Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [198,12,Mod(1,198)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(198, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("198.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 198 = 2 \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 198.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: | \(152.131949751\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 331687x - 40657734 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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
\( x^{3} - 331687x - 40657734 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 667\nu - 221198 ) / 220 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{2} + 639\nu + 663374 ) / 220 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -667\beta_{2} + 639\beta _1 + 2653709 ) / 12 \)
|
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 |
|
32.0000 | 0 | 1024.00 | 490.681 | 0 | −85589.8 | 32768.0 | 0 | 15701.8 | |||||||||||||||||||||||||||
| 1.2 | 32.0000 | 0 | 1024.00 | 553.980 | 0 | 22407.2 | 32768.0 | 0 | 17727.4 | ||||||||||||||||||||||||||||
| 1.3 | 32.0000 | 0 | 1024.00 | 4579.34 | 0 | 32606.6 | 32768.0 | 0 | 146539. | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
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 | 198.12.a.l | 3 | |
| 3.b | odd | 2 | 1 | 22.12.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 176.12.a.d | 3 | ||
| 33.d | even | 2 | 1 | 242.12.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.12.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 176.12.a.d | 3 | 12.b | even | 2 | 1 | ||
| 198.12.a.l | 3 | 1.a | even | 1 | 1 | trivial | |
| 242.12.a.c | 3 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 5624T_{5}^{2} + 5055685T_{5} - 1244790450 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(198))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 32)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 1244790450 \)
|
| $7$ |
\( T^{3} + \cdots + 62533813132000 \)
|
| $11$ |
\( (T + 161051)^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 32\!\cdots\!88 \)
|
| $17$ |
\( T^{3} + \cdots + 15\!\cdots\!72 \)
|
| $19$ |
\( T^{3} + \cdots - 78\!\cdots\!60 \)
|
| $23$ |
\( T^{3} + \cdots + 27\!\cdots\!16 \)
|
| $29$ |
\( T^{3} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 54\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots + 28\!\cdots\!94 \)
|
| $41$ |
\( T^{3} + \cdots - 73\!\cdots\!64 \)
|
| $43$ |
\( T^{3} + \cdots + 70\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots - 87\!\cdots\!12 \)
|
| $53$ |
\( T^{3} + \cdots + 94\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots + 26\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 59\!\cdots\!20 \)
|
| $67$ |
\( T^{3} + \cdots + 47\!\cdots\!72 \)
|
| $71$ |
\( T^{3} + \cdots - 25\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots - 42\!\cdots\!16 \)
|
| $79$ |
\( T^{3} + \cdots - 96\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 54\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots + 92\!\cdots\!30 \)
|
| $97$ |
\( T^{3} + \cdots - 13\!\cdots\!02 \)
|
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