Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [799,1,Mod(93,799)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([5, 4]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("799.93");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 799 = 17 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 799.h (of order \(8\), degree \(4\), 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: | \(0.398752945094\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
| Coefficient field: | \(\Q(\zeta_{40})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{40}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(377\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{40}^{15}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 93.1 |
|
−1.34500 | + | 1.34500i | −0.652583 | + | 1.57547i | − | 2.61803i | 0 | −1.24129 | − | 2.99673i | 1.84206 | − | 0.763007i | 2.17625 | + | 2.17625i | −1.34915 | − | 1.34915i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.2 | −0.831254 | + | 0.831254i | 0.763007 | − | 1.84206i | − | 0.381966i | 0 | 0.896969 | + | 2.16547i | 0.431351 | − | 0.178671i | −0.513743 | − | 0.513743i | −2.10391 | − | 2.10391i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.3 | 0.831254 | − | 0.831254i | −0.581990 | + | 1.40505i | − | 0.381966i | 0 | 0.684170 | + | 1.65173i | −1.57547 | + | 0.652583i | 0.513743 | + | 0.513743i | −0.928339 | − | 0.928339i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 93.4 | 1.34500 | − | 1.34500i | 0.178671 | − | 0.431351i | − | 2.61803i | 0 | −0.339853 | − | 0.820478i | −1.40505 | + | 0.581990i | −2.17625 | − | 2.17625i | 0.552967 | + | 0.552967i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.1 | −1.34500 | + | 1.34500i | −1.79671 | − | 0.744220i | − | 2.61803i | 0 | 3.41754 | − | 1.41559i | −0.497066 | − | 1.20002i | 2.17625 | + | 2.17625i | 1.96718 | + | 1.96718i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.2 | −0.831254 | + | 0.831254i | 1.20002 | + | 0.497066i | − | 0.381966i | 0 | −1.41071 | + | 0.584336i | 0.399903 | + | 0.965451i | −0.513743 | − | 0.513743i | 0.485875 | + | 0.485875i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.3 | 0.831254 | − | 0.831254i | −0.144974 | − | 0.0600500i | − | 0.381966i | 0 | −0.170427 | + | 0.0705930i | 0.744220 | + | 1.79671i | 0.513743 | + | 0.513743i | −0.689695 | − | 0.689695i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 281.4 | 1.34500 | − | 1.34500i | −0.965451 | − | 0.399903i | − | 2.61803i | 0 | −1.83640 | + | 0.760661i | 0.0600500 | + | 0.144974i | −2.17625 | − | 2.17625i | 0.0650673 | + | 0.0650673i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 563.1 | −1.34500 | − | 1.34500i | −1.79671 | + | 0.744220i | 2.61803i | 0 | 3.41754 | + | 1.41559i | −0.497066 | + | 1.20002i | 2.17625 | − | 2.17625i | 1.96718 | − | 1.96718i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 563.2 | −0.831254 | − | 0.831254i | 1.20002 | − | 0.497066i | 0.381966i | 0 | −1.41071 | − | 0.584336i | 0.399903 | − | 0.965451i | −0.513743 | + | 0.513743i | 0.485875 | − | 0.485875i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 563.3 | 0.831254 | + | 0.831254i | −0.144974 | + | 0.0600500i | 0.381966i | 0 | −0.170427 | − | 0.0705930i | 0.744220 | − | 1.79671i | 0.513743 | − | 0.513743i | −0.689695 | + | 0.689695i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 563.4 | 1.34500 | + | 1.34500i | −0.965451 | + | 0.399903i | 2.61803i | 0 | −1.83640 | − | 0.760661i | 0.0600500 | − | 0.144974i | −2.17625 | + | 2.17625i | 0.0650673 | − | 0.0650673i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.1 | −1.34500 | − | 1.34500i | −0.652583 | − | 1.57547i | 2.61803i | 0 | −1.24129 | + | 2.99673i | 1.84206 | + | 0.763007i | 2.17625 | − | 2.17625i | −1.34915 | + | 1.34915i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.2 | −0.831254 | − | 0.831254i | 0.763007 | + | 1.84206i | 0.381966i | 0 | 0.896969 | − | 2.16547i | 0.431351 | + | 0.178671i | −0.513743 | + | 0.513743i | −2.10391 | + | 2.10391i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.3 | 0.831254 | + | 0.831254i | −0.581990 | − | 1.40505i | 0.381966i | 0 | 0.684170 | − | 1.65173i | −1.57547 | − | 0.652583i | 0.513743 | − | 0.513743i | −0.928339 | + | 0.928339i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 610.4 | 1.34500 | + | 1.34500i | 0.178671 | + | 0.431351i | 2.61803i | 0 | −0.339853 | + | 0.820478i | −1.40505 | − | 0.581990i | −2.17625 | + | 2.17625i | 0.552967 | − | 0.552967i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
| 17.d | even | 8 | 1 | inner |
| 799.h | odd | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 799.1.h.b | ✓ | 16 |
| 17.d | even | 8 | 1 | inner | 799.1.h.b | ✓ | 16 |
| 47.b | odd | 2 | 1 | CM | 799.1.h.b | ✓ | 16 |
| 799.h | odd | 8 | 1 | inner | 799.1.h.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 799.1.h.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 799.1.h.b | ✓ | 16 | 17.d | even | 8 | 1 | inner |
| 799.1.h.b | ✓ | 16 | 47.b | odd | 2 | 1 | CM |
| 799.1.h.b | ✓ | 16 | 799.h | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 15T_{2}^{4} + 25 \)
acting on \(S_{1}^{\mathrm{new}}(799, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + 15 T^{4} + 25)^{2} \)
|
| $3$ |
\( T^{16} + 4 T^{15} + \cdots + 1 \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} - 2 T^{14} + \cdots + 1 \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( T^{16} \)
|
| $17$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $19$ |
\( T^{16} \)
|
| $23$ |
\( T^{16} \)
|
| $29$ |
\( T^{16} \)
|
| $31$ |
\( T^{16} \)
|
| $37$ |
\( T^{16} - 2 T^{14} + \cdots + 1 \)
|
| $41$ |
\( T^{16} \)
|
| $43$ |
\( T^{16} \)
|
| $47$ |
\( (T^{2} + 1)^{8} \)
|
| $53$ |
\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \)
|
| $59$ |
\( (T^{8} + 7 T^{4} + 1)^{2} \)
|
| $61$ |
\( T^{16} - 4 T^{15} + \cdots + 1 \)
|
| $67$ |
\( T^{16} \)
|
| $71$ |
\( T^{16} + 4 T^{15} + \cdots + 1 \)
|
| $73$ |
\( T^{16} \)
|
| $79$ |
\( T^{16} + 4 T^{15} + \cdots + 1 \)
|
| $83$ |
\( (T^{2} + 2 T + 2)^{8} \)
|
| $89$ |
\( (T^{4} + 5 T^{2} + 5)^{4} \)
|
| $97$ |
\( T^{16} - 2 T^{14} + \cdots + 1 \)
|
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