Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3249,2,Mod(1,3249)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3249, 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("3249.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3249 = 3^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3249.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: | \(25.9433956167\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.6357609.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 57) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} - \nu^{2} + 14\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{2} - \nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 23\nu^{3} - 3\nu^{2} + 26\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} - 2\nu^{4} + 23\nu^{3} + 21\nu^{2} - 24\nu - 24 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 7\beta_{4} + 9\beta_{3} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{4} + \beta_{3} - 23\beta_{2} + 22\beta _1 + 3 \)
|
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 |
|
−2.37441 | 0 | 3.63780 | −2.82462 | 0 | −2.11513 | −3.88880 | 0 | 6.70680 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.59848 | 0 | 0.555147 | 1.00416 | 0 | 2.56963 | 2.30957 | 0 | −1.60514 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.758254 | 0 | −1.42505 | −3.16171 | 0 | 3.79543 | 2.59706 | 0 | 2.39738 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.842316 | 0 | −1.29050 | −1.70747 | 0 | 3.93033 | −2.77164 | 0 | −1.43823 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.25119 | 0 | −0.434532 | −4.35146 | 0 | −1.79631 | −3.04605 | 0 | −5.44449 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.63764 | 0 | 4.95714 | 2.04110 | 0 | 2.61605 | 7.79987 | 0 | 5.38368 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(19\) | \(1\) |
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 | 3249.2.a.bh | 6 | |
| 3.b | odd | 2 | 1 | 1083.2.a.q | 6 | ||
| 19.b | odd | 2 | 1 | 3249.2.a.bg | 6 | ||
| 19.e | even | 9 | 2 | 171.2.u.e | 12 | ||
| 57.d | even | 2 | 1 | 1083.2.a.p | 6 | ||
| 57.l | odd | 18 | 2 | 57.2.i.b | ✓ | 12 | |
| 228.v | even | 18 | 2 | 912.2.bo.j | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 57.2.i.b | ✓ | 12 | 57.l | odd | 18 | 2 | |
| 171.2.u.e | 12 | 19.e | even | 9 | 2 | ||
| 912.2.bo.j | 12 | 228.v | even | 18 | 2 | ||
| 1083.2.a.p | 6 | 57.d | even | 2 | 1 | ||
| 1083.2.a.q | 6 | 3.b | odd | 2 | 1 | ||
| 3249.2.a.bg | 6 | 19.b | odd | 2 | 1 | ||
| 3249.2.a.bh | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):
|
\( T_{2}^{6} - 9T_{2}^{4} - T_{2}^{3} + 18T_{2}^{2} - 8 \)
|
|
\( T_{5}^{6} + 9T_{5}^{5} + 18T_{5}^{4} - 37T_{5}^{3} - 126T_{5}^{2} + 136 \)
|
|
\( T_{13}^{6} - 3T_{13}^{5} - 12T_{13}^{4} + 39T_{13}^{3} + 18T_{13}^{2} - 108T_{13} + 57 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{4} + \cdots - 8 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 9 T^{5} + \cdots + 136 \)
|
| $7$ |
\( T^{6} - 9 T^{5} + \cdots + 381 \)
|
| $11$ |
\( T^{6} + 9 T^{5} + \cdots - 456 \)
|
| $13$ |
\( T^{6} - 3 T^{5} + \cdots + 57 \)
|
| $17$ |
\( T^{6} + 15 T^{5} + \cdots + 6408 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 1576 \)
|
| $29$ |
\( T^{6} - 15 T^{5} + \cdots - 296 \)
|
| $31$ |
\( T^{6} - 36 T^{4} + \cdots - 431 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 2467 \)
|
| $41$ |
\( T^{6} + 6 T^{5} + \cdots - 296 \)
|
| $43$ |
\( T^{6} - 15 T^{5} + \cdots + 24247 \)
|
| $47$ |
\( T^{6} + 9 T^{5} + \cdots - 216 \)
|
| $53$ |
\( T^{6} - 6 T^{5} + \cdots - 1368 \)
|
| $59$ |
\( T^{6} - 15 T^{5} + \cdots + 1272 \)
|
| $61$ |
\( T^{6} - 3 T^{5} + \cdots + 73 \)
|
| $67$ |
\( T^{6} + 18 T^{5} + \cdots + 35416 \)
|
| $71$ |
\( T^{6} - 15 T^{5} + \cdots + 15528 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 32507 \)
|
| $79$ |
\( T^{6} - 21 T^{5} + \cdots - 67869 \)
|
| $83$ |
\( T^{6} - 3 T^{5} + \cdots - 275336 \)
|
| $89$ |
\( T^{6} - 24 T^{5} + \cdots + 146584 \)
|
| $97$ |
\( T^{6} + 3 T^{5} + \cdots - 359 \)
|
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