Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [473,2,Mod(1,473)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(473, 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("473.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 473 = 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 473.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: | \(3.77692401561\) |
| 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: |
\( x^{9} - 4x^{8} - 5x^{7} + 36x^{6} - 20x^{5} - 65x^{4} + 66x^{3} + 4x^{2} - 8x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{9} - 4x^{8} - 5x^{7} + 36x^{6} - 20x^{5} - 65x^{4} + 66x^{3} + 4x^{2} - 8x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 18\nu^{4} + 16\nu^{3} - 32\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 9\nu^{5} + 18\nu^{4} + 16\nu^{3} - 33\nu^{2} + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{8} - 3\nu^{7} - 7\nu^{6} + 27\nu^{5} - 2\nu^{4} - 49\nu^{3} + 33\nu^{2} + 4\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{8} - 4\nu^{7} - 5\nu^{6} + 37\nu^{5} - 20\nu^{4} - 74\nu^{3} + 66\nu^{2} + 21\nu - 6 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{8} - 6\nu^{7} - \nu^{6} + 55\nu^{5} - 56\nu^{4} - 107\nu^{3} + 131\nu^{2} + 26\nu - 11 \)
|
| \(\beta_{7}\) | \(=\) |
\( -2\nu^{8} + 8\nu^{7} + 11\nu^{6} - 74\nu^{5} + 31\nu^{4} + 148\nu^{3} - 115\nu^{2} - 40\nu + 11 \)
|
| \(\beta_{8}\) | \(=\) |
\( 5\nu^{8} - 18\nu^{7} - 31\nu^{6} + 165\nu^{5} - 45\nu^{4} - 320\nu^{3} + 224\nu^{2} + 69\nu - 21 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + \beta_{5} - 2\beta_{4} - 6\beta_{3} + 6\beta_{2} - 2\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{6} + 10\beta_{5} - \beta_{4} - 8\beta_{3} - 9\beta_{2} + 28\beta _1 - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{8} + 19\beta_{7} + 11\beta_{5} - 18\beta_{4} - 37\beta_{3} + 37\beta_{2} - 20\beta _1 + 85 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{7} - 65\beta_{6} + 78\beta_{5} - 9\beta_{4} - 54\beta_{3} - 66\beta_{2} + 168\beta _1 - 23 \)
|
| \(\nu^{8}\) | \(=\) |
\( 65\beta_{8} + 143\beta_{7} - \beta_{6} + 92\beta_{5} - 129\beta_{4} - 233\beta_{3} + 234\beta_{2} - 155\beta _1 + 515 \)
|
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 |
|
−2.55763 | 1.30520 | 4.54148 | −2.39446 | −3.33822 | 2.20064 | −6.50016 | −1.29645 | 6.12414 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.54520 | 1.46462 | 0.387644 | 3.07032 | −2.26313 | 4.74509 | 2.49141 | −0.854893 | −4.74426 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.376792 | 2.53736 | −1.85803 | 0.306222 | −0.956057 | −1.03665 | 1.45367 | 3.43820 | −0.115382 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.182211 | −1.46902 | −1.96680 | 2.94405 | −0.267670 | 1.19142 | −0.722793 | −0.841992 | 0.536438 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.241366 | −0.787232 | −1.94174 | −4.17979 | −0.190011 | 5.22604 | −0.951402 | −2.38027 | −1.00886 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.54975 | 1.78787 | 0.401739 | 1.27940 | 2.77077 | 3.41428 | −2.47691 | 0.196492 | 1.98275 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.78034 | 2.99541 | 1.16961 | −2.25125 | 5.33285 | 1.73713 | −1.47837 | 5.97247 | −4.00799 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.14073 | −2.78642 | 2.58273 | −0.146187 | −5.96499 | 2.76134 | 1.24747 | 4.76416 | −0.312948 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.58522 | −0.0477867 | 4.68336 | 1.37168 | −0.123539 | −1.23928 | 6.93708 | −2.99772 | 3.54611 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(43\) | \(1\) |
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 | 473.2.a.f | ✓ | 9 |
| 3.b | odd | 2 | 1 | 4257.2.a.r | 9 | ||
| 4.b | odd | 2 | 1 | 7568.2.a.bm | 9 | ||
| 11.b | odd | 2 | 1 | 5203.2.a.j | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 473.2.a.f | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 4257.2.a.r | 9 | 3.b | odd | 2 | 1 | ||
| 5203.2.a.j | 9 | 11.b | odd | 2 | 1 | ||
| 7568.2.a.bm | 9 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(473))\):
|
\( T_{2}^{9} - 4T_{2}^{8} - 5T_{2}^{7} + 36T_{2}^{6} - 20T_{2}^{5} - 65T_{2}^{4} + 66T_{2}^{3} + 4T_{2}^{2} - 8T_{2} + 1 \)
|
|
\( T_{3}^{9} - 5T_{3}^{8} - 4T_{3}^{7} + 52T_{3}^{6} - 47T_{3}^{5} - 108T_{3}^{4} + 148T_{3}^{3} + 43T_{3}^{2} - 82T_{3} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 4 T^{8} + \cdots + 1 \)
|
| $3$ |
\( T^{9} - 5 T^{8} + \cdots - 4 \)
|
| $5$ |
\( T^{9} - 25 T^{7} + \cdots - 16 \)
|
| $7$ |
\( T^{9} - 19 T^{8} + \cdots - 1368 \)
|
| $11$ |
\( (T - 1)^{9} \)
|
| $13$ |
\( T^{9} - 11 T^{8} + \cdots + 992 \)
|
| $17$ |
\( T^{9} - 3 T^{8} + \cdots - 27264 \)
|
| $19$ |
\( T^{9} - 17 T^{8} + \cdots + 9728 \)
|
| $23$ |
\( T^{9} - 72 T^{7} + \cdots - 1756 \)
|
| $29$ |
\( T^{9} - 18 T^{8} + \cdots - 51699 \)
|
| $31$ |
\( T^{9} - 14 T^{8} + \cdots + 93116 \)
|
| $37$ |
\( T^{9} - 15 T^{8} + \cdots - 864 \)
|
| $41$ |
\( T^{9} - 4 T^{8} + \cdots - 10800 \)
|
| $43$ |
\( (T + 1)^{9} \)
|
| $47$ |
\( T^{9} + 9 T^{8} + \cdots + 39416 \)
|
| $53$ |
\( T^{9} + 13 T^{8} + \cdots - 13039511 \)
|
| $59$ |
\( T^{9} + 13 T^{8} + \cdots - 901608 \)
|
| $61$ |
\( T^{9} - 16 T^{8} + \cdots + 44501 \)
|
| $67$ |
\( T^{9} - 3 T^{8} + \cdots + 4476 \)
|
| $71$ |
\( T^{9} - T^{8} + \cdots + 7012928 \)
|
| $73$ |
\( T^{9} - 32 T^{8} + \cdots - 29607124 \)
|
| $79$ |
\( T^{9} - 4 T^{8} + \cdots - 862948 \)
|
| $83$ |
\( T^{9} - 31 T^{8} + \cdots - 5611728 \)
|
| $89$ |
\( T^{9} + 24 T^{8} + \cdots - 55966544 \)
|
| $97$ |
\( T^{9} - 2 T^{8} + \cdots - 6371003 \)
|
| show more | |
| show less |