Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [768,3,Mod(319,768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("768.319");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.l (of order \(4\), degree \(2\), 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: | \(20.9264843029\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\zeta_{24}^{7} - \zeta_{24}^{3} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{4} + 2\zeta_{24}^{2} - 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{2} + 1 \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{6} + \beta_{4} + 2\beta_1 ) / 4 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} + 2\beta_{2} ) / 4 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( -\beta_{6} + \beta_{4} ) / 2 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( -\beta_{7} + \beta_{5} + 2 ) / 4 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{6} - \beta_{4} + 2\beta_1 ) / 4 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{6} + \beta_{4} + 2\beta_{3} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).
| \(n\) | \(257\) | \(511\) | \(517\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 319.1 |
|
0 | −1.22474 | − | 1.22474i | 0 | −0.732051 | − | 0.732051i | 0 | −11.9700 | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −1.22474 | − | 1.22474i | 0 | 2.73205 | + | 2.73205i | 0 | 2.17209 | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | 1.22474 | + | 1.22474i | 0 | −0.732051 | − | 0.732051i | 0 | 11.9700 | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | 1.22474 | + | 1.22474i | 0 | 2.73205 | + | 2.73205i | 0 | −2.17209 | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 703.1 | 0 | −1.22474 | + | 1.22474i | 0 | −0.732051 | + | 0.732051i | 0 | −11.9700 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 703.2 | 0 | −1.22474 | + | 1.22474i | 0 | 2.73205 | − | 2.73205i | 0 | 2.17209 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 703.3 | 0 | 1.22474 | − | 1.22474i | 0 | −0.732051 | + | 0.732051i | 0 | 11.9700 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 703.4 | 0 | 1.22474 | − | 1.22474i | 0 | 2.73205 | − | 2.73205i | 0 | −2.17209 | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 16.e | even | 4 | 1 | inner |
| 16.f | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 768.3.l.c | yes | 8 |
| 4.b | odd | 2 | 1 | inner | 768.3.l.c | yes | 8 |
| 8.b | even | 2 | 1 | 768.3.l.b | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 768.3.l.b | ✓ | 8 | |
| 16.e | even | 4 | 1 | 768.3.l.b | ✓ | 8 | |
| 16.e | even | 4 | 1 | inner | 768.3.l.c | yes | 8 |
| 16.f | odd | 4 | 1 | 768.3.l.b | ✓ | 8 | |
| 16.f | odd | 4 | 1 | inner | 768.3.l.c | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 768.3.l.b | ✓ | 8 | 8.b | even | 2 | 1 | |
| 768.3.l.b | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 768.3.l.b | ✓ | 8 | 16.e | even | 4 | 1 | |
| 768.3.l.b | ✓ | 8 | 16.f | odd | 4 | 1 | |
| 768.3.l.c | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 768.3.l.c | yes | 8 | 4.b | odd | 2 | 1 | inner |
| 768.3.l.c | yes | 8 | 16.e | even | 4 | 1 | inner |
| 768.3.l.c | yes | 8 | 16.f | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{3} + 8T_{5}^{2} + 16T_{5} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(768, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 9)^{2} \)
|
| $5$ |
\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 16)^{2} \)
|
| $7$ |
\( (T^{4} - 148 T^{2} + 676)^{2} \)
|
| $11$ |
\( T^{8} + 172928 T^{4} + 4096 \)
|
| $13$ |
\( (T^{4} + 12 T^{3} + \cdots + 39204)^{2} \)
|
| $17$ |
\( (T^{2} + 32 T + 148)^{4} \)
|
| $19$ |
\( T^{8} + 82208 T^{4} + 7311616 \)
|
| $23$ |
\( (T^{4} - 1776 T^{2} + 97344)^{2} \)
|
| $29$ |
\( (T^{4} + 4 T^{3} + \cdots + 2704)^{2} \)
|
| $31$ |
\( (T^{4} + 1236 T^{2} + 338724)^{2} \)
|
| $37$ |
\( (T^{4} - 44 T^{3} + \cdots + 21316)^{2} \)
|
| $41$ |
\( (T^{4} + 2072 T^{2} + 824464)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 68415660933376 \)
|
| $47$ |
\( (T^{4} + 11376 T^{2} + 26132544)^{2} \)
|
| $53$ |
\( (T^{4} + 60 T^{3} + \cdots + 318096)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 81867462148096 \)
|
| $61$ |
\( (T^{4} + 36 T^{3} + \cdots + 20629764)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 385832677605376 \)
|
| $71$ |
\( (T^{4} - 2896 T^{2} + 1577536)^{2} \)
|
| $73$ |
\( (T^{4} + 7328 T^{2} + 43264)^{2} \)
|
| $79$ |
\( (T^{4} + 4468 T^{2} + 13924)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!76 \)
|
| $89$ |
\( (T^{4} + 23336 T^{2} + 116726416)^{2} \)
|
| $97$ |
\( (T^{2} - 80 T + 1552)^{4} \)
|
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