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: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{889}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 222 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{889})\). 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 | −144.672 | 256.000 | −1706.58 | 2314.76 | −8664.66 | −4096.00 | 1247.12 | 27305.3 | ||||||||||||||||||||||||
| 1.2 | −16.0000 | 123.672 | 256.000 | 1185.58 | −1978.76 | 1174.66 | −4096.00 | −4388.12 | −18969.3 | |||||||||||||||||||||||||
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.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 198.10.a.n | 2 | ||
| 4.b | odd | 2 | 1 | 176.10.a.e | 2 | ||
| 11.b | odd | 2 | 1 | 242.10.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.10.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 176.10.a.e | 2 | 4.b | odd | 2 | 1 | ||
| 198.10.a.n | 2 | 3.b | odd | 2 | 1 | ||
| 242.10.a.e | 2 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 21T_{3} - 17892 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{2} \)
|
| $3$ |
\( T^{2} + 21T - 17892 \)
|
| $5$ |
\( T^{2} + 521 T - 2023290 \)
|
| $7$ |
\( T^{2} + 7490 T - 10178000 \)
|
| $11$ |
\( (T - 14641)^{2} \)
|
| $13$ |
\( T^{2} + \cdots + 5595212424 \)
|
| $17$ |
\( T^{2} + \cdots + 110050366092 \)
|
| $19$ |
\( T^{2} + \cdots - 235841735968 \)
|
| $23$ |
\( T^{2} + \cdots - 251963912952 \)
|
| $29$ |
\( T^{2} + \cdots - 11964147063840 \)
|
| $31$ |
\( T^{2} + \cdots - 21505055373504 \)
|
| $37$ |
\( T^{2} + \cdots - 34431390323214 \)
|
| $41$ |
\( T^{2} + \cdots + 33438413932128 \)
|
| $43$ |
\( T^{2} + \cdots - 200309296545160 \)
|
| $47$ |
\( T^{2} + \cdots - 168165989315712 \)
|
| $53$ |
\( T^{2} + \cdots + 866157473672220 \)
|
| $59$ |
\( T^{2} + \cdots - 11\!\cdots\!48 \)
|
| $61$ |
\( T^{2} + \cdots - 25\!\cdots\!60 \)
|
| $67$ |
\( T^{2} + \cdots + 15\!\cdots\!68 \)
|
| $71$ |
\( T^{2} + \cdots + 26\!\cdots\!60 \)
|
| $73$ |
\( T^{2} + \cdots + 15\!\cdots\!08 \)
|
| $79$ |
\( T^{2} + \cdots - 87\!\cdots\!20 \)
|
| $83$ |
\( T^{2} + \cdots - 34\!\cdots\!84 \)
|
| $89$ |
\( T^{2} + \cdots + 41\!\cdots\!62 \)
|
| $97$ |
\( T^{2} + \cdots + 13\!\cdots\!02 \)
|
| show more | |
| show less |