Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [66,4,Mod(1,66)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 66 = 2 \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 66.a (trivial) |
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: | yes |
| Analytic conductor: | \(3.89412606038\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{97}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \sqrt{97}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | 3.00000 | 4.00000 | −4.84886 | 6.00000 | 28.5466 | 8.00000 | 9.00000 | −9.69772 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | 14.8489 | 6.00000 | −30.5466 | 8.00000 | 9.00000 | 29.6977 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 66.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 198.4.a.h | 2 | ||
| 4.b | odd | 2 | 1 | 528.4.a.n | 2 | ||
| 5.b | even | 2 | 1 | 1650.4.a.s | 2 | ||
| 5.c | odd | 4 | 2 | 1650.4.c.u | 4 | ||
| 8.b | even | 2 | 1 | 2112.4.a.bb | 2 | ||
| 8.d | odd | 2 | 1 | 2112.4.a.bi | 2 | ||
| 11.b | odd | 2 | 1 | 726.4.a.o | 2 | ||
| 12.b | even | 2 | 1 | 1584.4.a.ba | 2 | ||
| 33.d | even | 2 | 1 | 2178.4.a.bf | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 66.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 198.4.a.h | 2 | 3.b | odd | 2 | 1 | ||
| 528.4.a.n | 2 | 4.b | odd | 2 | 1 | ||
| 726.4.a.o | 2 | 11.b | odd | 2 | 1 | ||
| 1584.4.a.ba | 2 | 12.b | even | 2 | 1 | ||
| 1650.4.a.s | 2 | 5.b | even | 2 | 1 | ||
| 1650.4.c.u | 4 | 5.c | odd | 4 | 2 | ||
| 2112.4.a.bb | 2 | 8.b | even | 2 | 1 | ||
| 2112.4.a.bi | 2 | 8.d | odd | 2 | 1 | ||
| 2178.4.a.bf | 2 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 10T_{5} - 72 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(66))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( (T - 3)^{2} \)
|
| $5$ |
\( T^{2} - 10T - 72 \)
|
| $7$ |
\( T^{2} + 2T - 872 \)
|
| $11$ |
\( (T + 11)^{2} \)
|
| $13$ |
\( T^{2} - 14T - 4704 \)
|
| $17$ |
\( T^{2} + 80T + 1212 \)
|
| $19$ |
\( T^{2} + 60T - 8800 \)
|
| $23$ |
\( T^{2} + 202T + 7776 \)
|
| $29$ |
\( T^{2} + 336T + 24732 \)
|
| $31$ |
\( T^{2} - 128T - 20736 \)
|
| $37$ |
\( T^{2} - 188T - 90492 \)
|
| $41$ |
\( T^{2} + 132T - 51516 \)
|
| $43$ |
\( T^{2} - 480T + 51392 \)
|
| $47$ |
\( T^{2} + 390T + 37152 \)
|
| $53$ |
\( T^{2} - 610T - 25800 \)
|
| $59$ |
\( T^{2} - 372T + 3168 \)
|
| $61$ |
\( T^{2} - 1050 T + 270872 \)
|
| $67$ |
\( T^{2} - 408T + 16784 \)
|
| $71$ |
\( T^{2} - 942T + 200016 \)
|
| $73$ |
\( T^{2} - 740T - 260412 \)
|
| $79$ |
\( T^{2} + 1646 T + 606616 \)
|
| $83$ |
\( T^{2} + 352T - 231312 \)
|
| $89$ |
\( T^{2} - 2036 T + 1011492 \)
|
| $97$ |
\( T^{2} - 336T - 447076 \)
|
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