Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [52,6,Mod(1,52)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52, 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("52.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 52.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: | \(8.33995863027\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.203961.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 62x + 96 \)
|
| 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 a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 62x + 96 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 3\nu - 42 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} - 3\beta _1 + 81 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −29.1356 | 0 | −78.6019 | 0 | 40.3466 | 0 | 605.881 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 14.4420 | 0 | −17.1549 | 0 | 211.335 | 0 | −34.4294 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 26.6936 | 0 | 103.757 | 0 | −113.682 | 0 | 469.548 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(13\) | \(1\) |
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 | 52.6.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 468.6.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 208.6.a.i | 3 | ||
| 8.b | even | 2 | 1 | 832.6.a.r | 3 | ||
| 8.d | odd | 2 | 1 | 832.6.a.u | 3 | ||
| 13.b | even | 2 | 1 | 676.6.a.d | 3 | ||
| 13.d | odd | 4 | 2 | 676.6.d.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.6.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 208.6.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 468.6.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 676.6.a.d | 3 | 13.b | even | 2 | 1 | ||
| 676.6.d.d | 6 | 13.d | odd | 4 | 2 | ||
| 832.6.a.r | 3 | 8.b | even | 2 | 1 | ||
| 832.6.a.u | 3 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 12T_{3}^{2} - 813T_{3} + 11232 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(52))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 12 T^{2} + \cdots + 11232 \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots - 139906 \)
|
| $7$ |
\( T^{3} - 138 T^{2} + \cdots + 969326 \)
|
| $11$ |
\( T^{3} + 290 T^{2} + \cdots - 25690184 \)
|
| $13$ |
\( (T + 169)^{3} \)
|
| $17$ |
\( T^{3} - 2592 T^{2} + \cdots + 581484906 \)
|
| $19$ |
\( T^{3} + \cdots + 3357970344 \)
|
| $23$ |
\( T^{3} + \cdots + 2219034752 \)
|
| $29$ |
\( T^{3} + \cdots - 8846516296 \)
|
| $31$ |
\( T^{3} + \cdots + 163408337504 \)
|
| $37$ |
\( T^{3} + \cdots + 64282255398 \)
|
| $41$ |
\( T^{3} + \cdots + 449466706944 \)
|
| $43$ |
\( T^{3} + \cdots - 489811424572 \)
|
| $47$ |
\( T^{3} + \cdots + 611415014446 \)
|
| $53$ |
\( T^{3} + \cdots - 2213570693088 \)
|
| $59$ |
\( T^{3} + \cdots + 33695919624536 \)
|
| $61$ |
\( T^{3} + \cdots - 16058441694592 \)
|
| $67$ |
\( T^{3} + \cdots - 54907579876888 \)
|
| $71$ |
\( T^{3} + \cdots + 85505995715626 \)
|
| $73$ |
\( T^{3} + \cdots - 11150608025096 \)
|
| $79$ |
\( T^{3} + \cdots + 2886179269312 \)
|
| $83$ |
\( T^{3} + \cdots - 85948017463296 \)
|
| $89$ |
\( T^{3} + \cdots - 55528884883944 \)
|
| $97$ |
\( T^{3} + \cdots + 621761777741368 \)
|
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