Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,4,Mod(361,630)]
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([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.k (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: | \(37.1712033036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{295})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 295x^{2} + 87025 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 210) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 295x^{2} + 87025 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 295 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 295 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 295\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 295\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 0 | −2.58778 | − | 18.3386i | −8.00000 | 0 | 5.00000 | − | 8.66025i | ||||||||||||||||||||
| 361.2 | 1.00000 | + | 1.73205i | 0 | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 0 | 14.5878 | + | 11.4104i | −8.00000 | 0 | 5.00000 | − | 8.66025i | |||||||||||||||||||||
| 541.1 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 0 | −2.58778 | + | 18.3386i | −8.00000 | 0 | 5.00000 | + | 8.66025i | |||||||||||||||||||||
| 541.2 | 1.00000 | − | 1.73205i | 0 | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 0 | 14.5878 | − | 11.4104i | −8.00000 | 0 | 5.00000 | + | 8.66025i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.4.k.m | 4 | |
| 3.b | odd | 2 | 1 | 210.4.i.i | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 630.4.k.m | 4 | |
| 21.g | even | 6 | 1 | 1470.4.a.bs | 2 | ||
| 21.h | odd | 6 | 1 | 210.4.i.i | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 1470.4.a.bn | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.4.i.i | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 210.4.i.i | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 630.4.k.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 630.4.k.m | 4 | 7.c | even | 3 | 1 | inner | |
| 1470.4.a.bn | 2 | 21.h | odd | 6 | 1 | ||
| 1470.4.a.bs | 2 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\):
|
\( T_{11}^{4} - 50T_{11}^{3} + 2170T_{11}^{2} - 16500T_{11} + 108900 \)
|
|
\( T_{13}^{2} + 100T_{13} + 2205 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} - 24 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} - 50 T^{3} + \cdots + 108900 \)
|
| $13$ |
\( (T^{2} + 100 T + 2205)^{2} \)
|
| $17$ |
\( T^{4} - 34 T^{3} + \cdots + 50211396 \)
|
| $19$ |
\( T^{4} - 46 T^{3} + \cdots + 17564481 \)
|
| $23$ |
\( T^{4} - 126 T^{3} + \cdots + 397045476 \)
|
| $29$ |
\( (T^{2} - 150 T + 2970)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 1690114321 \)
|
| $37$ |
\( T^{4} - 176 T^{3} + \cdots + 260854801 \)
|
| $41$ |
\( (T^{2} + 10 T - 85230)^{2} \)
|
| $43$ |
\( (T^{2} - 132 T - 10099)^{2} \)
|
| $47$ |
\( T^{4} - 316 T^{3} + \cdots + 409819536 \)
|
| $53$ |
\( T^{4} - 236 T^{3} + \cdots + 242611776 \)
|
| $59$ |
\( T^{4} + \cdots + 500491162116 \)
|
| $61$ |
\( T^{4} + \cdots + 23133193216 \)
|
| $67$ |
\( T^{4} + \cdots + 19182527001 \)
|
| $71$ |
\( (T^{2} - 386 T - 147126)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 93467775625 \)
|
| $79$ |
\( T^{4} + \cdots + 325345892881 \)
|
| $83$ |
\( (T^{2} - 182 T - 1089414)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 12627915876 \)
|
| $97$ |
\( (T^{2} + 392 T - 4064)^{2} \)
|
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