Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,8,Mod(12,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.12");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.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: | \(4.06100533129\) |
| 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: |
\( x^{6} + 449x^{4} + 37224x^{2} + 205776 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} + 449x^{4} + 37224x^{2} + 205776 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 365\nu^{2} + 12324 ) / 240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 365\nu^{3} + 12084\nu ) / 240 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{2} - 150 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 445\nu^{3} - 34644\nu ) / 160 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} - 150 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} - 3\beta_{3} - 282\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 365\beta_{4} + 240\beta_{2} + 42426 \)
|
| \(\nu^{5}\) | \(=\) |
\( 730\beta_{5} + 1335\beta_{3} + 90846\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 12.1 |
|
− | 18.4900i | −27.4160 | −213.880 | − | 70.5606i | 506.922i | 454.852i | 1587.93i | −1435.36 | −1304.67 | ||||||||||||||||||||||||||||||||||
| 12.2 | − | 10.0583i | 50.8656 | 26.8297 | − | 8.88672i | − | 511.623i | − | 510.452i | − | 1557.33i | 400.306 | −89.3858 | ||||||||||||||||||||||||||||||||
| 12.3 | − | 2.43912i | −51.4496 | 122.051 | 488.312i | 125.492i | 616.243i | − | 609.904i | 460.056 | 1191.05 | |||||||||||||||||||||||||||||||||||
| 12.4 | 2.43912i | −51.4496 | 122.051 | − | 488.312i | − | 125.492i | − | 616.243i | 609.904i | 460.056 | 1191.05 | ||||||||||||||||||||||||||||||||||
| 12.5 | 10.0583i | 50.8656 | 26.8297 | 8.88672i | 511.623i | 510.452i | 1557.33i | 400.306 | −89.3858 | |||||||||||||||||||||||||||||||||||||
| 12.6 | 18.4900i | −27.4160 | −213.880 | 70.5606i | − | 506.922i | − | 454.852i | − | 1587.93i | −1435.36 | −1304.67 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 13.8.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 117.8.b.b | 6 | ||
| 4.b | odd | 2 | 1 | 208.8.f.a | 6 | ||
| 13.b | even | 2 | 1 | inner | 13.8.b.a | ✓ | 6 |
| 13.d | odd | 4 | 2 | 169.8.a.d | 6 | ||
| 39.d | odd | 2 | 1 | 117.8.b.b | 6 | ||
| 52.b | odd | 2 | 1 | 208.8.f.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.8.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 13.8.b.a | ✓ | 6 | 13.b | even | 2 | 1 | inner |
| 117.8.b.b | 6 | 3.b | odd | 2 | 1 | ||
| 117.8.b.b | 6 | 39.d | odd | 2 | 1 | ||
| 169.8.a.d | 6 | 13.d | odd | 4 | 2 | ||
| 208.8.f.a | 6 | 4.b | odd | 2 | 1 | ||
| 208.8.f.a | 6 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(13, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 449 T^{4} + \cdots + 205776 \)
|
| $3$ |
\( (T^{3} + 28 T^{2} + \cdots - 71748)^{2} \)
|
| $5$ |
\( T^{6} + \cdots + 93756690000 \)
|
| $7$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots + 24\!\cdots\!13 \)
|
| $17$ |
\( (T^{3} - 6576 T^{2} + \cdots - 976631438250)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 62\!\cdots\!36 \)
|
| $23$ |
\( (T^{3} + \cdots - 38560806996192)^{2} \)
|
| $29$ |
\( (T^{3} + \cdots + 16\!\cdots\!80)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 30\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots + 11\!\cdots\!88)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 32\!\cdots\!96 \)
|
| $53$ |
\( (T^{3} + \cdots + 18\!\cdots\!92)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 36\!\cdots\!96 \)
|
| $61$ |
\( (T^{3} + \cdots + 10\!\cdots\!88)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 54\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!04 \)
|
| $79$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 14\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots + 22\!\cdots\!96 \)
|
| $97$ |
\( T^{6} + \cdots + 17\!\cdots\!56 \)
|
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