Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,7,Mod(511,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.511");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.g (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: | \(176.681536220\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 384) |
| 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
\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -9\nu^{3} + 45\nu^{2} - 45\nu + 54 ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( -24\nu^{3} + 24\nu^{2} - 216\nu - 48 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -864\nu^{3} - 5616 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 9\beta_{2} + 48\beta _1 + 432 ) / 1728 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} - 9\beta_{2} + 432\beta _1 - 3888 ) / 1728 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{3} - 5616 ) / 864 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).
| \(n\) | \(257\) | \(511\) | \(517\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 511.1 |
|
0 | − | 15.5885i | 0 | −20.0000 | 0 | − | 155.727i | 0 | −243.000 | 0 | ||||||||||||||||||||||||||||
| 511.2 | 0 | − | 15.5885i | 0 | −20.0000 | 0 | 529.850i | 0 | −243.000 | 0 | ||||||||||||||||||||||||||||||
| 511.3 | 0 | 15.5885i | 0 | −20.0000 | 0 | − | 529.850i | 0 | −243.000 | 0 | ||||||||||||||||||||||||||||||
| 511.4 | 0 | 15.5885i | 0 | −20.0000 | 0 | 155.727i | 0 | −243.000 | 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 | 768.7.g.c | 4 | |
| 4.b | odd | 2 | 1 | inner | 768.7.g.c | 4 | |
| 8.b | even | 2 | 1 | 768.7.g.e | 4 | ||
| 8.d | odd | 2 | 1 | 768.7.g.e | 4 | ||
| 16.e | even | 4 | 2 | 384.7.b.d | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 384.7.b.d | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.7.b.d | ✓ | 8 | 16.e | even | 4 | 2 | |
| 384.7.b.d | ✓ | 8 | 16.f | odd | 4 | 2 | |
| 768.7.g.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 768.7.g.c | 4 | 4.b | odd | 2 | 1 | inner | |
| 768.7.g.e | 4 | 8.b | even | 2 | 1 | ||
| 768.7.g.e | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 20 \)
acting on \(S_{7}^{\mathrm{new}}(768, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 243)^{2} \)
|
| $5$ |
\( (T + 20)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 6808230144 \)
|
| $11$ |
\( T^{4} + \cdots + 1010555709696 \)
|
| $13$ |
\( (T^{2} - 2880 T - 1099008)^{2} \)
|
| $17$ |
\( (T^{2} - 1116 T - 50450364)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 285122947021056 \)
|
| $23$ |
\( T^{4} + \cdots + 30\!\cdots\!04 \)
|
| $29$ |
\( (T^{2} + 25304 T - 867851888)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $37$ |
\( (T^{2} - 106128 T + 2736472896)^{2} \)
|
| $41$ |
\( (T^{2} - 24732 T - 303937596)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 62\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 96\!\cdots\!24 \)
|
| $53$ |
\( (T^{2} - 128200 T - 21589314800)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 48\!\cdots\!24 \)
|
| $61$ |
\( (T^{2} + 40464 T - 3933966528)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 44\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots + 27\!\cdots\!24 \)
|
| $73$ |
\( (T^{2} + 19228 T - 147928769852)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 88\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots + 37\!\cdots\!84 \)
|
| $89$ |
\( (T^{2} + 1659708 T + 683581488516)^{2} \)
|
| $97$ |
\( (T^{2} + 1232420 T - 31455232700)^{2} \)
|
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