Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,3,Mod(134,189)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(189, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.134");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 189.b (of order \(2\), degree \(1\), 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: | \(8\) |
| Coefficient field: | 8.0.1997017344.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 11\nu^{4} + 16\nu^{2} - 20 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} + 33\nu^{4} + 72\nu^{2} + 28 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 13\nu^{5} + 42\nu^{3} + 36\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 14\nu^{4} + 49\nu^{2} + 28 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} + 39\nu^{5} + 118\nu^{3} + 40\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} + 71\nu^{5} + 272\nu^{3} + 260\nu ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{6} + 3\beta_{4} - 2\beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} - 11 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{6} - 12\beta_{4} + 17\beta_1 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} - 11\beta_{3} + 25\beta_{2} + 73 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12\beta_{7} - 56\beta_{6} + 54\beta_{4} - 149\beta_1 ) / 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -22\beta_{5} + 105\beta_{3} - 203\beta_{2} - 567 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -156\beta_{7} + 464\beta_{6} - 270\beta_{4} + 1295\beta_1 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(136\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 134.1 |
|
− | 3.92812i | 0 | −11.4301 | − | 7.06404i | 0 | −2.64575 | 29.1863i | 0 | −27.7484 | ||||||||||||||||||||||||||||||||||||||||
| 134.2 | − | 2.27733i | 0 | −1.18625 | 4.94527i | 0 | 2.64575 | − | 6.40785i | 0 | 11.2620 | |||||||||||||||||||||||||||||||||||||||||
| 134.3 | − | 0.928117i | 0 | 3.13860 | − | 4.06404i | 0 | −2.64575 | − | 6.62545i | 0 | −3.77190 | ||||||||||||||||||||||||||||||||||||||||
| 134.4 | − | 0.722666i | 0 | 3.47775 | − | 7.94527i | 0 | 2.64575 | − | 5.40392i | 0 | −5.74177 | ||||||||||||||||||||||||||||||||||||||||
| 134.5 | 0.722666i | 0 | 3.47775 | 7.94527i | 0 | 2.64575 | 5.40392i | 0 | −5.74177 | |||||||||||||||||||||||||||||||||||||||||||
| 134.6 | 0.928117i | 0 | 3.13860 | 4.06404i | 0 | −2.64575 | 6.62545i | 0 | −3.77190 | |||||||||||||||||||||||||||||||||||||||||||
| 134.7 | 2.27733i | 0 | −1.18625 | − | 4.94527i | 0 | 2.64575 | 6.40785i | 0 | 11.2620 | ||||||||||||||||||||||||||||||||||||||||||
| 134.8 | 3.92812i | 0 | −11.4301 | 7.06404i | 0 | −2.64575 | − | 29.1863i | 0 | −27.7484 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 189.3.b.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 189.3.b.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | 3024.3.d.j | 8 | ||
| 9.c | even | 3 | 2 | 567.3.r.e | 16 | ||
| 9.d | odd | 6 | 2 | 567.3.r.e | 16 | ||
| 12.b | even | 2 | 1 | 3024.3.d.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.3.b.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 189.3.b.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 567.3.r.e | 16 | 9.c | even | 3 | 2 | ||
| 567.3.r.e | 16 | 9.d | odd | 6 | 2 | ||
| 3024.3.d.j | 8 | 4.b | odd | 2 | 1 | ||
| 3024.3.d.j | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 22T_{2}^{6} + 109T_{2}^{4} + 120T_{2}^{2} + 36 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 22 T^{6} + \cdots + 36 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 154 T^{6} + \cdots + 1272384 \)
|
| $7$ |
\( (T^{2} - 7)^{4} \)
|
| $11$ |
\( T^{8} + 616 T^{6} + \cdots + 84492864 \)
|
| $13$ |
\( (T^{4} - 18 T^{3} + \cdots - 26654)^{2} \)
|
| $17$ |
\( T^{8} + 876 T^{6} + \cdots + 57562569 \)
|
| $19$ |
\( (T^{4} - 6 T^{3} + \cdots + 42256)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 4882655376 \)
|
| $29$ |
\( T^{8} + \cdots + 3638261124 \)
|
| $31$ |
\( (T^{4} + 14 T^{3} + \cdots + 32634)^{2} \)
|
| $37$ |
\( (T^{4} + 2 T^{3} + \cdots + 814032)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 4052217312144 \)
|
| $43$ |
\( (T^{4} + 76 T^{3} + \cdots - 2939831)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 3631749696 \)
|
| $53$ |
\( T^{8} + \cdots + 52968101904 \)
|
| $59$ |
\( T^{8} + \cdots + 68515632191889 \)
|
| $61$ |
\( (T^{4} - 90 T^{3} + \cdots - 526784)^{2} \)
|
| $67$ |
\( (T^{4} + 66 T^{3} + \cdots + 24597736)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 259415011584 \)
|
| $73$ |
\( (T^{4} - 136 T^{3} + \cdots + 9014688)^{2} \)
|
| $79$ |
\( (T^{4} - 158 T^{3} + \cdots - 3444668)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 129353590650384 \)
|
| $89$ |
\( T^{8} + \cdots + 4824840688704 \)
|
| $97$ |
\( (T^{4} + 182 T^{3} + \cdots - 59939928)^{2} \)
|
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