Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8001,2,Mod(1,8001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, 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("8001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8001 = 3^{2} \cdot 7 \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8001.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: | \(63.8883066572\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.118870813.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2667) |
| 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
\( x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} - 4\nu^{2} + 7\nu + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} - 3\nu^{2} + 11\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} + 8\nu^{4} + 3\nu^{3} - 12\nu^{2} - 4\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} + 2\nu^{5} - 8\nu^{4} - 18\nu^{3} + 6\nu^{2} + 22\nu + 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} + \nu^{5} - 15\nu^{4} - 14\nu^{3} + 15\nu^{2} + 16\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 2\beta_{3} + 4\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 6\beta_{5} + 8\beta_{4} - 7\beta_{3} - 6\beta_{2} + 3\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{5} + 11\beta_{4} - 18\beta_{3} - 3\beta_{2} + 21\beta _1 + 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} + 39\beta_{5} + 54\beta_{4} - 50\beta_{3} - 36\beta_{2} + 32\beta _1 + 92 \)
|
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 |
|
−2.09968 | 0 | 2.40865 | 3.29054 | 0 | 1.00000 | −0.858029 | 0 | −6.90907 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.840819 | 0 | −1.29302 | −2.74724 | 0 | 1.00000 | 2.76884 | 0 | 2.30993 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.692358 | 0 | −1.52064 | 2.78145 | 0 | 1.00000 | 2.43754 | 0 | −1.92576 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.246202 | 0 | −1.93938 | 0.318209 | 0 | 1.00000 | 0.969884 | 0 | −0.0783436 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.24280 | 0 | −0.455452 | 3.33400 | 0 | 1.00000 | −3.05163 | 0 | 4.14349 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.15625 | 0 | 2.64943 | 0.239094 | 0 | 1.00000 | 1.40032 | 0 | 0.515547 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.48001 | 0 | 4.15043 | 0.783950 | 0 | 1.00000 | 5.33307 | 0 | 1.94420 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(127\) | \(1\) |
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 | 8001.2.a.l | 7 | |
| 3.b | odd | 2 | 1 | 2667.2.a.j | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2667.2.a.j | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 8001.2.a.l | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
|
\( T_{2}^{7} - 2T_{2}^{6} - 7T_{2}^{5} + 11T_{2}^{4} + 14T_{2}^{3} - 9T_{2}^{2} - 11T_{2} - 2 \)
|
|
\( T_{5}^{7} - 8T_{5}^{6} + 13T_{5}^{5} + 42T_{5}^{4} - 149T_{5}^{3} + 138T_{5}^{2} - 46T_{5} + 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 2 T^{6} + \cdots - 2 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} - 8 T^{6} + \cdots + 5 \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} - 3 T^{6} + \cdots + 1130 \)
|
| $13$ |
\( T^{7} + 23 T^{6} + \cdots - 5519 \)
|
| $17$ |
\( T^{7} + 3 T^{6} + \cdots - 3898 \)
|
| $19$ |
\( T^{7} + 9 T^{6} + \cdots + 248 \)
|
| $23$ |
\( T^{7} + 12 T^{6} + \cdots + 31373 \)
|
| $29$ |
\( T^{7} - 9 T^{6} + \cdots + 53173 \)
|
| $31$ |
\( T^{7} + 33 T^{6} + \cdots - 15973 \)
|
| $37$ |
\( T^{7} + 33 T^{6} + \cdots - 1549 \)
|
| $41$ |
\( T^{7} - 3 T^{6} + \cdots - 5198 \)
|
| $43$ |
\( T^{7} + 9 T^{6} + \cdots - 52792 \)
|
| $47$ |
\( T^{7} + 11 T^{6} + \cdots - 1279202 \)
|
| $53$ |
\( T^{7} + T^{6} + \cdots - 24569 \)
|
| $59$ |
\( T^{7} - 30 T^{6} + \cdots + 108929 \)
|
| $61$ |
\( T^{7} + 19 T^{6} + \cdots - 37165 \)
|
| $67$ |
\( T^{7} + 30 T^{6} + \cdots - 386 \)
|
| $71$ |
\( T^{7} + 8 T^{6} + \cdots - 981910 \)
|
| $73$ |
\( T^{7} + 20 T^{6} + \cdots + 65465 \)
|
| $79$ |
\( T^{7} - 8 T^{6} + \cdots - 546464 \)
|
| $83$ |
\( T^{7} - 34 T^{6} + \cdots - 6353 \)
|
| $89$ |
\( T^{7} - 12 T^{6} + \cdots - 9440161 \)
|
| $97$ |
\( T^{7} - 7 T^{6} + \cdots + 674 \)
|
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