Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,6,Mod(1,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.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: | \(3.52844403589\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{793}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 198 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{793})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
4.00000 | 0.419872 | 16.0000 | 63.9006 | 1.67949 | 77.4808 | 64.0000 | −242.824 | 255.603 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 28.5801 | 16.0000 | −76.9006 | 114.321 | −91.4808 | 64.0000 | 573.824 | −307.603 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
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 | 22.6.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 198.6.a.k | 2 | ||
| 4.b | odd | 2 | 1 | 176.6.a.f | 2 | ||
| 5.b | even | 2 | 1 | 550.6.a.h | 2 | ||
| 5.c | odd | 4 | 2 | 550.6.b.j | 4 | ||
| 7.b | odd | 2 | 1 | 1078.6.a.h | 2 | ||
| 8.b | even | 2 | 1 | 704.6.a.k | 2 | ||
| 8.d | odd | 2 | 1 | 704.6.a.p | 2 | ||
| 11.b | odd | 2 | 1 | 242.6.a.g | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.6.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 176.6.a.f | 2 | 4.b | odd | 2 | 1 | ||
| 198.6.a.k | 2 | 3.b | odd | 2 | 1 | ||
| 242.6.a.g | 2 | 11.b | odd | 2 | 1 | ||
| 550.6.a.h | 2 | 5.b | even | 2 | 1 | ||
| 550.6.b.j | 4 | 5.c | odd | 4 | 2 | ||
| 704.6.a.k | 2 | 8.b | even | 2 | 1 | ||
| 704.6.a.p | 2 | 8.d | odd | 2 | 1 | ||
| 1078.6.a.h | 2 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 29T_{3} + 12 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(22))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} - 29T + 12 \)
|
| $5$ |
\( T^{2} + 13T - 4914 \)
|
| $7$ |
\( T^{2} + 14T - 7088 \)
|
| $11$ |
\( (T + 121)^{2} \)
|
| $13$ |
\( T^{2} + 646T + 84504 \)
|
| $17$ |
\( T^{2} + 208 T - 3037476 \)
|
| $19$ |
\( T^{2} + 2148 T + 769664 \)
|
| $23$ |
\( T^{2} - 349T + 20736 \)
|
| $29$ |
\( T^{2} - 4422 T - 21668256 \)
|
| $31$ |
\( T^{2} - 14381 T + 50132952 \)
|
| $37$ |
\( T^{2} + 4267 T + 150474 \)
|
| $41$ |
\( T^{2} + 10110 T - 178286832 \)
|
| $43$ |
\( T^{2} + 9798 T - 19793224 \)
|
| $47$ |
\( T^{2} - 21144 T + 92468736 \)
|
| $53$ |
\( T^{2} - 39584 T + 361877916 \)
|
| $59$ |
\( T^{2} + \cdots + 2046037356 \)
|
| $61$ |
\( T^{2} + 29550 T + 163449608 \)
|
| $67$ |
\( T^{2} + 64149 T + 585679844 \)
|
| $71$ |
\( T^{2} + \cdots - 3313271952 \)
|
| $73$ |
\( T^{2} + \cdots - 1315930776 \)
|
| $79$ |
\( T^{2} + \cdots - 1348802144 \)
|
| $83$ |
\( T^{2} + \cdots - 5297916504 \)
|
| $89$ |
\( T^{2} + 18047 T - 117223146 \)
|
| $97$ |
\( T^{2} + \cdots - 13142971534 \)
|
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