Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,4,Mod(25,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.b (of order \(2\), degree \(1\), 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: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{17})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 9x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 17\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 6\nu^{2} + 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 27 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{2} + 17\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
− | 2.00000i | 3.00000 | −4.00000 | − | 3.36932i | − | 6.00000i | − | 24.7386i | 8.00000i | 9.00000 | −6.73863 | ||||||||||||||||||||||||||
| 25.2 | − | 2.00000i | 3.00000 | −4.00000 | 21.3693i | − | 6.00000i | 24.7386i | 8.00000i | 9.00000 | 42.7386 | |||||||||||||||||||||||||||||
| 25.3 | 2.00000i | 3.00000 | −4.00000 | − | 21.3693i | 6.00000i | − | 24.7386i | − | 8.00000i | 9.00000 | 42.7386 | ||||||||||||||||||||||||||||
| 25.4 | 2.00000i | 3.00000 | −4.00000 | 3.36932i | 6.00000i | 24.7386i | − | 8.00000i | 9.00000 | −6.73863 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.4.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 234.4.b.c | 4 | ||
| 4.b | odd | 2 | 1 | 624.4.c.d | 4 | ||
| 13.b | even | 2 | 1 | inner | 78.4.b.b | ✓ | 4 |
| 13.d | odd | 4 | 1 | 1014.4.a.o | 2 | ||
| 13.d | odd | 4 | 1 | 1014.4.a.q | 2 | ||
| 39.d | odd | 2 | 1 | 234.4.b.c | 4 | ||
| 52.b | odd | 2 | 1 | 624.4.c.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.4.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 78.4.b.b | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 234.4.b.c | 4 | 3.b | odd | 2 | 1 | ||
| 234.4.b.c | 4 | 39.d | odd | 2 | 1 | ||
| 624.4.c.d | 4 | 4.b | odd | 2 | 1 | ||
| 624.4.c.d | 4 | 52.b | odd | 2 | 1 | ||
| 1014.4.a.o | 2 | 13.d | odd | 4 | 1 | ||
| 1014.4.a.q | 2 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 468T_{5}^{2} + 5184 \)
acting on \(S_{4}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 468T^{2} + 5184 \)
|
| $7$ |
\( (T^{2} + 612)^{2} \)
|
| $11$ |
\( T^{4} + 2484 T^{2} + 876096 \)
|
| $13$ |
\( T^{4} - 32 T^{3} + \cdots + 4826809 \)
|
| $17$ |
\( (T^{2} + 48 T - 4932)^{2} \)
|
| $19$ |
\( T^{4} + 2376 T^{2} + 1296 \)
|
| $23$ |
\( (T^{2} - 204 T + 4896)^{2} \)
|
| $29$ |
\( (T - 126)^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 1033493904 \)
|
| $37$ |
\( T^{4} + \cdots + 4349666304 \)
|
| $41$ |
\( T^{4} + \cdots + 1434288384 \)
|
| $43$ |
\( (T^{2} - 472 T - 32432)^{2} \)
|
| $47$ |
\( T^{4} + 17604 T^{2} + 72182016 \)
|
| $53$ |
\( (T^{2} + 120 T - 134100)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 51307686144 \)
|
| $61$ |
\( (T^{2} - 176 T - 262148)^{2} \)
|
| $67$ |
\( T^{4} + 394056 T^{2} + 431475984 \)
|
| $71$ |
\( T^{4} + \cdots + 1935296064 \)
|
| $73$ |
\( T^{4} + \cdots + 30964737024 \)
|
| $79$ |
\( (T^{2} + 1408 T + 143104)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 48667889664 \)
|
| $89$ |
\( T^{4} + \cdots + 3806619515136 \)
|
| $97$ |
\( T^{4} + \cdots + 4693011664896 \)
|
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