Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7098,2,Mod(1,7098)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7098, 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("7098.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7098.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: | \(56.6778153547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6148961.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 11x^{4} + 12x^{3} + 32x^{2} - 16x - 29 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{6} - 2x^{5} - 11x^{4} + 12x^{3} + 32x^{2} - 16x - 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 9\nu^{2} + 7\nu + 15 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 9\nu^{2} + 13\nu - 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} - 3\nu^{3} - 19\nu^{2} + 8\nu + 29 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 12\nu^{3} - 12\nu^{2} + 23\nu + 31 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 3\beta_{4} + \beta_{3} - 4\beta_{2} + 7\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{5} + 15\beta_{4} + 11\beta_{3} - 6\beta_{2} + 16\beta _1 + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( -22\beta_{5} + 48\beta_{4} + 24\beta_{3} - 48\beta_{2} + 73\beta _1 + 77 \)
|
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 |
|
1.00000 | 1.00000 | 1.00000 | −0.937626 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −0.937626 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 1.00000 | 1.00000 | −0.867888 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −0.867888 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 1.00000 | 1.00000 | 0.404061 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 0.404061 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.00000 | 1.00000 | 2.59594 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 2.59594 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.00000 | 1.00000 | 3.86789 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 3.86789 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.00000 | 1.00000 | 3.93763 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | 3.93763 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
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 | 7098.2.a.cu | yes | 6 |
| 13.b | even | 2 | 1 | 7098.2.a.cq | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7098.2.a.cq | ✓ | 6 | 13.b | even | 2 | 1 | |
| 7098.2.a.cu | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7098))\):
|
\( T_{5}^{6} - 9T_{5}^{5} + 21T_{5}^{4} + 9T_{5}^{3} - 49T_{5}^{2} - 15T_{5} + 13 \)
|
|
\( T_{11}^{6} - 10T_{11}^{5} - 9T_{11}^{4} + 296T_{11}^{3} - 342T_{11}^{2} - 1630T_{11} + 937 \)
|
|
\( T_{17}^{6} - 4T_{17}^{5} - 55T_{17}^{4} + 256T_{17}^{3} + 452T_{17}^{2} - 2486T_{17} + 337 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} - 9 T^{5} + \cdots + 13 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} - 10 T^{5} + \cdots + 937 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 337 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots - 1 \)
|
| $23$ |
\( T^{6} - 67 T^{4} + \cdots + 547 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 64 \)
|
| $31$ |
\( T^{6} - 9 T^{5} + \cdots - 5447 \)
|
| $37$ |
\( T^{6} - 9 T^{5} + \cdots - 211 \)
|
| $41$ |
\( T^{6} - 25 T^{5} + \cdots - 111007 \)
|
| $43$ |
\( T^{6} - 19 T^{5} + \cdots + 8728 \)
|
| $47$ |
\( T^{6} - 21 T^{5} + \cdots - 392 \)
|
| $53$ |
\( T^{6} + 4 T^{5} + \cdots + 6728 \)
|
| $59$ |
\( T^{6} - 20 T^{5} + \cdots + 2456 \)
|
| $61$ |
\( T^{6} - 3 T^{5} + \cdots + 7288 \)
|
| $67$ |
\( T^{6} - 24 T^{5} + \cdots + 87688 \)
|
| $71$ |
\( T^{6} - 13 T^{5} + \cdots + 203624 \)
|
| $73$ |
\( T^{6} + 9 T^{5} + \cdots - 14792 \)
|
| $79$ |
\( T^{6} - 28 T^{5} + \cdots - 169000 \)
|
| $83$ |
\( T^{6} - 15 T^{5} + \cdots + 106952 \)
|
| $89$ |
\( T^{6} - 11 T^{5} + \cdots + 1163 \)
|
| $97$ |
\( T^{6} - 2 T^{5} + \cdots - 569192 \)
|
| show more | |
| show less |