Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,2,Mod(211,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.211");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.j (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: | \(5.03057532734\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.954288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 45 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 2\nu^{3} - 6\nu - 18 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} - 24\nu - 72 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} - 10\beta_{3} - 4\beta_{2} + 6\beta _1 + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 211.1 |
|
−0.500000 | − | 0.866025i | −1.62241 | − | 0.606458i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 0.285997 | + | 1.70828i | −0.500000 | − | 0.866025i | 1.00000 | 2.26442 | + | 1.96784i | 1.00000 | ||||||||||||||||||||||
| 211.2 | −0.500000 | − | 0.866025i | 0.403374 | + | 1.68443i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 1.25707 | − | 1.19154i | −0.500000 | − | 0.866025i | 1.00000 | −2.67458 | + | 1.35891i | 1.00000 | |||||||||||||||||||||||
| 211.3 | −0.500000 | − | 0.866025i | 1.71903 | − | 0.211943i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −1.04307 | − | 1.38276i | −0.500000 | − | 0.866025i | 1.00000 | 2.91016 | − | 0.728674i | 1.00000 | |||||||||||||||||||||||
| 421.1 | −0.500000 | + | 0.866025i | −1.62241 | + | 0.606458i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 0.285997 | − | 1.70828i | −0.500000 | + | 0.866025i | 1.00000 | 2.26442 | − | 1.96784i | 1.00000 | |||||||||||||||||||||||
| 421.2 | −0.500000 | + | 0.866025i | 0.403374 | − | 1.68443i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 1.25707 | + | 1.19154i | −0.500000 | + | 0.866025i | 1.00000 | −2.67458 | − | 1.35891i | 1.00000 | |||||||||||||||||||||||
| 421.3 | −0.500000 | + | 0.866025i | 1.71903 | + | 0.211943i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −1.04307 | + | 1.38276i | −0.500000 | + | 0.866025i | 1.00000 | 2.91016 | + | 0.728674i | 1.00000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.2.j.k | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1890.2.j.j | 6 | ||
| 9.c | even | 3 | 1 | inner | 630.2.j.k | ✓ | 6 |
| 9.c | even | 3 | 1 | 5670.2.a.bt | 3 | ||
| 9.d | odd | 6 | 1 | 1890.2.j.j | 6 | ||
| 9.d | odd | 6 | 1 | 5670.2.a.bp | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 630.2.j.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 630.2.j.k | ✓ | 6 | 9.c | even | 3 | 1 | inner |
| 1890.2.j.j | 6 | 3.b | odd | 2 | 1 | ||
| 1890.2.j.j | 6 | 9.d | odd | 6 | 1 | ||
| 5670.2.a.bp | 3 | 9.d | odd | 6 | 1 | ||
| 5670.2.a.bt | 3 | 9.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):
|
\( T_{11}^{6} - T_{11}^{5} + 6T_{11}^{4} - T_{11}^{3} + 28T_{11}^{2} - 15T_{11} + 9 \)
|
|
\( T_{13}^{6} - 2T_{13}^{5} + 8T_{13}^{4} + 4T_{13}^{3} + 20T_{13}^{2} - 8T_{13} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( T^{6} - T^{5} + \cdots + 27 \)
|
| $5$ |
\( (T^{2} + T + 1)^{3} \)
|
| $7$ |
\( (T^{2} + T + 1)^{3} \)
|
| $11$ |
\( T^{6} - T^{5} + 6 T^{4} + \cdots + 9 \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 4 \)
|
| $17$ |
\( (T^{3} - 7 T^{2} + 7 T + 3)^{2} \)
|
| $19$ |
\( (T^{3} + 3 T^{2} - 21 T - 59)^{2} \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots + 12996 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 324 \)
|
| $31$ |
\( T^{6} + 4 T^{5} + \cdots + 256 \)
|
| $37$ |
\( (T^{3} - 10 T^{2} + \cdots + 346)^{2} \)
|
| $41$ |
\( T^{6} + 7 T^{5} + \cdots + 826281 \)
|
| $43$ |
\( T^{6} + 17 T^{5} + \cdots + 2601 \)
|
| $47$ |
\( T^{6} + 14 T^{5} + \cdots + 576 \)
|
| $53$ |
\( (T^{3} + 6 T^{2} - 6 T - 54)^{2} \)
|
| $59$ |
\( T^{6} + 29 T^{5} + \cdots + 651249 \)
|
| $61$ |
\( T^{6} - 2 T^{5} + \cdots + 4 \)
|
| $67$ |
\( T^{6} - T^{5} + \cdots + 19881 \)
|
| $71$ |
\( (T^{3} - 10 T^{2} + \cdots + 2502)^{2} \)
|
| $73$ |
\( (T^{3} - T^{2} - 37 T + 61)^{2} \)
|
| $79$ |
\( T^{6} + 2 T^{5} + \cdots + 576 \)
|
| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 186624 \)
|
| $89$ |
\( (T^{3} - 22 T^{2} + \cdots + 312)^{2} \)
|
| $97$ |
\( T^{6} - 5 T^{5} + \cdots + 12769 \)
|
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