Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,17,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, 17, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.8");
S:= CuspForms(chi, 17);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 17 \) |
| 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: | \(14.6092089471\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 2218x^{4} + 1229881x^{2} + 4304178 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{29} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 2218x^{4} + 1229881x^{2} + 4304178 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 1776\nu^{3} - 713297\nu ) / 3423 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{5} - 189285\nu^{3} - 173484742\nu ) / 3423 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -550\nu^{5} - 1002882\nu^{3} - 474472784\nu ) / 3423 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1854\nu^{4} - 2029680\nu^{2} + 19522836 ) / 1141 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 17118\nu^{4} + 25717392\nu^{2} + 4978323180 ) / 1141 \)
|
| \(\nu\) | \(=\) |
\( ( -243\beta_{3} + 46\beta_{2} + 132316\beta_1 ) / 3779136 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 103\beta_{5} + 951\beta_{4} - 465673536 ) / 629856 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 269487\beta_{3} - 144902\beta_{2} - 144015692\beta_1 ) / 3779136 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -14095\beta_{5} - 178593\beta_{4} + 64553993928 ) / 78732 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -305277741\beta_{3} + 224534290\beta_{2} + 148455280612\beta_1 ) / 3779136 \)
|
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 |
|
− | 478.345i | 0 | −163278. | − | 554730.i | 0 | −4.51382e6 | 4.67543e7i | 0 | −2.65352e8 | ||||||||||||||||||||||||||||||||||
| 8.2 | − | 387.651i | 0 | −84737.3 | 469186.i | 0 | 9.51468e6 | 7.44340e6i | 0 | 1.81880e8 | ||||||||||||||||||||||||||||||||||||
| 8.3 | − | 40.8280i | 0 | 63869.1 | 283843.i | 0 | −2.30051e6 | − | 5.28335e6i | 0 | 1.15887e7 | |||||||||||||||||||||||||||||||||||
| 8.4 | 40.8280i | 0 | 63869.1 | − | 283843.i | 0 | −2.30051e6 | 5.28335e6i | 0 | 1.15887e7 | ||||||||||||||||||||||||||||||||||||
| 8.5 | 387.651i | 0 | −84737.3 | − | 469186.i | 0 | 9.51468e6 | − | 7.44340e6i | 0 | 1.81880e8 | |||||||||||||||||||||||||||||||||||
| 8.6 | 478.345i | 0 | −163278. | 554730.i | 0 | −4.51382e6 | − | 4.67543e7i | 0 | −2.65352e8 | ||||||||||||||||||||||||||||||||||||
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 | 9.17.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 9.17.b.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.17.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.17.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(9, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 57316583964672 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots - 98\!\cdots\!48)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 10\!\cdots\!52 \)
|
| $13$ |
\( (T^{3} + \cdots - 61\!\cdots\!72)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 57\!\cdots\!72 \)
|
| $19$ |
\( (T^{3} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 73\!\cdots\!68 \)
|
| $29$ |
\( T^{6} + \cdots + 51\!\cdots\!28 \)
|
| $31$ |
\( (T^{3} + \cdots - 38\!\cdots\!16)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 41\!\cdots\!84)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 14\!\cdots\!68 \)
|
| $43$ |
\( (T^{3} + \cdots - 38\!\cdots\!36)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 58\!\cdots\!48 \)
|
| $53$ |
\( T^{6} + \cdots + 76\!\cdots\!08 \)
|
| $59$ |
\( T^{6} + \cdots + 32\!\cdots\!72 \)
|
| $61$ |
\( (T^{3} + \cdots - 10\!\cdots\!72)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 41\!\cdots\!76)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 35\!\cdots\!08 \)
|
| $73$ |
\( (T^{3} + \cdots - 54\!\cdots\!52)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 44\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 16\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 41\!\cdots\!32 \)
|
| $97$ |
\( (T^{3} + \cdots - 55\!\cdots\!36)^{2} \)
|
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