Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,5,Mod(20,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 6]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.20");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.j (of order \(10\), 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: | \(7.95948715746\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | 8.0.37515625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{10}]$ |
$q$-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 in terms of a root \(\nu\) of
\( x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - 5\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 7\nu^{2} ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 11\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 3\nu^{6} - 3\nu^{5} + \nu^{4} - 3\nu^{3} + 18\nu^{2} - 12\nu - 24 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{7} - 4\nu^{2} \)
|
| \(\beta_{7}\) | \(=\) |
\( -3\nu^{5} - 17 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - 4\beta_{3} ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 8\beta_{5} + \beta_{4} - 8\beta_{3} + 7\beta_{2} + \beta _1 - 7 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} - 17 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -5\beta_{4} - 11\beta_{2} ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -7\beta_{6} + 16\beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{2} - \beta_{3} - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 20.1 |
|
−1.69169 | + | 5.20648i | 0 | −11.3014 | − | 8.21092i | 0 | 0 | −39.6418 | − | 28.8015i | −8.99398 | + | 6.53451i | 25.0304 | − | 77.0356i | 0 | ||||||||||||||||||||||||||||||||
| 20.2 | 1.19169 | − | 3.66764i | 0 | 0.912817 | + | 0.663200i | 0 | 0 | −39.6418 | − | 28.8015i | 53.4383 | − | 38.8252i | 25.0304 | − | 77.0356i | 0 | |||||||||||||||||||||||||||||||||
| 27.1 | −1.69169 | − | 5.20648i | 0 | −11.3014 | + | 8.21092i | 0 | 0 | −39.6418 | + | 28.8015i | −8.99398 | − | 6.53451i | 25.0304 | + | 77.0356i | 0 | |||||||||||||||||||||||||||||||||
| 27.2 | 1.19169 | + | 3.66764i | 0 | 0.912817 | − | 0.663200i | 0 | 0 | −39.6418 | + | 28.8015i | 53.4383 | + | 38.8252i | 25.0304 | + | 77.0356i | 0 | |||||||||||||||||||||||||||||||||
| 48.1 | −6.35709 | − | 4.61870i | 0 | 14.1360 | + | 43.5060i | 0 | 0 | 15.1418 | + | 46.6018i | 72.2264 | − | 222.290i | −65.5304 | − | 47.6106i | 0 | |||||||||||||||||||||||||||||||||
| 48.2 | 5.85709 | + | 4.25542i | 0 | 11.2526 | + | 34.6319i | 0 | 0 | 15.1418 | + | 46.6018i | −45.6707 | + | 140.560i | −65.5304 | − | 47.6106i | 0 | |||||||||||||||||||||||||||||||||
| 69.1 | −6.35709 | + | 4.61870i | 0 | 14.1360 | − | 43.5060i | 0 | 0 | 15.1418 | − | 46.6018i | 72.2264 | + | 222.290i | −65.5304 | + | 47.6106i | 0 | |||||||||||||||||||||||||||||||||
| 69.2 | 5.85709 | − | 4.25542i | 0 | 11.2526 | − | 34.6319i | 0 | 0 | 15.1418 | − | 46.6018i | −45.6707 | − | 140.560i | −65.5304 | + | 47.6106i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 11.c | even | 5 | 1 | inner |
| 77.j | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 77.5.j.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | CM | 77.5.j.a | ✓ | 8 |
| 11.c | even | 5 | 1 | inner | 77.5.j.a | ✓ | 8 |
| 77.j | odd | 10 | 1 | inner | 77.5.j.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.5.j.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 77.5.j.a | ✓ | 8 | 7.b | odd | 2 | 1 | CM |
| 77.5.j.a | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 77.5.j.a | ✓ | 8 | 77.j | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 3T_{2}^{6} - 76T_{2}^{5} + 2325T_{2}^{4} + 2324T_{2}^{3} + 104723T_{2}^{2} - 93678T_{2} + 1442401 \)
acting on \(S_{5}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 1442401 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} + 49 T^{3} + \cdots + 5764801)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 45\!\cdots\!61 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( (T^{4} - 734 T^{3} + \cdots - 72017135039)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 31\!\cdots\!21 \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} + \cdots + 21\!\cdots\!41 \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( (T^{4} + \cdots + 56546507307361)^{2} \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} + \cdots + 27\!\cdots\!21 \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( (T^{4} + \cdots + 163996745362081)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 49\!\cdots\!81 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} + \cdots + 26\!\cdots\!61 \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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