Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1350,4,Mod(649,1350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1350.649");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1350.c (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: | \(79.6525785077\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).
| \(n\) | \(1001\) | \(1027\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 2.00000i | 0 | −4.00000 | 0 | 0 | − | 24.6969i | 8.00000i | 0 | 0 | ||||||||||||||||||||||||||||
| 649.2 | − | 2.00000i | 0 | −4.00000 | 0 | 0 | 4.69694i | 8.00000i | 0 | 0 | ||||||||||||||||||||||||||||||
| 649.3 | 2.00000i | 0 | −4.00000 | 0 | 0 | − | 4.69694i | − | 8.00000i | 0 | 0 | |||||||||||||||||||||||||||||
| 649.4 | 2.00000i | 0 | −4.00000 | 0 | 0 | 24.6969i | − | 8.00000i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1350.4.c.z | 4 | |
| 3.b | odd | 2 | 1 | 1350.4.c.w | 4 | ||
| 5.b | even | 2 | 1 | inner | 1350.4.c.z | 4 | |
| 5.c | odd | 4 | 1 | 1350.4.a.bc | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1350.4.a.bp | yes | 2 | |
| 15.d | odd | 2 | 1 | 1350.4.c.w | 4 | ||
| 15.e | even | 4 | 1 | 1350.4.a.bi | yes | 2 | |
| 15.e | even | 4 | 1 | 1350.4.a.bj | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1350.4.a.bc | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 1350.4.a.bi | yes | 2 | 15.e | even | 4 | 1 | |
| 1350.4.a.bj | yes | 2 | 15.e | even | 4 | 1 | |
| 1350.4.a.bp | yes | 2 | 5.c | odd | 4 | 1 | |
| 1350.4.c.w | 4 | 3.b | odd | 2 | 1 | ||
| 1350.4.c.w | 4 | 15.d | odd | 2 | 1 | ||
| 1350.4.c.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 1350.4.c.z | 4 | 5.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 \(S_{4}^{\mathrm{new}}(1350, [\chi])\):
|
\( T_{7}^{4} + 632T_{7}^{2} + 13456 \)
|
|
\( T_{11}^{2} - 30T_{11} - 639 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 632 T^{2} + 13456 \)
|
| $11$ |
\( (T^{2} - 30 T - 639)^{2} \)
|
| $13$ |
\( T^{4} + 10850 T^{2} + 28890625 \)
|
| $17$ |
\( T^{4} + 11448 T^{2} + 25765776 \)
|
| $19$ |
\( (T^{2} - 32 T - 5144)^{2} \)
|
| $23$ |
\( T^{4} + 10962 T^{2} + 28291761 \)
|
| $29$ |
\( (T^{2} + 120 T - 1800)^{2} \)
|
| $31$ |
\( (T^{2} - 244 T - 6716)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 1606486561 \)
|
| $41$ |
\( (T^{2} - 420 T + 36324)^{2} \)
|
| $43$ |
\( T^{4} + 72368 T^{2} + 225480256 \)
|
| $47$ |
\( T^{4} + \cdots + 66584125521 \)
|
| $53$ |
\( T^{4} + \cdots + 20179907136 \)
|
| $59$ |
\( (T^{2} - 330 T - 15111)^{2} \)
|
| $61$ |
\( (T^{2} + 458 T + 3841)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 33472897936 \)
|
| $71$ |
\( (T^{2} - 1110 T + 259425)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 31372911376 \)
|
| $79$ |
\( (T^{2} + 580 T - 353300)^{2} \)
|
| $83$ |
\( T^{4} + 1411200 T^{2} + 207360000 \)
|
| $89$ |
\( (T^{2} - 660 T + 91404)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 149528382721 \)
|
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