Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1682,2,Mod(1681,1682)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1682.1681");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1682 = 2 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1682.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: | \(13.4308376200\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 58) |
| 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 \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).
| \(n\) | \(843\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1681.1 |
|
− | 1.00000i | − | 2.24698i | −1.00000 | −1.55496 | −2.24698 | 3.49396 | 1.00000i | −2.04892 | 1.55496i | ||||||||||||||||||||||||||||||||||
| 1681.2 | − | 1.00000i | − | 0.554958i | −1.00000 | −0.198062 | −0.554958 | 0.109916 | 1.00000i | 2.69202 | 0.198062i | |||||||||||||||||||||||||||||||||||
| 1681.3 | − | 1.00000i | 0.801938i | −1.00000 | −3.24698 | 0.801938 | −2.60388 | 1.00000i | 2.35690 | 3.24698i | ||||||||||||||||||||||||||||||||||||
| 1681.4 | 1.00000i | − | 0.801938i | −1.00000 | −3.24698 | 0.801938 | −2.60388 | − | 1.00000i | 2.35690 | − | 3.24698i | ||||||||||||||||||||||||||||||||||
| 1681.5 | 1.00000i | 0.554958i | −1.00000 | −0.198062 | −0.554958 | 0.109916 | − | 1.00000i | 2.69202 | − | 0.198062i | |||||||||||||||||||||||||||||||||||
| 1681.6 | 1.00000i | 2.24698i | −1.00000 | −1.55496 | −2.24698 | 3.49396 | − | 1.00000i | −2.04892 | − | 1.55496i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1682.2.b.g | 6 | |
| 29.b | even | 2 | 1 | inner | 1682.2.b.g | 6 | |
| 29.c | odd | 4 | 1 | 1682.2.a.m | 3 | ||
| 29.c | odd | 4 | 1 | 1682.2.a.n | 3 | ||
| 29.f | odd | 28 | 2 | 58.2.d.a | ✓ | 6 | |
| 87.k | even | 28 | 2 | 522.2.k.c | 6 | ||
| 116.l | even | 28 | 2 | 464.2.u.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.2.d.a | ✓ | 6 | 29.f | odd | 28 | 2 | |
| 464.2.u.b | 6 | 116.l | even | 28 | 2 | ||
| 522.2.k.c | 6 | 87.k | even | 28 | 2 | ||
| 1682.2.a.m | 3 | 29.c | odd | 4 | 1 | ||
| 1682.2.a.n | 3 | 29.c | odd | 4 | 1 | ||
| 1682.2.b.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 1682.2.b.g | 6 | 29.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1682, [\chi])\):
|
\( T_{3}^{6} + 6T_{3}^{4} + 5T_{3}^{2} + 1 \)
|
|
\( T_{5}^{3} + 5T_{5}^{2} + 6T_{5} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T^{3} + 5 T^{2} + 6 T + 1)^{2} \)
|
| $7$ |
\( (T^{3} - T^{2} - 9 T + 1)^{2} \)
|
| $11$ |
\( T^{6} + 45 T^{4} + \cdots + 729 \)
|
| $13$ |
\( (T^{3} + 2 T^{2} - 15 T - 29)^{2} \)
|
| $17$ |
\( T^{6} + 41 T^{4} + \cdots + 1681 \)
|
| $19$ |
\( T^{6} + 66 T^{4} + \cdots + 1681 \)
|
| $23$ |
\( (T^{3} - 49 T - 91)^{2} \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( T^{6} + 97 T^{4} + \cdots + 1849 \)
|
| $37$ |
\( T^{6} + 101 T^{4} + \cdots + 19321 \)
|
| $41$ |
\( T^{6} + 110 T^{4} + \cdots + 1 \)
|
| $43$ |
\( T^{6} + 189 T^{4} + \cdots + 41209 \)
|
| $47$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $53$ |
\( (T^{3} - 24 T^{2} + \cdots - 337)^{2} \)
|
| $59$ |
\( (T^{3} + 19 T^{2} + \cdots + 169)^{2} \)
|
| $61$ |
\( T^{6} + 202 T^{4} + \cdots + 169 \)
|
| $67$ |
\( (T - 11)^{6} \)
|
| $71$ |
\( (T^{3} + T^{2} - 86 T + 251)^{2} \)
|
| $73$ |
\( T^{6} + 194 T^{4} + \cdots + 241081 \)
|
| $79$ |
\( T^{6} + 84 T^{4} + \cdots + 3136 \)
|
| $83$ |
\( (T^{3} + 14 T^{2} + \cdots - 889)^{2} \)
|
| $89$ |
\( T^{6} + 227 T^{4} + \cdots + 253009 \)
|
| $97$ |
\( T^{6} + 136 T^{4} + \cdots + 10816 \)
|
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