Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [483,2,Mod(160,483)]
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([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("483.160");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 483 = 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 483.h (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: | 8.0.342102016.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 3\nu^{5} + 5\nu^{4} + 2\nu^{3} - 2\nu^{2} + 8 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{4} + 8\nu^{3} - 10\nu^{2} + 24\nu - 8 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + \nu^{5} - 3\nu^{4} - 6\nu^{3} - 10\nu^{2} + 8\nu - 8 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - \nu^{5} - 3\nu^{4} + 6\nu^{3} - 10\nu^{2} - 8\nu - 8 ) / 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 3\nu^{4} + 6\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} - 3\nu^{5} + 5\nu^{4} - 2\nu^{3} - 2\nu^{2} + 8 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} - \beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{2} - 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -4\beta_{7} - \beta_{5} + 2\beta_{4} - \beta_{3} + 4\beta_{2} + 5\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -6\beta_{7} - 7\beta_{6} - 3\beta_{5} - 3\beta_{4} - 6\beta_{2} - 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -4\beta_{7} + \beta_{5} - 2\beta_{4} + \beta_{3} + 4\beta_{2} - 21\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 160.1 |
|
−1.00000 | − | 1.00000i | −1.00000 | −3.33513 | 1.00000i | −2.60399 | + | 0.468213i | 3.00000 | −1.00000 | 3.33513 | |||||||||||||||||||||||||||||||||||||||
| 160.2 | −1.00000 | − | 1.00000i | −1.00000 | −1.69614 | 1.00000i | 2.17238 | − | 1.51022i | 3.00000 | −1.00000 | 1.69614 | ||||||||||||||||||||||||||||||||||||||||
| 160.3 | −1.00000 | − | 1.00000i | −1.00000 | 1.69614 | 1.00000i | −2.17238 | + | 1.51022i | 3.00000 | −1.00000 | −1.69614 | ||||||||||||||||||||||||||||||||||||||||
| 160.4 | −1.00000 | − | 1.00000i | −1.00000 | 3.33513 | 1.00000i | 2.60399 | − | 0.468213i | 3.00000 | −1.00000 | −3.33513 | ||||||||||||||||||||||||||||||||||||||||
| 160.5 | −1.00000 | 1.00000i | −1.00000 | −3.33513 | − | 1.00000i | −2.60399 | − | 0.468213i | 3.00000 | −1.00000 | 3.33513 | ||||||||||||||||||||||||||||||||||||||||
| 160.6 | −1.00000 | 1.00000i | −1.00000 | −1.69614 | − | 1.00000i | 2.17238 | + | 1.51022i | 3.00000 | −1.00000 | 1.69614 | ||||||||||||||||||||||||||||||||||||||||
| 160.7 | −1.00000 | 1.00000i | −1.00000 | 1.69614 | − | 1.00000i | −2.17238 | − | 1.51022i | 3.00000 | −1.00000 | −1.69614 | ||||||||||||||||||||||||||||||||||||||||
| 160.8 | −1.00000 | 1.00000i | −1.00000 | 3.33513 | − | 1.00000i | 2.60399 | + | 0.468213i | 3.00000 | −1.00000 | −3.33513 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
| 161.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.h.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1449.2.h.d | 8 | ||
| 7.b | odd | 2 | 1 | inner | 483.2.h.a | ✓ | 8 |
| 21.c | even | 2 | 1 | 1449.2.h.d | 8 | ||
| 23.b | odd | 2 | 1 | inner | 483.2.h.a | ✓ | 8 |
| 69.c | even | 2 | 1 | 1449.2.h.d | 8 | ||
| 161.c | even | 2 | 1 | inner | 483.2.h.a | ✓ | 8 |
| 483.c | odd | 2 | 1 | 1449.2.h.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.2.h.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 483.2.h.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 483.2.h.a | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
| 483.2.h.a | ✓ | 8 | 161.c | even | 2 | 1 | inner |
| 1449.2.h.d | 8 | 3.b | odd | 2 | 1 | ||
| 1449.2.h.d | 8 | 21.c | even | 2 | 1 | ||
| 1449.2.h.d | 8 | 69.c | even | 2 | 1 | ||
| 1449.2.h.d | 8 | 483.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( (T^{4} - 14 T^{2} + 32)^{2} \)
|
| $7$ |
\( T^{8} - 18 T^{6} + \cdots + 2401 \)
|
| $11$ |
\( (T^{4} + 20 T^{2} + 32)^{2} \)
|
| $13$ |
\( (T^{4} + 36 T^{2} + 256)^{2} \)
|
| $17$ |
\( (T^{4} - 14 T^{2} + 32)^{2} \)
|
| $19$ |
\( (T^{4} - 62 T^{2} + 128)^{2} \)
|
| $23$ |
\( (T^{4} + 2 T^{3} + \cdots + 529)^{2} \)
|
| $29$ |
\( (T^{2} - 68)^{4} \)
|
| $31$ |
\( (T^{4} + 144 T^{2} + 4096)^{2} \)
|
| $37$ |
\( (T^{4} + 40 T^{2} + 128)^{2} \)
|
| $41$ |
\( (T^{4} + 132 T^{2} + 1024)^{2} \)
|
| $43$ |
\( (T^{4} + 10 T^{2} + 8)^{2} \)
|
| $47$ |
\( (T^{4} + 52 T^{2} + 64)^{2} \)
|
| $53$ |
\( (T^{4} + 126 T^{2} + 2592)^{2} \)
|
| $59$ |
\( (T^{4} + 52 T^{2} + 64)^{2} \)
|
| $61$ |
\( (T^{4} - 40 T^{2} + 128)^{2} \)
|
| $67$ |
\( (T^{4} + 250 T^{2} + 5000)^{2} \)
|
| $71$ |
\( (T^{2} + 14 T + 32)^{4} \)
|
| $73$ |
\( (T^{4} + 144 T^{2} + 4096)^{2} \)
|
| $79$ |
\( (T^{4} + 266 T^{2} + 1352)^{2} \)
|
| $83$ |
\( (T^{4} - 320 T^{2} + 8192)^{2} \)
|
| $89$ |
\( (T^{4} - 126 T^{2} + 2592)^{2} \)
|
| $97$ |
\( (T^{4} - 40 T^{2} + 128)^{2} \)
|
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