Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,11,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, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.8");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| 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: | \(5.71821527406\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{385})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 187x^{2} + 188x + 9606 \)
|
| 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
\( x^{4} - 2x^{3} - 187x^{2} + 188x + 9606 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9\nu^{3} + 183\nu^{2} - 1002\nu - 18066 ) / 131 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -108\nu^{3} + 162\nu^{2} + 30888\nu - 15471 ) / 131 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -135\nu^{3} + 792\nu^{2} + 11493\nu - 61488 ) / 131 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 3\beta_{2} + 6\beta _1 + 243 ) / 486 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 16\beta_{3} + 3\beta_{2} + 276\beta _1 + 45927 ) / 486 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -548\beta_{3} + 273\beta_{2} + 2130\beta _1 + 68769 ) / 486 \)
|
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 |
|
− | 60.7152i | 0 | −2662.33 | − | 994.474i | 0 | 1598.68 | 99471.8i | 0 | −60379.7 | ||||||||||||||||||||||||||||
| 8.2 | − | 22.5314i | 0 | 516.335 | − | 3667.34i | 0 | −23830.7 | − | 34705.9i | 0 | −82630.3 | ||||||||||||||||||||||||||||
| 8.3 | 22.5314i | 0 | 516.335 | 3667.34i | 0 | −23830.7 | 34705.9i | 0 | −82630.3 | |||||||||||||||||||||||||||||||
| 8.4 | 60.7152i | 0 | −2662.33 | 994.474i | 0 | 1598.68 | − | 99471.8i | 0 | −60379.7 | ||||||||||||||||||||||||||||||
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 | 9.11.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 9.11.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 144.11.e.d | 4 | ||
| 5.b | even | 2 | 1 | 225.11.c.a | 4 | ||
| 5.c | odd | 4 | 2 | 225.11.d.a | 8 | ||
| 9.c | even | 3 | 2 | 81.11.d.f | 8 | ||
| 9.d | odd | 6 | 2 | 81.11.d.f | 8 | ||
| 12.b | even | 2 | 1 | 144.11.e.d | 4 | ||
| 15.d | odd | 2 | 1 | 225.11.c.a | 4 | ||
| 15.e | even | 4 | 2 | 225.11.d.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.11.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 9.11.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 81.11.d.f | 8 | 9.c | even | 3 | 2 | ||
| 81.11.d.f | 8 | 9.d | odd | 6 | 2 | ||
| 144.11.e.d | 4 | 4.b | odd | 2 | 1 | ||
| 144.11.e.d | 4 | 12.b | even | 2 | 1 | ||
| 225.11.c.a | 4 | 5.b | even | 2 | 1 | ||
| 225.11.c.a | 4 | 15.d | odd | 2 | 1 | ||
| 225.11.d.a | 8 | 5.c | odd | 4 | 2 | ||
| 225.11.d.a | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(9, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4194 T^{2} + 1871424 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 13301119584900 \)
|
| $7$ |
\( (T^{2} + 22232 T - 38097584)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 48\!\cdots\!84 \)
|
| $13$ |
\( (T^{2} + 86912 T - 196148800064)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 55\!\cdots\!24 \)
|
| $19$ |
\( (T^{2} + \cdots - 4504522991360)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 15\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 14\!\cdots\!04 \)
|
| $31$ |
\( (T^{2} + \cdots + 12\!\cdots\!36)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 628794332830076)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 93\!\cdots\!04 \)
|
| $43$ |
\( (T^{2} + \cdots - 15\!\cdots\!36)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 29\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 90\!\cdots\!44 \)
|
| $61$ |
\( (T^{2} + \cdots + 45\!\cdots\!44)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 37\!\cdots\!56)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 33\!\cdots\!84 \)
|
| $73$ |
\( (T^{2} + \cdots - 43\!\cdots\!84)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 20\!\cdots\!64)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 52\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $97$ |
\( (T^{2} + \cdots + 54\!\cdots\!04)^{2} \)
|
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