Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,1,Mod(14,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([15, 15, 22]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("465.14");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 465.bl (of order \(30\), degree \(8\), 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: | \(0.232065230874\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{15})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{15}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(406\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\zeta_{30}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 14.1 |
|
0.0646021 | − | 0.198825i | −0.669131 | + | 0.743145i | 0.773659 | + | 0.562096i | 0.500000 | − | 0.866025i | 0.104528 | + | 0.181049i | 0 | 0.330869 | − | 0.240391i | −0.104528 | − | 0.994522i | −0.139886 | − | 0.155360i | ||||||||||||||||||||||||||
| 59.1 | −1.58268 | + | 1.14988i | 0.104528 | − | 0.994522i | 0.873619 | − | 2.68872i | 0.500000 | − | 0.866025i | 0.978148 | + | 1.69420i | 0 | 1.10453 | + | 3.39939i | −0.978148 | − | 0.207912i | 0.204489 | + | 1.94558i | |||||||||||||||||||||||||||
| 134.1 | −1.58268 | − | 1.14988i | 0.104528 | + | 0.994522i | 0.873619 | + | 2.68872i | 0.500000 | + | 0.866025i | 0.978148 | − | 1.69420i | 0 | 1.10453 | − | 3.39939i | −0.978148 | + | 0.207912i | 0.204489 | − | 1.94558i | |||||||||||||||||||||||||||
| 164.1 | −0.564602 | + | 1.73767i | 0.978148 | + | 0.207912i | −1.89169 | − | 1.37440i | 0.500000 | + | 0.866025i | −0.913545 | + | 1.58231i | 0 | 1.97815 | − | 1.43721i | 0.913545 | + | 0.406737i | −1.78716 | + | 0.379874i | |||||||||||||||||||||||||||
| 224.1 | −0.564602 | − | 1.73767i | 0.978148 | − | 0.207912i | −1.89169 | + | 1.37440i | 0.500000 | − | 0.866025i | −0.913545 | − | 1.58231i | 0 | 1.97815 | + | 1.43721i | 0.913545 | − | 0.406737i | −1.78716 | − | 0.379874i | |||||||||||||||||||||||||||
| 299.1 | 0.0646021 | + | 0.198825i | −0.669131 | − | 0.743145i | 0.773659 | − | 0.562096i | 0.500000 | + | 0.866025i | 0.104528 | − | 0.181049i | 0 | 0.330869 | + | 0.240391i | −0.104528 | + | 0.994522i | −0.139886 | + | 0.155360i | |||||||||||||||||||||||||||
| 329.1 | 1.08268 | + | 0.786610i | −0.913545 | − | 0.406737i | 0.244415 | + | 0.752232i | 0.500000 | − | 0.866025i | −0.669131 | − | 1.15897i | 0 | 0.0864545 | − | 0.266080i | 0.669131 | + | 0.743145i | 1.22256 | − | 0.544320i | |||||||||||||||||||||||||||
| 359.1 | 1.08268 | − | 0.786610i | −0.913545 | + | 0.406737i | 0.244415 | − | 0.752232i | 0.500000 | + | 0.866025i | −0.669131 | + | 1.15897i | 0 | 0.0864545 | + | 0.266080i | 0.669131 | − | 0.743145i | 1.22256 | + | 0.544320i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
| 31.g | even | 15 | 1 | inner |
| 465.bl | odd | 30 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 465.1.bl.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 465.1.bl.b | yes | 8 | |
| 5.b | even | 2 | 1 | 465.1.bl.b | yes | 8 | |
| 5.c | odd | 4 | 2 | 2325.1.fd.c | 16 | ||
| 15.d | odd | 2 | 1 | CM | 465.1.bl.a | ✓ | 8 |
| 15.e | even | 4 | 2 | 2325.1.fd.c | 16 | ||
| 31.g | even | 15 | 1 | inner | 465.1.bl.a | ✓ | 8 |
| 93.o | odd | 30 | 1 | 465.1.bl.b | yes | 8 | |
| 155.u | even | 30 | 1 | 465.1.bl.b | yes | 8 | |
| 155.w | odd | 60 | 2 | 2325.1.fd.c | 16 | ||
| 465.bl | odd | 30 | 1 | inner | 465.1.bl.a | ✓ | 8 |
| 465.bt | even | 60 | 2 | 2325.1.fd.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.1.bl.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 465.1.bl.a | ✓ | 8 | 15.d | odd | 2 | 1 | CM |
| 465.1.bl.a | ✓ | 8 | 31.g | even | 15 | 1 | inner |
| 465.1.bl.a | ✓ | 8 | 465.bl | odd | 30 | 1 | inner |
| 465.1.bl.b | yes | 8 | 3.b | odd | 2 | 1 | |
| 465.1.bl.b | yes | 8 | 5.b | even | 2 | 1 | |
| 465.1.bl.b | yes | 8 | 93.o | odd | 30 | 1 | |
| 465.1.bl.b | yes | 8 | 155.u | even | 30 | 1 | |
| 2325.1.fd.c | 16 | 5.c | odd | 4 | 2 | ||
| 2325.1.fd.c | 16 | 15.e | even | 4 | 2 | ||
| 2325.1.fd.c | 16 | 155.w | odd | 60 | 2 | ||
| 2325.1.fd.c | 16 | 465.bt | even | 60 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 3T_{2}^{6} - T_{2}^{5} - T_{2}^{3} + 23T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(465, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $3$ |
\( T^{8} + T^{7} - T^{5} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} - T + 1)^{4} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} - T^{7} + T^{5} + \cdots + 1 \)
|
| $19$ |
\( T^{8} + 6 T^{7} + \cdots + 1 \)
|
| $23$ |
\( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $53$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( (T^{4} - T^{3} - 4 T^{2} + \cdots + 1)^{2} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( T^{8} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \)
|
| $83$ |
\( T^{8} - T^{7} - 4 T^{5} + \cdots + 1 \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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