Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,3,Mod(134,135)]
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, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.134");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.d (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: | yes |
| Analytic conductor: | \(3.67848356886\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 134.1 |
|
−2.85410 | 0 | 4.14590 | −5.00000 | 0 | 0 | −0.416408 | 0 | 14.2705 | ||||||||||||||||||||||||
| 134.2 | 3.85410 | 0 | 10.8541 | −5.00000 | 0 | 0 | 26.4164 | 0 | −19.2705 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 135.3.d.d | yes | 2 |
| 3.b | odd | 2 | 1 | 135.3.d.c | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 2160.3.c.a | 2 | ||
| 5.b | even | 2 | 1 | 135.3.d.c | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 675.3.c.n | 4 | ||
| 9.c | even | 3 | 2 | 405.3.h.e | 4 | ||
| 9.d | odd | 6 | 2 | 405.3.h.f | 4 | ||
| 12.b | even | 2 | 1 | 2160.3.c.f | 2 | ||
| 15.d | odd | 2 | 1 | CM | 135.3.d.d | yes | 2 |
| 15.e | even | 4 | 2 | 675.3.c.n | 4 | ||
| 20.d | odd | 2 | 1 | 2160.3.c.f | 2 | ||
| 45.h | odd | 6 | 2 | 405.3.h.e | 4 | ||
| 45.j | even | 6 | 2 | 405.3.h.f | 4 | ||
| 60.h | even | 2 | 1 | 2160.3.c.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.3.d.c | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 135.3.d.c | ✓ | 2 | 5.b | even | 2 | 1 | |
| 135.3.d.d | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 135.3.d.d | yes | 2 | 15.d | odd | 2 | 1 | CM |
| 405.3.h.e | 4 | 9.c | even | 3 | 2 | ||
| 405.3.h.e | 4 | 45.h | odd | 6 | 2 | ||
| 405.3.h.f | 4 | 9.d | odd | 6 | 2 | ||
| 405.3.h.f | 4 | 45.j | even | 6 | 2 | ||
| 675.3.c.n | 4 | 5.c | odd | 4 | 2 | ||
| 675.3.c.n | 4 | 15.e | even | 4 | 2 | ||
| 2160.3.c.a | 2 | 4.b | odd | 2 | 1 | ||
| 2160.3.c.a | 2 | 60.h | even | 2 | 1 | ||
| 2160.3.c.f | 2 | 12.b | even | 2 | 1 | ||
| 2160.3.c.f | 2 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(135, [\chi])\):
|
\( T_{2}^{2} - T_{2} - 11 \)
|
|
\( T_{7} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T - 11 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 14T - 671 \)
|
| $19$ |
\( T^{2} - 22T - 599 \)
|
| $23$ |
\( T^{2} - 34T - 431 \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} + 2T - 2879 \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( T^{2} \)
|
| $43$ |
\( T^{2} \)
|
| $47$ |
\( (T - 14)^{2} \)
|
| $53$ |
\( T^{2} + 86T - 1031 \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( T^{2} - 118T + 2761 \)
|
| $67$ |
\( T^{2} \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} \)
|
| $79$ |
\( T^{2} + 98T - 9119 \)
|
| $83$ |
\( T^{2} - 154T + 3049 \)
|
| $89$ |
\( T^{2} \)
|
| $97$ |
\( T^{2} \)
|
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