Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1521,2,Mod(1351,1521)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1521.1351");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1521 = 3^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1521.b (of order \(2\), degree \(1\), not 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: | \(12.1452461474\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.153664.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 507) |
| 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
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 3\nu^{2} + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 4\nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1351.1 |
|
− | 1.80194i | 0 | −1.24698 | − | 1.44504i | 0 | − | 3.44504i | − | 1.35690i | 0 | −2.60388 | ||||||||||||||||||||||||||||||||
| 1351.2 | − | 1.24698i | 0 | 0.445042 | 2.80194i | 0 | 4.80194i | − | 3.04892i | 0 | 3.49396 | |||||||||||||||||||||||||||||||||||
| 1351.3 | − | 0.445042i | 0 | 1.80194 | 0.246980i | 0 | − | 1.75302i | − | 1.69202i | 0 | 0.109916 | ||||||||||||||||||||||||||||||||||
| 1351.4 | 0.445042i | 0 | 1.80194 | − | 0.246980i | 0 | 1.75302i | 1.69202i | 0 | 0.109916 | ||||||||||||||||||||||||||||||||||||
| 1351.5 | 1.24698i | 0 | 0.445042 | − | 2.80194i | 0 | − | 4.80194i | 3.04892i | 0 | 3.49396 | |||||||||||||||||||||||||||||||||||
| 1351.6 | 1.80194i | 0 | −1.24698 | 1.44504i | 0 | 3.44504i | 1.35690i | 0 | −2.60388 | |||||||||||||||||||||||||||||||||||||
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 | 1521.2.b.m | 6 | |
| 3.b | odd | 2 | 1 | 507.2.b.g | 6 | ||
| 13.b | even | 2 | 1 | inner | 1521.2.b.m | 6 | |
| 13.d | odd | 4 | 1 | 1521.2.a.p | 3 | ||
| 13.d | odd | 4 | 1 | 1521.2.a.q | 3 | ||
| 39.d | odd | 2 | 1 | 507.2.b.g | 6 | ||
| 39.f | even | 4 | 1 | 507.2.a.j | ✓ | 3 | |
| 39.f | even | 4 | 1 | 507.2.a.k | yes | 3 | |
| 39.h | odd | 6 | 2 | 507.2.j.h | 12 | ||
| 39.i | odd | 6 | 2 | 507.2.j.h | 12 | ||
| 39.k | even | 12 | 2 | 507.2.e.j | 6 | ||
| 39.k | even | 12 | 2 | 507.2.e.k | 6 | ||
| 156.l | odd | 4 | 1 | 8112.2.a.by | 3 | ||
| 156.l | odd | 4 | 1 | 8112.2.a.cf | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 507.2.a.j | ✓ | 3 | 39.f | even | 4 | 1 | |
| 507.2.a.k | yes | 3 | 39.f | even | 4 | 1 | |
| 507.2.b.g | 6 | 3.b | odd | 2 | 1 | ||
| 507.2.b.g | 6 | 39.d | odd | 2 | 1 | ||
| 507.2.e.j | 6 | 39.k | even | 12 | 2 | ||
| 507.2.e.k | 6 | 39.k | even | 12 | 2 | ||
| 507.2.j.h | 12 | 39.h | odd | 6 | 2 | ||
| 507.2.j.h | 12 | 39.i | odd | 6 | 2 | ||
| 1521.2.a.p | 3 | 13.d | odd | 4 | 1 | ||
| 1521.2.a.q | 3 | 13.d | odd | 4 | 1 | ||
| 1521.2.b.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 1521.2.b.m | 6 | 13.b | even | 2 | 1 | inner | |
| 8112.2.a.by | 3 | 156.l | odd | 4 | 1 | ||
| 8112.2.a.cf | 3 | 156.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1521, [\chi])\):
|
\( T_{2}^{6} + 5T_{2}^{4} + 6T_{2}^{2} + 1 \)
|
|
\( T_{5}^{6} + 10T_{5}^{4} + 17T_{5}^{2} + 1 \)
|
|
\( T_{7}^{6} + 38T_{7}^{4} + 381T_{7}^{2} + 841 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 5 T^{4} + \cdots + 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 10 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{6} + 38 T^{4} + \cdots + 841 \)
|
| $11$ |
\( T^{6} + 61 T^{4} + \cdots + 1849 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{3} + 7 T^{2} + 14 T + 7)^{2} \)
|
| $19$ |
\( T^{6} + 101 T^{4} + \cdots + 12769 \)
|
| $23$ |
\( (T^{3} - 2 T^{2} - 43 T - 83)^{2} \)
|
| $29$ |
\( (T^{3} - 8 T^{2} + 5 T + 43)^{2} \)
|
| $31$ |
\( T^{6} + 110 T^{4} + \cdots + 38809 \)
|
| $37$ |
\( T^{6} + 70 T^{4} + \cdots + 8281 \)
|
| $41$ |
\( T^{6} + 5 T^{4} + \cdots + 1 \)
|
| $43$ |
\( (T^{3} - 3 T^{2} - 25 T - 29)^{2} \)
|
| $47$ |
\( T^{6} + 321 T^{4} + \cdots + 829921 \)
|
| $53$ |
\( (T^{3} - 13 T^{2} + \cdots - 29)^{2} \)
|
| $59$ |
\( T^{6} + 196 T^{4} + \cdots + 3136 \)
|
| $61$ |
\( (T^{3} + 13 T^{2} + \cdots - 223)^{2} \)
|
| $67$ |
\( T^{6} + 69 T^{4} + \cdots + 9409 \)
|
| $71$ |
\( T^{6} + 194 T^{4} + \cdots + 212521 \)
|
| $73$ |
\( T^{6} + 122 T^{4} + \cdots + 27889 \)
|
| $79$ |
\( (T^{3} + 9 T^{2} + \cdots - 169)^{2} \)
|
| $83$ |
\( T^{6} + 146 T^{4} + \cdots + 1849 \)
|
| $89$ |
\( T^{6} + 41 T^{4} + \cdots + 1 \)
|
| $97$ |
\( T^{6} + 363 T^{4} + \cdots + 1413721 \)
|
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