Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [402,2,Mod(37,402)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(402, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("402.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 402 = 2 \cdot 3 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 402.e (of order \(3\), degree \(2\), 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: | \(3.20998616126\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.48843675.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 7x^{4} - 2x^{3} + 49x^{2} - 7x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} + 7x^{4} - 2x^{3} + 49x^{2} - 7x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 1 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 7\nu^{2} - \nu + 35 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 7\nu^{3} - \nu^{2} + 49\nu ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} - 35\nu^{3} + 12\nu^{2} - 245\nu + 35 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 5\beta_{4} - 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{5} - 35\beta_{4} + 7\beta_{3} + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 12\beta_{4} - 49\beta_{2} - 49\beta _1 - 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/402\mathbb{Z}\right)^\times\).
| \(n\) | \(269\) | \(337\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−0.500000 | − | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | −2.71448 | −0.500000 | − | 0.866025i | 1.18420 | − | 2.05109i | 1.00000 | 1.00000 | 1.35724 | + | 2.35081i | ||||||||||||||||||||||||||
| 37.2 | −0.500000 | − | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 0.143277 | −0.500000 | − | 0.866025i | −2.48974 | + | 4.31235i | 1.00000 | 1.00000 | −0.0716387 | − | 0.124082i | |||||||||||||||||||||||||||
| 37.3 | −0.500000 | − | 0.866025i | 1.00000 | −0.500000 | + | 0.866025i | 2.57120 | −0.500000 | − | 0.866025i | 0.805538 | − | 1.39523i | 1.00000 | 1.00000 | −1.28560 | − | 2.22673i | |||||||||||||||||||||||||||
| 163.1 | −0.500000 | + | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | −2.71448 | −0.500000 | + | 0.866025i | 1.18420 | + | 2.05109i | 1.00000 | 1.00000 | 1.35724 | − | 2.35081i | |||||||||||||||||||||||||||
| 163.2 | −0.500000 | + | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 0.143277 | −0.500000 | + | 0.866025i | −2.48974 | − | 4.31235i | 1.00000 | 1.00000 | −0.0716387 | + | 0.124082i | |||||||||||||||||||||||||||
| 163.3 | −0.500000 | + | 0.866025i | 1.00000 | −0.500000 | − | 0.866025i | 2.57120 | −0.500000 | + | 0.866025i | 0.805538 | + | 1.39523i | 1.00000 | 1.00000 | −1.28560 | + | 2.22673i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 67.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 402.2.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1206.2.h.f | 6 | ||
| 67.c | even | 3 | 1 | inner | 402.2.e.b | ✓ | 6 |
| 201.g | odd | 6 | 1 | 1206.2.h.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 402.2.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 402.2.e.b | ✓ | 6 | 67.c | even | 3 | 1 | inner |
| 1206.2.h.f | 6 | 3.b | odd | 2 | 1 | ||
| 1206.2.h.f | 6 | 201.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 7T_{5} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T^{3} - 7 T + 1)^{2} \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots + 361 \)
|
| $11$ |
\( T^{6} - 3 T^{5} + \cdots + 25 \)
|
| $13$ |
\( T^{6} - 9 T^{5} + \cdots + 25 \)
|
| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 60025 \)
|
| $19$ |
\( (T^{2} + T + 1)^{3} \)
|
| $23$ |
\( T^{6} + 63 T^{4} + \cdots + 729 \)
|
| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 4225 \)
|
| $31$ |
\( T^{6} - 13 T^{5} + \cdots + 198025 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots + 18769 \)
|
| $41$ |
\( T^{6} - T^{5} + \cdots + 625 \)
|
| $43$ |
\( (T^{3} + 13 T^{2} + \cdots - 619)^{2} \)
|
| $47$ |
\( T^{6} - 19 T^{5} + \cdots + 388129 \)
|
| $53$ |
\( (T^{3} + 3 T^{2} + \cdots - 557)^{2} \)
|
| $59$ |
\( (T^{3} - 3 T^{2} - 60 T + 35)^{2} \)
|
| $61$ |
\( T^{6} + 7 T^{4} + \cdots + 1 \)
|
| $67$ |
\( T^{6} + 18 T^{5} + \cdots + 300763 \)
|
| $71$ |
\( T^{6} + 12 T^{5} + \cdots + 1301881 \)
|
| $73$ |
\( T^{6} - T^{5} + \cdots + 47089 \)
|
| $79$ |
\( T^{6} + 10 T^{5} + \cdots + 961 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots + 707281 \)
|
| $89$ |
\( (T^{3} + 5 T^{2} - 70 T - 25)^{2} \)
|
| $97$ |
\( T^{6} + T^{5} + \cdots + 49 \)
|
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