)
|
| $19$ |
\( (T^{2} - 133 T - 1484)^{2} \)
|
| $23$ |
\( T^{4} + 69 T^{3} + \cdots + 910116 \)
|
| $29$ |
\( T^{4} + 237 T^{3} + \cdots + 6081156 \)
|
| $31$ |
\( T^{4} + 211 T^{3} + \cdots + 81072016 \)
|
| $37$ |
\( (T^{2} - 262 T - 29144)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 2895623721 \)
|
| $43$ |
\( T^{4} + \cdots + 18017961361 \)
|
| $47$ |
\( T^{4} + 483 T^{3} + \cdots + 26687556 \)
|
| $53$ |
\( (T^{2} - 150 T - 40680)^{2} \)
|
| $59$ |
\( T^{4} + 168 T^{3} + \cdots + 37344321 \)
|
| $61$ |
\( T^{4} + \cdots + 49259139136 \)
|
| $67$ |
\( T^{4} + \cdots + 93555845161 \)
|
| $71$ |
\( (T^{2} + 312 T - 217584)^{2} \)
|
| $73$ |
\( (T^{2} + 311 T - 80006)^{2} \)
|
| $79$ |
\( T^{4} + 349 T^{3} + \cdots + 89794576 \)
|
| $83$ |
\( T^{4} + \cdots + 72105138576 \)
|
| $89$ |
\( (T^{2} + 492 T - 317484)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 12154842001 \)
|
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