Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.67");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(45.9205987462\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 203x^{4} + 10528x^{2} + 23232 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 203x^{4} + 10528x^{2} + 23232 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 115\nu^{3} + 1464\nu + 1056 ) / 2112 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 88\nu^{4} - 115\nu^{3} + 9064\nu^{2} + 1704\nu + 22176 ) / 2112 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - 88\nu^{4} - 115\nu^{3} - 9064\nu^{2} + 1704\nu - 22176 ) / 2112 \)
|
| \(\beta_{4}\) | \(=\) |
\( -3\nu^{2} - 203 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -61\nu^{5} - 3847\nu^{3} + 3168\nu^{2} + 227496\nu + 214368 ) / 2112 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2\beta _1 - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 203 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} - 100\beta_{3} - 100\beta_{2} - 78\beta _1 + 39 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 103\beta_{4} - 36\beta_{3} + 36\beta_{2} + 20153 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -230\beta_{5} - 115\beta_{4} + 10036\beta_{3} + 10036\beta_{2} + 12378\beta _1 - 6189 ) / 3 \)
|
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 |
|
−8.58260 | − | 14.8655i | −13.5000 | + | 23.3827i | −83.3221 | + | 144.318i | −34.6774 | − | 60.0630i | 463.461 | 0 | 663.336 | −364.500 | − | 631.333i | −595.245 | + | 1030.99i | ||||||||||||||||||||||||
| 67.2 | 1.31575 | + | 2.27894i | −13.5000 | + | 23.3827i | 60.5376 | − | 104.854i | −164.023 | − | 284.097i | −71.0503 | 0 | 655.440 | −364.500 | − | 631.333i | 431.626 | − | 747.599i | |||||||||||||||||||||||||
| 67.3 | 8.76686 | + | 15.1846i | −13.5000 | + | 23.3827i | −89.7155 | + | 155.392i | 255.701 | + | 442.887i | −473.410 | 0 | −901.776 | −364.500 | − | 631.333i | −4483.38 | + | 7765.44i | |||||||||||||||||||||||||
| 79.1 | −8.58260 | + | 14.8655i | −13.5000 | − | 23.3827i | −83.3221 | − | 144.318i | −34.6774 | + | 60.0630i | 463.461 | 0 | 663.336 | −364.500 | + | 631.333i | −595.245 | − | 1030.99i | |||||||||||||||||||||||||
| 79.2 | 1.31575 | − | 2.27894i | −13.5000 | − | 23.3827i | 60.5376 | + | 104.854i | −164.023 | + | 284.097i | −71.0503 | 0 | 655.440 | −364.500 | + | 631.333i | 431.626 | + | 747.599i | |||||||||||||||||||||||||
| 79.3 | 8.76686 | − | 15.1846i | −13.5000 | − | 23.3827i | −89.7155 | − | 155.392i | 255.701 | − | 442.887i | −473.410 | 0 | −901.776 | −364.500 | + | 631.333i | −4483.38 | − | 7765.44i | |||||||||||||||||||||||||
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 | 147.8.e.i | 6 | |
| 7.b | odd | 2 | 1 | 147.8.e.j | 6 | ||
| 7.c | even | 3 | 1 | 21.8.a.d | ✓ | 3 | |
| 7.c | even | 3 | 1 | inner | 147.8.e.i | 6 | |
| 7.d | odd | 6 | 1 | 147.8.a.e | 3 | ||
| 7.d | odd | 6 | 1 | 147.8.e.j | 6 | ||
| 21.g | even | 6 | 1 | 441.8.a.n | 3 | ||
| 21.h | odd | 6 | 1 | 63.8.a.g | 3 | ||
| 28.g | odd | 6 | 1 | 336.8.a.r | 3 | ||
| 35.j | even | 6 | 1 | 525.8.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.a.d | ✓ | 3 | 7.c | even | 3 | 1 | |
| 63.8.a.g | 3 | 21.h | odd | 6 | 1 | ||
| 147.8.a.e | 3 | 7.d | odd | 6 | 1 | ||
| 147.8.e.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 147.8.e.i | 6 | 7.c | even | 3 | 1 | inner | |
| 147.8.e.j | 6 | 7.b | odd | 2 | 1 | ||
| 147.8.e.j | 6 | 7.d | odd | 6 | 1 | ||
| 336.8.a.r | 3 | 28.g | odd | 6 | 1 | ||
| 441.8.a.n | 3 | 21.g | even | 6 | 1 | ||
| 525.8.a.g | 3 | 35.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{6} - 3T_{2}^{5} + 309T_{2}^{4} - 684T_{2}^{3} + 92376T_{2}^{2} - 237600T_{2} + 627264 \)
|
|
\( T_{5}^{6} - 114 T_{5}^{5} + 193476 T_{5}^{4} + 43845120 T_{5}^{3} + 31246617600 T_{5}^{2} + \cdots + 135377879040000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 627264 \)
|
| $3$ |
\( (T^{2} + 27 T + 729)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 135377879040000 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 44\!\cdots\!84 \)
|
| $13$ |
\( (T^{3} - 12762 T^{2} + \cdots + 755769903784)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 15\!\cdots\!24 \)
|
| $19$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!04 \)
|
| $29$ |
\( (T^{3} + \cdots + 208394349111720)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 61\!\cdots\!96 \)
|
| $37$ |
\( T^{6} + \cdots + 20\!\cdots\!64 \)
|
| $41$ |
\( (T^{3} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots - 30\!\cdots\!36)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 50\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 32\!\cdots\!24 \)
|
| $59$ |
\( T^{6} + \cdots + 96\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
|
| $67$ |
\( T^{6} + \cdots + 17\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots - 56\!\cdots\!08)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 80\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + \cdots + 79\!\cdots\!52)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots + 70\!\cdots\!88)^{2} \)
|
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