Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4730,2,Mod(1,4730)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, 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("4730.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4730.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: | \(37.7692401561\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 12x^{6} + 7x^{5} + 41x^{4} - 6x^{3} - 28x^{2} + 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 12x^{6} + 7x^{5} + 41x^{4} - 6x^{3} - 28x^{2} + 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 10\nu^{4} - \nu^{3} + 24\nu^{2} + 2\nu - 8 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 7\nu^{4} + 39\nu^{3} - 4\nu^{2} - 16\nu - 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 12\nu^{5} - 5\nu^{4} - 39\nu^{3} - 8\nu^{2} + 12\nu + 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 7\nu^{4} + 39\nu^{3} - 6\nu^{2} - 16\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 7\nu^{4} + 41\nu^{3} - 6\nu^{2} - 26\nu + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 17\nu^{4} + 26\nu^{3} - 32\nu^{2} - 8\nu + 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + \beta_{4} + 7\beta_{3} + 2\beta _1 + 17 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 7\beta_{6} - 9\beta_{5} + \beta_{3} + \beta_{2} + 30\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - 37\beta_{5} + 10\beta_{4} + 46\beta_{3} + 2\beta_{2} + 23\beta _1 + 107 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\beta_{7} + 46\beta_{6} - 68\beta_{5} + 3\beta_{4} + 15\beta_{3} + 14\beta_{2} + 190\beta _1 + 107 \)
|
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 | −3.42597 | 1.00000 | 1.00000 | 3.42597 | −2.60155 | −1.00000 | 8.73725 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.86277 | 1.00000 | 1.00000 | 2.86277 | −1.78911 | −1.00000 | 5.19548 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.84371 | 1.00000 | 1.00000 | 1.84371 | 0.526782 | −1.00000 | 0.399257 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.40279 | 1.00000 | 1.00000 | 1.40279 | 3.56264 | −1.00000 | −1.03219 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.515357 | 1.00000 | 1.00000 | 0.515357 | −4.64210 | −1.00000 | −2.73441 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −0.228053 | 1.00000 | 1.00000 | 0.228053 | −2.81890 | −1.00000 | −2.94799 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 1.57663 | 1.00000 | 1.00000 | −1.57663 | 0.800428 | −1.00000 | −0.514223 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 1.70201 | 1.00000 | 1.00000 | −1.70201 | 0.961819 | −1.00000 | −0.103166 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
| \(43\) | \(1\) |
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 | 4730.2.a.w | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4730.2.a.w | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):
|
\( T_{3}^{8} + 7T_{3}^{7} + 9T_{3}^{6} - 30T_{3}^{5} - 69T_{3}^{4} + 9T_{3}^{3} + 97T_{3}^{2} + 56T_{3} + 8 \)
|
|
\( T_{7}^{8} + 6T_{7}^{7} - 9T_{7}^{6} - 93T_{7}^{5} - 56T_{7}^{4} + 243T_{7}^{3} + 78T_{7}^{2} - 260T_{7} + 88 \)
|
|
\( T_{13}^{8} + 2T_{13}^{7} - 42T_{13}^{6} - 90T_{13}^{5} + 280T_{13}^{4} + 620T_{13}^{3} - 256T_{13}^{2} - 624T_{13} - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} + 7 T^{7} + \cdots + 8 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots + 88 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( T^{8} + 2 T^{7} + \cdots - 64 \)
|
| $17$ |
\( T^{8} + 8 T^{7} + \cdots - 8 \)
|
| $19$ |
\( T^{8} - 83 T^{6} + \cdots - 5216 \)
|
| $23$ |
\( T^{8} + 18 T^{7} + \cdots + 2048 \)
|
| $29$ |
\( T^{8} - 8 T^{7} + \cdots - 27992 \)
|
| $31$ |
\( T^{8} + 11 T^{7} + \cdots + 16384 \)
|
| $37$ |
\( T^{8} + 17 T^{7} + \cdots - 280576 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots + 174208 \)
|
| $43$ |
\( (T + 1)^{8} \)
|
| $47$ |
\( T^{8} + 19 T^{7} + \cdots - 17493536 \)
|
| $53$ |
\( T^{8} + 7 T^{7} + \cdots - 82804 \)
|
| $59$ |
\( T^{8} - T^{7} + \cdots - 54976 \)
|
| $61$ |
\( T^{8} - 6 T^{7} + \cdots - 1952 \)
|
| $67$ |
\( T^{8} + 22 T^{7} + \cdots + 1741568 \)
|
| $71$ |
\( T^{8} + 14 T^{7} + \cdots + 13255096 \)
|
| $73$ |
\( T^{8} + 13 T^{7} + \cdots + 24704 \)
|
| $79$ |
\( T^{8} + 8 T^{7} + \cdots + 4480312 \)
|
| $83$ |
\( T^{8} + 4 T^{7} + \cdots + 267952 \)
|
| $89$ |
\( T^{8} - 5 T^{7} + \cdots + 32521856 \)
|
| $97$ |
\( T^{8} + 23 T^{7} + \cdots + 677632 \)
|
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