Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.55");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(50.6063741284\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-643})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 160x^{2} - 161x + 25921 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 160x^{2} - 161x + 25921 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 160\nu^{2} - 160\nu - 25921 ) / 25760 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -12\nu^{3} + 12\nu^{2} + 3852\nu + 966 ) / 161 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 969\nu^{3} + 480\nu^{2} + 154080\nu - 310569 ) / 12880 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 18\beta_1 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 5778\beta _1 + 5778 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 80\beta_{3} - 40\beta_{2} + 2169 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-1 - \beta_{1}\) |
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 |
|
−4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −103.261 | − | 178.854i | 0 | 457.045 | − | 791.624i | 512.000 | 0 | 1652.18 | ||||||||||||||||||||||
| 55.2 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 160.261 | + | 277.580i | 0 | −597.045 | + | 1034.11i | 512.000 | 0 | −2564.18 | |||||||||||||||||||||||
| 109.1 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −103.261 | + | 178.854i | 0 | 457.045 | + | 791.624i | 512.000 | 0 | 1652.18 | |||||||||||||||||||||||
| 109.2 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 160.261 | − | 277.580i | 0 | −597.045 | − | 1034.11i | 512.000 | 0 | −2564.18 | |||||||||||||||||||||||
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 | 162.8.c.n | 4 | |
| 3.b | odd | 2 | 1 | 162.8.c.o | 4 | ||
| 9.c | even | 3 | 1 | 162.8.a.d | yes | 2 | |
| 9.c | even | 3 | 1 | inner | 162.8.c.n | 4 | |
| 9.d | odd | 6 | 1 | 162.8.a.c | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 162.8.c.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.8.a.c | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 162.8.a.d | yes | 2 | 9.c | even | 3 | 1 | |
| 162.8.c.n | 4 | 1.a | even | 1 | 1 | trivial | |
| 162.8.c.n | 4 | 9.c | even | 3 | 1 | inner | |
| 162.8.c.o | 4 | 3.b | odd | 2 | 1 | ||
| 162.8.c.o | 4 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 114T_{5}^{3} + 79191T_{5}^{2} + 7546230T_{5} + 4381778025 \)
acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 4381778025 \)
|
| $7$ |
\( T^{4} + \cdots + 1191380982016 \)
|
| $11$ |
\( T^{4} + \cdots + 122305904640000 \)
|
| $13$ |
\( T^{4} + \cdots + 11\!\cdots\!09 \)
|
| $17$ |
\( (T^{2} + 36510 T + 66302289)^{2} \)
|
| $19$ |
\( (T^{2} - 6472 T - 207304688)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 47\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 44\!\cdots\!25 \)
|
| $31$ |
\( T^{4} + \cdots + 31\!\cdots\!64 \)
|
| $37$ |
\( (T^{2} + 286958 T - 119637948563)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 22\!\cdots\!04 \)
|
| $43$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $53$ |
\( (T^{2} + 2373852 T + 947978500932)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!25 \)
|
| $67$ |
\( T^{4} + \cdots + 35\!\cdots\!16 \)
|
| $71$ |
\( (T^{2} + 2776920 T - 608963703984)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots + 24356425782073)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 26\!\cdots\!84 \)
|
| $83$ |
\( T^{4} + \cdots + 24\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + \cdots - 24042892404087)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 70\!\cdots\!76 \)
|
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