Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(11.3307883956\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{463}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 463 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(\beta = 8\sqrt{463}\). 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 |
|
−16.0000 | −155.139 | 256.000 | 982.395 | 2482.23 | 5991.53 | −4096.00 | 4385.26 | −15718.3 | ||||||||||||||||||||||||
| 1.2 | −16.0000 | 189.139 | 256.000 | −2460.39 | −3026.23 | 2204.47 | −4096.00 | 16090.7 | 39366.3 | |||||||||||||||||||||||||
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 | 22.10.a.e | ✓ | 2 |
| 3.b | odd | 2 | 1 | 198.10.a.o | 2 | ||
| 4.b | odd | 2 | 1 | 176.10.a.d | 2 | ||
| 11.b | odd | 2 | 1 | 242.10.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.10.a.e | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 176.10.a.d | 2 | 4.b | odd | 2 | 1 | ||
| 198.10.a.o | 2 | 3.b | odd | 2 | 1 | ||
| 242.10.a.f | 2 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 34T_{3} - 29343 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{2} \)
|
| $3$ |
\( T^{2} - 34T - 29343 \)
|
| $5$ |
\( T^{2} + 1478 T - 2417079 \)
|
| $7$ |
\( T^{2} - 8196 T + 13208132 \)
|
| $11$ |
\( (T + 14641)^{2} \)
|
| $13$ |
\( T^{2} + \cdots + 7854369104 \)
|
| $17$ |
\( T^{2} + \cdots + 70527212004 \)
|
| $19$ |
\( T^{2} + \cdots - 52821000576 \)
|
| $23$ |
\( T^{2} + \cdots + 3836788876521 \)
|
| $29$ |
\( T^{2} + \cdots + 106368524544 \)
|
| $31$ |
\( T^{2} + \cdots - 21525297147463 \)
|
| $37$ |
\( T^{2} + \cdots - 157355194445311 \)
|
| $41$ |
\( T^{2} + \cdots - 795139013725632 \)
|
| $43$ |
\( T^{2} + \cdots - 539973897275164 \)
|
| $47$ |
\( T^{2} + \cdots + 377986493834496 \)
|
| $53$ |
\( T^{2} + \cdots + 41\!\cdots\!36 \)
|
| $59$ |
\( T^{2} + \cdots - 33\!\cdots\!39 \)
|
| $61$ |
\( T^{2} + \cdots + 31\!\cdots\!68 \)
|
| $67$ |
\( T^{2} + \cdots - 59\!\cdots\!51 \)
|
| $71$ |
\( T^{2} + \cdots + 30\!\cdots\!93 \)
|
| $73$ |
\( T^{2} + \cdots + 60\!\cdots\!64 \)
|
| $79$ |
\( T^{2} + \cdots + 15\!\cdots\!96 \)
|
| $83$ |
\( T^{2} + \cdots + 71\!\cdots\!16 \)
|
| $89$ |
\( T^{2} + \cdots - 11\!\cdots\!11 \)
|
| $97$ |
\( T^{2} + \cdots + 98765015761841 \)
|
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