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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.5061125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 5x^{4} + 14x^{3} + 9x^{2} - 10x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 5x^{4} + 14x^{3} + 9x^{2} - 10x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + \nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 6\nu^{2} - 2\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 6\nu^{3} - 2\nu^{2} - 6\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + \nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 8\beta_{2} + 9\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} + 6\beta_{4} + 10\beta_{3} + 18\beta_{2} + 31\beta _1 + 28 \)
|
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 |
|
−1.66326 | −1.00000 | 0.766426 | −1.00000 | 1.66326 | 1.11641 | 2.05175 | 1.00000 | 1.66326 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57790 | −1.00000 | 0.489760 | −1.00000 | 1.57790 | 3.72751 | 2.38300 | 1.00000 | 1.57790 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.0483725 | −1.00000 | −1.99766 | −1.00000 | −0.0483725 | −2.92858 | −0.193377 | 1.00000 | −0.0483725 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.46133 | −1.00000 | 0.135477 | −1.00000 | −1.46133 | 3.43745 | −2.72468 | 1.00000 | −1.46133 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.91149 | −1.00000 | 1.65380 | −1.00000 | −1.91149 | −0.0349904 | −0.661762 | 1.00000 | −1.91149 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.81996 | −1.00000 | 5.95220 | −1.00000 | −2.81996 | 0.682206 | 11.1451 | 1.00000 | −2.81996 | |||||||||||||||||||||||||||||||||||||
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.bi | yes | 6 |
| 19.b | odd | 2 | 1 | 5415.2.a.bc | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5415.2.a.bc | ✓ | 6 | 19.b | odd | 2 | 1 | |
| 5415.2.a.bi | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{6} - 3T_{2}^{5} - 5T_{2}^{4} + 16T_{2}^{3} + 6T_{2}^{2} - 21T_{2} + 1 \)
|
|
\( T_{7}^{6} - 6T_{7}^{5} + 49T_{7}^{3} - 72T_{7}^{2} + 26T_{7} + 1 \)
|
|
\( T_{11}^{6} + 4T_{11}^{5} - 17T_{11}^{4} - 35T_{11}^{3} + 88T_{11}^{2} + 49T_{11} - 109 \)
|
|
\( T_{13}^{6} - 8T_{13}^{5} - 15T_{13}^{4} + 224T_{13}^{3} - 341T_{13}^{2} - 280T_{13} + 355 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots + 1 \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots - 109 \)
|
| $13$ |
\( T^{6} - 8 T^{5} + \cdots + 355 \)
|
| $17$ |
\( T^{6} + 11 T^{5} + \cdots - 71 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots + 241 \)
|
| $29$ |
\( T^{6} - 7 T^{5} + \cdots - 281 \)
|
| $31$ |
\( T^{6} + T^{5} + \cdots - 905 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots + 509 \)
|
| $41$ |
\( T^{6} - 19 T^{5} + \cdots + 56995 \)
|
| $43$ |
\( T^{6} - 8 T^{5} + \cdots - 46931 \)
|
| $47$ |
\( T^{6} + 4 T^{5} + \cdots + 551 \)
|
| $53$ |
\( T^{6} - 24 T^{5} + \cdots + 569 \)
|
| $59$ |
\( T^{6} - 25 T^{5} + \cdots + 104479 \)
|
| $61$ |
\( T^{6} - 8 T^{5} + \cdots + 131 \)
|
| $67$ |
\( T^{6} - 17 T^{5} + \cdots - 29761 \)
|
| $71$ |
\( T^{6} + 5 T^{5} + \cdots - 1823219 \)
|
| $73$ |
\( T^{6} - 30 T^{5} + \cdots - 141691 \)
|
| $79$ |
\( T^{6} + 11 T^{5} + \cdots - 591475 \)
|
| $83$ |
\( T^{6} + 35 T^{5} + \cdots - 53695 \)
|
| $89$ |
\( T^{6} - T^{5} + \cdots - 51395 \)
|
| $97$ |
\( T^{6} - 35 T^{5} + \cdots + 6691 \)
|
| show more | |
| show less |