Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,6,Mod(17,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.17");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.f (of order \(4\), degree \(2\), 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: | \(14.4345437832\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(-\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
−2.82843 | + | 2.82843i | 0 | − | 16.0000i | 17.6777 | − | 53.0330i | 0 | −22.0000 | − | 22.0000i | 45.2548 | + | 45.2548i | 0 | 100.000 | + | 200.000i | |||||||||||||||||||
| 17.2 | 2.82843 | − | 2.82843i | 0 | − | 16.0000i | −17.6777 | + | 53.0330i | 0 | −22.0000 | − | 22.0000i | −45.2548 | − | 45.2548i | 0 | 100.000 | + | 200.000i | ||||||||||||||||||||
| 53.1 | −2.82843 | − | 2.82843i | 0 | 16.0000i | 17.6777 | + | 53.0330i | 0 | −22.0000 | + | 22.0000i | 45.2548 | − | 45.2548i | 0 | 100.000 | − | 200.000i | |||||||||||||||||||||
| 53.2 | 2.82843 | + | 2.82843i | 0 | 16.0000i | −17.6777 | − | 53.0330i | 0 | −22.0000 | + | 22.0000i | −45.2548 | + | 45.2548i | 0 | 100.000 | − | 200.000i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 90.6.f.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 90.6.f.b | ✓ | 4 |
| 5.b | even | 2 | 1 | 450.6.f.b | 4 | ||
| 5.c | odd | 4 | 1 | inner | 90.6.f.b | ✓ | 4 |
| 5.c | odd | 4 | 1 | 450.6.f.b | 4 | ||
| 15.d | odd | 2 | 1 | 450.6.f.b | 4 | ||
| 15.e | even | 4 | 1 | inner | 90.6.f.b | ✓ | 4 |
| 15.e | even | 4 | 1 | 450.6.f.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.6.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 90.6.f.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 90.6.f.b | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 90.6.f.b | ✓ | 4 | 15.e | even | 4 | 1 | inner |
| 450.6.f.b | 4 | 5.b | even | 2 | 1 | ||
| 450.6.f.b | 4 | 5.c | odd | 4 | 1 | ||
| 450.6.f.b | 4 | 15.d | odd | 2 | 1 | ||
| 450.6.f.b | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 44T_{7} + 968 \)
acting on \(S_{6}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 256 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 5000 T^{2} + 9765625 \)
|
| $7$ |
\( (T^{2} + 44 T + 968)^{2} \)
|
| $11$ |
\( (T^{2} + 29768)^{2} \)
|
| $13$ |
\( (T^{2} + 1674 T + 1401138)^{2} \)
|
| $17$ |
\( T^{4} + 6690585616 \)
|
| $19$ |
\( (T^{2} + 1024144)^{2} \)
|
| $23$ |
\( T^{4} + 194817277628416 \)
|
| $29$ |
\( (T^{2} - 7612802)^{2} \)
|
| $31$ |
\( (T + 4136)^{4} \)
|
| $37$ |
\( (T^{2} + 5334 T + 14225778)^{2} \)
|
| $41$ |
\( (T^{2} + 107839298)^{2} \)
|
| $43$ |
\( (T^{2} + 21120 T + 223027200)^{2} \)
|
| $47$ |
\( T^{4} + 24\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + 13\!\cdots\!36 \)
|
| $59$ |
\( (T^{2} - 605659208)^{2} \)
|
| $61$ |
\( (T + 11040)^{4} \)
|
| $67$ |
\( (T^{2} - 55888 T + 1561734272)^{2} \)
|
| $71$ |
\( (T^{2} + 1799520032)^{2} \)
|
| $73$ |
\( (T^{2} - 5678 T + 16119842)^{2} \)
|
| $79$ |
\( (T^{2} + 9739321344)^{2} \)
|
| $83$ |
\( T^{4} + 47\!\cdots\!16 \)
|
| $89$ |
\( (T^{2} - 13207350338)^{2} \)
|
| $97$ |
\( (T^{2} - 46086 T + 1061959698)^{2} \)
|
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