Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1638,2,Mod(755,1638)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1638, 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("1638.755");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1638.g (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: | \(13.0794958511\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.836829184.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} + 7\nu^{2} + 6 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 11\nu^{3} + 26\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{5} - 10\nu^{4} - 18\nu^{3} - 27\nu^{2} - 32\nu - 18 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} + 10\nu^{4} - 18\nu^{3} + 27\nu^{2} - 32\nu + 18 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 14\nu^{5} + 10\nu^{4} + 59\nu^{3} + 19\nu^{2} + 78\nu - 14 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 14\nu^{5} - 10\nu^{4} + 59\nu^{3} - 19\nu^{2} + 78\nu + 14 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{7} - 5\beta_{6} - 3\beta_{5} - 3\beta_{4} + 10\beta_{3} + 4\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -7\beta_{7} + 7\beta_{6} - 7\beta_{5} + 7\beta_{4} + 8\beta _1 + 44 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 29\beta_{7} + 29\beta_{6} + 7\beta_{5} + 7\beta_{4} - 58\beta_{3} - 36\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 43\beta_{7} - 43\beta_{6} + 51\beta_{5} - 51\beta_{4} - 80\beta _1 - 260 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -181\beta_{7} - 181\beta_{6} + \beta_{5} + \beta_{4} + 378\beta_{3} + 268\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).
| \(n\) | \(379\) | \(703\) | \(911\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 755.1 |
|
− | 1.00000i | 0 | −1.00000 | −3.72844 | 0 | 1.15711 | − | 2.37930i | 1.00000i | 0 | 3.72844i | |||||||||||||||||||||||||||||||||||||||
| 755.2 | − | 1.00000i | 0 | −1.00000 | −2.92238 | 0 | 2.16830 | − | 1.51608i | 1.00000i | 0 | 2.92238i | ||||||||||||||||||||||||||||||||||||||||
| 755.3 | − | 1.00000i | 0 | −1.00000 | 0.314226 | 0 | −0.864220 | + | 2.50062i | 1.00000i | 0 | − | 0.314226i | |||||||||||||||||||||||||||||||||||||||
| 755.4 | − | 1.00000i | 0 | −1.00000 | 2.33660 | 0 | −0.461191 | − | 2.60525i | 1.00000i | 0 | − | 2.33660i | |||||||||||||||||||||||||||||||||||||||
| 755.5 | 1.00000i | 0 | −1.00000 | −3.72844 | 0 | 1.15711 | + | 2.37930i | − | 1.00000i | 0 | − | 3.72844i | |||||||||||||||||||||||||||||||||||||||
| 755.6 | 1.00000i | 0 | −1.00000 | −2.92238 | 0 | 2.16830 | + | 1.51608i | − | 1.00000i | 0 | − | 2.92238i | |||||||||||||||||||||||||||||||||||||||
| 755.7 | 1.00000i | 0 | −1.00000 | 0.314226 | 0 | −0.864220 | − | 2.50062i | − | 1.00000i | 0 | 0.314226i | ||||||||||||||||||||||||||||||||||||||||
| 755.8 | 1.00000i | 0 | −1.00000 | 2.33660 | 0 | −0.461191 | + | 2.60525i | − | 1.00000i | 0 | 2.33660i | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 1638.2.g.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1638.2.g.d | yes | 8 | |
| 7.b | odd | 2 | 1 | 1638.2.g.d | yes | 8 | |
| 21.c | even | 2 | 1 | inner | 1638.2.g.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1638.2.g.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1638.2.g.b | ✓ | 8 | 21.c | even | 2 | 1 | inner |
| 1638.2.g.d | yes | 8 | 3.b | odd | 2 | 1 | |
| 1638.2.g.d | yes | 8 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 4T_{5}^{3} - 6T_{5}^{2} - 24T_{5} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 4 T^{3} - 6 T^{2} + \cdots + 8)^{2} \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} + 56 T^{6} + \cdots + 4096 \)
|
| $13$ |
\( (T^{2} + 1)^{4} \)
|
| $17$ |
\( (T^{4} + 4 T^{3} - 6 T^{2} + \cdots + 8)^{2} \)
|
| $19$ |
\( T^{8} + 44 T^{6} + \cdots + 1024 \)
|
| $23$ |
\( T^{8} + 76 T^{6} + \cdots + 1024 \)
|
| $29$ |
\( T^{8} + 144 T^{6} + \cdots + 35344 \)
|
| $31$ |
\( T^{8} + 116 T^{6} + \cdots + 64 \)
|
| $37$ |
\( (T^{4} - 4 T^{3} - 60 T^{2} + \cdots - 32)^{2} \)
|
| $41$ |
\( (T^{4} - 90 T^{2} + \cdots - 376)^{2} \)
|
| $43$ |
\( (T^{4} + 12 T^{3} + \cdots - 544)^{2} \)
|
| $47$ |
\( (T^{4} - 8 T^{3} + \cdots - 1472)^{2} \)
|
| $53$ |
\( T^{8} + 176 T^{6} + \cdots + 80656 \)
|
| $59$ |
\( (T^{4} + 16 T^{3} + 30 T^{2} + \cdots - 8)^{2} \)
|
| $61$ |
\( T^{8} + 160 T^{6} + \cdots + 256 \)
|
| $67$ |
\( (T^{4} + 8 T^{3} + \cdots - 1352)^{2} \)
|
| $71$ |
\( T^{8} + 400 T^{6} + \cdots + 16777216 \)
|
| $73$ |
\( T^{8} + 168 T^{6} + \cdots + 2262016 \)
|
| $79$ |
\( (T^{4} + 20 T^{3} + \cdots + 32)^{2} \)
|
| $83$ |
\( (T^{4} + 32 T^{3} + \cdots - 584)^{2} \)
|
| $89$ |
\( (T^{4} - 8 T^{3} + \cdots + 184)^{2} \)
|
| $97$ |
\( T^{8} + 160 T^{6} + \cdots + 256 \)
|
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