Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [270,3,Mod(269,270)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("270.269");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 270.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: | \(7.35696713773\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(217\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 269.1 |
|
−1.41421 | 0 | 2.00000 | −3.53553 | − | 3.53553i | 0 | 5.00000i | −2.82843 | 0 | 5.00000 | + | 5.00000i | ||||||||||||||||||||||||||
| 269.2 | −1.41421 | 0 | 2.00000 | −3.53553 | + | 3.53553i | 0 | − | 5.00000i | −2.82843 | 0 | 5.00000 | − | 5.00000i | ||||||||||||||||||||||||||
| 269.3 | 1.41421 | 0 | 2.00000 | 3.53553 | − | 3.53553i | 0 | − | 5.00000i | 2.82843 | 0 | 5.00000 | − | 5.00000i | ||||||||||||||||||||||||||
| 269.4 | 1.41421 | 0 | 2.00000 | 3.53553 | + | 3.53553i | 0 | 5.00000i | 2.82843 | 0 | 5.00000 | + | 5.00000i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 270.3.b.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 270.3.b.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 2160.3.c.i | 4 | ||
| 5.b | even | 2 | 1 | inner | 270.3.b.c | ✓ | 4 |
| 5.c | odd | 4 | 1 | 1350.3.d.f | 2 | ||
| 5.c | odd | 4 | 1 | 1350.3.d.g | 2 | ||
| 9.c | even | 3 | 2 | 810.3.j.e | 8 | ||
| 9.d | odd | 6 | 2 | 810.3.j.e | 8 | ||
| 12.b | even | 2 | 1 | 2160.3.c.i | 4 | ||
| 15.d | odd | 2 | 1 | inner | 270.3.b.c | ✓ | 4 |
| 15.e | even | 4 | 1 | 1350.3.d.f | 2 | ||
| 15.e | even | 4 | 1 | 1350.3.d.g | 2 | ||
| 20.d | odd | 2 | 1 | 2160.3.c.i | 4 | ||
| 45.h | odd | 6 | 2 | 810.3.j.e | 8 | ||
| 45.j | even | 6 | 2 | 810.3.j.e | 8 | ||
| 60.h | even | 2 | 1 | 2160.3.c.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.3.b.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 270.3.b.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 270.3.b.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 270.3.b.c | ✓ | 4 | 15.d | odd | 2 | 1 | inner |
| 810.3.j.e | 8 | 9.c | even | 3 | 2 | ||
| 810.3.j.e | 8 | 9.d | odd | 6 | 2 | ||
| 810.3.j.e | 8 | 45.h | odd | 6 | 2 | ||
| 810.3.j.e | 8 | 45.j | even | 6 | 2 | ||
| 1350.3.d.f | 2 | 5.c | odd | 4 | 1 | ||
| 1350.3.d.f | 2 | 15.e | even | 4 | 1 | ||
| 1350.3.d.g | 2 | 5.c | odd | 4 | 1 | ||
| 1350.3.d.g | 2 | 15.e | even | 4 | 1 | ||
| 2160.3.c.i | 4 | 4.b | odd | 2 | 1 | ||
| 2160.3.c.i | 4 | 12.b | even | 2 | 1 | ||
| 2160.3.c.i | 4 | 20.d | odd | 2 | 1 | ||
| 2160.3.c.i | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\):
|
\( T_{7}^{2} + 25 \)
|
|
\( T_{17}^{2} - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 625 \)
|
| $7$ |
\( (T^{2} + 25)^{2} \)
|
| $11$ |
\( (T^{2} + 2)^{2} \)
|
| $13$ |
\( (T^{2} + 81)^{2} \)
|
| $17$ |
\( (T^{2} - 128)^{2} \)
|
| $19$ |
\( (T - 21)^{4} \)
|
| $23$ |
\( (T^{2} - 2)^{2} \)
|
| $29$ |
\( (T^{2} + 1458)^{2} \)
|
| $31$ |
\( (T - 40)^{4} \)
|
| $37$ |
\( (T^{2} + 625)^{2} \)
|
| $41$ |
\( (T^{2} + 2738)^{2} \)
|
| $43$ |
\( (T^{2} + 4096)^{2} \)
|
| $47$ |
\( (T^{2} - 512)^{2} \)
|
| $53$ |
\( (T^{2} - 5202)^{2} \)
|
| $59$ |
\( (T^{2} + 8192)^{2} \)
|
| $61$ |
\( (T + 97)^{4} \)
|
| $67$ |
\( (T^{2} + 17161)^{2} \)
|
| $71$ |
\( (T^{2} + 7938)^{2} \)
|
| $73$ |
\( (T^{2} + 289)^{2} \)
|
| $79$ |
\( (T + 117)^{4} \)
|
| $83$ |
\( (T^{2} - 3362)^{2} \)
|
| $89$ |
\( (T^{2} + 21632)^{2} \)
|
| $97$ |
\( (T^{2} + 1681)^{2} \)
|
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