Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,10,Mod(55,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.55");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.c (of order \(3\), degree \(2\), not 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: | \(83.4358054585\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{3329})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 833x^{2} + 832x + 692224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 54) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 833x^{2} + 832x + 692224 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 833\nu^{2} - 833\nu + 692224 ) / 693056 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{3} - 22491\nu^{2} + 37447515\nu - 18690048 ) / 693056 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 54\nu^{3} + 67419 ) / 833 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 27\beta_1 ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 44955\beta _1 - 44955 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 833\beta_{3} - 67419 ) / 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
8.00000 | − | 13.8564i | 0 | −128.000 | − | 221.703i | −550.916 | − | 954.215i | 0 | −5506.58 | + | 9537.68i | −4096.00 | 0 | −17629.3 | ||||||||||||||||||||||
| 55.2 | 8.00000 | − | 13.8564i | 0 | −128.000 | − | 221.703i | 1006.92 | + | 1744.03i | 0 | 2282.58 | − | 3953.55i | −4096.00 | 0 | 32221.3 | |||||||||||||||||||||||
| 109.1 | 8.00000 | + | 13.8564i | 0 | −128.000 | + | 221.703i | −550.916 | + | 954.215i | 0 | −5506.58 | − | 9537.68i | −4096.00 | 0 | −17629.3 | |||||||||||||||||||||||
| 109.2 | 8.00000 | + | 13.8564i | 0 | −128.000 | + | 221.703i | 1006.92 | − | 1744.03i | 0 | 2282.58 | + | 3953.55i | −4096.00 | 0 | 32221.3 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.10.c.o | 4 | |
| 3.b | odd | 2 | 1 | 162.10.c.l | 4 | ||
| 9.c | even | 3 | 1 | 54.10.a.f | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 162.10.c.o | 4 | |
| 9.d | odd | 6 | 1 | 54.10.a.g | yes | 2 | |
| 9.d | odd | 6 | 1 | 162.10.c.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.10.a.f | ✓ | 2 | 9.c | even | 3 | 1 | |
| 54.10.a.g | yes | 2 | 9.d | odd | 6 | 1 | |
| 162.10.c.l | 4 | 3.b | odd | 2 | 1 | ||
| 162.10.c.l | 4 | 9.d | odd | 6 | 1 | ||
| 162.10.c.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 162.10.c.o | 4 | 9.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 912T_{5}^{3} + 3050649T_{5}^{2} + 2023641360T_{5} + 4923539399025 \)
acting on \(S_{10}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 4923539399025 \)
|
| $7$ |
\( T^{4} + \cdots + 25\!\cdots\!01 \)
|
| $11$ |
\( T^{4} + \cdots + 14\!\cdots\!61 \)
|
| $13$ |
\( T^{4} + \cdots + 58\!\cdots\!96 \)
|
| $17$ |
\( (T^{2} + 399960 T - 77467104000)^{2} \)
|
| $19$ |
\( (T^{2} - 8104 T - 582668105396)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 16\!\cdots\!16 \)
|
| $29$ |
\( T^{4} + \cdots + 32\!\cdots\!96 \)
|
| $31$ |
\( T^{4} + \cdots + 49\!\cdots\!25 \)
|
| $37$ |
\( (T^{2} + \cdots + 296981605917364)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 93\!\cdots\!36 \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $53$ |
\( (T^{2} + \cdots - 44\!\cdots\!89)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 22\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 14\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 32\!\cdots\!91)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 32\!\cdots\!76 \)
|
| $83$ |
\( T^{4} + \cdots + 50\!\cdots\!01 \)
|
| $89$ |
\( (T^{2} + \cdots + 34\!\cdots\!24)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 94\!\cdots\!61 \)
|
| show more | |
| show less |