Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,4,Mod(46,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.46");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.e (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: | \(7.96525785077\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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 \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−1.68614 | + | 2.92048i | 0 | −1.68614 | − | 2.92048i | 2.50000 | + | 4.33013i | 0 | −0.813859 | + | 1.40965i | −15.6060 | 0 | −16.8614 | ||||||||||||||||||||||
| 46.2 | 1.18614 | − | 2.05446i | 0 | 1.18614 | + | 2.05446i | 2.50000 | + | 4.33013i | 0 | −3.68614 | + | 6.38458i | 24.6060 | 0 | 11.8614 | |||||||||||||||||||||||
| 91.1 | −1.68614 | − | 2.92048i | 0 | −1.68614 | + | 2.92048i | 2.50000 | − | 4.33013i | 0 | −0.813859 | − | 1.40965i | −15.6060 | 0 | −16.8614 | |||||||||||||||||||||||
| 91.2 | 1.18614 | + | 2.05446i | 0 | 1.18614 | − | 2.05446i | 2.50000 | − | 4.33013i | 0 | −3.68614 | − | 6.38458i | 24.6060 | 0 | 11.8614 | |||||||||||||||||||||||
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 | 135.4.e.a | 4 | |
| 3.b | odd | 2 | 1 | 45.4.e.a | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 135.4.e.a | 4 | |
| 9.c | even | 3 | 1 | 405.4.a.e | 2 | ||
| 9.d | odd | 6 | 1 | 45.4.e.a | ✓ | 4 | |
| 9.d | odd | 6 | 1 | 405.4.a.d | 2 | ||
| 15.d | odd | 2 | 1 | 225.4.e.a | 4 | ||
| 15.e | even | 4 | 2 | 225.4.k.a | 8 | ||
| 45.h | odd | 6 | 1 | 225.4.e.a | 4 | ||
| 45.h | odd | 6 | 1 | 2025.4.a.l | 2 | ||
| 45.j | even | 6 | 1 | 2025.4.a.j | 2 | ||
| 45.l | even | 12 | 2 | 225.4.k.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.e.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 45.4.e.a | ✓ | 4 | 9.d | odd | 6 | 1 | |
| 135.4.e.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 135.4.e.a | 4 | 9.c | even | 3 | 1 | inner | |
| 225.4.e.a | 4 | 15.d | odd | 2 | 1 | ||
| 225.4.e.a | 4 | 45.h | odd | 6 | 1 | ||
| 225.4.k.a | 8 | 15.e | even | 4 | 2 | ||
| 225.4.k.a | 8 | 45.l | even | 12 | 2 | ||
| 405.4.a.d | 2 | 9.d | odd | 6 | 1 | ||
| 405.4.a.e | 2 | 9.c | even | 3 | 1 | ||
| 2025.4.a.j | 2 | 45.j | even | 6 | 1 | ||
| 2025.4.a.l | 2 | 45.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} + 9T_{2}^{2} - 8T_{2} + 64 \)
acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + \cdots + 64 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} + 9 T^{3} + \cdots + 144 \)
|
| $11$ |
\( T^{4} + 37 T^{3} + \cdots + 18496 \)
|
| $13$ |
\( T^{4} - 112 T^{3} + \cdots + 6801664 \)
|
| $17$ |
\( (T^{2} + 77 T - 3674)^{2} \)
|
| $19$ |
\( (T^{2} - 35 T - 4850)^{2} \)
|
| $23$ |
\( T^{4} + 267 T^{3} + \cdots + 181117764 \)
|
| $29$ |
\( T^{4} - 325 T^{3} + \cdots + 192044164 \)
|
| $31$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $37$ |
\( (T^{2} + 638 T + 100936)^{2} \)
|
| $41$ |
\( T^{4} - 238 T^{3} + \cdots + 241460521 \)
|
| $43$ |
\( T^{4} + \cdots + 3611048464 \)
|
| $47$ |
\( T^{4} + \cdots + 40784610304 \)
|
| $53$ |
\( (T^{2} + 224 T - 69956)^{2} \)
|
| $59$ |
\( T^{4} + 85 T^{3} + \cdots + 324072004 \)
|
| $61$ |
\( T^{4} + \cdots + 89074790116 \)
|
| $67$ |
\( T^{4} + \cdots + 35157375009 \)
|
| $71$ |
\( (T^{2} + 394 T - 61016)^{2} \)
|
| $73$ |
\( (T^{2} + 811 T - 182276)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 16576047504 \)
|
| $83$ |
\( T^{4} + \cdots + 2740786203024 \)
|
| $89$ |
\( (T^{2} - 1065 T - 535050)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 15416202244 \)
|
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