Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,4,Mod(7,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.7");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 19.c (of order \(3\), 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: | \(1.12103629011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{73})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 19x^{2} + 18x + 324 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 19x^{2} + 18x + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 19\nu^{2} - 19\nu + 324 ) / 342 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 37 ) / 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 18\beta_{2} + \beta _1 - 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( 19\beta_{3} - 37 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−2.38600 | − | 4.13267i | −0.500000 | − | 0.866025i | −7.38600 | + | 12.7929i | −6.88600 | − | 11.9269i | −2.38600 | + | 4.13267i | 27.0880 | 32.3160 | 13.0000 | − | 22.5167i | −32.8600 | + | 56.9152i | ||||||||||||||||
| 7.2 | 1.88600 | + | 3.26665i | −0.500000 | − | 0.866025i | −3.11400 | + | 5.39360i | −2.61400 | − | 4.52758i | 1.88600 | − | 3.26665i | −7.08801 | 6.68399 | 13.0000 | − | 22.5167i | 9.86001 | − | 17.0780i | |||||||||||||||||
| 11.1 | −2.38600 | + | 4.13267i | −0.500000 | + | 0.866025i | −7.38600 | − | 12.7929i | −6.88600 | + | 11.9269i | −2.38600 | − | 4.13267i | 27.0880 | 32.3160 | 13.0000 | + | 22.5167i | −32.8600 | − | 56.9152i | |||||||||||||||||
| 11.2 | 1.88600 | − | 3.26665i | −0.500000 | + | 0.866025i | −3.11400 | − | 5.39360i | −2.61400 | + | 4.52758i | 1.88600 | + | 3.26665i | −7.08801 | 6.68399 | 13.0000 | + | 22.5167i | 9.86001 | + | 17.0780i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 19.4.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 171.4.f.d | 4 | ||
| 4.b | odd | 2 | 1 | 304.4.i.d | 4 | ||
| 19.c | even | 3 | 1 | inner | 19.4.c.b | ✓ | 4 |
| 19.c | even | 3 | 1 | 361.4.a.f | 2 | ||
| 19.d | odd | 6 | 1 | 361.4.a.e | 2 | ||
| 57.h | odd | 6 | 1 | 171.4.f.d | 4 | ||
| 76.g | odd | 6 | 1 | 304.4.i.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.4.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 19.4.c.b | ✓ | 4 | 19.c | even | 3 | 1 | inner |
| 171.4.f.d | 4 | 3.b | odd | 2 | 1 | ||
| 171.4.f.d | 4 | 57.h | odd | 6 | 1 | ||
| 304.4.i.d | 4 | 4.b | odd | 2 | 1 | ||
| 304.4.i.d | 4 | 76.g | odd | 6 | 1 | ||
| 361.4.a.e | 2 | 19.d | odd | 6 | 1 | ||
| 361.4.a.f | 2 | 19.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + T_{2}^{3} + 19T_{2}^{2} - 18T_{2} + 324 \)
acting on \(S_{4}^{\mathrm{new}}(19, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} + \cdots + 324 \)
|
| $3$ |
\( (T^{2} + T + 1)^{2} \)
|
| $5$ |
\( T^{4} + 19 T^{3} + \cdots + 5184 \)
|
| $7$ |
\( (T^{2} - 20 T - 192)^{2} \)
|
| $11$ |
\( (T^{2} + 33 T + 108)^{2} \)
|
| $13$ |
\( T^{4} + 101 T^{3} + \cdots + 4384836 \)
|
| $17$ |
\( T^{4} - 75 T^{3} + \cdots + 44116164 \)
|
| $19$ |
\( T^{4} - 57 T^{3} + \cdots + 47045881 \)
|
| $23$ |
\( T^{4} + T^{3} + \cdots + 27815076 \)
|
| $29$ |
\( T^{4} - 85 T^{3} + \cdots + 833592384 \)
|
| $31$ |
\( (T^{2} - 22 T - 536)^{2} \)
|
| $37$ |
\( (T^{2} - 448 T + 26524)^{2} \)
|
| $41$ |
\( T^{4} + 124 T^{3} + \cdots + 14220441 \)
|
| $43$ |
\( T^{4} - 311 T^{3} + \cdots + 260241424 \)
|
| $47$ |
\( T^{4} + 411 T^{3} + \cdots + 209438784 \)
|
| $53$ |
\( T^{4} + \cdots + 26191538244 \)
|
| $59$ |
\( T^{4} + \cdots + 32209121961 \)
|
| $61$ |
\( T^{4} + \cdots + 4843603216 \)
|
| $67$ |
\( T^{4} + \cdots + 161337985561 \)
|
| $71$ |
\( T^{4} + \cdots + 372805494084 \)
|
| $73$ |
\( T^{4} + \cdots + 16291714321 \)
|
| $79$ |
\( T^{4} - 331 T^{3} + \cdots + 97061904 \)
|
| $83$ |
\( (T^{2} - 1459 T + 59112)^{2} \)
|
| $89$ |
\( T^{4} + 601 T^{3} + \cdots + 556865604 \)
|
| $97$ |
\( T^{4} + \cdots + 1806048395449 \)
|
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