Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,3,Mod(71,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.71");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.e (of order \(2\), degree \(1\), 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: | \(17.1662566547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.98344960000.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} - \nu^{4} + 11\nu^{2} + 270 ) / 90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{7} + 19\nu^{5} + 151\nu^{3} - 1155\nu ) / 450 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 7\nu^{4} + 55\nu^{2} - 108 ) / 18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{7} + 52\nu^{5} + 358\nu^{3} - 615\nu ) / 450 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{6} - 34\nu^{4} - 196\nu^{2} + 405 ) / 45 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} + 17\nu^{5} + 113\nu^{3} - 195\nu ) / 90 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 6\nu^{5} - 24\nu^{3} + 235\nu ) / 30 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{4} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + 2\beta _1 - 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{7} + 2\beta_{6} - \beta_{4} + 8\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -11\beta_{5} - 16\beta_{3} + 8\beta _1 - 21 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -15\beta_{7} + 65\beta_{6} - 97\beta_{4} - 49\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{5} + 19\beta_{3} - 83\beta _1 + 264 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -42\beta_{7} - 673\beta_{6} + 1076\beta_{4} - 133\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 71.1 |
|
− | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −2.64575 | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||
| 71.2 | − | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 2.64575 | 2.82843i | 0 | −3.16228 | |||||||||||||||||||||||||||||||||||||||||
| 71.3 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −2.64575 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
| 71.4 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 2.64575 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
| 71.5 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −2.64575 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
| 71.6 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 2.64575 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
| 71.7 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −2.64575 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
| 71.8 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 2.64575 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.3.e.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 630.3.e.b | ✓ | 8 |
| 5.b | even | 2 | 1 | 3150.3.e.f | 8 | ||
| 5.c | odd | 4 | 2 | 3150.3.c.f | 16 | ||
| 15.d | odd | 2 | 1 | 3150.3.e.f | 8 | ||
| 15.e | even | 4 | 2 | 3150.3.c.f | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 630.3.e.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 630.3.e.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 3150.3.c.f | 16 | 5.c | odd | 4 | 2 | ||
| 3150.3.c.f | 16 | 15.e | even | 4 | 2 | ||
| 3150.3.e.f | 8 | 5.b | even | 2 | 1 | ||
| 3150.3.e.f | 8 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{8} + 384T_{11}^{6} + 44832T_{11}^{4} + 1673216T_{11}^{2} + 15872256 \)
acting on \(S_{3}^{\mathrm{new}}(630, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{2} + 5)^{4} \)
|
| $7$ |
\( (T^{2} - 7)^{4} \)
|
| $11$ |
\( T^{8} + 384 T^{6} + \cdots + 15872256 \)
|
| $13$ |
\( (T^{4} - 16 T^{3} + \cdots - 3804)^{2} \)
|
| $17$ |
\( T^{8} + 1152 T^{6} + \cdots + 331776 \)
|
| $19$ |
\( (T^{4} + 40 T^{3} + \cdots + 24900)^{2} \)
|
| $23$ |
\( T^{8} + 616 T^{6} + \cdots + 80353296 \)
|
| $29$ |
\( T^{8} + \cdots + 182633150736 \)
|
| $31$ |
\( (T^{4} - 56 T^{3} + \cdots - 53724)^{2} \)
|
| $37$ |
\( (T^{4} + 64 T^{3} + \cdots - 4205424)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 11776317808896 \)
|
| $43$ |
\( (T^{4} - 32 T^{3} + \cdots + 4862736)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 147161235456 \)
|
| $53$ |
\( T^{8} + \cdots + 8432658441216 \)
|
| $59$ |
\( T^{8} + \cdots + 7297000079616 \)
|
| $61$ |
\( (T^{4} + 72 T^{3} + \cdots + 5776)^{2} \)
|
| $67$ |
\( (T^{4} - 32 T^{3} + \cdots - 956016)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 13463498301696 \)
|
| $73$ |
\( (T^{4} - 10604 T^{2} + \cdots + 6015844)^{2} \)
|
| $79$ |
\( (T^{4} + 32 T^{3} + \cdots - 14801856)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 36\!\cdots\!96 \)
|
| $89$ |
\( T^{8} + \cdots + 416365629696 \)
|
| $97$ |
\( (T^{4} - 16 T^{3} + \cdots + 24802724)^{2} \)
|
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