Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8575,2,Mod(1,8575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8575, 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("8575.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8575 = 5^{2} \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8575.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: | \(68.4717197332\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.1279733.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 343) |
| 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
\( x^{6} - 2x^{5} - 6x^{4} + 10x^{3} + 10x^{2} - 11x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 3\nu^{2} + 5\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} + 7\beta_{2} + 18\beta _1 + 8 \)
|
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 |
|
−1.71083 | 1.04055 | 0.926937 | 0 | −1.78020 | 0 | 1.83583 | −1.91726 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.54570 | 2.37250 | 0.389182 | 0 | −3.66716 | 0 | 2.48984 | 2.62874 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.0849355 | −1.40003 | −1.99279 | 0 | 0.118912 | 0 | 0.339129 | −1.03992 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.908891 | 2.20643 | −1.17392 | 0 | 2.00541 | 0 | −2.88475 | 1.86834 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.10066 | −2.17443 | 2.41276 | 0 | −4.56774 | 0 | 0.867058 | 1.72816 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.33192 | 2.95499 | 3.43783 | 0 | 6.89078 | 0 | 3.35289 | 5.73194 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 8575.2.a.o | 6 | |
| 5.b | even | 2 | 1 | 343.2.a.c | ✓ | 6 | |
| 7.b | odd | 2 | 1 | 8575.2.a.n | 6 | ||
| 15.d | odd | 2 | 1 | 3087.2.a.k | 6 | ||
| 20.d | odd | 2 | 1 | 5488.2.a.p | 6 | ||
| 35.c | odd | 2 | 1 | 343.2.a.d | yes | 6 | |
| 35.i | odd | 6 | 2 | 343.2.c.d | 12 | ||
| 35.j | even | 6 | 2 | 343.2.c.e | 12 | ||
| 105.g | even | 2 | 1 | 3087.2.a.j | 6 | ||
| 140.c | even | 2 | 1 | 5488.2.a.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 343.2.a.c | ✓ | 6 | 5.b | even | 2 | 1 | |
| 343.2.a.d | yes | 6 | 35.c | odd | 2 | 1 | |
| 343.2.c.d | 12 | 35.i | odd | 6 | 2 | ||
| 343.2.c.e | 12 | 35.j | even | 6 | 2 | ||
| 3087.2.a.j | 6 | 105.g | even | 2 | 1 | ||
| 3087.2.a.k | 6 | 15.d | odd | 2 | 1 | ||
| 5488.2.a.h | 6 | 140.c | even | 2 | 1 | ||
| 5488.2.a.p | 6 | 20.d | odd | 2 | 1 | ||
| 8575.2.a.n | 6 | 7.b | odd | 2 | 1 | ||
| 8575.2.a.o | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8575))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 10T_{2}^{3} + 10T_{2}^{2} - 11T_{2} - 1 \)
|
|
\( T_{3}^{6} - 5T_{3}^{5} - T_{3}^{4} + 34T_{3}^{3} - 28T_{3}^{2} - 49T_{3} + 49 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 1 \)
|
| $3$ |
\( T^{6} - 5 T^{5} + \cdots + 49 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + T^{5} + \cdots - 239 \)
|
| $13$ |
\( T^{6} - 7 T^{5} + \cdots + 343 \)
|
| $17$ |
\( T^{6} - 26 T^{5} + \cdots + 3479 \)
|
| $19$ |
\( T^{6} - 3 T^{5} + \cdots + 343 \)
|
| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 113 \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots - 1777 \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots + 343 \)
|
| $37$ |
\( T^{6} - 2 T^{5} + \cdots - 1093 \)
|
| $41$ |
\( T^{6} + 28 T^{5} + \cdots + 9947 \)
|
| $43$ |
\( T^{6} + 5 T^{5} + \cdots - 83 \)
|
| $47$ |
\( T^{6} - 18 T^{5} + \cdots - 4067 \)
|
| $53$ |
\( T^{6} - T^{5} + \cdots + 41 \)
|
| $59$ |
\( T^{6} + 5 T^{5} + \cdots + 4753 \)
|
| $61$ |
\( T^{6} - 17 T^{5} + \cdots - 2107 \)
|
| $67$ |
\( T^{6} - 22 T^{5} + \cdots + 1861 \)
|
| $71$ |
\( T^{6} + 2 T^{5} + \cdots - 1189 \)
|
| $73$ |
\( T^{6} - 12 T^{5} + \cdots + 5537 \)
|
| $79$ |
\( T^{6} + 2 T^{5} + \cdots - 25019 \)
|
| $83$ |
\( T^{6} - 7 T^{5} + \cdots - 431837 \)
|
| $89$ |
\( T^{6} + 39 T^{5} + \cdots - 2107 \)
|
| $97$ |
\( T^{6} - 35 T^{5} + \cdots + 218491 \)
|
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