Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,4,Mod(67,147)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.67");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.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: | \(8.67328077084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−1.20711 | − | 2.09077i | 1.50000 | − | 2.59808i | 1.08579 | − | 1.88064i | −9.94975 | − | 17.2335i | −7.24264 | 0 | −24.5563 | −4.50000 | − | 7.79423i | −24.0208 | + | 41.6053i | ||||||||||||||||||
| 67.2 | 0.207107 | + | 0.358719i | 1.50000 | − | 2.59808i | 3.91421 | − | 6.77962i | −0.0502525 | − | 0.0870399i | 1.24264 | 0 | 6.55635 | −4.50000 | − | 7.79423i | 0.0208153 | − | 0.0360531i | |||||||||||||||||||
| 79.1 | −1.20711 | + | 2.09077i | 1.50000 | + | 2.59808i | 1.08579 | + | 1.88064i | −9.94975 | + | 17.2335i | −7.24264 | 0 | −24.5563 | −4.50000 | + | 7.79423i | −24.0208 | − | 41.6053i | |||||||||||||||||||
| 79.2 | 0.207107 | − | 0.358719i | 1.50000 | + | 2.59808i | 3.91421 | + | 6.77962i | −0.0502525 | + | 0.0870399i | 1.24264 | 0 | 6.55635 | −4.50000 | + | 7.79423i | 0.0208153 | + | 0.0360531i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 147.4.e.k | 4 | |
| 3.b | odd | 2 | 1 | 441.4.e.v | 4 | ||
| 7.b | odd | 2 | 1 | 147.4.e.j | 4 | ||
| 7.c | even | 3 | 1 | 147.4.a.j | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 147.4.e.k | 4 | |
| 7.d | odd | 6 | 1 | 147.4.a.k | yes | 2 | |
| 7.d | odd | 6 | 1 | 147.4.e.j | 4 | ||
| 21.c | even | 2 | 1 | 441.4.e.u | 4 | ||
| 21.g | even | 6 | 1 | 441.4.a.o | 2 | ||
| 21.g | even | 6 | 1 | 441.4.e.u | 4 | ||
| 21.h | odd | 6 | 1 | 441.4.a.n | 2 | ||
| 21.h | odd | 6 | 1 | 441.4.e.v | 4 | ||
| 28.f | even | 6 | 1 | 2352.4.a.bl | 2 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.cf | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.4.a.j | ✓ | 2 | 7.c | even | 3 | 1 | |
| 147.4.a.k | yes | 2 | 7.d | odd | 6 | 1 | |
| 147.4.e.j | 4 | 7.b | odd | 2 | 1 | ||
| 147.4.e.j | 4 | 7.d | odd | 6 | 1 | ||
| 147.4.e.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 147.4.e.k | 4 | 7.c | even | 3 | 1 | inner | |
| 441.4.a.n | 2 | 21.h | odd | 6 | 1 | ||
| 441.4.a.o | 2 | 21.g | even | 6 | 1 | ||
| 441.4.e.u | 4 | 21.c | even | 2 | 1 | ||
| 441.4.e.u | 4 | 21.g | even | 6 | 1 | ||
| 441.4.e.v | 4 | 3.b | odd | 2 | 1 | ||
| 441.4.e.v | 4 | 21.h | odd | 6 | 1 | ||
| 2352.4.a.bl | 2 | 28.f | even | 6 | 1 | ||
| 2352.4.a.cf | 2 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \)
|
|
\( T_{5}^{4} + 20T_{5}^{3} + 398T_{5}^{2} + 40T_{5} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $5$ |
\( T^{4} + 20 T^{3} + \cdots + 4 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 20 T^{3} + \cdots + 1106704 \)
|
| $13$ |
\( (T^{2} - 104 T + 1454)^{2} \)
|
| $17$ |
\( T^{4} + 116 T^{3} + \cdots + 470596 \)
|
| $19$ |
\( T^{4} + 192 T^{3} + \cdots + 68029504 \)
|
| $23$ |
\( T^{4} + 28 T^{3} + \cdots + 1882384 \)
|
| $29$ |
\( (T^{2} - 296 T + 14216)^{2} \)
|
| $31$ |
\( T^{4} - 104 T^{3} + \cdots + 5271616 \)
|
| $37$ |
\( T^{4} - 248 T^{3} + \cdots + 115949824 \)
|
| $41$ |
\( (T^{2} - 20 T - 95822)^{2} \)
|
| $43$ |
\( (T^{2} + 720 T + 109600)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 15054308416 \)
|
| $53$ |
\( T^{4} + \cdots + 58198667536 \)
|
| $59$ |
\( T^{4} - 616 T^{3} + \cdots + 53114944 \)
|
| $61$ |
\( T^{4} + 16 T^{3} + \cdots + 2116 \)
|
| $67$ |
\( T^{4} + \cdots + 295901185024 \)
|
| $71$ |
\( (T^{2} - 988 T + 136388)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 101695934404 \)
|
| $79$ |
\( T^{4} + \cdots + 12408177664 \)
|
| $83$ |
\( (T^{2} - 1016 T + 245264)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 36943146436 \)
|
| $97$ |
\( (T^{2} - 488 T - 658066)^{2} \)
|
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