Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4592,2,Mod(1,4592)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4592.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4592 = 2^{4} \cdot 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4592.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: | \(36.6673046082\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 13x^{5} + 25x^{4} + 60x^{3} - 29x^{2} - 66x - 19 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2296) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 13x^{5} + 25x^{4} + 60x^{3} - 29x^{2} - 66x - 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 17\nu^{4} + 22\nu^{3} + 76\nu^{2} - 37\nu - 55 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 4\nu^{5} - 8\nu^{4} + 31\nu^{3} + 22\nu^{2} - 49\nu - 16 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} + 12\nu^{5} + 29\nu^{4} - 108\nu^{3} - 96\nu^{2} + 197\nu + 83 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + 11\nu^{5} + 31\nu^{4} - 91\nu^{3} - 118\nu^{2} + 136\nu + 100 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 10\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} - \beta_{5} + 6\beta_{4} + 3\beta_{3} + 12\beta_{2} + 32\beta _1 + 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( 18\beta_{6} - 9\beta_{5} + 29\beta_{4} + 23\beta_{3} + 36\beta_{2} + 129\beta _1 + 78 \)
|
| \(\nu^{6}\) | \(=\) |
\( 65\beta_{6} - 13\beta_{5} + 138\beta_{4} + 85\beta_{3} + 156\beta_{2} + 467\beta _1 + 341 \)
|
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 |
|
0 | −2.86028 | 0 | 2.98665 | 0 | 1.00000 | 0 | 5.18121 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.01742 | 0 | −1.50384 | 0 | 1.00000 | 0 | 1.06999 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.341828 | 0 | 3.73680 | 0 | 1.00000 | 0 | −2.88315 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.41545 | 0 | −1.85652 | 0 | 1.00000 | 0 | −0.996510 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.70941 | 0 | 0.782319 | 0 | 1.00000 | 0 | −0.0779052 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.78862 | 0 | 3.37948 | 0 | 1.00000 | 0 | 4.77639 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.30605 | 0 | −2.52490 | 0 | 1.00000 | 0 | 7.92999 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(41\) | \(1\) |
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 | 4592.2.a.bh | 7 | |
| 4.b | odd | 2 | 1 | 2296.2.a.j | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2296.2.a.j | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 4592.2.a.bh | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4592))\):
|
\( T_{3}^{7} - 4T_{3}^{6} - 10T_{3}^{5} + 50T_{3}^{4} + 5T_{3}^{3} - 147T_{3}^{2} + 80T_{3} + 44 \)
|
|
\( T_{5}^{7} - 5T_{5}^{6} - 11T_{5}^{5} + 66T_{5}^{4} + 46T_{5}^{3} - 256T_{5}^{2} - 120T_{5} + 208 \)
|
|
\( T_{11}^{7} - 37T_{11}^{5} - 46T_{11}^{4} + 214T_{11}^{3} + 220T_{11}^{2} - 256T_{11} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 4 T^{6} + \cdots + 44 \)
|
| $5$ |
\( T^{7} - 5 T^{6} + \cdots + 208 \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} - 37 T^{5} + \cdots + 16 \)
|
| $13$ |
\( T^{7} - 13 T^{6} + \cdots + 5824 \)
|
| $17$ |
\( T^{7} - 9 T^{6} + \cdots - 3976 \)
|
| $19$ |
\( T^{7} - 10 T^{6} + \cdots - 28 \)
|
| $23$ |
\( T^{7} - 4 T^{6} + \cdots + 44 \)
|
| $29$ |
\( T^{7} - 21 T^{6} + \cdots - 14464 \)
|
| $31$ |
\( T^{7} - 11 T^{6} + \cdots + 3136 \)
|
| $37$ |
\( T^{7} - 9 T^{6} + \cdots - 11504 \)
|
| $41$ |
\( (T + 1)^{7} \)
|
| $43$ |
\( T^{7} + T^{6} + \cdots - 2104 \)
|
| $47$ |
\( T^{7} - 5 T^{6} + \cdots + 125584 \)
|
| $53$ |
\( T^{7} - 11 T^{6} + \cdots - 2432 \)
|
| $59$ |
\( T^{7} + 7 T^{6} + \cdots - 1998832 \)
|
| $61$ |
\( T^{7} - 26 T^{6} + \cdots - 601984 \)
|
| $67$ |
\( T^{7} + 9 T^{6} + \cdots - 10096 \)
|
| $71$ |
\( T^{7} + 12 T^{6} + \cdots - 10816 \)
|
| $73$ |
\( T^{7} - 12 T^{6} + \cdots - 252512 \)
|
| $79$ |
\( T^{7} - 18 T^{6} + \cdots - 684032 \)
|
| $83$ |
\( T^{7} + 6 T^{6} + \cdots - 176912 \)
|
| $89$ |
\( T^{7} - 12 T^{6} + \cdots + 19204 \)
|
| $97$ |
\( T^{7} - 26 T^{6} + \cdots + 102772 \)
|
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