Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9027,2,Mod(1,9027)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9027, 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("9027.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9027 = 3^{2} \cdot 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9027.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: | \(72.0809579046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3x^{9} - 10x^{8} + 34x^{7} + 28x^{6} - 129x^{5} - 3x^{4} + 178x^{3} - 56x^{2} - 56x + 15 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1003) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3x^{9} - 10x^{8} + 34x^{7} + 28x^{6} - 129x^{5} - 3x^{4} + 178x^{3} - 56x^{2} - 56x + 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{9} + 2\nu^{8} + 45\nu^{7} - 13\nu^{6} - 162\nu^{5} + 20\nu^{4} + 188\nu^{3} - 11\nu^{2} - 31\nu + 3 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{9} + \nu^{8} + 26\nu^{7} - 10\nu^{6} - 116\nu^{5} + 31\nu^{4} + 199\nu^{3} - 30\nu^{2} - 96\nu + 5 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} - 4\nu^{8} - 13\nu^{7} + 47\nu^{6} + 58\nu^{5} - 173\nu^{4} - 96\nu^{3} + 197\nu^{2} + 34\nu - 20 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{9} - 5\nu^{8} - 32\nu^{7} + 50\nu^{6} + 111\nu^{5} - 162\nu^{4} - 134\nu^{3} + 178\nu^{2} + 39\nu - 25 ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 6\nu^{9} - 3\nu^{8} - 71\nu^{7} + 30\nu^{6} + 271\nu^{5} - 107\nu^{4} - 352\nu^{3} + 160\nu^{2} + 99\nu - 43 ) / 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 5\nu^{9} - 6\nu^{8} - 58\nu^{7} + 60\nu^{6} + 227\nu^{5} - 200\nu^{4} - 326\nu^{3} + 243\nu^{2} + 107\nu - 51 ) / 7 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 5\nu^{9} - 6\nu^{8} - 58\nu^{7} + 60\nu^{6} + 227\nu^{5} - 200\nu^{4} - 333\nu^{3} + 243\nu^{2} + 135\nu - 51 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} + \beta_{8} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + \beta_{6} - \beta_{4} + 5\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{9} + 7\beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 18\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{9} + 2\beta_{8} + \beta_{7} + 9\beta_{6} - \beta_{5} - 8\beta_{4} + 2\beta_{3} + 23\beta_{2} + 2\beta _1 + 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( -44\beta_{9} + 42\beta_{8} + \beta_{7} + 13\beta_{6} - 11\beta_{5} - 10\beta_{4} + 11\beta_{3} + 85\beta _1 + 9 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 76 \beta_{9} + 25 \beta_{8} + 12 \beta_{7} + 63 \beta_{6} - 14 \beta_{5} - 52 \beta_{4} + 24 \beta_{3} + \cdots + 223 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 269 \beta_{9} + 242 \beta_{8} + 14 \beta_{7} + 113 \beta_{6} - 87 \beta_{5} - 77 \beta_{4} + \cdots + 59 \)
|
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 |
|
−2.30116 | 0 | 3.29536 | 3.47217 | 0 | −4.18349 | −2.98084 | 0 | −7.99004 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.03516 | 0 | 2.14188 | 1.25900 | 0 | 0.391654 | −0.288756 | 0 | −2.56227 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.86166 | 0 | 1.46576 | 1.05085 | 0 | 1.46101 | 0.994564 | 0 | −1.95633 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.26414 | 0 | −0.401948 | 0.0160405 | 0 | −4.37637 | 3.03640 | 0 | −0.0202774 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.755134 | 0 | −1.42977 | −1.80730 | 0 | −1.15845 | 2.58994 | 0 | 1.36475 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.441455 | 0 | −1.80512 | 1.49805 | 0 | −0.804687 | −1.67979 | 0 | 0.661322 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.85341 | 0 | 1.43512 | 4.08879 | 0 | 1.71195 | −1.04695 | 0 | 7.57819 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.86330 | 0 | 1.47190 | −0.335552 | 0 | −1.78229 | −0.984011 | 0 | −0.625236 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.40787 | 0 | 3.79784 | −0.977935 | 0 | −2.76854 | 4.32896 | 0 | −2.35474 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.65122 | 0 | 5.02897 | 3.73588 | 0 | 2.50921 | 8.03048 | 0 | 9.90463 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(17\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 9027.2.a.j | 10 | |
| 3.b | odd | 2 | 1 | 1003.2.a.g | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1003.2.a.g | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 9027.2.a.j | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9027))\):
|
\( T_{2}^{10} - T_{2}^{9} - 17T_{2}^{8} + 12T_{2}^{7} + 108T_{2}^{6} - 44T_{2}^{5} - 308T_{2}^{4} + 37T_{2}^{3} + 357T_{2}^{2} + 45T_{2} - 81 \)
|
|
\( T_{5}^{10} - 12T_{5}^{9} + 46T_{5}^{8} - 30T_{5}^{7} - 182T_{5}^{6} + 321T_{5}^{5} + 40T_{5}^{4} - 332T_{5}^{3} + 89T_{5}^{2} + 61T_{5} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - T^{9} + \cdots - 81 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - 12 T^{9} + \cdots - 1 \)
|
| $7$ |
\( T^{10} + 9 T^{9} + \cdots + 207 \)
|
| $11$ |
\( T^{10} + 12 T^{9} + \cdots - 28013 \)
|
| $13$ |
\( T^{10} + 11 T^{9} + \cdots + 5269 \)
|
| $17$ |
\( (T - 1)^{10} \)
|
| $19$ |
\( T^{10} + 5 T^{9} + \cdots + 113 \)
|
| $23$ |
\( T^{10} - 7 T^{9} + \cdots + 856847 \)
|
| $29$ |
\( T^{10} + 10 T^{9} + \cdots + 2389713 \)
|
| $31$ |
\( T^{10} - 13 T^{9} + \cdots - 18031693 \)
|
| $37$ |
\( T^{10} - 12 T^{9} + \cdots + 1125833 \)
|
| $41$ |
\( T^{10} - 29 T^{9} + \cdots - 7290189 \)
|
| $43$ |
\( T^{10} + 18 T^{9} + \cdots - 1294115 \)
|
| $47$ |
\( T^{10} - 18 T^{9} + \cdots - 96741 \)
|
| $53$ |
\( T^{10} - 250 T^{8} + \cdots + 6717541 \)
|
| $59$ |
\( (T - 1)^{10} \)
|
| $61$ |
\( T^{10} - 8 T^{9} + \cdots - 74895799 \)
|
| $67$ |
\( T^{10} + \cdots - 299435985 \)
|
| $71$ |
\( T^{10} + \cdots - 120838269 \)
|
| $73$ |
\( T^{10} + \cdots + 370174307 \)
|
| $79$ |
\( T^{10} - 3 T^{9} + \cdots - 95891 \)
|
| $83$ |
\( T^{10} - 461 T^{8} + \cdots - 13159551 \)
|
| $89$ |
\( T^{10} - 45 T^{9} + \cdots - 14047979 \)
|
| $97$ |
\( T^{10} + \cdots - 847560813 \)
|
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