Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,3,Mod(77,78)]
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([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.77");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.d (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: | \(2.12534606201\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 4x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 7\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 7\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 77.1 |
|
−1.41421 | −2.00000 | − | 2.23607i | 2.00000 | −4.24264 | 2.82843 | + | 3.16228i | 9.48683i | −2.82843 | −1.00000 | + | 8.94427i | 6.00000 | ||||||||||||||||||||||||
| 77.2 | −1.41421 | −2.00000 | + | 2.23607i | 2.00000 | −4.24264 | 2.82843 | − | 3.16228i | − | 9.48683i | −2.82843 | −1.00000 | − | 8.94427i | 6.00000 | ||||||||||||||||||||||||
| 77.3 | 1.41421 | −2.00000 | − | 2.23607i | 2.00000 | 4.24264 | −2.82843 | − | 3.16228i | − | 9.48683i | 2.82843 | −1.00000 | + | 8.94427i | 6.00000 | ||||||||||||||||||||||||
| 77.4 | 1.41421 | −2.00000 | + | 2.23607i | 2.00000 | 4.24264 | −2.82843 | + | 3.16228i | 9.48683i | 2.82843 | −1.00000 | − | 8.94427i | 6.00000 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.3.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 78.3.d.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 624.3.l.g | 4 | ||
| 12.b | even | 2 | 1 | 624.3.l.g | 4 | ||
| 13.b | even | 2 | 1 | inner | 78.3.d.a | ✓ | 4 |
| 39.d | odd | 2 | 1 | inner | 78.3.d.a | ✓ | 4 |
| 52.b | odd | 2 | 1 | 624.3.l.g | 4 | ||
| 156.h | even | 2 | 1 | 624.3.l.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.3.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 78.3.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 78.3.d.a | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 78.3.d.a | ✓ | 4 | 39.d | odd | 2 | 1 | inner |
| 624.3.l.g | 4 | 4.b | odd | 2 | 1 | ||
| 624.3.l.g | 4 | 12.b | even | 2 | 1 | ||
| 624.3.l.g | 4 | 52.b | odd | 2 | 1 | ||
| 624.3.l.g | 4 | 156.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 18 \)
acting on \(S_{3}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( (T^{2} + 4 T + 9)^{2} \)
|
| $5$ |
\( (T^{2} - 18)^{2} \)
|
| $7$ |
\( (T^{2} + 90)^{2} \)
|
| $11$ |
\( (T^{2} - 72)^{2} \)
|
| $13$ |
\( (T + 13)^{4} \)
|
| $17$ |
\( (T^{2} + 720)^{2} \)
|
| $19$ |
\( (T^{2} + 810)^{2} \)
|
| $23$ |
\( (T^{2} + 180)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} + 90)^{2} \)
|
| $37$ |
\( (T^{2} + 1440)^{2} \)
|
| $41$ |
\( (T^{2} - 3042)^{2} \)
|
| $43$ |
\( (T + 40)^{4} \)
|
| $47$ |
\( (T^{2} - 648)^{2} \)
|
| $53$ |
\( (T^{2} + 720)^{2} \)
|
| $59$ |
\( (T^{2} - 72)^{2} \)
|
| $61$ |
\( (T - 40)^{4} \)
|
| $67$ |
\( (T^{2} + 15210)^{2} \)
|
| $71$ |
\( (T^{2} - 8712)^{2} \)
|
| $73$ |
\( (T^{2} + 5760)^{2} \)
|
| $79$ |
\( (T - 40)^{4} \)
|
| $83$ |
\( (T^{2} - 648)^{2} \)
|
| $89$ |
\( (T^{2} - 13122)^{2} \)
|
| $97$ |
\( T^{4} \)
|
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