Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [65,2,Mod(16,65)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.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: | \(0.519027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 7 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} - 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 16.1 |
|
−0.651388 | − | 1.12824i | −0.500000 | − | 0.866025i | 0.151388 | − | 0.262211i | −1.00000 | −0.651388 | + | 1.12824i | 0.500000 | − | 0.866025i | −3.00000 | 1.00000 | − | 1.73205i | 0.651388 | + | 1.12824i | ||||||||||||||||
| 16.2 | 1.15139 | + | 1.99426i | −0.500000 | − | 0.866025i | −1.65139 | + | 2.86029i | −1.00000 | 1.15139 | − | 1.99426i | 0.500000 | − | 0.866025i | −3.00000 | 1.00000 | − | 1.73205i | −1.15139 | − | 1.99426i | |||||||||||||||||
| 61.1 | −0.651388 | + | 1.12824i | −0.500000 | + | 0.866025i | 0.151388 | + | 0.262211i | −1.00000 | −0.651388 | − | 1.12824i | 0.500000 | + | 0.866025i | −3.00000 | 1.00000 | + | 1.73205i | 0.651388 | − | 1.12824i | |||||||||||||||||
| 61.2 | 1.15139 | − | 1.99426i | −0.500000 | + | 0.866025i | −1.65139 | − | 2.86029i | −1.00000 | 1.15139 | + | 1.99426i | 0.500000 | + | 0.866025i | −3.00000 | 1.00000 | + | 1.73205i | −1.15139 | + | 1.99426i | |||||||||||||||||
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 | 65.2.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 585.2.j.d | 4 | ||
| 4.b | odd | 2 | 1 | 1040.2.q.o | 4 | ||
| 5.b | even | 2 | 1 | 325.2.e.a | 4 | ||
| 5.c | odd | 4 | 2 | 325.2.o.b | 8 | ||
| 13.b | even | 2 | 1 | 845.2.e.d | 4 | ||
| 13.c | even | 3 | 1 | inner | 65.2.e.b | ✓ | 4 |
| 13.c | even | 3 | 1 | 845.2.a.c | 2 | ||
| 13.d | odd | 4 | 2 | 845.2.m.d | 8 | ||
| 13.e | even | 6 | 1 | 845.2.a.f | 2 | ||
| 13.e | even | 6 | 1 | 845.2.e.d | 4 | ||
| 13.f | odd | 12 | 2 | 845.2.c.d | 4 | ||
| 13.f | odd | 12 | 2 | 845.2.m.d | 8 | ||
| 39.h | odd | 6 | 1 | 7605.2.a.bb | 2 | ||
| 39.i | odd | 6 | 1 | 585.2.j.d | 4 | ||
| 39.i | odd | 6 | 1 | 7605.2.a.bg | 2 | ||
| 52.j | odd | 6 | 1 | 1040.2.q.o | 4 | ||
| 65.l | even | 6 | 1 | 4225.2.a.t | 2 | ||
| 65.n | even | 6 | 1 | 325.2.e.a | 4 | ||
| 65.n | even | 6 | 1 | 4225.2.a.x | 2 | ||
| 65.q | odd | 12 | 2 | 325.2.o.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 65.2.e.b | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 325.2.e.a | 4 | 5.b | even | 2 | 1 | ||
| 325.2.e.a | 4 | 65.n | even | 6 | 1 | ||
| 325.2.o.b | 8 | 5.c | odd | 4 | 2 | ||
| 325.2.o.b | 8 | 65.q | odd | 12 | 2 | ||
| 585.2.j.d | 4 | 3.b | odd | 2 | 1 | ||
| 585.2.j.d | 4 | 39.i | odd | 6 | 1 | ||
| 845.2.a.c | 2 | 13.c | even | 3 | 1 | ||
| 845.2.a.f | 2 | 13.e | even | 6 | 1 | ||
| 845.2.c.d | 4 | 13.f | odd | 12 | 2 | ||
| 845.2.e.d | 4 | 13.b | even | 2 | 1 | ||
| 845.2.e.d | 4 | 13.e | even | 6 | 1 | ||
| 845.2.m.d | 8 | 13.d | odd | 4 | 2 | ||
| 845.2.m.d | 8 | 13.f | odd | 12 | 2 | ||
| 1040.2.q.o | 4 | 4.b | odd | 2 | 1 | ||
| 1040.2.q.o | 4 | 52.j | odd | 6 | 1 | ||
| 4225.2.a.t | 2 | 65.l | even | 6 | 1 | ||
| 4225.2.a.x | 2 | 65.n | even | 6 | 1 | ||
| 7605.2.a.bb | 2 | 39.h | odd | 6 | 1 | ||
| 7605.2.a.bg | 2 | 39.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \)
|
| $3$ |
\( (T^{2} + T + 1)^{2} \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( (T^{2} - T + 1)^{2} \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 81 \)
|
| $13$ |
\( (T^{2} - 13)^{2} \)
|
| $17$ |
\( T^{4} + 8 T^{3} + \cdots + 9 \)
|
| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 81 \)
|
| $23$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $29$ |
\( T^{4} + 2 T^{3} + \cdots + 2601 \)
|
| $31$ |
\( (T + 4)^{4} \)
|
| $37$ |
\( T^{4} + 13T^{2} + 169 \)
|
| $41$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 1849 \)
|
| $47$ |
\( (T^{2} - 4 T - 48)^{2} \)
|
| $53$ |
\( (T^{2} - 8 T - 36)^{2} \)
|
| $59$ |
\( T^{4} + 117 T^{2} + 13689 \)
|
| $61$ |
\( (T^{2} - T + 1)^{2} \)
|
| $67$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $71$ |
\( T^{4} - 12 T^{3} + \cdots + 6561 \)
|
| $73$ |
\( (T^{2} + 16 T + 12)^{2} \)
|
| $79$ |
\( (T^{2} + 4 T - 48)^{2} \)
|
| $83$ |
\( (T^{2} + 4 T - 48)^{2} \)
|
| $89$ |
\( T^{4} + 2 T^{3} + \cdots + 2601 \)
|
| $97$ |
\( T^{4} - 24 T^{3} + \cdots + 17161 \)
|
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