Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,5,Mod(21,22)]
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([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.21");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.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: | \(2.27413918784\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{553})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 824\nu + 132 ) / 561 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} - 526\nu + 264 ) / 561 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 558\nu^{2} - 824\nu - 76164 ) / 561 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta _1 + 136 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} + 412\beta_{2} - 266\beta _1 + 276 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.1 |
|
− | 2.82843i | −11.2580 | −8.00000 | −19.2580 | 31.8424i | − | 33.9411i | 22.6274i | 45.7420 | 54.4698i | ||||||||||||||||||||||||||||
| 21.2 | − | 2.82843i | 12.2580 | −8.00000 | 4.25798 | − | 34.6708i | − | 33.9411i | 22.6274i | 69.2580 | − | 12.0434i | |||||||||||||||||||||||||||
| 21.3 | 2.82843i | −11.2580 | −8.00000 | −19.2580 | − | 31.8424i | 33.9411i | − | 22.6274i | 45.7420 | − | 54.4698i | ||||||||||||||||||||||||||||
| 21.4 | 2.82843i | 12.2580 | −8.00000 | 4.25798 | 34.6708i | 33.9411i | − | 22.6274i | 69.2580 | 12.0434i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 22.5.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 198.5.d.a | 4 | ||
| 4.b | odd | 2 | 1 | 176.5.h.e | 4 | ||
| 5.b | even | 2 | 1 | 550.5.d.a | 4 | ||
| 5.c | odd | 4 | 2 | 550.5.c.a | 8 | ||
| 8.b | even | 2 | 1 | 704.5.h.i | 4 | ||
| 8.d | odd | 2 | 1 | 704.5.h.j | 4 | ||
| 11.b | odd | 2 | 1 | inner | 22.5.b.a | ✓ | 4 |
| 33.d | even | 2 | 1 | 198.5.d.a | 4 | ||
| 44.c | even | 2 | 1 | 176.5.h.e | 4 | ||
| 55.d | odd | 2 | 1 | 550.5.d.a | 4 | ||
| 55.e | even | 4 | 2 | 550.5.c.a | 8 | ||
| 88.b | odd | 2 | 1 | 704.5.h.i | 4 | ||
| 88.g | even | 2 | 1 | 704.5.h.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.5.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 22.5.b.a | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 176.5.h.e | 4 | 4.b | odd | 2 | 1 | ||
| 176.5.h.e | 4 | 44.c | even | 2 | 1 | ||
| 198.5.d.a | 4 | 3.b | odd | 2 | 1 | ||
| 198.5.d.a | 4 | 33.d | even | 2 | 1 | ||
| 550.5.c.a | 8 | 5.c | odd | 4 | 2 | ||
| 550.5.c.a | 8 | 55.e | even | 4 | 2 | ||
| 550.5.d.a | 4 | 5.b | even | 2 | 1 | ||
| 550.5.d.a | 4 | 55.d | odd | 2 | 1 | ||
| 704.5.h.i | 4 | 8.b | even | 2 | 1 | ||
| 704.5.h.i | 4 | 88.b | odd | 2 | 1 | ||
| 704.5.h.j | 4 | 8.d | odd | 2 | 1 | ||
| 704.5.h.j | 4 | 88.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(22, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( (T^{2} - T - 138)^{2} \)
|
| $5$ |
\( (T^{2} + 15 T - 82)^{2} \)
|
| $7$ |
\( (T^{2} + 1152)^{2} \)
|
| $11$ |
\( T^{4} + 180 T^{3} + \cdots + 214358881 \)
|
| $13$ |
\( T^{4} + 112032 T^{2} + 557715456 \)
|
| $17$ |
\( T^{4} + \cdots + 21069103104 \)
|
| $19$ |
\( T^{4} + 169632 T^{2} + 26873856 \)
|
| $23$ |
\( (T^{2} + 783 T - 178666)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 443364212736 \)
|
| $31$ |
\( (T^{2} - 2209 T + 1069366)^{2} \)
|
| $37$ |
\( (T^{2} + 191 T - 1694258)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 6140246114304 \)
|
| $43$ |
\( T^{4} + \cdots + 5615307472896 \)
|
| $47$ |
\( (T^{2} - 2844 T - 534988)^{2} \)
|
| $53$ |
\( (T^{2} + 4284 T + 1048964)^{2} \)
|
| $59$ |
\( (T^{2} + 1695 T - 582538)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 74192451158016 \)
|
| $67$ |
\( (T^{2} + 4367 T + 4736566)^{2} \)
|
| $71$ |
\( (T^{2} - 1761 T - 10612234)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 33350347800576 \)
|
| $79$ |
\( T^{4} + \cdots + 205902754873344 \)
|
| $83$ |
\( T^{4} + \cdots + 39\!\cdots\!36 \)
|
| $89$ |
\( (T^{2} + 4383 T - 48856162)^{2} \)
|
| $97$ |
\( (T^{2} - 8641 T + 14335486)^{2} \)
|
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