Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8649,2,Mod(1,8649)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8649, 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("8649.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8649 = 3^{2} \cdot 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8649.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: | \(69.0626127082\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.1697203125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} - 5x^{6} + 12x^{5} + 9x^{4} - 12x^{3} - 5x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 93) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{7} - 5x^{6} + 12x^{5} + 9x^{4} - 12x^{3} - 5x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} + 4\nu^{4} - 9\nu^{3} - 5\nu^{2} + 3\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 9\nu^{3} + 6\nu^{2} - 5\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{7} + 2\nu^{6} + 8\nu^{5} - 8\nu^{4} - 17\nu^{3} + 4\nu^{2} + 6\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 4\nu^{4} + 17\nu^{3} + 5\nu^{2} - 9\nu - 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 5\nu^{5} + 12\nu^{4} + 9\nu^{3} - 12\nu^{2} - 4\nu + 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} - 6\nu^{6} - 9\nu^{5} + 21\nu^{4} + 14\nu^{3} - 15\nu^{2} - 4\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + 3\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} - \beta_{5} + 3\beta_{4} + 12\beta_{3} + 8\beta_{2} + 21\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} + 8\beta_{6} - 3\beta_{5} + 13\beta_{4} + 39\beta_{3} + 23\beta_{2} + 69\beta _1 + 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 16\beta_{7} + 23\beta_{6} - 13\beta_{5} + 42\beta_{4} + 133\beta_{3} + 77\beta_{2} + 221\beta _1 + 123 \)
|
| \(\nu^{7}\) | \(=\) |
\( 56\beta_{7} + 77\beta_{6} - 42\beta_{5} + 146\beta_{4} + 435\beta_{3} + 244\beta_{2} + 721\beta _1 + 388 \)
|
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 |
|
−2.26278 | 0 | 3.12018 | 0.935927 | 0 | −3.32300 | −2.53473 | 0 | −2.11780 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.773055 | 0 | −1.40239 | 1.75478 | 0 | 1.30487 | 2.63023 | 0 | −1.35654 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0.154929 | 0 | −1.97600 | 3.16206 | 0 | −2.70562 | −0.615997 | 0 | 0.489894 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.331409 | 0 | −1.89017 | −1.92600 | 0 | −1.41388 | −1.28924 | 0 | −0.638292 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.30649 | 0 | −0.293092 | 1.68211 | 0 | 2.45220 | −2.99589 | 0 | 2.19765 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.56400 | 0 | 0.446091 | −1.37281 | 0 | −0.139513 | −2.43031 | 0 | −2.14707 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.18333 | 0 | 2.76694 | −0.544022 | 0 | 0.576411 | 1.67449 | 0 | −1.18778 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.49568 | 0 | 4.22843 | 2.30796 | 0 | −2.75147 | 5.56145 | 0 | 5.75994 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 8649.2.a.bj | 8 | |
| 3.b | odd | 2 | 1 | 2883.2.a.m | 8 | ||
| 31.b | odd | 2 | 1 | 8649.2.a.bi | 8 | ||
| 31.g | even | 15 | 2 | 279.2.y.b | 16 | ||
| 93.c | even | 2 | 1 | 2883.2.a.n | 8 | ||
| 93.o | odd | 30 | 2 | 93.2.m.a | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.2.m.a | ✓ | 16 | 93.o | odd | 30 | 2 | |
| 279.2.y.b | 16 | 31.g | even | 15 | 2 | ||
| 2883.2.a.m | 8 | 3.b | odd | 2 | 1 | ||
| 2883.2.a.n | 8 | 93.c | even | 2 | 1 | ||
| 8649.2.a.bi | 8 | 31.b | odd | 2 | 1 | ||
| 8649.2.a.bj | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8649))\):
|
\( T_{2}^{8} - 5T_{2}^{7} + 2T_{2}^{6} + 25T_{2}^{5} - 41T_{2}^{4} + 5T_{2}^{3} + 23T_{2}^{2} - 10T_{2} + 1 \)
|
|
\( T_{5}^{8} - 6T_{5}^{7} + 4T_{5}^{6} + 33T_{5}^{5} - 51T_{5}^{4} - 39T_{5}^{3} + 91T_{5}^{2} - 3T_{5} - 29 \)
|
|
\( T_{7}^{8} + 6T_{7}^{7} - 51T_{7}^{5} - 51T_{7}^{4} + 99T_{7}^{3} + 90T_{7}^{2} - 54T_{7} - 9 \)
|
|
\( T_{11}^{8} - 38T_{11}^{6} - 15T_{11}^{5} + 354T_{11}^{4} + 75T_{11}^{3} - 752T_{11}^{2} + 75T_{11} + 211 \)
|
|
\( T_{13}^{8} - 12T_{13}^{7} + 36T_{13}^{6} + 39T_{13}^{5} - 246T_{13}^{4} + 27T_{13}^{3} + 324T_{13}^{2} + 81T_{13} - 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5 T^{7} + \cdots + 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 6 T^{7} + \cdots - 29 \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots - 9 \)
|
| $11$ |
\( T^{8} - 38 T^{6} + \cdots + 211 \)
|
| $13$ |
\( T^{8} - 12 T^{7} + \cdots - 9 \)
|
| $17$ |
\( T^{8} + 2 T^{7} + \cdots - 659 \)
|
| $19$ |
\( T^{8} + 8 T^{7} + \cdots + 24721 \)
|
| $23$ |
\( T^{8} + 4 T^{7} + \cdots - 16619 \)
|
| $29$ |
\( T^{8} - 6 T^{7} + \cdots + 51991 \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots + 388051 \)
|
| $41$ |
\( T^{8} - 20 T^{7} + \cdots + 1655401 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots + 9811 \)
|
| $47$ |
\( T^{8} + 6 T^{7} + \cdots + 523801 \)
|
| $53$ |
\( T^{8} - 30 T^{7} + \cdots + 665811 \)
|
| $59$ |
\( T^{8} - 20 T^{7} + \cdots - 12597749 \)
|
| $61$ |
\( T^{8} - 13 T^{7} + \cdots - 452849 \)
|
| $67$ |
\( T^{8} - 32 T^{7} + \cdots - 1287089 \)
|
| $71$ |
\( T^{8} - 17 T^{7} + \cdots - 19949 \)
|
| $73$ |
\( T^{8} + 12 T^{7} + \cdots + 48344841 \)
|
| $79$ |
\( T^{8} - 30 T^{7} + \cdots + 8491221 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots + 6271 \)
|
| $89$ |
\( T^{8} + 6 T^{7} + \cdots - 117659 \)
|
| $97$ |
\( T^{8} + 39 T^{7} + \cdots - 2834199 \)
|
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