Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [483,2,Mod(461,483)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(483, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("483.461");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 483 = 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 483.d (of order \(2\), degree \(1\), 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: | \(3.85677441763\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| 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: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).
| \(n\) | \(323\) | \(346\) | \(442\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 461.1 |
|
− | 1.00000i | −0.923880 | + | 1.46508i | 1.00000 | 3.37849 | 1.46508 | + | 0.923880i | −2.41421 | − | 1.08239i | − | 3.00000i | −1.29289 | − | 2.70711i | − | 3.37849i | |||||||||||||||||||||||||||||||
| 461.2 | − | 1.00000i | −0.382683 | + | 1.68925i | 1.00000 | −2.93015 | 1.68925 | + | 0.382683i | 0.414214 | − | 2.61313i | − | 3.00000i | −2.70711 | − | 1.29289i | 2.93015i | |||||||||||||||||||||||||||||||||
| 461.3 | − | 1.00000i | 0.382683 | − | 1.68925i | 1.00000 | 2.93015 | −1.68925 | − | 0.382683i | 0.414214 | + | 2.61313i | − | 3.00000i | −2.70711 | − | 1.29289i | − | 2.93015i | ||||||||||||||||||||||||||||||||
| 461.4 | − | 1.00000i | 0.923880 | − | 1.46508i | 1.00000 | −3.37849 | −1.46508 | − | 0.923880i | −2.41421 | + | 1.08239i | − | 3.00000i | −1.29289 | − | 2.70711i | 3.37849i | |||||||||||||||||||||||||||||||||
| 461.5 | 1.00000i | −0.923880 | − | 1.46508i | 1.00000 | 3.37849 | 1.46508 | − | 0.923880i | −2.41421 | + | 1.08239i | 3.00000i | −1.29289 | + | 2.70711i | 3.37849i | |||||||||||||||||||||||||||||||||||
| 461.6 | 1.00000i | −0.382683 | − | 1.68925i | 1.00000 | −2.93015 | 1.68925 | − | 0.382683i | 0.414214 | + | 2.61313i | 3.00000i | −2.70711 | + | 1.29289i | − | 2.93015i | ||||||||||||||||||||||||||||||||||
| 461.7 | 1.00000i | 0.382683 | + | 1.68925i | 1.00000 | 2.93015 | −1.68925 | + | 0.382683i | 0.414214 | − | 2.61313i | 3.00000i | −2.70711 | + | 1.29289i | 2.93015i | |||||||||||||||||||||||||||||||||||
| 461.8 | 1.00000i | 0.923880 | + | 1.46508i | 1.00000 | −3.37849 | −1.46508 | + | 0.923880i | −2.41421 | − | 1.08239i | 3.00000i | −1.29289 | + | 2.70711i | − | 3.37849i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 483.2.d.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 483.2.d.c | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 483.2.d.c | ✓ | 8 |
| 21.c | even | 2 | 1 | inner | 483.2.d.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.2.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 483.2.d.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 483.2.d.c | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 483.2.d.c | ✓ | 8 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):
|
\( T_{2}^{2} + 1 \)
|
|
\( T_{5}^{4} - 20T_{5}^{2} + 98 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
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| $3$ |
\( T^{8} + 8 T^{6} + \cdots + 81 \)
|
| $5$ |
\( (T^{4} - 20 T^{2} + 98)^{2} \)
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| $7$ |
\( (T^{4} + 4 T^{3} + 10 T^{2} + \cdots + 49)^{2} \)
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| $11$ |
\( (T^{4} + 12 T^{2} + 4)^{2} \)
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| $13$ |
\( (T^{4} + 16 T^{2} + 32)^{2} \)
|
| $17$ |
\( (T^{4} - 20 T^{2} + 2)^{2} \)
|
| $19$ |
\( (T^{4} + 32 T^{2} + 128)^{2} \)
|
| $23$ |
\( (T^{2} + 1)^{4} \)
|
| $29$ |
\( (T^{2} + 18)^{4} \)
|
| $31$ |
\( (T^{4} + 68 T^{2} + 1058)^{2} \)
|
| $37$ |
\( (T^{2} - 4 T - 68)^{4} \)
|
| $41$ |
\( (T^{4} - 160 T^{2} + 6272)^{2} \)
|
| $43$ |
\( (T^{2} + 8 T - 16)^{4} \)
|
| $47$ |
\( (T^{4} - 52 T^{2} + 578)^{2} \)
|
| $53$ |
\( (T^{4} + 24 T^{2} + 16)^{2} \)
|
| $59$ |
\( (T^{4} - 100 T^{2} + 1922)^{2} \)
|
| $61$ |
\( (T^{4} + 244 T^{2} + 10082)^{2} \)
|
| $67$ |
\( (T^{2} - 4 T - 158)^{4} \)
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| $71$ |
\( (T^{4} + 96 T^{2} + 256)^{2} \)
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| $73$ |
\( (T^{4} + 80 T^{2} + 1568)^{2} \)
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| $79$ |
\( (T^{2} - 4 T - 158)^{4} \)
|
| $83$ |
\( (T^{4} - 80 T^{2} + 32)^{2} \)
|
| $89$ |
\( (T^{4} - 180 T^{2} + 7938)^{2} \)
|
| $97$ |
\( (T^{4} + 148 T^{2} + 1058)^{2} \)
|
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