Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,15,Mod(8,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.8");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.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: | \(11.1896071337\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{3745})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 1867x^{2} + 1868x + 879846 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{10} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 1867x^{2} + 1868x + 879846 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -18\nu^{3} + 27\nu^{2} + 16731\nu - 8370 ) / 139 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -12\nu^{3} + 18\nu^{2} + 33672\nu - 16839 ) / 139 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 207\nu^{2} - 1138\nu - 194274 ) / 139 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{2} - 2\beta _1 + 243 ) / 486 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 324\beta_{3} + 3\beta_{2} + 16\beta _1 + 454167 ) / 486 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 486\beta_{3} + 2793\beta_{2} - 5588\beta _1 + 681129 ) / 486 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
− | 148.909i | 0 | −5789.91 | − | 41671.5i | 0 | −1.41648e6 | − | 1.57756e6i | 0 | −6.20526e6 | |||||||||||||||||||||||||||
| 8.2 | − | 110.725i | 0 | 4123.91 | − | 35180.3i | 0 | 883527. | − | 2.27074e6i | 0 | −3.89535e6 | ||||||||||||||||||||||||||||
| 8.3 | 110.725i | 0 | 4123.91 | 35180.3i | 0 | 883527. | 2.27074e6i | 0 | −3.89535e6 | |||||||||||||||||||||||||||||||
| 8.4 | 148.909i | 0 | −5789.91 | 41671.5i | 0 | −1.41648e6 | 1.57756e6i | 0 | −6.20526e6 | |||||||||||||||||||||||||||||||
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 | 9.15.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 9.15.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 144.15.e.d | 4 | ||
| 12.b | even | 2 | 1 | 144.15.e.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.15.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.15.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 144.15.e.d | 4 | 4.b | odd | 2 | 1 | ||
| 144.15.e.d | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(9, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 34434 T^{2} + 271854144 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} + \cdots - 1251497085104)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 92\!\cdots\!84 \)
|
| $13$ |
\( (T^{2} + \cdots - 11\!\cdots\!64)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 67\!\cdots\!64 \)
|
| $19$ |
\( (T^{2} + \cdots - 60\!\cdots\!20)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 50\!\cdots\!04 \)
|
| $29$ |
\( T^{4} + \cdots + 25\!\cdots\!84 \)
|
| $31$ |
\( (T^{2} + \cdots - 12\!\cdots\!64)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots + 40\!\cdots\!44)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 68\!\cdots\!44 \)
|
| $43$ |
\( (T^{2} + \cdots + 40\!\cdots\!24)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 37\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots + 14\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} + \cdots - 29\!\cdots\!96)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 13\!\cdots\!84)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 22\!\cdots\!24 \)
|
| $73$ |
\( (T^{2} + \cdots + 99\!\cdots\!36)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 22\!\cdots\!64)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 35\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $97$ |
\( (T^{2} + \cdots + 62\!\cdots\!44)^{2} \)
|
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