Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3549,2,Mod(1,3549)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3549, 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("3549.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3549 = 3 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3549.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: | \(28.3389076774\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.121819537.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 25x^{4} + 55x^{3} + 224x^{2} - 252x - 728 \)
|
| 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
\( x^{6} - 3x^{5} - 25x^{4} + 55x^{3} + 224x^{2} - 252x - 728 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 10\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 17\nu^{2} + 18\nu + 76 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 17\nu^{3} + 39\nu^{2} + 72\nu - 92 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 11\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 4\beta_{3} + 38\beta_{2} + 21\beta _1 + 114 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 12\beta_{4} + 46\beta_{3} + 70\beta_{2} + 139\beta _1 + 214 \)
|
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 |
|
−1.24698 | −1.00000 | −0.445042 | −4.06987 | 1.24698 | 1.00000 | 3.04892 | 1.00000 | 5.07504 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.24698 | −1.00000 | −0.445042 | 3.06987 | 1.24698 | 1.00000 | 3.04892 | 1.00000 | −3.82806 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.445042 | −1.00000 | −1.80194 | −3.55940 | −0.445042 | 1.00000 | −1.69202 | 1.00000 | −1.58408 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.445042 | −1.00000 | −1.80194 | 2.55940 | −0.445042 | 1.00000 | −1.69202 | 1.00000 | 1.13904 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.80194 | −1.00000 | 1.24698 | −3.07801 | −1.80194 | 1.00000 | −1.35690 | 1.00000 | −5.54638 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.80194 | −1.00000 | 1.24698 | 2.07801 | −1.80194 | 1.00000 | −1.35690 | 1.00000 | 3.74444 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 3549.2.a.z | yes | 6 |
| 13.b | even | 2 | 1 | 3549.2.a.y | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3549.2.a.y | ✓ | 6 | 13.b | even | 2 | 1 | |
| 3549.2.a.z | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):
|
\( T_{2}^{3} - T_{2}^{2} - 2T_{2} + 1 \)
|
|
\( T_{5}^{6} + 3T_{5}^{5} - 25T_{5}^{4} - 55T_{5}^{3} + 224T_{5}^{2} + 252T_{5} - 728 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - T^{2} - 2 T + 1)^{2} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 728 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 47 T^{4} + \cdots - 281 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + 9 T^{5} + \cdots - 8 \)
|
| $19$ |
\( T^{6} + 11 T^{5} + \cdots + 64 \)
|
| $23$ |
\( T^{6} + 11 T^{5} + \cdots + 2353 \)
|
| $29$ |
\( T^{6} + 10 T^{5} + \cdots + 12649 \)
|
| $31$ |
\( T^{6} + 7 T^{5} + \cdots + 4472 \)
|
| $37$ |
\( T^{6} - 20 T^{5} + \cdots + 7267 \)
|
| $41$ |
\( T^{6} - 10 T^{5} + \cdots - 1856 \)
|
| $43$ |
\( T^{6} + 9 T^{5} + \cdots - 108352 \)
|
| $47$ |
\( T^{6} + 36 T^{5} + \cdots - 302184 \)
|
| $53$ |
\( T^{6} + 12 T^{5} + \cdots - 39403 \)
|
| $59$ |
\( T^{6} - 41 T^{5} + \cdots + 498568 \)
|
| $61$ |
\( T^{6} + 24 T^{5} + \cdots - 7384 \)
|
| $67$ |
\( T^{6} + 15 T^{5} + \cdots - 39733 \)
|
| $71$ |
\( T^{6} + 13 T^{5} + \cdots + 16457 \)
|
| $73$ |
\( T^{6} + 25 T^{5} + \cdots + 58024 \)
|
| $79$ |
\( T^{6} + 16 T^{5} + \cdots - 91141 \)
|
| $83$ |
\( T^{6} - 2 T^{5} + \cdots - 218792 \)
|
| $89$ |
\( T^{6} + 15 T^{5} + \cdots - 51064 \)
|
| $97$ |
\( T^{6} + 10 T^{5} + \cdots + 11752 \)
|
| show more | |
| show less |