Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,11,Mod(31,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.31");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.g (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: | \(30.4971481283\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 6937x^{2} + 6936x + 48108096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{7} \) |
| 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
\( x^{4} - x^{3} + 6937x^{2} + 6936x + 48108096 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 27\nu^{3} - 187299\nu^{2} + 187299\nu - 649365660 ) / 8019172 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48\nu^{3} + 499416 ) / 6937 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41619\nu^{3} - 20811\nu^{2} + 577401195\nu + 216496836 ) / 2004793 \)
|
| \(\nu\) | \(=\) |
\( ( 9\beta_{3} - 27\beta_{2} - 4\beta _1 + 648 ) / 2592 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{3} + 27\beta_{2} - 110980\beta _1 - 8989704 ) / 2592 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6937\beta_{2} - 499416 ) / 48 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(31\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 140.296i | 0 | −5887.64 | 0 | − | 29688.9i | 0 | −19683.0 | 0 | ||||||||||||||||||||||||||||
| 31.2 | 0 | − | 140.296i | 0 | 2107.64 | 0 | 11855.7i | 0 | −19683.0 | 0 | ||||||||||||||||||||||||||||||
| 31.3 | 0 | 140.296i | 0 | −5887.64 | 0 | 29688.9i | 0 | −19683.0 | 0 | |||||||||||||||||||||||||||||||
| 31.4 | 0 | 140.296i | 0 | 2107.64 | 0 | − | 11855.7i | 0 | −19683.0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.11.g.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 144.11.g.h | 4 | ||
| 4.b | odd | 2 | 1 | inner | 48.11.g.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 192.11.g.c | 4 | ||
| 8.d | odd | 2 | 1 | 192.11.g.c | 4 | ||
| 12.b | even | 2 | 1 | 144.11.g.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.11.g.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 48.11.g.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 144.11.g.h | 4 | 3.b | odd | 2 | 1 | ||
| 144.11.g.h | 4 | 12.b | even | 2 | 1 | ||
| 192.11.g.c | 4 | 8.b | even | 2 | 1 | ||
| 192.11.g.c | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 3780T_{5} - 12409020 \)
acting on \(S_{11}^{\mathrm{new}}(48, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 19683)^{2} \)
|
| $5$ |
\( (T^{2} + 3780 T - 12409020)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 12\!\cdots\!84 \)
|
| $11$ |
\( T^{4} + \cdots + 17\!\cdots\!44 \)
|
| $13$ |
\( (T^{2} + 268156 T - 28624035836)^{2} \)
|
| $17$ |
\( (T^{2} + 1928556 T + 642875070564)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 29\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 29\!\cdots\!44 \)
|
| $29$ |
\( (T^{2} + \cdots - 290157183392796)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 389375050303580)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 14\!\cdots\!44 \)
|
| $47$ |
\( T^{4} + \cdots + 47\!\cdots\!64 \)
|
| $53$ |
\( (T^{2} + \cdots + 30\!\cdots\!16)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 24\!\cdots\!64 \)
|
| $61$ |
\( (T^{2} + \cdots + 41\!\cdots\!16)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 56\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots + 30\!\cdots\!24 \)
|
| $73$ |
\( (T^{2} + \cdots - 12\!\cdots\!36)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 78\!\cdots\!24 \)
|
| $83$ |
\( T^{4} + \cdots + 10\!\cdots\!64 \)
|
| $89$ |
\( (T^{2} + \cdots + 63\!\cdots\!56)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 19\!\cdots\!44)^{2} \)
|
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