Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1045,1,Mod(284,1045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 6, 5]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1045.284");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1045 = 5 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1045.w (of order \(10\), 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: | \(0.521522938201\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\Q(\zeta_{40})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{20}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).
| \(n\) | \(496\) | \(761\) | \(837\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{40}^{8}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 284.1 |
|
−0.610425 | + | 1.87869i | −0.734572 | + | 0.533698i | −2.34786 | − | 1.70582i | −0.309017 | − | 0.951057i | −0.554254 | − | 1.70582i | 0 | 3.03979 | − | 2.20854i | −0.0542543 | + | 0.166977i | 1.97538 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 284.2 | −0.0966818 | + | 0.297556i | 1.44168 | − | 1.04744i | 0.729825 | + | 0.530249i | −0.309017 | − | 0.951057i | 0.172288 | + | 0.530249i | 0 | −0.481456 | + | 0.349798i | 0.672288 | − | 2.06909i | 0.312869 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 284.3 | 0.0966818 | − | 0.297556i | −1.44168 | + | 1.04744i | 0.729825 | + | 0.530249i | −0.309017 | − | 0.951057i | 0.172288 | + | 0.530249i | 0 | 0.481456 | − | 0.349798i | 0.672288 | − | 2.06909i | −0.312869 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 284.4 | 0.610425 | − | 1.87869i | 0.734572 | − | 0.533698i | −2.34786 | − | 1.70582i | −0.309017 | − | 0.951057i | −0.554254 | − | 1.70582i | 0 | −3.03979 | + | 2.20854i | −0.0542543 | + | 0.166977i | −1.97538 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.1 | −0.610425 | − | 1.87869i | −0.734572 | − | 0.533698i | −2.34786 | + | 1.70582i | −0.309017 | + | 0.951057i | −0.554254 | + | 1.70582i | 0 | 3.03979 | + | 2.20854i | −0.0542543 | − | 0.166977i | 1.97538 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.2 | −0.0966818 | − | 0.297556i | 1.44168 | + | 1.04744i | 0.729825 | − | 0.530249i | −0.309017 | + | 0.951057i | 0.172288 | − | 0.530249i | 0 | −0.481456 | − | 0.349798i | 0.672288 | + | 2.06909i | 0.312869 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.3 | 0.0966818 | + | 0.297556i | −1.44168 | − | 1.04744i | 0.729825 | − | 0.530249i | −0.309017 | + | 0.951057i | 0.172288 | − | 0.530249i | 0 | 0.481456 | + | 0.349798i | 0.672288 | + | 2.06909i | −0.312869 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.4 | 0.610425 | + | 1.87869i | 0.734572 | + | 0.533698i | −2.34786 | + | 1.70582i | −0.309017 | + | 0.951057i | −0.554254 | + | 1.70582i | 0 | −3.03979 | − | 2.20854i | −0.0542543 | − | 0.166977i | −1.97538 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 664.1 | −1.44168 | − | 1.04744i | 0.610425 | − | 1.87869i | 0.672288 | + | 2.06909i | 0.809017 | − | 0.587785i | −2.84786 | + | 2.06909i | 0 | 0.647354 | − | 1.99235i | −2.34786 | − | 1.70582i | −1.78201 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 664.2 | −0.734572 | − | 0.533698i | −0.0966818 | + | 0.297556i | −0.0542543 | − | 0.166977i | 0.809017 | − | 0.587785i | 0.229825 | − | 0.166977i | 0 | −0.329843 | + | 1.01515i | 0.729825 | + | 0.530249i | −0.907981 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 664.3 | 0.734572 | + | 0.533698i | 0.0966818 | − | 0.297556i | −0.0542543 | − | 0.166977i | 0.809017 | − | 0.587785i | 0.229825 | − | 0.166977i | 0 | 0.329843 | − | 1.01515i | 0.729825 | + | 0.530249i | 0.907981 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 664.4 | 1.44168 | + | 1.04744i | −0.610425 | + | 1.87869i | 0.672288 | + | 2.06909i | 0.809017 | − | 0.587785i | −2.84786 | + | 2.06909i | 0 | −0.647354 | + | 1.99235i | −2.34786 | − | 1.70582i | 1.78201 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 949.1 | −1.44168 | + | 1.04744i | 0.610425 | + | 1.87869i | 0.672288 | − | 2.06909i | 0.809017 | + | 0.587785i | −2.84786 | − | 2.06909i | 0 | 0.647354 | + | 1.99235i | −2.34786 | + | 1.70582i | −1.78201 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 949.2 | −0.734572 | + | 0.533698i | −0.0966818 | − | 0.297556i | −0.0542543 | + | 0.166977i | 0.809017 | + | 0.587785i | 0.229825 | + | 0.166977i | 0 | −0.329843 | − | 1.01515i | 0.729825 | − | 0.530249i | −0.907981 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 949.3 | 0.734572 | − | 0.533698i | 0.0966818 | + | 0.297556i | −0.0542543 | + | 0.166977i | 0.809017 | + | 0.587785i | 0.229825 | + | 0.166977i | 0 | 0.329843 | + | 1.01515i | 0.729825 | − | 0.530249i | 0.907981 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 949.4 | 1.44168 | − | 1.04744i | −0.610425 | − | 1.87869i | 0.672288 | − | 2.06909i | 0.809017 | + | 0.587785i | −2.84786 | − | 2.06909i | 0 | −0.647354 | − | 1.99235i | −2.34786 | + | 1.70582i | 1.78201 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 95.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-95}) \) |
| 5.b | even | 2 | 1 | inner |
| 11.c | even | 5 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
| 55.j | even | 10 | 1 | inner |
| 209.m | odd | 10 | 1 | inner |
| 1045.w | odd | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1045.1.w.f | ✓ | 16 |
| 5.b | even | 2 | 1 | inner | 1045.1.w.f | ✓ | 16 |
| 11.c | even | 5 | 1 | inner | 1045.1.w.f | ✓ | 16 |
| 19.b | odd | 2 | 1 | inner | 1045.1.w.f | ✓ | 16 |
| 55.j | even | 10 | 1 | inner | 1045.1.w.f | ✓ | 16 |
| 95.d | odd | 2 | 1 | CM | 1045.1.w.f | ✓ | 16 |
| 209.m | odd | 10 | 1 | inner | 1045.1.w.f | ✓ | 16 |
| 1045.w | odd | 10 | 1 | inner | 1045.1.w.f | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1045.1.w.f | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 1045.1.w.f | ✓ | 16 | 5.b | even | 2 | 1 | inner |
| 1045.1.w.f | ✓ | 16 | 11.c | even | 5 | 1 | inner |
| 1045.1.w.f | ✓ | 16 | 19.b | odd | 2 | 1 | inner |
| 1045.1.w.f | ✓ | 16 | 55.j | even | 10 | 1 | inner |
| 1045.1.w.f | ✓ | 16 | 95.d | odd | 2 | 1 | CM |
| 1045.1.w.f | ✓ | 16 | 209.m | odd | 10 | 1 | inner |
| 1045.1.w.f | ✓ | 16 | 1045.w | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1045, [\chi])\):
|
\( T_{2}^{16} + 4T_{2}^{14} + 12T_{2}^{12} + 32T_{2}^{10} + 150T_{2}^{8} - 32T_{2}^{6} + 97T_{2}^{4} + 16T_{2}^{2} + 1 \)
|
|
\( T_{7} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $3$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $5$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \)
|
| $13$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $17$ |
\( T^{16} \)
|
| $19$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \)
|
| $23$ |
\( T^{16} \)
|
| $29$ |
\( T^{16} \)
|
| $31$ |
\( T^{16} \)
|
| $37$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
| $41$ |
\( T^{16} \)
|
| $43$ |
\( T^{16} \)
|
| $47$ |
\( T^{16} \)
|
| $53$ |
\( (T^{8} + 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \)
|
| $59$ |
\( T^{16} \)
|
| $61$ |
\( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \)
|
| $67$ |
\( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \)
|
| $71$ |
\( T^{16} \)
|
| $73$ |
\( T^{16} \)
|
| $79$ |
\( T^{16} \)
|
| $83$ |
\( T^{16} \)
|
| $89$ |
\( T^{16} \)
|
| $97$ |
\( T^{16} + 4 T^{14} + \cdots + 1 \)
|
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