Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,20,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, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| 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: | \(50.3396732424\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{51151}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 51151 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(\beta = 4\sqrt{51151}\). 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 |
|
512.000 | 345.029 | 262144. | 2.58768e6 | 176655. | −6.04760e6 | 1.34218e8 | −1.16214e9 | 1.32489e9 | ||||||||||||||||||||||||
| 1.2 | 512.000 | 16629.0 | 262144. | −5.53619e6 | 8.51403e6 | 9.09776e7 | 1.34218e8 | −8.85739e8 | −2.83453e9 | |||||||||||||||||||||||||
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 | 22.20.a.b | ✓ | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.20.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 16974T_{3} + 5737473 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(22))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 512)^{2} \)
|
| $3$ |
\( T^{2} - 16974 T + 5737473 \)
|
| $5$ |
\( T^{2} + \cdots - 14325920596575 \)
|
| $7$ |
\( T^{2} + \cdots - 550195941209516 \)
|
| $11$ |
\( (T + 2357947691)^{2} \)
|
| $13$ |
\( T^{2} + \cdots - 29\!\cdots\!60 \)
|
| $17$ |
\( T^{2} + \cdots + 24\!\cdots\!80 \)
|
| $19$ |
\( T^{2} + \cdots + 51\!\cdots\!52 \)
|
| $23$ |
\( T^{2} + \cdots - 11\!\cdots\!07 \)
|
| $29$ |
\( T^{2} + \cdots + 21\!\cdots\!08 \)
|
| $31$ |
\( T^{2} + \cdots + 31\!\cdots\!25 \)
|
| $37$ |
\( T^{2} + \cdots - 16\!\cdots\!15 \)
|
| $41$ |
\( T^{2} + \cdots - 45\!\cdots\!12 \)
|
| $43$ |
\( T^{2} + \cdots + 96\!\cdots\!00 \)
|
| $47$ |
\( T^{2} + \cdots - 32\!\cdots\!20 \)
|
| $53$ |
\( T^{2} + \cdots - 73\!\cdots\!16 \)
|
| $59$ |
\( T^{2} + \cdots - 12\!\cdots\!83 \)
|
| $61$ |
\( T^{2} + \cdots + 12\!\cdots\!00 \)
|
| $67$ |
\( T^{2} + \cdots + 39\!\cdots\!57 \)
|
| $71$ |
\( T^{2} + \cdots + 10\!\cdots\!33 \)
|
| $73$ |
\( T^{2} + \cdots + 32\!\cdots\!20 \)
|
| $79$ |
\( T^{2} + \cdots + 13\!\cdots\!24 \)
|
| $83$ |
\( T^{2} + \cdots - 11\!\cdots\!60 \)
|
| $89$ |
\( T^{2} + \cdots - 86\!\cdots\!75 \)
|
| $97$ |
\( T^{2} + \cdots + 56\!\cdots\!41 \)
|
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