Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,2,Mod(34,55)]
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([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("55.34");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.b (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: | \(0.439177211117\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu + 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - 3\beta_{2} + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).
| \(n\) | \(12\) | \(46\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 34.1 |
|
− | 2.52434i | 0.792287i | −4.37228 | 0.686141 | − | 2.12819i | 2.00000 | 3.46410i | 5.98844i | 2.37228 | −5.37228 | − | 1.73205i | |||||||||||||||||||||||||
| 34.2 | − | 0.792287i | 2.52434i | 1.37228 | −2.18614 | + | 0.469882i | 2.00000 | − | 3.46410i | − | 2.67181i | −3.37228 | 0.372281 | + | 1.73205i | ||||||||||||||||||||||||
| 34.3 | 0.792287i | − | 2.52434i | 1.37228 | −2.18614 | − | 0.469882i | 2.00000 | 3.46410i | 2.67181i | −3.37228 | 0.372281 | − | 1.73205i | ||||||||||||||||||||||||||
| 34.4 | 2.52434i | − | 0.792287i | −4.37228 | 0.686141 | + | 2.12819i | 2.00000 | − | 3.46410i | − | 5.98844i | 2.37228 | −5.37228 | + | 1.73205i | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 55.2.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 495.2.c.a | 4 | ||
| 4.b | odd | 2 | 1 | 880.2.b.h | 4 | ||
| 5.b | even | 2 | 1 | inner | 55.2.b.a | ✓ | 4 |
| 5.c | odd | 4 | 2 | 275.2.a.h | 4 | ||
| 11.b | odd | 2 | 1 | 605.2.b.c | 4 | ||
| 11.c | even | 5 | 4 | 605.2.j.i | 16 | ||
| 11.d | odd | 10 | 4 | 605.2.j.j | 16 | ||
| 15.d | odd | 2 | 1 | 495.2.c.a | 4 | ||
| 15.e | even | 4 | 2 | 2475.2.a.bi | 4 | ||
| 20.d | odd | 2 | 1 | 880.2.b.h | 4 | ||
| 20.e | even | 4 | 2 | 4400.2.a.cc | 4 | ||
| 55.d | odd | 2 | 1 | 605.2.b.c | 4 | ||
| 55.e | even | 4 | 2 | 3025.2.a.ba | 4 | ||
| 55.h | odd | 10 | 4 | 605.2.j.j | 16 | ||
| 55.j | even | 10 | 4 | 605.2.j.i | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.2.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 55.2.b.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 275.2.a.h | 4 | 5.c | odd | 4 | 2 | ||
| 495.2.c.a | 4 | 3.b | odd | 2 | 1 | ||
| 495.2.c.a | 4 | 15.d | odd | 2 | 1 | ||
| 605.2.b.c | 4 | 11.b | odd | 2 | 1 | ||
| 605.2.b.c | 4 | 55.d | odd | 2 | 1 | ||
| 605.2.j.i | 16 | 11.c | even | 5 | 4 | ||
| 605.2.j.i | 16 | 55.j | even | 10 | 4 | ||
| 605.2.j.j | 16 | 11.d | odd | 10 | 4 | ||
| 605.2.j.j | 16 | 55.h | odd | 10 | 4 | ||
| 880.2.b.h | 4 | 4.b | odd | 2 | 1 | ||
| 880.2.b.h | 4 | 20.d | odd | 2 | 1 | ||
| 2475.2.a.bi | 4 | 15.e | even | 4 | 2 | ||
| 3025.2.a.ba | 4 | 55.e | even | 4 | 2 | ||
| 4400.2.a.cc | 4 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(55, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 7T^{2} + 4 \)
|
| $3$ |
\( T^{4} + 7T^{2} + 4 \)
|
| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 25 \)
|
| $7$ |
\( (T^{2} + 12)^{2} \)
|
| $11$ |
\( (T + 1)^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 28T^{2} + 64 \)
|
| $19$ |
\( (T + 4)^{4} \)
|
| $23$ |
\( T^{4} + 7T^{2} + 4 \)
|
| $29$ |
\( (T^{2} + 6 T - 24)^{2} \)
|
| $31$ |
\( (T^{2} - T - 8)^{2} \)
|
| $37$ |
\( T^{4} + 123T^{2} + 144 \)
|
| $41$ |
\( (T^{2} - 6 T - 24)^{2} \)
|
| $43$ |
\( (T^{2} + 12)^{2} \)
|
| $47$ |
\( (T^{2} + 44)^{2} \)
|
| $53$ |
\( T^{4} + 112T^{2} + 1024 \)
|
| $59$ |
\( (T^{2} - 9 T + 12)^{2} \)
|
| $61$ |
\( (T^{2} - 10 T - 8)^{2} \)
|
| $67$ |
\( T^{4} + 87T^{2} + 36 \)
|
| $71$ |
\( (T^{2} + 3 T - 72)^{2} \)
|
| $73$ |
\( (T^{2} + 48)^{2} \)
|
| $79$ |
\( (T^{2} + 14 T + 16)^{2} \)
|
| $83$ |
\( (T^{2} + 44)^{2} \)
|
| $89$ |
\( (T^{2} - 3 T - 6)^{2} \)
|
| $97$ |
\( T^{4} + 51T^{2} + 576 \)
|
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