Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6498,2,Mod(1,6498)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6498, 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("6498.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6498.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: | \(51.8867912334\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.53327808.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 18x^{4} + 87x^{2} - 127 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 342) |
| 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
\( x^{6} - 18x^{4} + 87x^{2} - 127 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 15\nu^{2} + 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{5} + 29\nu^{3} - 73\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( 3\nu^{4} - 44\nu^{2} + 114 \)
|
| \(\beta_{5}\) | \(=\) |
\( 3\nu^{5} - 44\nu^{3} + 114\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} - 3\beta_{3} + 9\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 15\beta_{4} - 44\beta_{2} + 50 \)
|
| \(\nu^{5}\) | \(=\) |
\( -29\beta_{5} - 44\beta_{3} + 94\beta_1 \)
|
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 |
|
1.00000 | 0 | 1.00000 | −2.40533 | 0 | 2.81194 | 1.00000 | 0 | −2.40533 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −1.83582 | 0 | −0.850952 | 1.00000 | 0 | −1.83582 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | −0.166981 | 0 | −2.36019 | 1.00000 | 0 | −0.166981 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 2.16698 | 0 | 4.36019 | 1.00000 | 0 | 2.16698 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 3.83582 | 0 | 2.85095 | 1.00000 | 0 | 3.83582 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 0 | 1.00000 | 4.40533 | 0 | −0.811938 | 1.00000 | 0 | 4.40533 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(19\) | \(1\) |
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 | 6498.2.a.ce | 6 | |
| 3.b | odd | 2 | 1 | 6498.2.a.cb | 6 | ||
| 19.b | odd | 2 | 1 | 6498.2.a.cc | 6 | ||
| 19.e | even | 9 | 2 | 342.2.u.f | ✓ | 12 | |
| 57.d | even | 2 | 1 | 6498.2.a.cd | 6 | ||
| 57.l | odd | 18 | 2 | 342.2.u.g | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 342.2.u.f | ✓ | 12 | 19.e | even | 9 | 2 | |
| 342.2.u.g | yes | 12 | 57.l | odd | 18 | 2 | |
| 6498.2.a.cb | 6 | 3.b | odd | 2 | 1 | ||
| 6498.2.a.cc | 6 | 19.b | odd | 2 | 1 | ||
| 6498.2.a.cd | 6 | 57.d | even | 2 | 1 | ||
| 6498.2.a.ce | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6498))\):
|
\( T_{5}^{6} - 6T_{5}^{5} - 6T_{5}^{4} + 64T_{5}^{3} + 9T_{5}^{2} - 162T_{5} - 27 \)
|
|
\( T_{7}^{6} - 6T_{7}^{5} - 3T_{7}^{4} + 52T_{7}^{3} - 6T_{7}^{2} - 108T_{7} - 57 \)
|
|
\( T_{11}^{6} - 6T_{11}^{5} - 42T_{11}^{4} + 334T_{11}^{3} - 45T_{11}^{2} - 3402T_{11} + 5751 \)
|
|
\( T_{13}^{6} - 12T_{13}^{5} + 24T_{13}^{4} + 128T_{13}^{3} - 255T_{13}^{2} - 516T_{13} - 179 \)
|
|
\( T_{29}^{6} + 6T_{29}^{5} - 42T_{29}^{4} - 64T_{29}^{3} + 117T_{29}^{2} - 27 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 6 T^{5} + \cdots - 27 \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots - 57 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots + 5751 \)
|
| $13$ |
\( T^{6} - 12 T^{5} + \cdots - 179 \)
|
| $17$ |
\( T^{6} - 12 T^{5} + \cdots + 1377 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 18 T^{5} + \cdots - 2889 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots - 27 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots - 37 \)
|
| $37$ |
\( T^{6} - 12 T^{5} + \cdots + 15101 \)
|
| $41$ |
\( T^{6} - 75 T^{4} + \cdots - 1431 \)
|
| $43$ |
\( T^{6} + 18 T^{5} + \cdots + 43019 \)
|
| $47$ |
\( T^{6} - 30 T^{5} + \cdots - 8721 \)
|
| $53$ |
\( T^{6} + 18 T^{5} + \cdots + 428193 \)
|
| $59$ |
\( T^{6} + 6 T^{5} + \cdots + 5751 \)
|
| $61$ |
\( T^{6} - 177 T^{4} + \cdots + 54721 \)
|
| $67$ |
\( T^{6} - 18 T^{5} + \cdots + 146799 \)
|
| $71$ |
\( T^{6} - 24 T^{5} + \cdots + 36423 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots + 44013 \)
|
| $79$ |
\( T^{6} + 18 T^{5} + \cdots + 15139 \)
|
| $83$ |
\( T^{6} - 42 T^{5} + \cdots + 13851 \)
|
| $89$ |
\( T^{6} + 18 T^{5} + \cdots + 48357 \)
|
| $97$ |
\( T^{6} - 24 T^{5} + \cdots + 253601 \)
|
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