Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1755,2,Mod(1,1755)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, 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("1755.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1755 = 3^{3} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1755.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: | \(14.0137455547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 10\nu^{4} - 7\nu^{3} - 21\nu^{2} + 2\nu + 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 9\nu^{3} + 21\nu^{2} - 12\nu - 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 12\nu^{4} - 7\nu^{3} - 37\nu^{2} + 18 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 9\nu^{3} + 8\nu^{2} + 15\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{3} + 8\beta_{2} + \beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 9\beta_{4} + 8\beta_{3} + 31\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 11\beta_{5} + 2\beta_{4} - 11\beta_{3} + 59\beta_{2} + 8\beta _1 + 161 \)
|
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 |
|
−2.72441 | 0 | 5.42239 | −1.00000 | 0 | −4.53684 | −9.32397 | 0 | 2.72441 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.36513 | 0 | 3.59384 | −1.00000 | 0 | 3.86012 | −3.76963 | 0 | 2.36513 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.931141 | 0 | −1.13298 | −1.00000 | 0 | −1.36709 | 2.91724 | 0 | 0.931141 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.401763 | 0 | −1.83859 | −1.00000 | 0 | 2.55731 | 1.54220 | 0 | 0.401763 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.18218 | 0 | −0.602453 | −1.00000 | 0 | −5.08623 | −3.07657 | 0 | −1.18218 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.58727 | 0 | 0.519412 | −1.00000 | 0 | 2.72171 | −2.35009 | 0 | −1.58727 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.65299 | 0 | 5.03838 | −1.00000 | 0 | 3.85102 | 8.06081 | 0 | −2.65299 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1755.2.a.u | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1755.2.a.v | yes | 7 | |
| 5.b | even | 2 | 1 | 8775.2.a.bx | 7 | ||
| 15.d | odd | 2 | 1 | 8775.2.a.bw | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1755.2.a.u | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1755.2.a.v | yes | 7 | 3.b | odd | 2 | 1 | |
| 8775.2.a.bw | 7 | 15.d | odd | 2 | 1 | ||
| 8775.2.a.bx | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1755))\):
|
\( T_{2}^{7} + T_{2}^{6} - 12T_{2}^{5} - 9T_{2}^{4} + 39T_{2}^{3} + 16T_{2}^{2} - 30T_{2} - 12 \)
|
|
\( T_{7}^{7} - 2T_{7}^{6} - 44T_{7}^{5} + 116T_{7}^{4} + 507T_{7}^{3} - 1678T_{7}^{2} - 420T_{7} + 3264 \)
|
|
\( T_{17}^{7} + 11T_{17}^{6} - 6T_{17}^{5} - 338T_{17}^{4} - 579T_{17}^{3} + 1295T_{17}^{2} + 1032T_{17} - 1032 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots - 12 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} - 2 T^{6} + \cdots + 3264 \)
|
| $11$ |
\( T^{7} + T^{6} + \cdots - 96 \)
|
| $13$ |
\( (T - 1)^{7} \)
|
| $17$ |
\( T^{7} + 11 T^{6} + \cdots - 1032 \)
|
| $19$ |
\( T^{7} - 2 T^{6} + \cdots + 12544 \)
|
| $23$ |
\( T^{7} - T^{6} + \cdots + 52512 \)
|
| $29$ |
\( T^{7} - 4 T^{6} + \cdots + 20736 \)
|
| $31$ |
\( T^{7} - 129 T^{5} + \cdots - 49936 \)
|
| $37$ |
\( T^{7} - 23 T^{6} + \cdots - 328752 \)
|
| $41$ |
\( T^{7} + 2 T^{6} + \cdots + 126432 \)
|
| $43$ |
\( T^{7} - 8 T^{6} + \cdots - 449024 \)
|
| $47$ |
\( T^{7} + 2 T^{6} + \cdots + 2304 \)
|
| $53$ |
\( T^{7} + 10 T^{6} + \cdots - 474174 \)
|
| $59$ |
\( T^{7} - 13 T^{6} + \cdots - 309564 \)
|
| $61$ |
\( T^{7} - 21 T^{6} + \cdots - 733976 \)
|
| $67$ |
\( T^{7} - 21 T^{6} + \cdots - 24844 \)
|
| $71$ |
\( T^{7} - 10 T^{6} + \cdots + 1152 \)
|
| $73$ |
\( T^{7} - 13 T^{6} + \cdots - 997008 \)
|
| $79$ |
\( T^{7} - 8 T^{6} + \cdots + 264644 \)
|
| $83$ |
\( T^{7} - 4 T^{6} + \cdots + 1152 \)
|
| $89$ |
\( T^{7} - 27 T^{6} + \cdots - 4780848 \)
|
| $97$ |
\( T^{7} + 15 T^{6} + \cdots - 2569536 \)
|
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