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: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 11x^{7} + 8x^{6} + 37x^{5} - 18x^{4} - 41x^{3} + 12x^{2} + 6x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 11x^{7} + 8x^{6} + 37x^{5} - 18x^{4} - 41x^{3} + 12x^{2} + 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 8\nu^{5} + 5\nu^{4} + 13\nu^{3} - 3\nu^{2} + 2\nu - 1 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{8} - \nu^{7} - 10\nu^{6} + 7\nu^{5} + 27\nu^{4} - 11\nu^{3} - 14\nu^{2} + \nu - 8 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 15\nu^{5} - 22\nu^{4} + 24\nu^{3} + 15\nu^{2} - 3\nu - 1 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 10\nu^{5} + 7\nu^{4} + 27\nu^{3} - 11\nu^{2} - 18\nu + 1 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} - 13\nu^{6} - \nu^{5} + 52\nu^{4} + 4\nu^{3} - 65\nu^{2} - 3\nu + 9 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{8} - 2\nu^{7} - 21\nu^{6} + 14\nu^{5} + 66\nu^{4} - 21\nu^{3} - 68\nu^{2} - \nu + 9 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 2\nu^{8} - 2\nu^{7} - 21\nu^{6} + 15\nu^{5} + 65\nu^{4} - 30\nu^{3} - 62\nu^{2} + 17\nu + 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 8\beta_{4} + 5\beta_{3} - 7\beta_{2} + 8\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{8} + 8\beta_{7} - \beta_{6} - 9\beta_{5} + 11\beta_{4} + 8\beta_{3} - \beta_{2} + 29\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{8} + 11\beta_{7} - 12\beta_{6} - \beta_{5} + 57\beta_{4} + 29\beta_{3} - 44\beta_{2} + 58\beta _1 + 41 \)
|
| \(\nu^{7}\) | \(=\) |
\( -44\beta_{8} + 57\beta_{7} - 15\beta_{6} - 60\beta_{5} + 95\beta_{4} + 58\beta_{3} - 16\beta_{2} + 186\beta _1 + 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 16 \beta_{8} + 95 \beta_{7} - 101 \beta_{6} - 18 \beta_{5} + 397 \beta_{4} + 186 \beta_{3} + \cdots + 239 \)
|
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 |
|
−2.31147 | −1.00000 | 3.34288 | −1.46052 | 2.31147 | −1.00000 | −3.10401 | 1.00000 | 3.37595 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.60159 | −1.00000 | 0.565099 | −2.27787 | 1.60159 | −1.00000 | 2.29813 | 1.00000 | 3.64823 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.36450 | −1.00000 | −0.138150 | −3.32059 | 1.36450 | −1.00000 | 2.91750 | 1.00000 | 4.53093 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.374817 | −1.00000 | −1.85951 | 1.32690 | 0.374817 | −1.00000 | 1.44661 | 1.00000 | −0.497345 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.146218 | −1.00000 | −1.97862 | 1.04369 | −0.146218 | −1.00000 | −0.581745 | 1.00000 | 0.152606 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.494186 | −1.00000 | −1.75578 | 1.64627 | −0.494186 | −1.00000 | −1.85605 | 1.00000 | 0.813565 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.43930 | −1.00000 | 0.0715948 | −2.86638 | −1.43930 | −1.00000 | −2.77556 | 1.00000 | −4.12559 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.90042 | −1.00000 | 1.61158 | −1.76582 | −1.90042 | −1.00000 | −0.738153 | 1.00000 | −3.35579 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.67225 | −1.00000 | 5.14091 | −1.32568 | −2.67225 | −1.00000 | 8.39329 | 1.00000 | −3.54256 | ||||||||||||||||||||||||||||||||||||||||||||||
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.bf | yes | 9 |
| 13.b | even | 2 | 1 | 3549.2.a.be | ✓ | 9 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3549.2.a.be | ✓ | 9 | 13.b | even | 2 | 1 | |
| 3549.2.a.bf | yes | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):
|
\( T_{2}^{9} - T_{2}^{8} - 11T_{2}^{7} + 8T_{2}^{6} + 37T_{2}^{5} - 18T_{2}^{4} - 41T_{2}^{3} + 12T_{2}^{2} + 6T_{2} - 1 \)
|
|
\( T_{5}^{9} + 9T_{5}^{8} + 22T_{5}^{7} - 20T_{5}^{6} - 141T_{5}^{5} - 83T_{5}^{4} + 249T_{5}^{3} + 255T_{5}^{2} - 131T_{5} - 169 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - T^{8} - 11 T^{7} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{9} \)
|
| $5$ |
\( T^{9} + 9 T^{8} + \cdots - 169 \)
|
| $7$ |
\( (T + 1)^{9} \)
|
| $11$ |
\( T^{9} - T^{8} + \cdots + 29 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( T^{9} - 11 T^{8} + \cdots - 2633 \)
|
| $19$ |
\( T^{9} + 7 T^{8} + \cdots + 1861 \)
|
| $23$ |
\( T^{9} - 22 T^{8} + \cdots - 1861 \)
|
| $29$ |
\( T^{9} - 11 T^{8} + \cdots + 31424 \)
|
| $31$ |
\( T^{9} + 7 T^{8} + \cdots + 8471 \)
|
| $37$ |
\( T^{9} - T^{8} + \cdots + 136487 \)
|
| $41$ |
\( T^{9} + 16 T^{8} + \cdots - 45907 \)
|
| $43$ |
\( T^{9} - 32 T^{8} + \cdots - 4296704 \)
|
| $47$ |
\( T^{9} - 12 T^{8} + \cdots - 8667968 \)
|
| $53$ |
\( T^{9} - 13 T^{8} + \cdots - 455104 \)
|
| $59$ |
\( T^{9} + 29 T^{8} + \cdots + 3519424 \)
|
| $61$ |
\( T^{9} + 12 T^{8} + \cdots - 33721792 \)
|
| $67$ |
\( T^{9} - 20 T^{8} + \cdots - 13046272 \)
|
| $71$ |
\( T^{9} - 2 T^{8} + \cdots + 80941127 \)
|
| $73$ |
\( T^{9} + T^{8} + \cdots + 75156992 \)
|
| $79$ |
\( T^{9} - 3 T^{8} + \cdots - 29684416 \)
|
| $83$ |
\( T^{9} + 24 T^{8} + \cdots - 53248 \)
|
| $89$ |
\( T^{9} + 11 T^{8} + \cdots - 296717 \)
|
| $97$ |
\( T^{9} + 20 T^{8} + \cdots + 231232 \)
|
| show more | |
| show less |