Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3087,2,Mod(1,3087)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3087, 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("3087.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3087 = 3^{2} \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3087.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: | \(24.6498191040\) |
| 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: |
\( x^{9} - 4x^{8} - 9x^{7} + 35x^{6} + 40x^{5} - 92x^{4} - 88x^{3} + 52x^{2} + 32x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1029) |
| 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
\( x^{9} - 4x^{8} - 9x^{7} + 35x^{6} + 40x^{5} - 92x^{4} - 88x^{3} + 52x^{2} + 32x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} - 5\nu^{7} - 3\nu^{6} + 34\nu^{5} - \nu^{4} - 60\nu^{3} - 10\nu^{2} - 4\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{8} + 11\nu^{7} + 2\nu^{6} - 75\nu^{5} + 29\nu^{4} + 154\nu^{3} - 42\nu^{2} - 64\nu + 12 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{8} - 17\nu^{7} + \nu^{6} + 106\nu^{5} - 59\nu^{4} - 196\nu^{3} + 80\nu^{2} + 64\nu - 20 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{8} - 16\nu^{7} - 5\nu^{6} + 109\nu^{5} - 30\nu^{4} - 218\nu^{3} + 44\nu^{2} + 72\nu - 28 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{8} - 16\nu^{7} - 5\nu^{6} + 109\nu^{5} - 30\nu^{4} - 218\nu^{3} + 40\nu^{2} + 80\nu - 16 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{8} - 21\nu^{7} - 8\nu^{6} + 141\nu^{5} - 19\nu^{4} - 288\nu^{3} - 4\nu^{2} + 108\nu - 4 ) / 4 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{8} + 28\nu^{7} + \nu^{6} - 179\nu^{5} + 80\nu^{4} + 344\nu^{3} - 92\nu^{2} - 116\nu + 24 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{6} + 2\beta_{5} - \beta_{3} + \beta_{2} + 9\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} - 14\beta_{6} + 9\beta_{5} + \beta_{4} - 5\beta_{3} + 3\beta_{2} + 27\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{8} + 4\beta_{7} - 50\beta_{6} + 27\beta_{5} + 6\beta_{4} - 25\beta_{3} + 15\beta_{2} + 99\beta _1 + 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( 31\beta_{8} + 21\beta_{7} - 193\beta_{6} + 99\beta_{5} + 33\beta_{4} - 100\beta_{3} + 54\beta_{2} + 336\beta _1 + 113 \)
|
| \(\nu^{7}\) | \(=\) |
\( 139 \beta_{8} + 85 \beta_{7} - 704 \beta_{6} + 336 \beta_{5} + 147 \beta_{4} - 402 \beta_{3} + \cdots + 300 \)
|
| \(\nu^{8}\) | \(=\) |
\( 585 \beta_{8} + 353 \beta_{7} - 2603 \beta_{6} + 1198 \beta_{5} + 631 \beta_{4} - 1525 \beta_{3} + \cdots + 969 \)
|
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 |
|
−2.72696 | 0 | 5.43631 | 3.08015 | 0 | 0 | −9.37069 | 0 | −8.39943 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.53612 | 0 | 4.43192 | −1.11704 | 0 | 0 | −6.16765 | 0 | 2.83295 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.49320 | 0 | 4.21606 | 0.321703 | 0 | 0 | −5.52510 | 0 | −0.802071 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.67129 | 0 | 0.793217 | −3.38417 | 0 | 0 | 2.01689 | 0 | 5.65594 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.790979 | 0 | −1.37435 | 3.94710 | 0 | 0 | 2.66904 | 0 | −3.12208 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.335177 | 0 | −1.88766 | −1.43430 | 0 | 0 | 1.30305 | 0 | 0.480743 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.15127 | 0 | −0.674571 | −0.942951 | 0 | 0 | −3.07916 | 0 | −1.08559 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.77214 | 0 | 1.14049 | −2.38502 | 0 | 0 | −1.52317 | 0 | −4.22660 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.63032 | 0 | 4.91857 | 2.91453 | 0 | 0 | 7.67677 | 0 | 7.66614 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
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 | 3087.2.a.m | 9 | |
| 3.b | odd | 2 | 1 | 1029.2.a.h | yes | 9 | |
| 7.b | odd | 2 | 1 | 3087.2.a.l | 9 | ||
| 21.c | even | 2 | 1 | 1029.2.a.g | ✓ | 9 | |
| 21.g | even | 6 | 2 | 1029.2.e.f | 18 | ||
| 21.h | odd | 6 | 2 | 1029.2.e.e | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1029.2.a.g | ✓ | 9 | 21.c | even | 2 | 1 | |
| 1029.2.a.h | yes | 9 | 3.b | odd | 2 | 1 | |
| 1029.2.e.e | 18 | 21.h | odd | 6 | 2 | ||
| 1029.2.e.f | 18 | 21.g | even | 6 | 2 | ||
| 3087.2.a.l | 9 | 7.b | odd | 2 | 1 | ||
| 3087.2.a.m | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3087))\):
|
\( T_{2}^{9} + 5T_{2}^{8} - 5T_{2}^{7} - 56T_{2}^{6} - 37T_{2}^{5} + 164T_{2}^{4} + 189T_{2}^{3} - 104T_{2}^{2} - 172T_{2} - 41 \)
|
|
\( T_{5}^{9} - T_{5}^{8} - 27T_{5}^{7} + 8T_{5}^{6} + 240T_{5}^{5} + 109T_{5}^{4} - 713T_{5}^{3} - 847T_{5}^{2} - 92T_{5} + 139 \)
|
|
\( T_{13}^{9} - 11T_{13}^{8} - 10T_{13}^{7} + 423T_{13}^{6} - 792T_{13}^{5} - 2844T_{13}^{4} + 4416T_{13}^{3} + 9296T_{13}^{2} - 3136 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 5 T^{8} + \cdots - 41 \)
|
| $3$ |
\( T^{9} \)
|
| $5$ |
\( T^{9} - T^{8} + \cdots + 139 \)
|
| $7$ |
\( T^{9} \)
|
| $11$ |
\( T^{9} + 21 T^{8} + \cdots - 3569 \)
|
| $13$ |
\( T^{9} - 11 T^{8} + \cdots - 3136 \)
|
| $17$ |
\( T^{9} - 6 T^{8} + \cdots + 56 \)
|
| $19$ |
\( T^{9} + 2 T^{8} + \cdots + 953 \)
|
| $23$ |
\( T^{9} + 9 T^{8} + \cdots - 14063 \)
|
| $29$ |
\( T^{9} + 12 T^{8} + \cdots - 1857472 \)
|
| $31$ |
\( T^{9} + 3 T^{8} + \cdots + 8827 \)
|
| $37$ |
\( T^{9} - 8 T^{8} + \cdots + 3704 \)
|
| $41$ |
\( T^{9} - 17 T^{8} + \cdots + 216881 \)
|
| $43$ |
\( T^{9} - 16 T^{8} + \cdots + 799168 \)
|
| $47$ |
\( T^{9} - 7 T^{8} + \cdots - 1320256 \)
|
| $53$ |
\( T^{9} - 8 T^{8} + \cdots - 2542016 \)
|
| $59$ |
\( T^{9} - 30 T^{8} + \cdots + 18752 \)
|
| $61$ |
\( T^{9} - 7 T^{8} + \cdots - 4126912 \)
|
| $67$ |
\( T^{9} - 33 T^{8} + \cdots + 860672 \)
|
| $71$ |
\( T^{9} + 30 T^{8} + \cdots - 1412957 \)
|
| $73$ |
\( T^{9} - 48 T^{8} + \cdots + 258637504 \)
|
| $79$ |
\( T^{9} - 7 T^{8} + \cdots - 2126144 \)
|
| $83$ |
\( T^{9} - 11 T^{8} + \cdots - 4798528 \)
|
| $89$ |
\( T^{9} - 25 T^{8} + \cdots + 14421673 \)
|
| $97$ |
\( T^{9} - 50 T^{8} + \cdots + 14149184 \)
|
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