Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5415,2,Mod(1,5415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.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: | \(43.2389926945\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1397493.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 285) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 2\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - \nu^{2} + 6\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 9\nu^{2} + 4\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 12\beta_{3} + 18\beta_{2} + 20\beta _1 + 18 \)
|
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 |
|
−2.40162 | 1.00000 | 3.76778 | 1.00000 | −2.40162 | −0.213698 | −4.24555 | 1.00000 | −2.40162 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.11662 | 1.00000 | 2.48009 | 1.00000 | −2.11662 | 0.335802 | −1.01617 | 1.00000 | −2.11662 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.801527 | 1.00000 | −1.35755 | 1.00000 | −0.801527 | −0.782248 | 2.69117 | 1.00000 | −0.801527 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.415466 | 1.00000 | −1.82739 | 1.00000 | −0.415466 | −4.56248 | 1.59015 | 1.00000 | −0.415466 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.05432 | 1.00000 | −0.888399 | 1.00000 | 1.05432 | 1.62517 | −3.04531 | 1.00000 | 1.05432 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.68091 | 1.00000 | 0.825466 | 1.00000 | 1.68091 | −2.40254 | −1.97429 | 1.00000 | 1.68091 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 5415.2.a.bb | 6 | |
| 19.b | odd | 2 | 1 | 5415.2.a.bh | 6 | ||
| 19.e | even | 9 | 2 | 285.2.u.a | ✓ | 12 | |
| 57.l | odd | 18 | 2 | 855.2.bs.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.2.u.a | ✓ | 12 | 19.e | even | 9 | 2 | |
| 855.2.bs.a | 12 | 57.l | odd | 18 | 2 | ||
| 5415.2.a.bb | 6 | 1.a | even | 1 | 1 | trivial | |
| 5415.2.a.bh | 6 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{6} + 3T_{2}^{5} - 3T_{2}^{4} - 12T_{2}^{3} + 9T_{2} + 3 \)
|
|
\( T_{7}^{6} + 6T_{7}^{5} + 3T_{7}^{4} - 19T_{7}^{3} - 12T_{7}^{2} + 3T_{7} + 1 \)
|
|
\( T_{11}^{6} + 12T_{11}^{5} + 51T_{11}^{4} + 87T_{11}^{3} + 36T_{11}^{2} - 27T_{11} + 3 \)
|
|
\( T_{13}^{6} - 3T_{13}^{5} - 15T_{13}^{4} + 44T_{13}^{3} - 3T_{13}^{2} - 6T_{13} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3 T^{5} + \cdots + 3 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{6} + 12 T^{5} + \cdots + 3 \)
|
| $13$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $17$ |
\( (T^{3} + 9 T^{2} + 18 T + 9)^{2} \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 267 \)
|
| $29$ |
\( T^{6} + 9 T^{5} + \cdots + 999 \)
|
| $31$ |
\( T^{6} - 54 T^{4} + \cdots - 683 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 5833 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 4647 \)
|
| $43$ |
\( T^{6} - 132 T^{4} + \cdots + 757 \)
|
| $47$ |
\( T^{6} + 9 T^{5} + \cdots + 2109 \)
|
| $53$ |
\( T^{6} + 6 T^{5} + \cdots + 4539 \)
|
| $59$ |
\( T^{6} + 12 T^{5} + \cdots - 54321 \)
|
| $61$ |
\( T^{6} + 33 T^{5} + \cdots + 17317 \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots - 25721 \)
|
| $71$ |
\( T^{6} + 6 T^{5} + \cdots - 6471 \)
|
| $73$ |
\( T^{6} + 21 T^{5} + \cdots - 7361 \)
|
| $79$ |
\( T^{6} - 6 T^{5} + \cdots - 97829 \)
|
| $83$ |
\( T^{6} - 3 T^{5} + \cdots + 52599 \)
|
| $89$ |
\( T^{6} - 6 T^{5} + \cdots - 44277 \)
|
| $97$ |
\( T^{6} - 12 T^{5} + \cdots - 214073 \)
|
| show more | |
| show less |