Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,4,Mod(1,55)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(55, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("55.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.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: | \(3.24510505032\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1539480.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 25x^{2} + 9x + 96 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - x^{3} - 25x^{2} + 9x + 96 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 15\nu + 16 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 17\beta _1 + 10 \)
|
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 |
|
−4.18087 | 9.33168 | 9.47970 | 5.00000 | −39.0146 | −14.2911 | −6.18645 | 60.0803 | −20.9044 | ||||||||||||||||||||||||||||||
| 1.2 | −2.03085 | −5.45953 | −3.87566 | 5.00000 | 11.0875 | 27.2109 | 24.1176 | 2.80648 | −10.1542 | |||||||||||||||||||||||||||||||
| 1.3 | 2.30365 | 6.23583 | −2.69320 | 5.00000 | 14.3652 | 13.1506 | −24.6334 | 11.8856 | 11.5182 | |||||||||||||||||||||||||||||||
| 1.4 | 4.90807 | −1.10798 | 16.0892 | 5.00000 | −5.43806 | −17.0703 | 39.7022 | −25.7724 | 24.5404 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 55.4.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 495.4.a.n | 4 | ||
| 4.b | odd | 2 | 1 | 880.4.a.z | 4 | ||
| 5.b | even | 2 | 1 | 275.4.a.e | 4 | ||
| 5.c | odd | 4 | 2 | 275.4.b.e | 8 | ||
| 11.b | odd | 2 | 1 | 605.4.a.j | 4 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.bc | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 275.4.a.e | 4 | 5.b | even | 2 | 1 | ||
| 275.4.b.e | 8 | 5.c | odd | 4 | 2 | ||
| 495.4.a.n | 4 | 3.b | odd | 2 | 1 | ||
| 605.4.a.j | 4 | 11.b | odd | 2 | 1 | ||
| 880.4.a.z | 4 | 4.b | odd | 2 | 1 | ||
| 2475.4.a.bc | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} - 25T_{2}^{2} + 9T_{2} + 96 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(55))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 96 \)
|
| $3$ |
\( T^{4} - 9 T^{3} + \cdots + 352 \)
|
| $5$ |
\( (T - 5)^{4} \)
|
| $7$ |
\( T^{4} - 9 T^{3} + \cdots + 87296 \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} - 70 T^{3} + \cdots - 2887232 \)
|
| $17$ |
\( T^{4} - 103 T^{3} + \cdots + 199152 \)
|
| $19$ |
\( T^{4} + 205 T^{3} + \cdots - 5224000 \)
|
| $23$ |
\( T^{4} + 56 T^{3} + \cdots - 221568 \)
|
| $29$ |
\( T^{4} + \cdots + 1106175480 \)
|
| $31$ |
\( T^{4} - 49 T^{3} + \cdots + 126259200 \)
|
| $37$ |
\( T^{4} + \cdots - 4092453032 \)
|
| $41$ |
\( T^{4} - 736 T^{3} + \cdots + 329735184 \)
|
| $43$ |
\( T^{4} + \cdots + 2589511936 \)
|
| $47$ |
\( T^{4} + \cdots + 19971136128 \)
|
| $53$ |
\( T^{4} + \cdots + 20698646424 \)
|
| $59$ |
\( T^{4} + \cdots - 123367943040 \)
|
| $61$ |
\( T^{4} + \cdots + 4578287464 \)
|
| $67$ |
\( T^{4} + \cdots + 316737807616 \)
|
| $71$ |
\( T^{4} + \cdots + 1139751168 \)
|
| $73$ |
\( T^{4} + \cdots + 138483587488 \)
|
| $79$ |
\( T^{4} + \cdots - 289419632640 \)
|
| $83$ |
\( T^{4} + \cdots + 340395563136 \)
|
| $89$ |
\( T^{4} + \cdots + 27515045400 \)
|
| $97$ |
\( T^{4} + \cdots + 144669247168 \)
|
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