Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1100,4,Mod(1,1100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1100.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1100.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: | \(64.9021010063\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 93x^{3} + 71x^{2} + 1873x - 2715 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 93x^{3} + 71x^{2} + 1873x - 2715 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 6\nu^{3} + 54\nu^{2} - 260\nu - 69 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 3\nu^{3} - 81\nu^{2} - 226\nu + 1050 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} + \beta_{3} + 3\beta_{2} + 54\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18\beta_{4} - 3\beta_{3} + 72\beta_{2} + 64\beta _1 + 1941 \)
|
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 |
|
0 | −9.44158 | 0 | 0 | 0 | −27.8042 | 0 | 62.1434 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.57551 | 0 | 0 | 0 | 20.7640 | 0 | 4.08629 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.53918 | 0 | 0 | 0 | −31.0503 | 0 | −20.5526 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.25411 | 0 | 0 | 0 | −7.12487 | 0 | 0.605666 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 6.30216 | 0 | 0 | 0 | −4.78461 | 0 | 12.7172 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 1100.4.a.j | ✓ | 5 |
| 5.b | even | 2 | 1 | 1100.4.a.m | yes | 5 | |
| 5.c | odd | 4 | 2 | 1100.4.b.j | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1100.4.a.j | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1100.4.a.m | yes | 5 | 5.b | even | 2 | 1 | |
| 1100.4.b.j | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1100))\):
|
\( T_{3}^{5} + 6T_{3}^{4} - 79T_{3}^{3} - 334T_{3}^{2} + 1461T_{3} + 4426 \)
|
|
\( T_{7}^{5} + 50T_{7}^{4} + 129T_{7}^{3} - 20900T_{7}^{2} - 225721T_{7} - 611100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 6 T^{4} + \cdots + 4426 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + 50 T^{4} + \cdots - 611100 \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} + 45 T^{4} + \cdots + 569084405 \)
|
| $17$ |
\( T^{5} + 62 T^{4} + \cdots + 396700488 \)
|
| $19$ |
\( T^{5} + \cdots - 6579183679 \)
|
| $23$ |
\( T^{5} + \cdots + 12723171957 \)
|
| $29$ |
\( T^{5} + \cdots + 10780618347 \)
|
| $31$ |
\( T^{5} + \cdots - 9310638465 \)
|
| $37$ |
\( T^{5} + \cdots - 81614105668 \)
|
| $41$ |
\( T^{5} + \cdots - 63070168860 \)
|
| $43$ |
\( T^{5} + \cdots - 610802827632 \)
|
| $47$ |
\( T^{5} + \cdots + 202438063440 \)
|
| $53$ |
\( T^{5} + \cdots + 2056072511250 \)
|
| $59$ |
\( T^{5} + \cdots - 2722880626524 \)
|
| $61$ |
\( T^{5} + \cdots - 1054285961106 \)
|
| $67$ |
\( T^{5} + \cdots + 46022015760384 \)
|
| $71$ |
\( T^{5} + \cdots + 90164424394224 \)
|
| $73$ |
\( T^{5} + \cdots + 2152972860072 \)
|
| $79$ |
\( T^{5} + \cdots - 235259377206772 \)
|
| $83$ |
\( T^{5} + \cdots + 207960430987629 \)
|
| $89$ |
\( T^{5} + \cdots - 749849927110839 \)
|
| $97$ |
\( T^{5} + \cdots - 1617345275343 \)
|
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