Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3703,2,Mod(1,3703)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3703, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("3703.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3703 = 7 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3703.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: | \(29.5686038685\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.220036.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - 2x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 5x^{3} + 11x^{2} - 2x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 6\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 6\nu^{2} - 6\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} + 6\beta_{3} + \beta_{2} - \beta _1 + 15 \)
|
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 |
|
−2.35468 | 1.78601 | 3.54450 | 3.23299 | −4.20547 | 1.00000 | −3.63680 | 0.189826 | −7.61264 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.34574 | −0.682113 | −0.188981 | −3.09319 | 0.917947 | 1.00000 | 2.94580 | −2.53472 | 4.16264 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.0996987 | 0.331117 | −1.99006 | 1.61928 | 0.0330120 | 1.00000 | −0.397804 | −2.89036 | 0.161440 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.52472 | −1.96202 | 0.324784 | −2.25728 | −2.99153 | 1.00000 | −2.55424 | 0.849509 | −3.44173 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.07599 | 2.52700 | 2.30975 | −3.50179 | 5.24605 | 1.00000 | 0.643048 | 3.38575 | −7.26970 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 3703.2.a.h | ✓ | 5 |
| 23.b | odd | 2 | 1 | 3703.2.a.i | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3703.2.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 3703.2.a.i | yes | 5 | 23.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3703))\):
|
\( T_{2}^{5} - 7T_{2}^{3} + T_{2}^{2} + 10T_{2} - 1 \)
|
|
\( T_{5}^{5} + 4T_{5}^{4} - 12T_{5}^{3} - 54T_{5}^{2} + 16T_{5} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 7 T^{3} + \cdots - 1 \)
|
| $3$ |
\( T^{5} - 2 T^{4} + \cdots - 2 \)
|
| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 128 \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} - 32 T^{3} + \cdots + 256 \)
|
| $13$ |
\( T^{5} + 8 T^{4} + \cdots - 184 \)
|
| $17$ |
\( T^{5} + 14 T^{4} + \cdots - 5248 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 1024 \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} - 4 T^{4} + \cdots - 5812 \)
|
| $31$ |
\( T^{5} + 4 T^{4} + \cdots - 3082 \)
|
| $37$ |
\( T^{5} - 128 T^{3} + \cdots + 1024 \)
|
| $41$ |
\( T^{5} - 4 T^{4} + \cdots - 8 \)
|
| $43$ |
\( T^{5} + 8 T^{4} + \cdots + 4096 \)
|
| $47$ |
\( T^{5} - 8 T^{4} + \cdots - 3998 \)
|
| $53$ |
\( T^{5} + 12 T^{4} + \cdots - 2048 \)
|
| $59$ |
\( T^{5} + 10 T^{4} + \cdots + 608 \)
|
| $61$ |
\( T^{5} - 32 T^{4} + \cdots - 2048 \)
|
| $67$ |
\( T^{5} - 12 T^{4} + \cdots + 13568 \)
|
| $71$ |
\( T^{5} + 28 T^{4} + \cdots - 17504 \)
|
| $73$ |
\( T^{5} - 247 T^{3} + \cdots - 5992 \)
|
| $79$ |
\( T^{5} + 8 T^{4} + \cdots + 11008 \)
|
| $83$ |
\( T^{5} - 12 T^{4} + \cdots - 7168 \)
|
| $89$ |
\( T^{5} + 18 T^{4} + \cdots + 160256 \)
|
| $97$ |
\( T^{5} + 32 T^{4} + \cdots - 74368 \)
|
| show more | |
| show less |