Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [77,3,Mod(32,77)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(77, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("77.32");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 77 = 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 77.h (of order \(6\), 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: | \(2.09809803557\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} + 3\nu - 4 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{3} + \nu^{2} + 3\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
−2.29129 | + | 1.32288i | −2.50000 | + | 4.33013i | 1.50000 | − | 2.59808i | 3.00000 | + | 5.19615i | − | 13.2288i | −2.29129 | + | 6.61438i | − | 2.64575i | −8.00000 | − | 13.8564i | −13.7477 | − | 7.93725i | ||||||||||||||
| 32.2 | 2.29129 | − | 1.32288i | −2.50000 | + | 4.33013i | 1.50000 | − | 2.59808i | 3.00000 | + | 5.19615i | 13.2288i | 2.29129 | − | 6.61438i | 2.64575i | −8.00000 | − | 13.8564i | 13.7477 | + | 7.93725i | |||||||||||||||||
| 65.1 | −2.29129 | − | 1.32288i | −2.50000 | − | 4.33013i | 1.50000 | + | 2.59808i | 3.00000 | − | 5.19615i | 13.2288i | −2.29129 | − | 6.61438i | 2.64575i | −8.00000 | + | 13.8564i | −13.7477 | + | 7.93725i | |||||||||||||||||
| 65.2 | 2.29129 | + | 1.32288i | −2.50000 | − | 4.33013i | 1.50000 | + | 2.59808i | 3.00000 | − | 5.19615i | − | 13.2288i | 2.29129 | + | 6.61438i | − | 2.64575i | −8.00000 | + | 13.8564i | 13.7477 | − | 7.93725i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 11.b | odd | 2 | 1 | inner |
| 77.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 77.3.h.a | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 77.3.h.a | ✓ | 4 |
| 7.c | even | 3 | 1 | 539.3.c.f | 2 | ||
| 7.d | odd | 6 | 1 | 539.3.c.b | 2 | ||
| 11.b | odd | 2 | 1 | inner | 77.3.h.a | ✓ | 4 |
| 77.h | odd | 6 | 1 | inner | 77.3.h.a | ✓ | 4 |
| 77.h | odd | 6 | 1 | 539.3.c.f | 2 | ||
| 77.i | even | 6 | 1 | 539.3.c.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.3.h.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 77.3.h.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 77.3.h.a | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 77.3.h.a | ✓ | 4 | 77.h | odd | 6 | 1 | inner |
| 539.3.c.b | 2 | 7.d | odd | 6 | 1 | ||
| 539.3.c.b | 2 | 77.i | even | 6 | 1 | ||
| 539.3.c.f | 2 | 7.c | even | 3 | 1 | ||
| 539.3.c.f | 2 | 77.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 7T_{2}^{2} + 49 \)
acting on \(S_{3}^{\mathrm{new}}(77, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 7T^{2} + 49 \)
|
| $3$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $5$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $7$ |
\( T^{4} + 77T^{2} + 2401 \)
|
| $11$ |
\( (T^{2} - 11 T + 121)^{2} \)
|
| $13$ |
\( (T^{2} + 7)^{2} \)
|
| $17$ |
\( T^{4} - 700 T^{2} + 490000 \)
|
| $19$ |
\( T^{4} - 700 T^{2} + 490000 \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( (T^{2} + 175)^{2} \)
|
| $31$ |
\( (T^{2} + 16 T + 256)^{2} \)
|
| $37$ |
\( (T^{2} + 10 T + 100)^{2} \)
|
| $41$ |
\( (T^{2} + 700)^{2} \)
|
| $43$ |
\( (T^{2} + 4732)^{2} \)
|
| $47$ |
\( (T^{2} + 20 T + 400)^{2} \)
|
| $53$ |
\( (T^{2} + 70 T + 4900)^{2} \)
|
| $59$ |
\( (T^{2} + 19 T + 361)^{2} \)
|
| $61$ |
\( T^{4} - 4375 T^{2} + 19140625 \)
|
| $67$ |
\( (T^{2} + 35 T + 1225)^{2} \)
|
| $71$ |
\( (T - 84)^{4} \)
|
| $73$ |
\( T^{4} - 1372 T^{2} + 1882384 \)
|
| $79$ |
\( T^{4} - 175 T^{2} + 30625 \)
|
| $83$ |
\( (T^{2} + 112)^{2} \)
|
| $89$ |
\( (T^{2} - 134 T + 17956)^{2} \)
|
| $97$ |
\( (T + 25)^{4} \)
|
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