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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{67})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 67x^{2} + 4489 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} + 67x^{2} + 4489 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 67 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} ) / 67 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 67\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 67\beta_{3} ) / 2 \)
|
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 |
|
−11.1854 | − | 19.3736i | −13.5000 | + | 23.3827i | −186.224 | + | 322.550i | 59.4828 | + | 103.027i | 604.009 | 0 | 5468.48 | −364.500 | − | 631.333i | 1330.67 | − | 2304.79i | ||||||||||||||||||
| 67.2 | 5.18535 | + | 8.98129i | −13.5000 | + | 23.3827i | 10.2242 | − | 17.7089i | −71.4828 | − | 123.812i | −280.009 | 0 | 1539.52 | −364.500 | − | 631.333i | 741.327 | − | 1284.02i | |||||||||||||||||||
| 79.1 | −11.1854 | + | 19.3736i | −13.5000 | − | 23.3827i | −186.224 | − | 322.550i | 59.4828 | − | 103.027i | 604.009 | 0 | 5468.48 | −364.500 | + | 631.333i | 1330.67 | + | 2304.79i | |||||||||||||||||||
| 79.2 | 5.18535 | − | 8.98129i | −13.5000 | − | 23.3827i | 10.2242 | + | 17.7089i | −71.4828 | + | 123.812i | −280.009 | 0 | 1539.52 | −364.500 | + | 631.333i | 741.327 | + | 1284.02i | |||||||||||||||||||
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.e | 4 | |
| 7.b | odd | 2 | 1 | 147.8.e.f | 4 | ||
| 7.c | even | 3 | 1 | 147.8.a.d | 2 | ||
| 7.c | even | 3 | 1 | inner | 147.8.e.e | 4 | |
| 7.d | odd | 6 | 1 | 21.8.a.c | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 147.8.e.f | 4 | ||
| 21.g | even | 6 | 1 | 63.8.a.c | 2 | ||
| 21.h | odd | 6 | 1 | 441.8.a.h | 2 | ||
| 28.f | even | 6 | 1 | 336.8.a.p | 2 | ||
| 35.i | odd | 6 | 1 | 525.8.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.a.c | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 63.8.a.c | 2 | 21.g | even | 6 | 1 | ||
| 147.8.a.d | 2 | 7.c | even | 3 | 1 | ||
| 147.8.e.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 147.8.e.e | 4 | 7.c | even | 3 | 1 | inner | |
| 147.8.e.f | 4 | 7.b | odd | 2 | 1 | ||
| 147.8.e.f | 4 | 7.d | odd | 6 | 1 | ||
| 336.8.a.p | 2 | 28.f | even | 6 | 1 | ||
| 441.8.a.h | 2 | 21.h | odd | 6 | 1 | ||
| 525.8.a.d | 2 | 35.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{4} + 12T_{2}^{3} + 376T_{2}^{2} - 2784T_{2} + 53824 \)
|
|
\( T_{5}^{4} + 24T_{5}^{3} + 17584T_{5}^{2} - 408192T_{5} + 289272064 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 12 T^{3} + \cdots + 53824 \)
|
| $3$ |
\( (T^{2} + 27 T + 729)^{2} \)
|
| $5$ |
\( T^{4} + 24 T^{3} + \cdots + 289272064 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 395347845822736 \)
|
| $13$ |
\( (T^{2} - 1084 T - 21935228)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 82\!\cdots\!04 \)
|
| $19$ |
\( T^{4} + \cdots + 25\!\cdots\!76 \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!16 \)
|
| $29$ |
\( (T^{2} - 211308 T + 1307845988)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 19\!\cdots\!04 \)
|
| $37$ |
\( T^{4} + \cdots + 78\!\cdots\!44 \)
|
| $41$ |
\( (T^{2} + 749760 T + 127701459200)^{2} \)
|
| $43$ |
\( (T^{2} - 397096 T - 71585337968)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 72\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 69\!\cdots\!56 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!96 \)
|
| $61$ |
\( T^{4} + \cdots + 25\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots + 13\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + \cdots - 3929864540796)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots + 71\!\cdots\!96 \)
|
| $83$ |
\( (T^{2} + \cdots - 7065815171184)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 47\!\cdots\!44 \)
|
| $97$ |
\( (T^{2} + \cdots + 164115180472068)^{2} \)
|
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