Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [507,2,Mod(22,507)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("507.22");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 507 = 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 507.e (of order \(3\), degree \(2\), 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: | \(4.04841538248\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.64827.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).
| \(n\) | \(170\) | \(340\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−1.34601 | + | 2.33136i | 0.500000 | − | 0.866025i | −2.62349 | − | 4.54402i | −1.04892 | 1.34601 | + | 2.33136i | −0.277479 | − | 0.480608i | 8.74094 | −0.500000 | − | 0.866025i | 1.41185 | − | 2.44540i | ||||||||||||||||||||||
| 22.2 | −1.17845 | + | 2.04113i | 0.500000 | − | 0.866025i | −1.77748 | − | 3.07868i | 3.69202 | 1.17845 | + | 2.04113i | 0.400969 | + | 0.694498i | 3.66487 | −0.500000 | − | 0.866025i | −4.35086 | + | 7.53590i | |||||||||||||||||||||||
| 22.3 | 1.02446 | − | 1.77441i | 0.500000 | − | 0.866025i | −1.09903 | − | 1.90358i | 3.35690 | −1.02446 | − | 1.77441i | −1.12349 | − | 1.94594i | −0.405813 | −0.500000 | − | 0.866025i | 3.43900 | − | 5.95652i | |||||||||||||||||||||||
| 484.1 | −1.34601 | − | 2.33136i | 0.500000 | + | 0.866025i | −2.62349 | + | 4.54402i | −1.04892 | 1.34601 | − | 2.33136i | −0.277479 | + | 0.480608i | 8.74094 | −0.500000 | + | 0.866025i | 1.41185 | + | 2.44540i | |||||||||||||||||||||||
| 484.2 | −1.17845 | − | 2.04113i | 0.500000 | + | 0.866025i | −1.77748 | + | 3.07868i | 3.69202 | 1.17845 | − | 2.04113i | 0.400969 | − | 0.694498i | 3.66487 | −0.500000 | + | 0.866025i | −4.35086 | − | 7.53590i | |||||||||||||||||||||||
| 484.3 | 1.02446 | + | 1.77441i | 0.500000 | + | 0.866025i | −1.09903 | + | 1.90358i | 3.35690 | −1.02446 | + | 1.77441i | −1.12349 | + | 1.94594i | −0.405813 | −0.500000 | + | 0.866025i | 3.43900 | + | 5.95652i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 507.2.e.i | 6 | |
| 13.b | even | 2 | 1 | 507.2.e.l | 6 | ||
| 13.c | even | 3 | 1 | 507.2.a.l | yes | 3 | |
| 13.c | even | 3 | 1 | inner | 507.2.e.i | 6 | |
| 13.d | odd | 4 | 2 | 507.2.j.i | 12 | ||
| 13.e | even | 6 | 1 | 507.2.a.i | ✓ | 3 | |
| 13.e | even | 6 | 1 | 507.2.e.l | 6 | ||
| 13.f | odd | 12 | 2 | 507.2.b.f | 6 | ||
| 13.f | odd | 12 | 2 | 507.2.j.i | 12 | ||
| 39.h | odd | 6 | 1 | 1521.2.a.s | 3 | ||
| 39.i | odd | 6 | 1 | 1521.2.a.n | 3 | ||
| 39.k | even | 12 | 2 | 1521.2.b.k | 6 | ||
| 52.i | odd | 6 | 1 | 8112.2.a.cg | 3 | ||
| 52.j | odd | 6 | 1 | 8112.2.a.cp | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 507.2.a.i | ✓ | 3 | 13.e | even | 6 | 1 | |
| 507.2.a.l | yes | 3 | 13.c | even | 3 | 1 | |
| 507.2.b.f | 6 | 13.f | odd | 12 | 2 | ||
| 507.2.e.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 507.2.e.i | 6 | 13.c | even | 3 | 1 | inner | |
| 507.2.e.l | 6 | 13.b | even | 2 | 1 | ||
| 507.2.e.l | 6 | 13.e | even | 6 | 1 | ||
| 507.2.j.i | 12 | 13.d | odd | 4 | 2 | ||
| 507.2.j.i | 12 | 13.f | odd | 12 | 2 | ||
| 1521.2.a.n | 3 | 39.i | odd | 6 | 1 | ||
| 1521.2.a.s | 3 | 39.h | odd | 6 | 1 | ||
| 1521.2.b.k | 6 | 39.k | even | 12 | 2 | ||
| 8112.2.a.cg | 3 | 52.i | odd | 6 | 1 | ||
| 8112.2.a.cp | 3 | 52.j | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):
|
\( T_{2}^{6} + 3T_{2}^{5} + 13T_{2}^{4} + 14T_{2}^{3} + 55T_{2}^{2} + 52T_{2} + 169 \)
|
|
\( T_{5}^{3} - 6T_{5}^{2} + 5T_{5} + 13 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 169 \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( (T^{3} - 6 T^{2} + 5 T + 13)^{2} \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{6} + 5 T^{5} + \cdots + 1681 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - T^{5} + \cdots + 169 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 49 \)
|
| $23$ |
\( T^{6} + 49 T^{4} + \cdots + 8281 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 841 \)
|
| $31$ |
\( (T^{3} + 16 T^{2} + \cdots - 197)^{2} \)
|
| $37$ |
\( T^{6} + 22 T^{5} + \cdots + 142129 \)
|
| $41$ |
\( T^{6} + 11 T^{5} + \cdots + 841 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 1681 \)
|
| $47$ |
\( (T^{3} - 7 T^{2} + 14 T - 7)^{2} \)
|
| $53$ |
\( (T^{3} + 17 T^{2} + \cdots - 41)^{2} \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots + 10816 \)
|
| $61$ |
\( T^{6} - 13 T^{5} + \cdots + 27889 \)
|
| $67$ |
\( T^{6} + 11 T^{5} + \cdots + 1681 \)
|
| $71$ |
\( T^{6} + 91 T^{4} + \cdots + 41209 \)
|
| $73$ |
\( (T^{3} - 6 T^{2} + \cdots + 923)^{2} \)
|
| $79$ |
\( (T^{3} - 3 T^{2} - 18 T + 27)^{2} \)
|
| $83$ |
\( (T^{3} - 12 T^{2} + \cdots - 43)^{2} \)
|
| $89$ |
\( T^{6} + T^{5} + \cdots + 12769 \)
|
| $97$ |
\( T^{6} - 5 T^{5} + \cdots + 2679769 \)
|
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