Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,3,Mod(44,189)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(189, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.44");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 189.j (of order \(6\), 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: | \(5.14987699641\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.63369648.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 63) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} - 24\nu^{4} + 288\nu^{3} - 236\nu^{2} - 66\nu + 2241 ) / 1449 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 44\nu^{5} - 45\nu^{4} + 540\nu^{3} + 604\nu^{2} + 5310\nu + 36 ) / 1449 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} + 36\nu^{3} + 16\nu^{2} + 354\nu + 99 ) / 63 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\nu^{5} - 30\nu^{4} + 843\nu^{3} + 2281\nu^{2} + 9819\nu + 5337 ) / 1449 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\nu^{5} - 62\nu^{4} + 422\nu^{3} + 249\nu^{2} + 3130\nu - 1335 ) / 483 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta_{3} - 7\beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 2\beta_{4} + 13\beta _1 - 33 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -24\beta_{5} - 12\beta_{4} - 19\beta_{3} + 94\beta_{2} + 19\beta _1 - 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -52\beta_{5} - 104\beta_{4} - 187\beta_{3} + 571\beta_{2} + 519 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(136\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 44.1 |
|
− | 3.09257i | 0 | −5.56399 | 5.67824 | − | 3.27834i | 0 | 6.63848 | − | 2.22048i | 4.83675i | 0 | −10.1385 | − | 17.5604i | |||||||||||||||||||||||||||||
| 44.2 | 1.03435i | 0 | 2.93011 | 2.10422 | − | 1.21487i | 0 | −4.75661 | + | 5.13563i | 7.16819i | 0 | 1.25661 | + | 2.17651i | |||||||||||||||||||||||||||||||
| 44.3 | 3.79027i | 0 | −10.3661 | −0.282467 | + | 0.163082i | 0 | −2.88187 | − | 6.37925i | − | 24.1293i | 0 | −0.618126 | − | 1.07063i | ||||||||||||||||||||||||||||||
| 116.1 | − | 3.79027i | 0 | −10.3661 | −0.282467 | − | 0.163082i | 0 | −2.88187 | + | 6.37925i | 24.1293i | 0 | −0.618126 | + | 1.07063i | ||||||||||||||||||||||||||||||
| 116.2 | − | 1.03435i | 0 | 2.93011 | 2.10422 | + | 1.21487i | 0 | −4.75661 | − | 5.13563i | − | 7.16819i | 0 | 1.25661 | − | 2.17651i | |||||||||||||||||||||||||||||
| 116.3 | 3.09257i | 0 | −5.56399 | 5.67824 | + | 3.27834i | 0 | 6.63848 | + | 2.22048i | − | 4.83675i | 0 | −10.1385 | + | 17.5604i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 63.j | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 189.3.j.a | 6 | |
| 3.b | odd | 2 | 1 | 63.3.j.a | ✓ | 6 | |
| 7.c | even | 3 | 1 | 189.3.n.a | 6 | ||
| 9.c | even | 3 | 1 | 63.3.n.a | yes | 6 | |
| 9.d | odd | 6 | 1 | 189.3.n.a | 6 | ||
| 21.c | even | 2 | 1 | 441.3.j.c | 6 | ||
| 21.g | even | 6 | 1 | 441.3.n.c | 6 | ||
| 21.g | even | 6 | 1 | 441.3.r.c | 6 | ||
| 21.h | odd | 6 | 1 | 63.3.n.a | yes | 6 | |
| 21.h | odd | 6 | 1 | 441.3.r.b | 6 | ||
| 63.g | even | 3 | 1 | 441.3.r.b | 6 | ||
| 63.h | even | 3 | 1 | 63.3.j.a | ✓ | 6 | |
| 63.j | odd | 6 | 1 | inner | 189.3.j.a | 6 | |
| 63.k | odd | 6 | 1 | 441.3.r.c | 6 | ||
| 63.l | odd | 6 | 1 | 441.3.n.c | 6 | ||
| 63.t | odd | 6 | 1 | 441.3.j.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.3.j.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 63.3.j.a | ✓ | 6 | 63.h | even | 3 | 1 | |
| 63.3.n.a | yes | 6 | 9.c | even | 3 | 1 | |
| 63.3.n.a | yes | 6 | 21.h | odd | 6 | 1 | |
| 189.3.j.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 189.3.j.a | 6 | 63.j | odd | 6 | 1 | inner | |
| 189.3.n.a | 6 | 7.c | even | 3 | 1 | ||
| 189.3.n.a | 6 | 9.d | odd | 6 | 1 | ||
| 441.3.j.c | 6 | 21.c | even | 2 | 1 | ||
| 441.3.j.c | 6 | 63.t | odd | 6 | 1 | ||
| 441.3.n.c | 6 | 21.g | even | 6 | 1 | ||
| 441.3.n.c | 6 | 63.l | odd | 6 | 1 | ||
| 441.3.r.b | 6 | 21.h | odd | 6 | 1 | ||
| 441.3.r.b | 6 | 63.g | even | 3 | 1 | ||
| 441.3.r.c | 6 | 21.g | even | 6 | 1 | ||
| 441.3.r.c | 6 | 63.k | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 25T_{2}^{4} + 163T_{2}^{2} + 147 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 25 T^{4} + \cdots + 147 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 15 T^{5} + \cdots + 27 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 117649 \)
|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots + 1982907 \)
|
| $13$ |
\( T^{6} - 11 T^{5} + \cdots + 499849 \)
|
| $17$ |
\( T^{6} + 33 T^{5} + \cdots + 31434507 \)
|
| $19$ |
\( T^{6} + 19 T^{5} + \cdots + 214369 \)
|
| $23$ |
\( T^{6} + 15 T^{5} + \cdots + 128547 \)
|
| $29$ |
\( T^{6} - 51 T^{5} + \cdots + 694083 \)
|
| $31$ |
\( (T^{3} + 46 T^{2} + \cdots - 4632)^{2} \)
|
| $37$ |
\( T^{6} - 7 T^{5} + \cdots + 564110001 \)
|
| $41$ |
\( T^{6} + \cdots + 1387137027 \)
|
| $43$ |
\( T^{6} + 99 T^{5} + \cdots + 432265681 \)
|
| $47$ |
\( T^{6} + \cdots + 29274835968 \)
|
| $53$ |
\( T^{6} + \cdots + 5141134827 \)
|
| $59$ |
\( T^{6} + 4896 T^{4} + \cdots + 813189888 \)
|
| $61$ |
\( (T^{3} - 22 T^{2} + \cdots + 160696)^{2} \)
|
| $67$ |
\( (T^{3} + 98 T^{2} + \cdots - 210392)^{2} \)
|
| $71$ |
\( T^{6} + 5568 T^{4} + \cdots + 109734912 \)
|
| $73$ |
\( T^{6} + \cdots + 653628591729 \)
|
| $79$ |
\( (T^{3} - 90 T^{2} + \cdots - 16712)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 165842832483 \)
|
| $89$ |
\( T^{6} + \cdots + 15857178627 \)
|
| $97$ |
\( T^{6} + \cdots + 36908941689 \)
|
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