Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [36,4,Mod(35,36)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36, 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("36.35");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 36.b (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: | \(2.12406876021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-15})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 13x^{2} - 12x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 2x^{3} + 13x^{2} - 12x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} + 7\nu + 16 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} - 21\nu + 10 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} + 56\nu - 27 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 4\beta _1 - 11 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{3} - 22\beta_{2} + 6\beta _1 - 17 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).
| \(n\) | \(19\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 35.1 |
|
−2.73861 | − | 0.707107i | 0 | 7.00000 | + | 3.87298i | − | 9.89949i | 0 | − | 30.9839i | −16.4317 | − | 15.5563i | 0 | −7.00000 | + | 27.1109i | ||||||||||||||||||||
| 35.2 | −2.73861 | + | 0.707107i | 0 | 7.00000 | − | 3.87298i | 9.89949i | 0 | 30.9839i | −16.4317 | + | 15.5563i | 0 | −7.00000 | − | 27.1109i | |||||||||||||||||||||||
| 35.3 | 2.73861 | − | 0.707107i | 0 | 7.00000 | − | 3.87298i | − | 9.89949i | 0 | 30.9839i | 16.4317 | − | 15.5563i | 0 | −7.00000 | − | 27.1109i | ||||||||||||||||||||||
| 35.4 | 2.73861 | + | 0.707107i | 0 | 7.00000 | + | 3.87298i | 9.89949i | 0 | − | 30.9839i | 16.4317 | + | 15.5563i | 0 | −7.00000 | + | 27.1109i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 36.4.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 36.4.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 36.4.b.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 576.4.c.e | 4 | ||
| 8.d | odd | 2 | 1 | 576.4.c.e | 4 | ||
| 12.b | even | 2 | 1 | inner | 36.4.b.b | ✓ | 4 |
| 16.e | even | 4 | 2 | 2304.4.f.g | 8 | ||
| 16.f | odd | 4 | 2 | 2304.4.f.g | 8 | ||
| 24.f | even | 2 | 1 | 576.4.c.e | 4 | ||
| 24.h | odd | 2 | 1 | 576.4.c.e | 4 | ||
| 48.i | odd | 4 | 2 | 2304.4.f.g | 8 | ||
| 48.k | even | 4 | 2 | 2304.4.f.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 36.4.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 36.4.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 36.4.b.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 36.4.b.b | ✓ | 4 | 12.b | even | 2 | 1 | inner |
| 576.4.c.e | 4 | 8.b | even | 2 | 1 | ||
| 576.4.c.e | 4 | 8.d | odd | 2 | 1 | ||
| 576.4.c.e | 4 | 24.f | even | 2 | 1 | ||
| 576.4.c.e | 4 | 24.h | odd | 2 | 1 | ||
| 2304.4.f.g | 8 | 16.e | even | 4 | 2 | ||
| 2304.4.f.g | 8 | 16.f | odd | 4 | 2 | ||
| 2304.4.f.g | 8 | 48.i | odd | 4 | 2 | ||
| 2304.4.f.g | 8 | 48.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 98 \)
acting on \(S_{4}^{\mathrm{new}}(36, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 14T^{2} + 64 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 98)^{2} \)
|
| $7$ |
\( (T^{2} + 960)^{2} \)
|
| $11$ |
\( (T^{2} - 1920)^{2} \)
|
| $13$ |
\( (T + 28)^{4} \)
|
| $17$ |
\( (T^{2} + 2450)^{2} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} - 1920)^{2} \)
|
| $29$ |
\( (T^{2} + 18818)^{2} \)
|
| $31$ |
\( (T^{2} + 47040)^{2} \)
|
| $37$ |
\( (T - 266)^{4} \)
|
| $41$ |
\( (T^{2} + 51842)^{2} \)
|
| $43$ |
\( (T^{2} + 3840)^{2} \)
|
| $47$ |
\( (T^{2} - 94080)^{2} \)
|
| $53$ |
\( (T^{2} + 14450)^{2} \)
|
| $59$ |
\( (T^{2} - 376320)^{2} \)
|
| $61$ |
\( (T - 350)^{4} \)
|
| $67$ |
\( (T^{2} + 188160)^{2} \)
|
| $71$ |
\( (T^{2} - 432000)^{2} \)
|
| $73$ |
\( (T + 112)^{4} \)
|
| $79$ |
\( (T^{2} + 47040)^{2} \)
|
| $83$ |
\( (T^{2} - 94080)^{2} \)
|
| $89$ |
\( (T^{2} + 1681778)^{2} \)
|
| $97$ |
\( (T + 616)^{4} \)
|
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