Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [198,3,Mod(109,198)]
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, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("198.109");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 198 = 2 \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 198.d (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: | \(5.39510923433\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/198\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(155\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
− | 1.41421i | 0 | −2.00000 | 1.00000 | 0 | − | 8.48528i | 2.82843i | 0 | − | 1.41421i | |||||||||||||||||||||
| 109.2 | 1.41421i | 0 | −2.00000 | 1.00000 | 0 | 8.48528i | − | 2.82843i | 0 | 1.41421i | ||||||||||||||||||||||||
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 | 198.3.d.b | 2 | |
| 3.b | odd | 2 | 1 | 22.3.b.a | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 1584.3.j.d | 2 | ||
| 11.b | odd | 2 | 1 | inner | 198.3.d.b | 2 | |
| 12.b | even | 2 | 1 | 176.3.h.c | 2 | ||
| 15.d | odd | 2 | 1 | 550.3.d.a | 2 | ||
| 15.e | even | 4 | 2 | 550.3.c.a | 4 | ||
| 21.c | even | 2 | 1 | 1078.3.d.a | 2 | ||
| 24.f | even | 2 | 1 | 704.3.h.e | 2 | ||
| 24.h | odd | 2 | 1 | 704.3.h.d | 2 | ||
| 33.d | even | 2 | 1 | 22.3.b.a | ✓ | 2 | |
| 33.f | even | 10 | 4 | 242.3.d.b | 8 | ||
| 33.h | odd | 10 | 4 | 242.3.d.b | 8 | ||
| 44.c | even | 2 | 1 | 1584.3.j.d | 2 | ||
| 132.d | odd | 2 | 1 | 176.3.h.c | 2 | ||
| 165.d | even | 2 | 1 | 550.3.d.a | 2 | ||
| 165.l | odd | 4 | 2 | 550.3.c.a | 4 | ||
| 231.h | odd | 2 | 1 | 1078.3.d.a | 2 | ||
| 264.m | even | 2 | 1 | 704.3.h.d | 2 | ||
| 264.p | odd | 2 | 1 | 704.3.h.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.3.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 22.3.b.a | ✓ | 2 | 33.d | even | 2 | 1 | |
| 176.3.h.c | 2 | 12.b | even | 2 | 1 | ||
| 176.3.h.c | 2 | 132.d | odd | 2 | 1 | ||
| 198.3.d.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 198.3.d.b | 2 | 11.b | odd | 2 | 1 | inner | |
| 242.3.d.b | 8 | 33.f | even | 10 | 4 | ||
| 242.3.d.b | 8 | 33.h | odd | 10 | 4 | ||
| 550.3.c.a | 4 | 15.e | even | 4 | 2 | ||
| 550.3.c.a | 4 | 165.l | odd | 4 | 2 | ||
| 550.3.d.a | 2 | 15.d | odd | 2 | 1 | ||
| 550.3.d.a | 2 | 165.d | even | 2 | 1 | ||
| 704.3.h.d | 2 | 24.h | odd | 2 | 1 | ||
| 704.3.h.d | 2 | 264.m | even | 2 | 1 | ||
| 704.3.h.e | 2 | 24.f | even | 2 | 1 | ||
| 704.3.h.e | 2 | 264.p | odd | 2 | 1 | ||
| 1078.3.d.a | 2 | 21.c | even | 2 | 1 | ||
| 1078.3.d.a | 2 | 231.h | odd | 2 | 1 | ||
| 1584.3.j.d | 2 | 4.b | odd | 2 | 1 | ||
| 1584.3.j.d | 2 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 1 \)
acting on \(S_{3}^{\mathrm{new}}(198, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T - 1)^{2} \)
|
| $7$ |
\( T^{2} + 72 \)
|
| $11$ |
\( T^{2} + 14T + 121 \)
|
| $13$ |
\( T^{2} + 72 \)
|
| $17$ |
\( T^{2} + 648 \)
|
| $19$ |
\( T^{2} + 648 \)
|
| $23$ |
\( (T + 17)^{2} \)
|
| $29$ |
\( T^{2} + 1152 \)
|
| $31$ |
\( (T - 17)^{2} \)
|
| $37$ |
\( (T - 47)^{2} \)
|
| $41$ |
\( T^{2} + 72 \)
|
| $43$ |
\( T^{2} + 288 \)
|
| $47$ |
\( (T - 58)^{2} \)
|
| $53$ |
\( (T + 2)^{2} \)
|
| $59$ |
\( (T - 55)^{2} \)
|
| $61$ |
\( T^{2} + 7200 \)
|
| $67$ |
\( (T - 89)^{2} \)
|
| $71$ |
\( (T - 7)^{2} \)
|
| $73$ |
\( T^{2} + 16200 \)
|
| $79$ |
\( T^{2} + 1152 \)
|
| $83$ |
\( T^{2} + 1152 \)
|
| $89$ |
\( (T - 97)^{2} \)
|
| $97$ |
\( (T + 121)^{2} \)
|
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