Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [16,11,Mod(15,16)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16.15");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 16.c (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: | \(10.1657160428\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{505})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{25}\cdot 3 \) |
| 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
\( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1072\nu^{3} - 8128\nu^{2} + 264160\nu - 376992 ) / 8001 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 384\nu^{3} + 72768 ) / 127 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2336\nu^{3} - 471424\nu^{2} - 1064768\nu - 29994048 ) / 8001 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{3} - 8\beta_{2} + 174\beta _1 + 1536 ) / 6144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -195\beta_{3} + 16\beta_{2} - 786\beta _1 - 777216 ) / 12288 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 127\beta_{2} - 72768 ) / 384 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(15\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
0 | − | 408.379i | 0 | 5364.66 | 0 | − | 9544.75i | 0 | −107724. | 0 | ||||||||||||||||||||||||||||
| 15.2 | 0 | − | 214.389i | 0 | −3264.66 | 0 | 5808.15i | 0 | 13086.3 | 0 | ||||||||||||||||||||||||||||||
| 15.3 | 0 | 214.389i | 0 | −3264.66 | 0 | − | 5808.15i | 0 | 13086.3 | 0 | ||||||||||||||||||||||||||||||
| 15.4 | 0 | 408.379i | 0 | 5364.66 | 0 | 9544.75i | 0 | −107724. | 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 | 16.11.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 144.11.g.d | 4 | ||
| 4.b | odd | 2 | 1 | inner | 16.11.c.b | ✓ | 4 |
| 8.b | even | 2 | 1 | 64.11.c.b | 4 | ||
| 8.d | odd | 2 | 1 | 64.11.c.b | 4 | ||
| 12.b | even | 2 | 1 | 144.11.g.d | 4 | ||
| 16.e | even | 4 | 2 | 256.11.d.g | 8 | ||
| 16.f | odd | 4 | 2 | 256.11.d.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 16.11.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 16.11.c.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 64.11.c.b | 4 | 8.b | even | 2 | 1 | ||
| 64.11.c.b | 4 | 8.d | odd | 2 | 1 | ||
| 144.11.g.d | 4 | 3.b | odd | 2 | 1 | ||
| 144.11.g.d | 4 | 12.b | even | 2 | 1 | ||
| 256.11.d.g | 8 | 16.e | even | 4 | 2 | ||
| 256.11.d.g | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 212736T_{3}^{2} + 7665352704 \)
acting on \(S_{11}^{\mathrm{new}}(16, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 7665352704 \)
|
| $5$ |
\( (T^{2} - 2100 T - 17513820)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 30\!\cdots\!44 \)
|
| $11$ |
\( T^{4} + \cdots + 75\!\cdots\!04 \)
|
| $13$ |
\( (T^{2} + 520492 T - 20904319004)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots + 1113290672004)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 76\!\cdots\!04 \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!04 \)
|
| $29$ |
\( (T^{2} + \cdots + 60592888577124)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 25\!\cdots\!44 \)
|
| $37$ |
\( (T^{2} + \cdots - 26\!\cdots\!20)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 84\!\cdots\!56)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 46\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots + 38\!\cdots\!04 \)
|
| $53$ |
\( (T^{2} + \cdots - 13\!\cdots\!36)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 37\!\cdots\!04 \)
|
| $61$ |
\( (T^{2} + \cdots + 23\!\cdots\!16)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 90\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots + 16\!\cdots\!04 \)
|
| $73$ |
\( (T^{2} + \cdots - 14\!\cdots\!16)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 67\!\cdots\!84 \)
|
| $89$ |
\( (T^{2} + \cdots - 35\!\cdots\!24)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 14\!\cdots\!04)^{2} \)
|
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