Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(169,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.j (of order \(5\), degree \(4\), 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: | \(3.68908857338\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.324000000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 27 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 27\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 27\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(211\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −2.26728 | + | 1.64728i | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −2.80252 | ||||||||||||||||||||||||||
| 169.2 | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | 2.26728 | − | 1.64728i | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | 2.80252 | |||||||||||||||||||||||||||
| 295.1 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | −0.330792 | − | 1.01807i | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 1.07047 | |||||||||||||||||||||||||||
| 295.2 | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.330792 | + | 1.01807i | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | −1.07047 | |||||||||||||||||||||||||||
| 379.1 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | −0.330792 | + | 1.01807i | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 1.07047 | |||||||||||||||||||||||||||
| 379.2 | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.330792 | − | 1.01807i | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | −1.07047 | |||||||||||||||||||||||||||
| 421.1 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −2.26728 | − | 1.64728i | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −2.80252 | |||||||||||||||||||||||||||
| 421.2 | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | 2.26728 | + | 1.64728i | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | 2.80252 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 462.2.j.g | ✓ | 8 |
| 11.c | even | 5 | 1 | inner | 462.2.j.g | ✓ | 8 |
| 11.c | even | 5 | 1 | 5082.2.a.bz | 4 | ||
| 11.d | odd | 10 | 1 | 5082.2.a.ce | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.2.j.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 462.2.j.g | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 5082.2.a.bz | 4 | 11.c | even | 5 | 1 | ||
| 5082.2.a.ce | 4 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 3T_{5}^{6} + 54T_{5}^{4} + 108T_{5}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( T^{8} - 3 T^{6} + \cdots + 81 \)
|
| $7$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{8} - 8 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} - 6 T^{7} + \cdots + 1296 \)
|
| $17$ |
\( T^{8} + 32 T^{6} + \cdots + 121 \)
|
| $19$ |
\( T^{8} + 4 T^{7} + \cdots + 421201 \)
|
| $23$ |
\( (T^{4} - 31 T^{2} + \cdots - 71)^{2} \)
|
| $29$ |
\( T^{8} - 2 T^{7} + \cdots + 274576 \)
|
| $31$ |
\( T^{8} + 4 T^{7} + \cdots + 421201 \)
|
| $37$ |
\( T^{8} + 24 T^{7} + \cdots + 4157521 \)
|
| $41$ |
\( T^{8} + 32 T^{6} + \cdots + 121 \)
|
| $43$ |
\( (T^{4} - 4 T^{3} + \cdots + 244)^{2} \)
|
| $47$ |
\( T^{8} + 2 T^{7} + \cdots + 62031376 \)
|
| $53$ |
\( T^{8} + 22 T^{7} + \cdots + 4648336 \)
|
| $59$ |
\( T^{8} - 10 T^{7} + \cdots + 59536 \)
|
| $61$ |
\( T^{8} - 26 T^{7} + \cdots + 56010256 \)
|
| $67$ |
\( (T^{4} - 2 T^{3} + \cdots + 2596)^{2} \)
|
| $71$ |
\( T^{8} + 16 T^{7} + \cdots + 952576 \)
|
| $73$ |
\( T^{8} + 2 T^{7} + \cdots + 28344976 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots + 156816 \)
|
| $83$ |
\( T^{8} + 38 T^{7} + \cdots + 2835856 \)
|
| $89$ |
\( (T^{4} + 6 T^{3} + \cdots + 229)^{2} \)
|
| $97$ |
\( T^{8} - 6 T^{7} + \cdots + 1008016 \)
|
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