Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,4,Mod(49,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.49");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.i (of order \(3\), degree \(2\), not 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: | \(8.49627504083\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 9) |
| 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −3.55842 | − | 3.78651i | 0 | −2.31386 | + | 4.00772i | 0 | 6.05842 | + | 10.4935i | 0 | −1.67527 | + | 26.9480i | 0 | ||||||||||||||||||||||
| 49.2 | 0 | 5.05842 | + | 1.18843i | 0 | −5.18614 | + | 8.98266i | 0 | −2.55842 | − | 4.43132i | 0 | 24.1753 | + | 12.0232i | 0 | |||||||||||||||||||||||
| 97.1 | 0 | −3.55842 | + | 3.78651i | 0 | −2.31386 | − | 4.00772i | 0 | 6.05842 | − | 10.4935i | 0 | −1.67527 | − | 26.9480i | 0 | |||||||||||||||||||||||
| 97.2 | 0 | 5.05842 | − | 1.18843i | 0 | −5.18614 | − | 8.98266i | 0 | −2.55842 | + | 4.43132i | 0 | 24.1753 | − | 12.0232i | 0 | |||||||||||||||||||||||
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 | 144.4.i.c | 4 | |
| 3.b | odd | 2 | 1 | 432.4.i.c | 4 | ||
| 4.b | odd | 2 | 1 | 9.4.c.a | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 144.4.i.c | 4 | |
| 9.c | even | 3 | 1 | 1296.4.a.u | 2 | ||
| 9.d | odd | 6 | 1 | 432.4.i.c | 4 | ||
| 9.d | odd | 6 | 1 | 1296.4.a.i | 2 | ||
| 12.b | even | 2 | 1 | 27.4.c.a | 4 | ||
| 20.d | odd | 2 | 1 | 225.4.e.b | 4 | ||
| 20.e | even | 4 | 2 | 225.4.k.b | 8 | ||
| 36.f | odd | 6 | 1 | 9.4.c.a | ✓ | 4 | |
| 36.f | odd | 6 | 1 | 81.4.a.d | 2 | ||
| 36.h | even | 6 | 1 | 27.4.c.a | 4 | ||
| 36.h | even | 6 | 1 | 81.4.a.a | 2 | ||
| 180.n | even | 6 | 1 | 2025.4.a.n | 2 | ||
| 180.p | odd | 6 | 1 | 225.4.e.b | 4 | ||
| 180.p | odd | 6 | 1 | 2025.4.a.g | 2 | ||
| 180.x | even | 12 | 2 | 225.4.k.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.4.c.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 9.4.c.a | ✓ | 4 | 36.f | odd | 6 | 1 | |
| 27.4.c.a | 4 | 12.b | even | 2 | 1 | ||
| 27.4.c.a | 4 | 36.h | even | 6 | 1 | ||
| 81.4.a.a | 2 | 36.h | even | 6 | 1 | ||
| 81.4.a.d | 2 | 36.f | odd | 6 | 1 | ||
| 144.4.i.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 144.4.i.c | 4 | 9.c | even | 3 | 1 | inner | |
| 225.4.e.b | 4 | 20.d | odd | 2 | 1 | ||
| 225.4.e.b | 4 | 180.p | odd | 6 | 1 | ||
| 225.4.k.b | 8 | 20.e | even | 4 | 2 | ||
| 225.4.k.b | 8 | 180.x | even | 12 | 2 | ||
| 432.4.i.c | 4 | 3.b | odd | 2 | 1 | ||
| 432.4.i.c | 4 | 9.d | odd | 6 | 1 | ||
| 1296.4.a.i | 2 | 9.d | odd | 6 | 1 | ||
| 1296.4.a.u | 2 | 9.c | even | 3 | 1 | ||
| 2025.4.a.g | 2 | 180.p | odd | 6 | 1 | ||
| 2025.4.a.n | 2 | 180.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 15T_{5}^{3} + 177T_{5}^{2} + 720T_{5} + 2304 \)
acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 729 \)
|
| $5$ |
\( T^{4} + 15 T^{3} + \cdots + 2304 \)
|
| $7$ |
\( T^{4} - 7 T^{3} + \cdots + 3844 \)
|
| $11$ |
\( T^{4} - 66 T^{3} + \cdots + 314721 \)
|
| $13$ |
\( T^{4} - 11 T^{3} + \cdots + 3334276 \)
|
| $17$ |
\( (T^{2} - 99 T + 1782)^{2} \)
|
| $19$ |
\( (T^{2} - 77 T - 4532)^{2} \)
|
| $23$ |
\( T^{4} - 33 T^{3} + \cdots + 7322436 \)
|
| $29$ |
\( T^{4} - 51 T^{3} + \cdots + 412164 \)
|
| $31$ |
\( T^{4} - 43 T^{3} + \cdots + 150544 \)
|
| $37$ |
\( (T^{2} + 50 T - 23432)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 5606565129 \)
|
| $43$ |
\( T^{4} - 88 T^{3} + \cdots + 2686321 \)
|
| $47$ |
\( T^{4} - 399 T^{3} + \cdots + 813276324 \)
|
| $53$ |
\( (T^{2} - 54 T - 215784)^{2} \)
|
| $59$ |
\( T^{4} - 798 T^{3} + \cdots + 43678881 \)
|
| $61$ |
\( T^{4} + \cdots + 1829786176 \)
|
| $67$ |
\( T^{4} + \cdots + 43305193801 \)
|
| $71$ |
\( (T^{2} + 1368 T + 296784)^{2} \)
|
| $73$ |
\( (T^{2} + 455 T - 435398)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 392522298256 \)
|
| $83$ |
\( T^{4} + \cdots + 5010940944 \)
|
| $89$ |
\( (T^{2} + 396 T - 3564)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 2459267281 \)
|
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