Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,4,Mod(55,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.55");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.e (of order \(3\), 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: | \(4.60214898045\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{61})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 16x^{2} + 15x + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 16x^{2} + 15x + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 16\nu^{2} - 16\nu + 225 ) / 240 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 16\nu^{2} + 496\nu - 225 ) / 240 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 23 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 31\beta _1 - 31 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{3} - 23 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −14.6205 | −3.00000 | + | 5.19615i | −8.40512 | + | 14.5581i | −8.00000 | −4.50000 | + | 7.79423i | −14.6205 | − | 25.3234i | ||||||||||||||||
| 55.2 | 1.00000 | + | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | 16.6205 | −3.00000 | + | 5.19615i | −0.594875 | + | 1.03035i | −8.00000 | −4.50000 | + | 7.79423i | 16.6205 | + | 28.7875i | |||||||||||||||||
| 61.1 | 1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −14.6205 | −3.00000 | − | 5.19615i | −8.40512 | − | 14.5581i | −8.00000 | −4.50000 | − | 7.79423i | −14.6205 | + | 25.3234i | |||||||||||||||||
| 61.2 | 1.00000 | − | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | 16.6205 | −3.00000 | − | 5.19615i | −0.594875 | − | 1.03035i | −8.00000 | −4.50000 | − | 7.79423i | 16.6205 | − | 28.7875i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.4.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 234.4.h.f | 4 | ||
| 4.b | odd | 2 | 1 | 624.4.q.d | 4 | ||
| 13.c | even | 3 | 1 | inner | 78.4.e.c | ✓ | 4 |
| 13.c | even | 3 | 1 | 1014.4.a.l | 2 | ||
| 13.e | even | 6 | 1 | 1014.4.a.p | 2 | ||
| 13.f | odd | 12 | 2 | 1014.4.b.i | 4 | ||
| 39.i | odd | 6 | 1 | 234.4.h.f | 4 | ||
| 52.j | odd | 6 | 1 | 624.4.q.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.4.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 78.4.e.c | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 234.4.h.f | 4 | 3.b | odd | 2 | 1 | ||
| 234.4.h.f | 4 | 39.i | odd | 6 | 1 | ||
| 624.4.q.d | 4 | 4.b | odd | 2 | 1 | ||
| 624.4.q.d | 4 | 52.j | odd | 6 | 1 | ||
| 1014.4.a.l | 2 | 13.c | even | 3 | 1 | ||
| 1014.4.a.p | 2 | 13.e | even | 6 | 1 | ||
| 1014.4.b.i | 4 | 13.f | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 2T_{5} - 243 \)
acting on \(S_{4}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $5$ |
\( (T^{2} - 2 T - 243)^{2} \)
|
| $7$ |
\( T^{4} + 18 T^{3} + \cdots + 400 \)
|
| $11$ |
\( T^{4} - 2 T^{3} + \cdots + 3600 \)
|
| $13$ |
\( T^{4} - 36 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( T^{4} - 36 T^{3} + \cdots + 21316689 \)
|
| $19$ |
\( T^{4} - 22 T^{3} + \cdots + 306530064 \)
|
| $23$ |
\( T^{4} + 14 T^{3} + \cdots + 309056400 \)
|
| $29$ |
\( T^{4} + 146 T^{3} + \cdots + 594441 \)
|
| $31$ |
\( (T^{2} - 492 T + 60272)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 12288387609 \)
|
| $41$ |
\( T^{4} + 240 T^{3} + \cdots + 455625 \)
|
| $43$ |
\( T^{4} + \cdots + 11420769424 \)
|
| $47$ |
\( (T^{2} + 170 T + 7164)^{2} \)
|
| $53$ |
\( (T^{2} + 726 T + 112005)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 8100000000 \)
|
| $61$ |
\( T^{4} + \cdots + 9867442225 \)
|
| $67$ |
\( T^{4} + \cdots + 1371665296 \)
|
| $71$ |
\( T^{4} + \cdots + 17828658576 \)
|
| $73$ |
\( (T^{2} + 530 T - 245999)^{2} \)
|
| $79$ |
\( (T + 316)^{4} \)
|
| $83$ |
\( (T^{2} - 414 T - 880020)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 503730867600 \)
|
| $97$ |
\( T^{4} + \cdots + 957462250000 \)
|
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