Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("55.34");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(3.24510505032\) |
| 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: |
\( x^{6} + 22x^{4} + 101x^{2} + 16 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 22x^{4} + 101x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 13\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 17\nu^{2} + 32 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 21\nu^{3} + 84\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 13\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} - 17\beta_{3} + 87 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} - 42\beta_{2} + 189\beta_1 \)
|
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 |
|
− | 3.94784i | − | 3.56498i | −7.58548 | −11.0740 | − | 1.53855i | −14.0740 | 2.30429i | − | 1.63647i | 14.2909 | −6.07397 | + | 43.7183i | |||||||||||||||||||||||||||||
| 34.2 | − | 2.50006i | 2.61871i | 1.74972 | 9.54691 | + | 5.81864i | 6.54691 | 4.41889i | − | 24.3748i | 20.1424 | 14.5469 | − | 23.8678i | |||||||||||||||||||||||||||||||
| 34.3 | − | 0.405276i | − | 8.56932i | 7.83575 | −0.472938 | + | 11.1703i | −3.47294 | − | 27.4984i | − | 6.41784i | −46.4333 | 4.52706 | + | 0.191670i | |||||||||||||||||||||||||||||
| 34.4 | 0.405276i | 8.56932i | 7.83575 | −0.472938 | − | 11.1703i | −3.47294 | 27.4984i | 6.41784i | −46.4333 | 4.52706 | − | 0.191670i | |||||||||||||||||||||||||||||||||
| 34.5 | 2.50006i | − | 2.61871i | 1.74972 | 9.54691 | − | 5.81864i | 6.54691 | − | 4.41889i | 24.3748i | 20.1424 | 14.5469 | + | 23.8678i | |||||||||||||||||||||||||||||||
| 34.6 | 3.94784i | 3.56498i | −7.58548 | −11.0740 | + | 1.53855i | −14.0740 | − | 2.30429i | 1.63647i | 14.2909 | −6.07397 | − | 43.7183i | ||||||||||||||||||||||||||||||||
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 | 55.4.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 495.4.c.a | 6 | ||
| 4.b | odd | 2 | 1 | 880.4.b.f | 6 | ||
| 5.b | even | 2 | 1 | inner | 55.4.b.a | ✓ | 6 |
| 5.c | odd | 4 | 2 | 275.4.a.j | 6 | ||
| 15.d | odd | 2 | 1 | 495.4.c.a | 6 | ||
| 15.e | even | 4 | 2 | 2475.4.a.bn | 6 | ||
| 20.d | odd | 2 | 1 | 880.4.b.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.4.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 55.4.b.a | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 275.4.a.j | 6 | 5.c | odd | 4 | 2 | ||
| 495.4.c.a | 6 | 3.b | odd | 2 | 1 | ||
| 495.4.c.a | 6 | 15.d | odd | 2 | 1 | ||
| 880.4.b.f | 6 | 4.b | odd | 2 | 1 | ||
| 880.4.b.f | 6 | 20.d | odd | 2 | 1 | ||
| 2475.4.a.bn | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 22T_{2}^{4} + 101T_{2}^{2} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(55, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 22 T^{4} + \cdots + 16 \)
|
| $3$ |
\( T^{6} + 93 T^{4} + \cdots + 6400 \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 1953125 \)
|
| $7$ |
\( T^{6} + 781 T^{4} + \cdots + 78400 \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( T^{6} + 3096 T^{4} + \cdots + 595360000 \)
|
| $17$ |
\( T^{6} + 5441 T^{4} + \cdots + 545502736 \)
|
| $19$ |
\( (T^{3} - 129 T^{2} + \cdots + 493744)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 994439750656 \)
|
| $29$ |
\( (T^{3} - 247 T^{2} + \cdots + 2047004)^{2} \)
|
| $31$ |
\( (T^{3} + 257 T^{2} + \cdots - 484864)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 294700670255104 \)
|
| $41$ |
\( (T^{3} + 412 T^{2} + \cdots + 143488)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 23\!\cdots\!96 \)
|
| $47$ |
\( T^{6} + \cdots + 5233772609536 \)
|
| $53$ |
\( T^{6} + \cdots + 14\!\cdots\!44 \)
|
| $59$ |
\( (T^{3} - 100 T^{2} + \cdots + 20068352)^{2} \)
|
| $61$ |
\( (T^{3} + 1105 T^{2} + \cdots - 114682660)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $71$ |
\( (T^{3} + 135 T^{2} + \cdots - 68294144)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 27232283023936 \)
|
| $79$ |
\( (T^{3} - 412 T^{2} + \cdots + 287752192)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 35\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} + 93 T^{2} + \cdots + 155639300)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
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