Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,3,Mod(26,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.26");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.c (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: | \(3.67848356886\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 5\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 5 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 5\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
− | 2.61803i | 0 | −2.85410 | − | 2.23607i | 0 | −9.70820 | − | 3.00000i | 0 | −5.85410 | |||||||||||||||||||||||||||
| 26.2 | − | 0.381966i | 0 | 3.85410 | 2.23607i | 0 | 3.70820 | − | 3.00000i | 0 | 0.854102 | |||||||||||||||||||||||||||||
| 26.3 | 0.381966i | 0 | 3.85410 | − | 2.23607i | 0 | 3.70820 | 3.00000i | 0 | 0.854102 | ||||||||||||||||||||||||||||||
| 26.4 | 2.61803i | 0 | −2.85410 | 2.23607i | 0 | −9.70820 | 3.00000i | 0 | −5.85410 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 135.3.c.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 135.3.c.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 2160.3.l.g | 4 | ||
| 5.b | even | 2 | 1 | 675.3.c.p | 4 | ||
| 5.c | odd | 4 | 1 | 675.3.d.e | 4 | ||
| 5.c | odd | 4 | 1 | 675.3.d.i | 4 | ||
| 9.c | even | 3 | 2 | 405.3.i.c | 8 | ||
| 9.d | odd | 6 | 2 | 405.3.i.c | 8 | ||
| 12.b | even | 2 | 1 | 2160.3.l.g | 4 | ||
| 15.d | odd | 2 | 1 | 675.3.c.p | 4 | ||
| 15.e | even | 4 | 1 | 675.3.d.e | 4 | ||
| 15.e | even | 4 | 1 | 675.3.d.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.3.c.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 135.3.c.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 405.3.i.c | 8 | 9.c | even | 3 | 2 | ||
| 405.3.i.c | 8 | 9.d | odd | 6 | 2 | ||
| 675.3.c.p | 4 | 5.b | even | 2 | 1 | ||
| 675.3.c.p | 4 | 15.d | odd | 2 | 1 | ||
| 675.3.d.e | 4 | 5.c | odd | 4 | 1 | ||
| 675.3.d.e | 4 | 15.e | even | 4 | 1 | ||
| 675.3.d.i | 4 | 5.c | odd | 4 | 1 | ||
| 675.3.d.i | 4 | 15.e | even | 4 | 1 | ||
| 2160.3.l.g | 4 | 4.b | odd | 2 | 1 | ||
| 2160.3.l.g | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 7T_{2}^{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 7T^{2} + 1 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5)^{2} \)
|
| $7$ |
\( (T^{2} + 6 T - 36)^{2} \)
|
| $11$ |
\( T^{4} + 268 T^{2} + 13456 \)
|
| $13$ |
\( (T^{2} - 2 T - 44)^{2} \)
|
| $17$ |
\( T^{4} + 178T^{2} + 5041 \)
|
| $19$ |
\( (T^{2} - 38 T + 181)^{2} \)
|
| $23$ |
\( T^{4} + 810T^{2} + 2025 \)
|
| $29$ |
\( T^{4} + 3148 T^{2} + 13456 \)
|
| $31$ |
\( (T^{2} - 18 T - 1539)^{2} \)
|
| $37$ |
\( (T^{2} + 80 T + 1420)^{2} \)
|
| $41$ |
\( T^{4} + 7948 T^{2} + 15713296 \)
|
| $43$ |
\( (T^{2} + 22 T - 3524)^{2} \)
|
| $47$ |
\( T^{4} + 7048 T^{2} + 10471696 \)
|
| $53$ |
\( T^{4} + 6202 T^{2} + 4414201 \)
|
| $59$ |
\( T^{4} + 17308 T^{2} + 74718736 \)
|
| $61$ |
\( (T - 19)^{4} \)
|
| $67$ |
\( (T^{2} - 108 T + 36)^{2} \)
|
| $71$ |
\( T^{4} + 12268 T^{2} + 37405456 \)
|
| $73$ |
\( (T^{2} + 46 T - 4916)^{2} \)
|
| $79$ |
\( (T^{2} + 150 T + 4905)^{2} \)
|
| $83$ |
\( T^{4} + 5922 T^{2} + 7834401 \)
|
| $89$ |
\( T^{4} + 8028 T^{2} + 15397776 \)
|
| $97$ |
\( (T^{2} - 292 T + 21136)^{2} \)
|
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