Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5328,2,Mod(5327,5328)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5328, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5328.5327");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5328.l (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: | \(42.5442941969\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{37})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 13x^{2} + 14x + 123 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{41}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} - 2x^{3} - 13x^{2} + 14x + 123 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 100\nu - 51 ) / 45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 5\nu - 3 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 48\nu^{2} - 40\nu - 318 ) / 45 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{2} + 3\beta _1 + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{3} - 4\beta_{2} + 3\beta _1 + 45 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{3} - 56\beta_{2} + 12\beta _1 + 66 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times\).
| \(n\) | \(1297\) | \(1333\) | \(1999\) | \(2369\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5327.1 |
|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 5327.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 5327.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 5327.4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 148.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-37}) \) |
| 3.b | odd | 2 | 1 | inner |
| 444.g | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5328.2.l.c | yes | 4 |
| 3.b | odd | 2 | 1 | inner | 5328.2.l.c | yes | 4 |
| 4.b | odd | 2 | 1 | 5328.2.l.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 5328.2.l.a | ✓ | 4 | |
| 37.b | even | 2 | 1 | 5328.2.l.a | ✓ | 4 | |
| 111.d | odd | 2 | 1 | 5328.2.l.a | ✓ | 4 | |
| 148.b | odd | 2 | 1 | CM | 5328.2.l.c | yes | 4 |
| 444.g | even | 2 | 1 | inner | 5328.2.l.c | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5328.2.l.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 5328.2.l.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 5328.2.l.a | ✓ | 4 | 37.b | even | 2 | 1 | |
| 5328.2.l.a | ✓ | 4 | 111.d | odd | 2 | 1 | |
| 5328.2.l.c | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 5328.2.l.c | yes | 4 | 3.b | odd | 2 | 1 | inner |
| 5328.2.l.c | yes | 4 | 148.b | odd | 2 | 1 | CM |
| 5328.2.l.c | yes | 4 | 444.g | 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 \(S_{2}^{\mathrm{new}}(5328, [\chi])\):
|
\( T_{5} \)
|
|
\( T_{19}^{2} - 2T_{19} - 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T^{2} - 2 T - 36)^{2} \)
|
| $23$ |
\( (T^{2} + 18)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} - 10 T - 12)^{2} \)
|
| $37$ |
\( (T^{2} - 37)^{2} \)
|
| $41$ |
\( T^{4} + 164T^{2} + 4356 \)
|
| $43$ |
\( (T^{2} + 14 T + 12)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + 212T^{2} + 1764 \)
|
| $59$ |
\( (T^{2} + 162)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} - 148)^{2} \)
|
| $79$ |
\( (T^{2} + 22 T + 84)^{2} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
| show more | |
| show less |