Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3721,2,Mod(1,3721)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3721.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3721 = 61^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3721.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: | \(29.7123345921\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.966125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 61) |
| 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
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 4\nu^{2} + 4\nu - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 9\nu^{2} + 4\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 5\beta_{3} + 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 16\beta_1 \)
|
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 |
|
−2.13053 | −1.14867 | 2.53915 | −0.408619 | 2.44727 | −1.76670 | −1.14867 | −1.68056 | 0.870575 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.83468 | 1.16309 | 1.36605 | 0.468627 | −2.13390 | 2.78112 | 1.16309 | −1.64723 | −0.859780 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.340199 | 1.32142 | −1.88426 | 2.22446 | −0.449547 | 0.703389 | 1.32142 | −1.25384 | −0.756760 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.437664 | −1.66682 | −1.80845 | 1.37079 | −0.729509 | −0.0487885 | −1.66682 | −0.221703 | 0.599944 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.852692 | −2.79079 | −1.27292 | 0.420224 | −2.37969 | −3.40882 | −2.79079 | 4.78851 | 0.358321 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.01505 | 0.121769 | 2.06043 | −4.07548 | 0.245370 | 1.73980 | 0.121769 | −2.98517 | −8.21230 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(61\) | \( +1 \) |
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 | 3721.2.a.g | 6 | |
| 61.b | even | 2 | 1 | 3721.2.a.h | 6 | ||
| 61.e | even | 5 | 2 | 61.2.e.a | ✓ | 12 | |
| 183.n | odd | 10 | 2 | 549.2.k.a | 12 | ||
| 244.n | odd | 10 | 2 | 976.2.v.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 61.2.e.a | ✓ | 12 | 61.e | even | 5 | 2 | |
| 549.2.k.a | 12 | 183.n | odd | 10 | 2 | ||
| 976.2.v.a | 12 | 244.n | odd | 10 | 2 | ||
| 3721.2.a.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 3721.2.a.h | 6 | 61.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} - 6T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3721))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 6 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} + 3 T^{5} + \cdots - 1 \)
|
| $5$ |
\( T^{6} - 12 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{6} - 13 T^{4} + \cdots - 1 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 505 \)
|
| $13$ |
\( T^{6} - T^{5} + \cdots - 341 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots - 1181 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots - 1111 \)
|
| $23$ |
\( T^{6} - 14 T^{5} + \cdots + 1051 \)
|
| $29$ |
\( T^{6} + 4 T^{5} + \cdots + 9221 \)
|
| $31$ |
\( T^{6} - 7 T^{5} + \cdots - 2105 \)
|
| $37$ |
\( T^{6} + 22 T^{5} + \cdots - 6781 \)
|
| $41$ |
\( T^{6} + 8 T^{5} + \cdots + 116005 \)
|
| $43$ |
\( T^{6} + 7 T^{5} + \cdots - 1759 \)
|
| $47$ |
\( T^{6} + 11 T^{5} + \cdots + 59 \)
|
| $53$ |
\( T^{6} - 25 T^{5} + \cdots - 135791 \)
|
| $59$ |
\( T^{6} - 11 T^{5} + \cdots - 208769 \)
|
| $61$ |
\( T^{6} \)
|
| $67$ |
\( T^{6} + 7 T^{5} + \cdots + 101 \)
|
| $71$ |
\( T^{6} + 22 T^{5} + \cdots + 29879 \)
|
| $73$ |
\( T^{6} + 17 T^{5} + \cdots + 850025 \)
|
| $79$ |
\( T^{6} - 12 T^{5} + \cdots + 2749 \)
|
| $83$ |
\( T^{6} + 36 T^{5} + \cdots - 77699 \)
|
| $89$ |
\( T^{6} + 21 T^{5} + \cdots - 319 \)
|
| $97$ |
\( T^{6} + 24 T^{5} + \cdots - 16771 \)
|
| show more | |
| show less |