Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [115,3,Mod(91,115)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("115.91");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(3.13352304014\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 62\nu^{5} - 5599\nu^{4} + 5812\nu^{3} + 221635\nu^{2} - 193350\nu - 3002445 ) / 249285 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{5} + 172\nu^{4} - 416\nu^{3} + 1167\nu^{2} + 4975\nu - 2585 ) / 16619 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 272\nu^{5} - 439\nu^{4} - 6668\nu^{3} + 7360\nu^{2} + 205185\nu - 88575 ) / 249285 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -272\nu^{5} + 439\nu^{4} + 6668\nu^{3} - 7360\nu^{2} + 44100\nu + 88575 ) / 249285 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -551\nu^{5} + 706\nu^{4} + 30371\nu^{3} - 32506\nu^{2} - 581535\nu + 262830 ) / 249285 \)
|
| \(\nu\) | \(=\) |
\( \beta_{4} + \beta_{3} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( 15\beta_{5} + 11\beta_{4} + 40\beta_{3} + 3\beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 20\beta_{5} + 11\beta_{4} + 20\beta_{3} + 84\beta_{2} - 4\beta _1 - 53 \)
|
| \(\nu^{5}\) | \(=\) |
\( 400\beta_{5} - 494\beta_{4} + 1175\beta_{3} + 155\beta_{2} - 9\beta _1 + 87 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(51\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−1.00000 | −4.73103 | −3.00000 | − | 2.23607i | 4.73103 | 7.39757i | 7.00000 | 13.3827 | 2.23607i | |||||||||||||||||||||||||||||||||||
| 91.2 | −1.00000 | −4.73103 | −3.00000 | 2.23607i | 4.73103 | − | 7.39757i | 7.00000 | 13.3827 | − | 2.23607i | |||||||||||||||||||||||||||||||||||
| 91.3 | −1.00000 | 0.396209 | −3.00000 | − | 2.23607i | −0.396209 | 8.16531i | 7.00000 | −8.84302 | 2.23607i | ||||||||||||||||||||||||||||||||||||
| 91.4 | −1.00000 | 0.396209 | −3.00000 | 2.23607i | −0.396209 | − | 8.16531i | 7.00000 | −8.84302 | − | 2.23607i | |||||||||||||||||||||||||||||||||||
| 91.5 | −1.00000 | 5.33482 | −3.00000 | − | 2.23607i | −5.33482 | − | 13.3268i | 7.00000 | 19.4603 | 2.23607i | |||||||||||||||||||||||||||||||||||
| 91.6 | −1.00000 | 5.33482 | −3.00000 | 2.23607i | −5.33482 | 13.3268i | 7.00000 | 19.4603 | − | 2.23607i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 115.3.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1035.3.g.a | 6 | ||
| 4.b | odd | 2 | 1 | 1840.3.k.a | 6 | ||
| 5.b | even | 2 | 1 | 575.3.d.e | 6 | ||
| 5.c | odd | 4 | 2 | 575.3.c.c | 12 | ||
| 23.b | odd | 2 | 1 | inner | 115.3.d.a | ✓ | 6 |
| 69.c | even | 2 | 1 | 1035.3.g.a | 6 | ||
| 92.b | even | 2 | 1 | 1840.3.k.a | 6 | ||
| 115.c | odd | 2 | 1 | 575.3.d.e | 6 | ||
| 115.e | even | 4 | 2 | 575.3.c.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.3.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 115.3.d.a | ✓ | 6 | 23.b | odd | 2 | 1 | inner |
| 575.3.c.c | 12 | 5.c | odd | 4 | 2 | ||
| 575.3.c.c | 12 | 115.e | even | 4 | 2 | ||
| 575.3.d.e | 6 | 5.b | even | 2 | 1 | ||
| 575.3.d.e | 6 | 115.c | odd | 2 | 1 | ||
| 1035.3.g.a | 6 | 3.b | odd | 2 | 1 | ||
| 1035.3.g.a | 6 | 69.c | even | 2 | 1 | ||
| 1840.3.k.a | 6 | 4.b | odd | 2 | 1 | ||
| 1840.3.k.a | 6 | 92.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(115, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( (T^{3} - T^{2} - 25 T + 10)^{2} \)
|
| $5$ |
\( (T^{2} + 5)^{3} \)
|
| $7$ |
\( T^{6} + 299 T^{4} + \cdots + 648000 \)
|
| $11$ |
\( T^{6} + 731 T^{4} + \cdots + 8632980 \)
|
| $13$ |
\( (T^{3} + 5 T^{2} - 257 T - 54)^{2} \)
|
| $17$ |
\( T^{6} + 971 T^{4} + \cdots + 524880 \)
|
| $19$ |
\( T^{6} + 779 T^{4} + \cdots + 714420 \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots + 148035889 \)
|
| $29$ |
\( (T^{3} - 36 T^{2} + \cdots + 8872)^{2} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} + \cdots - 30050)^{2} \)
|
| $37$ |
\( T^{6} + 1856 T^{4} + \cdots + 123405120 \)
|
| $41$ |
\( (T^{3} - 71 T^{2} + \cdots + 8462)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 2411208000 \)
|
| $47$ |
\( (T^{3} - 56 T^{2} + \cdots + 16376)^{2} \)
|
| $53$ |
\( (T^{2} + 1620)^{3} \)
|
| $59$ |
\( (T^{3} - 118 T^{2} + \cdots + 118680)^{2} \)
|
| $61$ |
\( T^{6} + 18819 T^{4} + \cdots + 649116180 \)
|
| $67$ |
\( T^{6} + \cdots + 5202247680 \)
|
| $71$ |
\( (T^{3} + 109 T^{2} + \cdots - 33718)^{2} \)
|
| $73$ |
\( (T^{3} - 912 T - 344)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 70054917120 \)
|
| $83$ |
\( T^{6} + \cdots + 29786849280 \)
|
| $89$ |
\( T^{6} + \cdots + 39983258880 \)
|
| $97$ |
\( T^{6} + \cdots + 25687244880 \)
|
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