Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,10,Mod(1,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.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: | \(54.0787627972\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1495x^{2} + 193x + 99862 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 1495x^{2} + 193x + 99862 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 1361\nu + 858 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 748 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 748 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 1363\beta _1 + 638 \)
|
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 |
|
−36.2973 | −81.0000 | 805.496 | −625.000 | 2940.08 | −2401.00 | −10653.1 | 6561.00 | 22685.8 | ||||||||||||||||||||||||||||||
| 1.2 | −7.32605 | −81.0000 | −458.329 | −625.000 | 593.410 | −2401.00 | 7108.68 | 6561.00 | 4578.78 | |||||||||||||||||||||||||||||||
| 1.3 | 9.41677 | −81.0000 | −423.324 | −625.000 | −762.758 | −2401.00 | −8807.73 | 6561.00 | −5885.48 | |||||||||||||||||||||||||||||||
| 1.4 | 39.2066 | −81.0000 | 1025.16 | −625.000 | −3175.73 | −2401.00 | 20119.2 | 6561.00 | −24504.1 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 105.10.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 315.10.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.10.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 315.10.a.d | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} - 1486T_{2}^{2} + 3176T_{2} + 98176 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(105))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 98176 \)
|
| $3$ |
\( (T + 81)^{4} \)
|
| $5$ |
\( (T + 625)^{4} \)
|
| $7$ |
\( (T + 2401)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 11\!\cdots\!40 \)
|
| $13$ |
\( T^{4} + \cdots + 20\!\cdots\!08 \)
|
| $17$ |
\( T^{4} + \cdots + 23\!\cdots\!68 \)
|
| $19$ |
\( T^{4} + \cdots + 14\!\cdots\!60 \)
|
| $23$ |
\( T^{4} + \cdots + 59\!\cdots\!88 \)
|
| $29$ |
\( T^{4} + \cdots + 37\!\cdots\!60 \)
|
| $31$ |
\( T^{4} + \cdots - 53\!\cdots\!96 \)
|
| $37$ |
\( T^{4} + \cdots - 50\!\cdots\!92 \)
|
| $41$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 13\!\cdots\!12 \)
|
| $47$ |
\( T^{4} + \cdots + 58\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots - 59\!\cdots\!20 \)
|
| $59$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 16\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots + 92\!\cdots\!12 \)
|
| $71$ |
\( T^{4} + \cdots - 48\!\cdots\!36 \)
|
| $73$ |
\( T^{4} + \cdots + 44\!\cdots\!12 \)
|
| $79$ |
\( T^{4} + \cdots + 51\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots - 68\!\cdots\!20 \)
|
| $97$ |
\( T^{4} + \cdots + 12\!\cdots\!52 \)
|
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