Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,6,Mod(1,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.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: | \(10.1041806482\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.358541904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 111x^{2} + 756 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| 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 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
\( x^{4} - 111x^{2} + 756 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 96\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 56 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 56 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 96\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−10.1838 | 0 | 71.7105 | −4.37743 | 0 | 49.0000 | −404.405 | 0 | 44.5790 | ||||||||||||||||||||||||||||||
| 1.2 | −2.69991 | 0 | −24.7105 | −87.9366 | 0 | 49.0000 | 153.113 | 0 | 237.421 | |||||||||||||||||||||||||||||||
| 1.3 | 2.69991 | 0 | −24.7105 | 87.9366 | 0 | 49.0000 | −153.113 | 0 | 237.421 | |||||||||||||||||||||||||||||||
| 1.4 | 10.1838 | 0 | 71.7105 | 4.37743 | 0 | 49.0000 | 404.405 | 0 | 44.5790 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 63.6.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 63.6.a.h | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1008.6.a.by | 4 | ||
| 7.b | odd | 2 | 1 | 441.6.a.x | 4 | ||
| 12.b | even | 2 | 1 | 1008.6.a.by | 4 | ||
| 21.c | even | 2 | 1 | 441.6.a.x | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.6.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 63.6.a.h | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 441.6.a.x | 4 | 7.b | odd | 2 | 1 | ||
| 441.6.a.x | 4 | 21.c | even | 2 | 1 | ||
| 1008.6.a.by | 4 | 4.b | odd | 2 | 1 | ||
| 1008.6.a.by | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 111T_{2}^{2} + 756 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 111T^{2} + 756 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 7752 T^{2} + 148176 \)
|
| $7$ |
\( (T - 49)^{4} \)
|
| $11$ |
\( T^{4} - 236424 T^{2} + 407299536 \)
|
| $13$ |
\( (T^{2} - 436 T - 547484)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 20712534557904 \)
|
| $19$ |
\( (T^{2} - 3400 T + 2294992)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 27015936275664 \)
|
| $29$ |
\( T^{4} + \cdots + 269206953394176 \)
|
| $31$ |
\( (T^{2} + 2360 T - 3962672)^{2} \)
|
| $37$ |
\( (T^{2} - 3028 T - 131584604)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 1390182638544 \)
|
| $43$ |
\( (T^{2} - 5272 T - 31132016)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 21\!\cdots\!56 \)
|
| $59$ |
\( T^{4} + \cdots + 49\!\cdots\!96 \)
|
| $61$ |
\( (T^{2} - 6652 T - 122814524)^{2} \)
|
| $67$ |
\( (T^{2} - 101752 T + 2566947088)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!64 \)
|
| $73$ |
\( (T^{2} - 8764 T - 2896337276)^{2} \)
|
| $79$ |
\( (T^{2} - 17200 T - 975634112)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 97\!\cdots\!56 \)
|
| $89$ |
\( T^{4} + \cdots + 81\!\cdots\!64 \)
|
| $97$ |
\( (T^{2} - 218428 T + 11811076228)^{2} \)
|
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