Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [222,4,Mod(1,222)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(222, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("222.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 222 = 2 \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 222.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: | \(13.0984240213\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.576860.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 175x - 113 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 175x - 113 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 115 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{2} + 115 \)
|
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 |
|
2.00000 | 3.00000 | 4.00000 | −9.03880 | 6.00000 | 21.6813 | 8.00000 | 9.00000 | −18.0776 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | 5.64969 | 6.00000 | −11.0963 | 8.00000 | 9.00000 | 11.2994 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.00000 | 4.00000 | 17.3891 | 6.00000 | 14.4150 | 8.00000 | 9.00000 | 34.7782 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(37\) | \(1\) |
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 | 222.4.a.i | ✓ | 3 |
| 3.b | odd | 2 | 1 | 666.4.a.k | 3 | ||
| 4.b | odd | 2 | 1 | 1776.4.a.l | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 222.4.a.i | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 666.4.a.k | 3 | 3.b | odd | 2 | 1 | ||
| 1776.4.a.l | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 14T_{5}^{2} - 110T_{5} + 888 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(222))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} - 14 T^{2} + \cdots + 888 \)
|
| $7$ |
\( T^{3} - 25 T^{2} + \cdots + 3468 \)
|
| $11$ |
\( T^{3} - 25 T^{2} + \cdots + 13400 \)
|
| $13$ |
\( T^{3} - 17 T^{2} + \cdots - 16420 \)
|
| $17$ |
\( T^{3} - 45 T^{2} + \cdots - 38502 \)
|
| $19$ |
\( T^{3} - 9 T^{2} + \cdots - 188520 \)
|
| $23$ |
\( T^{3} - 53 T^{2} + \cdots - 159238 \)
|
| $29$ |
\( T^{3} - 58 T^{2} + \cdots - 933880 \)
|
| $31$ |
\( T^{3} - 2 T^{2} + \cdots - 598944 \)
|
| $37$ |
\( (T + 37)^{3} \)
|
| $41$ |
\( T^{3} + 34 T^{2} + \cdots - 153224 \)
|
| $43$ |
\( T^{3} + 108 T^{2} + \cdots - 3796224 \)
|
| $47$ |
\( T^{3} + 218 T^{2} + \cdots - 78637920 \)
|
| $53$ |
\( T^{3} - 21 T^{2} + \cdots + 10304236 \)
|
| $59$ |
\( T^{3} - 26 T^{2} + \cdots + 62940552 \)
|
| $61$ |
\( T^{3} + 594 T^{2} + \cdots - 20641000 \)
|
| $67$ |
\( T^{3} + 562 T^{2} + \cdots - 32998088 \)
|
| $71$ |
\( T^{3} + 360 T^{2} + \cdots - 150233856 \)
|
| $73$ |
\( T^{3} + 1143 T^{2} + \cdots - 5607720 \)
|
| $79$ |
\( T^{3} + 778 T^{2} + \cdots - 368980560 \)
|
| $83$ |
\( T^{3} - 41 T^{2} + \cdots + 7919160 \)
|
| $89$ |
\( T^{3} + 75 T^{2} + \cdots - 455365542 \)
|
| $97$ |
\( T^{3} + 2006 T^{2} + \cdots - 329946472 \)
|
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