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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 |
|
−3.00000 | − | 5.19615i | −13.5000 | + | 23.3827i | 46.0000 | − | 79.6743i | 195.000 | + | 337.750i | 162.000 | 0 | −1320.00 | −364.500 | − | 631.333i | 1170.00 | − | 2026.50i | ||||||||||||
| 79.1 | −3.00000 | + | 5.19615i | −13.5000 | − | 23.3827i | 46.0000 | + | 79.6743i | 195.000 | − | 337.750i | 162.000 | 0 | −1320.00 | −364.500 | + | 631.333i | 1170.00 | + | 2026.50i | |||||||||||||
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.a | 2 | |
| 7.b | odd | 2 | 1 | 147.8.e.b | 2 | ||
| 7.c | even | 3 | 1 | 147.8.a.b | 1 | ||
| 7.c | even | 3 | 1 | inner | 147.8.e.a | 2 | |
| 7.d | odd | 6 | 1 | 3.8.a.a | ✓ | 1 | |
| 7.d | odd | 6 | 1 | 147.8.e.b | 2 | ||
| 21.g | even | 6 | 1 | 9.8.a.a | 1 | ||
| 21.h | odd | 6 | 1 | 441.8.a.a | 1 | ||
| 28.f | even | 6 | 1 | 48.8.a.g | 1 | ||
| 35.i | odd | 6 | 1 | 75.8.a.a | 1 | ||
| 35.k | even | 12 | 2 | 75.8.b.c | 2 | ||
| 56.j | odd | 6 | 1 | 192.8.a.i | 1 | ||
| 56.m | even | 6 | 1 | 192.8.a.a | 1 | ||
| 63.i | even | 6 | 1 | 81.8.c.c | 2 | ||
| 63.k | odd | 6 | 1 | 81.8.c.a | 2 | ||
| 63.s | even | 6 | 1 | 81.8.c.c | 2 | ||
| 63.t | odd | 6 | 1 | 81.8.c.a | 2 | ||
| 77.i | even | 6 | 1 | 363.8.a.b | 1 | ||
| 84.j | odd | 6 | 1 | 144.8.a.b | 1 | ||
| 91.s | odd | 6 | 1 | 507.8.a.a | 1 | ||
| 105.p | even | 6 | 1 | 225.8.a.i | 1 | ||
| 105.w | odd | 12 | 2 | 225.8.b.f | 2 | ||
| 168.ba | even | 6 | 1 | 576.8.a.w | 1 | ||
| 168.be | odd | 6 | 1 | 576.8.a.x | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.8.a.a | ✓ | 1 | 7.d | odd | 6 | 1 | |
| 9.8.a.a | 1 | 21.g | even | 6 | 1 | ||
| 48.8.a.g | 1 | 28.f | even | 6 | 1 | ||
| 75.8.a.a | 1 | 35.i | odd | 6 | 1 | ||
| 75.8.b.c | 2 | 35.k | even | 12 | 2 | ||
| 81.8.c.a | 2 | 63.k | odd | 6 | 1 | ||
| 81.8.c.a | 2 | 63.t | odd | 6 | 1 | ||
| 81.8.c.c | 2 | 63.i | even | 6 | 1 | ||
| 81.8.c.c | 2 | 63.s | even | 6 | 1 | ||
| 144.8.a.b | 1 | 84.j | odd | 6 | 1 | ||
| 147.8.a.b | 1 | 7.c | even | 3 | 1 | ||
| 147.8.e.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 147.8.e.a | 2 | 7.c | even | 3 | 1 | inner | |
| 147.8.e.b | 2 | 7.b | odd | 2 | 1 | ||
| 147.8.e.b | 2 | 7.d | odd | 6 | 1 | ||
| 192.8.a.a | 1 | 56.m | even | 6 | 1 | ||
| 192.8.a.i | 1 | 56.j | odd | 6 | 1 | ||
| 225.8.a.i | 1 | 105.p | even | 6 | 1 | ||
| 225.8.b.f | 2 | 105.w | odd | 12 | 2 | ||
| 363.8.a.b | 1 | 77.i | even | 6 | 1 | ||
| 441.8.a.a | 1 | 21.h | odd | 6 | 1 | ||
| 507.8.a.a | 1 | 91.s | odd | 6 | 1 | ||
| 576.8.a.w | 1 | 168.ba | even | 6 | 1 | ||
| 576.8.a.x | 1 | 168.be | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{2} + 6T_{2} + 36 \)
|
|
\( T_{5}^{2} - 390T_{5} + 152100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 6T + 36 \)
|
| $3$ |
\( T^{2} + 27T + 729 \)
|
| $5$ |
\( T^{2} - 390T + 152100 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 948T + 898704 \)
|
| $13$ |
\( (T - 5098)^{2} \)
|
| $17$ |
\( T^{2} - 28386 T + 805764996 \)
|
| $19$ |
\( T^{2} + 8620 T + 74304400 \)
|
| $23$ |
\( T^{2} - 15288 T + 233722944 \)
|
| $29$ |
\( (T - 36510)^{2} \)
|
| $31$ |
\( T^{2} + \cdots + 76622668864 \)
|
| $37$ |
\( T^{2} + \cdots + 72106212676 \)
|
| $41$ |
\( (T - 629718)^{2} \)
|
| $43$ |
\( (T - 685772)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 340234223616 \)
|
| $53$ |
\( T^{2} + \cdots + 183233651364 \)
|
| $59$ |
\( T^{2} + \cdots + 1706628704400 \)
|
| $61$ |
\( T^{2} + \cdots + 90397638244 \)
|
| $67$ |
\( T^{2} + \cdots + 257296475536 \)
|
| $71$ |
\( (T - 5560632)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 1874385522724 \)
|
| $79$ |
\( T^{2} + \cdots + 47799524238400 \)
|
| $83$ |
\( (T - 4376748)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 72732071456100 \)
|
| $97$ |
\( (T - 8826814)^{2} \)
|
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