Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,2,Mod(97,490)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("490.97");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.g (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: | \(3.91266969904\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| 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 \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{16}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
−0.707107 | − | 0.707107i | −0.765367 | − | 0.765367i | 1.00000i | 0.158513 | + | 2.23044i | 1.08239i | 0 | 0.707107 | − | 0.707107i | − | 1.82843i | 1.46508 | − | 1.68925i | |||||||||||||||||||||||||||||||
| 97.2 | −0.707107 | − | 0.707107i | 0.765367 | + | 0.765367i | 1.00000i | −0.158513 | − | 2.23044i | − | 1.08239i | 0 | 0.707107 | − | 0.707107i | − | 1.82843i | −1.46508 | + | 1.68925i | |||||||||||||||||||||||||||||||
| 97.3 | 0.707107 | + | 0.707107i | −1.84776 | − | 1.84776i | 1.00000i | −2.23044 | + | 0.158513i | − | 2.61313i | 0 | −0.707107 | + | 0.707107i | 3.82843i | −1.68925 | − | 1.46508i | ||||||||||||||||||||||||||||||||
| 97.4 | 0.707107 | + | 0.707107i | 1.84776 | + | 1.84776i | 1.00000i | 2.23044 | − | 0.158513i | 2.61313i | 0 | −0.707107 | + | 0.707107i | 3.82843i | 1.68925 | + | 1.46508i | |||||||||||||||||||||||||||||||||
| 293.1 | −0.707107 | + | 0.707107i | −0.765367 | + | 0.765367i | − | 1.00000i | 0.158513 | − | 2.23044i | − | 1.08239i | 0 | 0.707107 | + | 0.707107i | 1.82843i | 1.46508 | + | 1.68925i | |||||||||||||||||||||||||||||||
| 293.2 | −0.707107 | + | 0.707107i | 0.765367 | − | 0.765367i | − | 1.00000i | −0.158513 | + | 2.23044i | 1.08239i | 0 | 0.707107 | + | 0.707107i | 1.82843i | −1.46508 | − | 1.68925i | ||||||||||||||||||||||||||||||||
| 293.3 | 0.707107 | − | 0.707107i | −1.84776 | + | 1.84776i | − | 1.00000i | −2.23044 | − | 0.158513i | 2.61313i | 0 | −0.707107 | − | 0.707107i | − | 3.82843i | −1.68925 | + | 1.46508i | |||||||||||||||||||||||||||||||
| 293.4 | 0.707107 | − | 0.707107i | 1.84776 | − | 1.84776i | − | 1.00000i | 2.23044 | + | 0.158513i | − | 2.61313i | 0 | −0.707107 | − | 0.707107i | − | 3.82843i | 1.68925 | − | 1.46508i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 490.2.g.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 490.2.g.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 490.2.g.a | ✓ | 8 |
| 7.c | even | 3 | 2 | 490.2.l.b | 16 | ||
| 7.d | odd | 6 | 2 | 490.2.l.b | 16 | ||
| 35.f | even | 4 | 1 | inner | 490.2.g.a | ✓ | 8 |
| 35.k | even | 12 | 2 | 490.2.l.b | 16 | ||
| 35.l | odd | 12 | 2 | 490.2.l.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.2.g.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 490.2.g.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 490.2.g.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 490.2.g.a | ✓ | 8 | 35.f | even | 4 | 1 | inner |
| 490.2.l.b | 16 | 7.c | even | 3 | 2 | ||
| 490.2.l.b | 16 | 7.d | odd | 6 | 2 | ||
| 490.2.l.b | 16 | 35.k | even | 12 | 2 | ||
| 490.2.l.b | 16 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 48T_{3}^{4} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( T^{8} + 48T^{4} + 64 \)
|
| $5$ |
\( T^{8} - 48T^{4} + 625 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{2} + 4 T - 4)^{4} \)
|
| $13$ |
\( T^{8} + 1548 T^{4} + 334084 \)
|
| $17$ |
\( T^{8} + 1548 T^{4} + 334084 \)
|
| $19$ |
\( (T^{4} - 72 T^{2} + 648)^{2} \)
|
| $23$ |
\( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
|
| $29$ |
\( (T^{4} + 44 T^{2} + 196)^{2} \)
|
| $31$ |
\( (T^{4} + 16 T^{2} + 32)^{2} \)
|
| $37$ |
\( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 64)^{2} \)
|
| $41$ |
\( (T^{4} + 116 T^{2} + 2)^{2} \)
|
| $43$ |
\( (T^{4} + 24 T^{3} + \cdots + 4624)^{2} \)
|
| $47$ |
\( T^{8} + 3264 T^{4} + 2458624 \)
|
| $53$ |
\( (T^{4} + 4 T^{3} + \cdots + 1156)^{2} \)
|
| $59$ |
\( (T^{4} - 72 T^{2} + 648)^{2} \)
|
| $61$ |
\( (T^{4} + 52 T^{2} + 578)^{2} \)
|
| $67$ |
\( (T^{2} - 12 T + 72)^{4} \)
|
| $71$ |
\( (T^{2} + 4 T - 68)^{4} \)
|
| $73$ |
\( T^{8} + 49932 T^{4} + 23059204 \)
|
| $79$ |
\( (T^{4} + 264 T^{2} + 4624)^{2} \)
|
| $83$ |
\( T^{8} + 35376 T^{4} + 5345344 \)
|
| $89$ |
\( (T^{4} - 340 T^{2} + 18818)^{2} \)
|
| $97$ |
\( T^{8} + 4428 T^{4} + 9604 \)
|
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