Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2499,2,Mod(1,2499)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2499, 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("2499.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2499.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: | \(19.9546154651\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.29722117.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 9x^{2} - 7x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 357) |
| 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
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 9x^{2} - 7x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 11\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 12\nu^{2} + 2\nu - 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 5\nu^{3} + 18\nu^{2} - 2\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + 7\beta_{4} + 6\beta_{3} + \beta_{2} + 7\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{5} + 3\beta_{4} + 8\beta_{2} + 30\beta _1 + 8 \)
|
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 |
|
−2.32797 | 1.00000 | 3.41944 | 0.803188 | −2.32797 | 0 | −3.30440 | 1.00000 | −1.86980 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.691813 | 1.00000 | −1.52139 | 2.80840 | −0.691813 | 0 | 2.43615 | 1.00000 | −1.94289 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.528483 | 1.00000 | −1.72071 | −1.45297 | −0.528483 | 0 | 1.96633 | 1.00000 | 0.767869 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.866560 | 1.00000 | −1.24907 | −0.673093 | 0.866560 | 0 | −2.81552 | 1.00000 | −0.583275 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.10362 | 1.00000 | 2.42523 | 3.34838 | 2.10362 | 0 | 0.894514 | 1.00000 | 7.04372 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.57808 | 1.00000 | 4.64651 | 2.16610 | 2.57808 | 0 | 6.82293 | 1.00000 | 5.58438 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 2499.2.a.bf | 6 | |
| 3.b | odd | 2 | 1 | 7497.2.a.bx | 6 | ||
| 7.b | odd | 2 | 1 | 2499.2.a.be | 6 | ||
| 7.d | odd | 6 | 2 | 357.2.i.g | ✓ | 12 | |
| 21.c | even | 2 | 1 | 7497.2.a.by | 6 | ||
| 21.g | even | 6 | 2 | 1071.2.i.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 357.2.i.g | ✓ | 12 | 7.d | odd | 6 | 2 | |
| 1071.2.i.h | 12 | 21.g | even | 6 | 2 | ||
| 2499.2.a.be | 6 | 7.b | odd | 2 | 1 | ||
| 2499.2.a.bf | 6 | 1.a | even | 1 | 1 | trivial | |
| 7497.2.a.bx | 6 | 3.b | odd | 2 | 1 | ||
| 7497.2.a.by | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2499))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 9T_{2}^{2} - 7T_{2} - 4 \)
|
|
\( T_{5}^{6} - 7T_{5}^{5} + 11T_{5}^{4} + 15T_{5}^{3} - 37T_{5}^{2} - 3T_{5} + 16 \)
|
|
\( T_{11}^{6} - T_{11}^{5} - 30T_{11}^{4} + 46T_{11}^{3} + 197T_{11}^{2} - 422T_{11} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 4 \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} - 7 T^{5} + \cdots + 16 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - T^{5} + \cdots + 128 \)
|
| $13$ |
\( T^{6} - 4 T^{5} + \cdots - 7 \)
|
| $17$ |
\( (T - 1)^{6} \)
|
| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 1376 \)
|
| $23$ |
\( T^{6} - 13 T^{5} + \cdots - 1016 \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots + 2194 \)
|
| $31$ |
\( T^{6} - 16 T^{5} + \cdots + 22831 \)
|
| $37$ |
\( T^{6} + 5 T^{5} + \cdots - 177484 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots - 13174 \)
|
| $43$ |
\( T^{6} + 7 T^{5} + \cdots + 59296 \)
|
| $47$ |
\( T^{6} - 3 T^{5} + \cdots - 58 \)
|
| $53$ |
\( T^{6} - 3 T^{5} + \cdots + 43192 \)
|
| $59$ |
\( T^{6} - 12 T^{5} + \cdots + 10048 \)
|
| $61$ |
\( T^{6} - 29 T^{5} + \cdots + 784 \)
|
| $67$ |
\( T^{6} + 9 T^{5} + \cdots - 7792 \)
|
| $71$ |
\( T^{6} - 29 T^{5} + \cdots - 16856 \)
|
| $73$ |
\( T^{6} + 2 T^{5} + \cdots + 40834 \)
|
| $79$ |
\( T^{6} + 6 T^{5} + \cdots - 110912 \)
|
| $83$ |
\( T^{6} - 29 T^{5} + \cdots + 22806 \)
|
| $89$ |
\( T^{6} - 12 T^{5} + \cdots - 19616 \)
|
| $97$ |
\( T^{6} - 7 T^{5} + \cdots + 27944 \)
|
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