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: | \(\Q(\sqrt{-3}, \sqrt{-355})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 88x^{2} - 89x + 7921 \)
|
| 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
\( x^{4} - x^{3} - 88x^{2} - 89x + 7921 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 88\nu^{2} - 88\nu - 7921 ) / 7832 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 177\nu ) / 89 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 88\nu - 177 ) / 88 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 265\beta _1 + 266 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 176\beta_{3} - 88\beta_{2} + 88\beta _1 + 443 ) / 3 \)
|
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 |
|
−5.90858 | − | 10.2340i | 13.5000 | − | 23.3827i | −5.82274 | + | 10.0853i | 253.172 | + | 438.506i | −319.064 | 0 | −1374.98 | −364.500 | − | 631.333i | 2991.77 | − | 5181.90i | ||||||||||||||||||
| 67.2 | 10.4086 | + | 18.0282i | 13.5000 | − | 23.3827i | −152.677 | + | 264.445i | −73.1717 | − | 126.737i | 562.064 | 0 | −3692.02 | −364.500 | − | 631.333i | 1523.23 | − | 2638.31i | |||||||||||||||||||
| 79.1 | −5.90858 | + | 10.2340i | 13.5000 | + | 23.3827i | −5.82274 | − | 10.0853i | 253.172 | − | 438.506i | −319.064 | 0 | −1374.98 | −364.500 | + | 631.333i | 2991.77 | + | 5181.90i | |||||||||||||||||||
| 79.2 | 10.4086 | − | 18.0282i | 13.5000 | + | 23.3827i | −152.677 | − | 264.445i | −73.1717 | + | 126.737i | 562.064 | 0 | −3692.02 | −364.500 | + | 631.333i | 1523.23 | + | 2638.31i | |||||||||||||||||||
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.h | 4 | |
| 7.b | odd | 2 | 1 | 147.8.e.g | 4 | ||
| 7.c | even | 3 | 1 | 21.8.a.b | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 147.8.e.h | 4 | |
| 7.d | odd | 6 | 1 | 147.8.a.c | 2 | ||
| 7.d | odd | 6 | 1 | 147.8.e.g | 4 | ||
| 21.g | even | 6 | 1 | 441.8.a.m | 2 | ||
| 21.h | odd | 6 | 1 | 63.8.a.f | 2 | ||
| 28.g | odd | 6 | 1 | 336.8.a.n | 2 | ||
| 35.j | even | 6 | 1 | 525.8.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.a.b | ✓ | 2 | 7.c | even | 3 | 1 | |
| 63.8.a.f | 2 | 21.h | odd | 6 | 1 | ||
| 147.8.a.c | 2 | 7.d | odd | 6 | 1 | ||
| 147.8.e.g | 4 | 7.b | odd | 2 | 1 | ||
| 147.8.e.g | 4 | 7.d | odd | 6 | 1 | ||
| 147.8.e.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 147.8.e.h | 4 | 7.c | even | 3 | 1 | inner | |
| 336.8.a.n | 2 | 28.g | odd | 6 | 1 | ||
| 441.8.a.m | 2 | 21.g | even | 6 | 1 | ||
| 525.8.a.e | 2 | 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 \(S_{8}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{4} - 9T_{2}^{3} + 327T_{2}^{2} + 2214T_{2} + 60516 \)
|
|
\( T_{5}^{4} - 360T_{5}^{3} + 203700T_{5}^{2} + 26676000T_{5} + 5490810000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 9 T^{3} + \cdots + 60516 \)
|
| $3$ |
\( (T^{2} - 27 T + 729)^{2} \)
|
| $5$ |
\( T^{4} + \cdots + 5490810000 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 367733703315456 \)
|
| $13$ |
\( (T^{2} - 7708 T - 7230524)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 14\!\cdots\!16 \)
|
| $19$ |
\( T^{4} + \cdots + 61\!\cdots\!96 \)
|
| $23$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + 435996 T + 46362346164)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 74\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + 25056 T - 416101387716)^{2} \)
|
| $43$ |
\( (T^{2} - 496216 T + 23523686224)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 64\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 15\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $71$ |
\( (T^{2} - 54540 T - 387209643840)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 50\!\cdots\!96 \)
|
| $79$ |
\( T^{4} + \cdots + 33\!\cdots\!56 \)
|
| $83$ |
\( (T^{2} + \cdots + 6817674434256)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 38\!\cdots\!96 \)
|
| $97$ |
\( (T^{2} + \cdots + 62174212264276)^{2} \)
|
| show more | |
| show less |