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: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−1.13746 | − | 1.97014i | −1.50000 | + | 2.59808i | 1.41238 | − | 2.44631i | 2.27492 | + | 3.94027i | 6.82475 | 0 | −24.6254 | −4.50000 | − | 7.79423i | 5.17525 | − | 8.96379i | ||||||||||||||||||
| 67.2 | 2.63746 | + | 4.56821i | −1.50000 | + | 2.59808i | −9.91238 | + | 17.1687i | −5.27492 | − | 9.13642i | −15.8248 | 0 | −62.3746 | −4.50000 | − | 7.79423i | 27.8248 | − | 48.1939i | |||||||||||||||||||
| 79.1 | −1.13746 | + | 1.97014i | −1.50000 | − | 2.59808i | 1.41238 | + | 2.44631i | 2.27492 | − | 3.94027i | 6.82475 | 0 | −24.6254 | −4.50000 | + | 7.79423i | 5.17525 | + | 8.96379i | |||||||||||||||||||
| 79.2 | 2.63746 | − | 4.56821i | −1.50000 | − | 2.59808i | −9.91238 | − | 17.1687i | −5.27492 | + | 9.13642i | −15.8248 | 0 | −62.3746 | −4.50000 | + | 7.79423i | 27.8248 | + | 48.1939i | |||||||||||||||||||
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.l | 4 | |
| 3.b | odd | 2 | 1 | 441.4.e.q | 4 | ||
| 7.b | odd | 2 | 1 | 147.4.e.m | 4 | ||
| 7.c | even | 3 | 1 | 21.4.a.c | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 147.4.e.l | 4 | |
| 7.d | odd | 6 | 1 | 147.4.a.i | 2 | ||
| 7.d | odd | 6 | 1 | 147.4.e.m | 4 | ||
| 21.c | even | 2 | 1 | 441.4.e.p | 4 | ||
| 21.g | even | 6 | 1 | 441.4.a.r | 2 | ||
| 21.g | even | 6 | 1 | 441.4.e.p | 4 | ||
| 21.h | odd | 6 | 1 | 63.4.a.e | 2 | ||
| 21.h | odd | 6 | 1 | 441.4.e.q | 4 | ||
| 28.f | even | 6 | 1 | 2352.4.a.bz | 2 | ||
| 28.g | odd | 6 | 1 | 336.4.a.m | 2 | ||
| 35.j | even | 6 | 1 | 525.4.a.n | 2 | ||
| 35.l | odd | 12 | 2 | 525.4.d.g | 4 | ||
| 56.k | odd | 6 | 1 | 1344.4.a.bo | 2 | ||
| 56.p | even | 6 | 1 | 1344.4.a.bg | 2 | ||
| 84.n | even | 6 | 1 | 1008.4.a.ba | 2 | ||
| 105.o | odd | 6 | 1 | 1575.4.a.p | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.a.c | ✓ | 2 | 7.c | even | 3 | 1 | |
| 63.4.a.e | 2 | 21.h | odd | 6 | 1 | ||
| 147.4.a.i | 2 | 7.d | odd | 6 | 1 | ||
| 147.4.e.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 147.4.e.l | 4 | 7.c | even | 3 | 1 | inner | |
| 147.4.e.m | 4 | 7.b | odd | 2 | 1 | ||
| 147.4.e.m | 4 | 7.d | odd | 6 | 1 | ||
| 336.4.a.m | 2 | 28.g | odd | 6 | 1 | ||
| 441.4.a.r | 2 | 21.g | even | 6 | 1 | ||
| 441.4.e.p | 4 | 21.c | even | 2 | 1 | ||
| 441.4.e.p | 4 | 21.g | even | 6 | 1 | ||
| 441.4.e.q | 4 | 3.b | odd | 2 | 1 | ||
| 441.4.e.q | 4 | 21.h | odd | 6 | 1 | ||
| 525.4.a.n | 2 | 35.j | even | 6 | 1 | ||
| 525.4.d.g | 4 | 35.l | odd | 12 | 2 | ||
| 1008.4.a.ba | 2 | 84.n | even | 6 | 1 | ||
| 1344.4.a.bg | 2 | 56.p | even | 6 | 1 | ||
| 1344.4.a.bo | 2 | 56.k | odd | 6 | 1 | ||
| 1575.4.a.p | 2 | 105.o | odd | 6 | 1 | ||
| 2352.4.a.bz | 2 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{4} - 3T_{2}^{3} + 21T_{2}^{2} + 36T_{2} + 144 \)
|
|
\( T_{5}^{4} + 6T_{5}^{3} + 84T_{5}^{2} - 288T_{5} + 2304 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 144 \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 2304 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 6 T^{3} + \cdots + 2005056 \)
|
| $13$ |
\( (T^{2} - 16 T - 1988)^{2} \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 2304 \)
|
| $19$ |
\( T^{4} + 64 T^{3} + \cdots + 51609856 \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 271063296 \)
|
| $29$ |
\( (T^{2} + 252 T + 7668)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 5398134784 \)
|
| $37$ |
\( T^{4} - 248 T^{3} + \cdots + 9560464 \)
|
| $41$ |
\( (T^{2} + 450 T + 37800)^{2} \)
|
| $43$ |
\( (T^{2} - 376 T + 2512)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 4337012736 \)
|
| $53$ |
\( T^{4} + \cdots + 92705634576 \)
|
| $59$ |
\( T^{4} + 804 T^{3} + \cdots + 908660736 \)
|
| $61$ |
\( T^{4} - 428 T^{3} + \cdots + 788261776 \)
|
| $67$ |
\( T^{4} + \cdots + 25836061696 \)
|
| $71$ |
\( (T^{2} - 954 T + 214704)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 81364139536 \)
|
| $79$ |
\( T^{4} + \cdots + 7126061056 \)
|
| $83$ |
\( (T^{2} - 1944 T + 813456)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 64438807104 \)
|
| $97$ |
\( (T^{2} - 808 T - 922292)^{2} \)
|
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