Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [31,2,Mod(5,31)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 31.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: | \(0.247536246266\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−2.41421 | 1.20711 | − | 2.09077i | 3.82843 | −0.500000 | − | 0.866025i | −2.91421 | + | 5.04757i | −1.20711 | + | 2.09077i | −4.41421 | −1.41421 | − | 2.44949i | 1.20711 | + | 2.09077i | ||||||||||||||||||
| 5.2 | 0.414214 | −0.207107 | + | 0.358719i | −1.82843 | −0.500000 | − | 0.866025i | −0.0857864 | + | 0.148586i | 0.207107 | − | 0.358719i | −1.58579 | 1.41421 | + | 2.44949i | −0.207107 | − | 0.358719i | |||||||||||||||||||
| 25.1 | −2.41421 | 1.20711 | + | 2.09077i | 3.82843 | −0.500000 | + | 0.866025i | −2.91421 | − | 5.04757i | −1.20711 | − | 2.09077i | −4.41421 | −1.41421 | + | 2.44949i | 1.20711 | − | 2.09077i | |||||||||||||||||||
| 25.2 | 0.414214 | −0.207107 | − | 0.358719i | −1.82843 | −0.500000 | + | 0.866025i | −0.0857864 | − | 0.148586i | 0.207107 | + | 0.358719i | −1.58579 | 1.41421 | − | 2.44949i | −0.207107 | + | 0.358719i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 31.2.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 279.2.h.c | 4 | ||
| 4.b | odd | 2 | 1 | 496.2.i.h | 4 | ||
| 5.b | even | 2 | 1 | 775.2.e.e | 4 | ||
| 5.c | odd | 4 | 2 | 775.2.o.d | 8 | ||
| 31.b | odd | 2 | 1 | 961.2.c.a | 4 | ||
| 31.c | even | 3 | 1 | inner | 31.2.c.a | ✓ | 4 |
| 31.c | even | 3 | 1 | 961.2.a.a | 2 | ||
| 31.d | even | 5 | 4 | 961.2.g.o | 16 | ||
| 31.e | odd | 6 | 1 | 961.2.a.c | 2 | ||
| 31.e | odd | 6 | 1 | 961.2.c.a | 4 | ||
| 31.f | odd | 10 | 4 | 961.2.g.r | 16 | ||
| 31.g | even | 15 | 4 | 961.2.d.l | 8 | ||
| 31.g | even | 15 | 4 | 961.2.g.o | 16 | ||
| 31.h | odd | 30 | 4 | 961.2.d.i | 8 | ||
| 31.h | odd | 30 | 4 | 961.2.g.r | 16 | ||
| 93.g | even | 6 | 1 | 8649.2.a.k | 2 | ||
| 93.h | odd | 6 | 1 | 279.2.h.c | 4 | ||
| 93.h | odd | 6 | 1 | 8649.2.a.l | 2 | ||
| 124.i | odd | 6 | 1 | 496.2.i.h | 4 | ||
| 155.j | even | 6 | 1 | 775.2.e.e | 4 | ||
| 155.o | odd | 12 | 2 | 775.2.o.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 31.2.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 31.2.c.a | ✓ | 4 | 31.c | even | 3 | 1 | inner |
| 279.2.h.c | 4 | 3.b | odd | 2 | 1 | ||
| 279.2.h.c | 4 | 93.h | odd | 6 | 1 | ||
| 496.2.i.h | 4 | 4.b | odd | 2 | 1 | ||
| 496.2.i.h | 4 | 124.i | odd | 6 | 1 | ||
| 775.2.e.e | 4 | 5.b | even | 2 | 1 | ||
| 775.2.e.e | 4 | 155.j | even | 6 | 1 | ||
| 775.2.o.d | 8 | 5.c | odd | 4 | 2 | ||
| 775.2.o.d | 8 | 155.o | odd | 12 | 2 | ||
| 961.2.a.a | 2 | 31.c | even | 3 | 1 | ||
| 961.2.a.c | 2 | 31.e | odd | 6 | 1 | ||
| 961.2.c.a | 4 | 31.b | odd | 2 | 1 | ||
| 961.2.c.a | 4 | 31.e | odd | 6 | 1 | ||
| 961.2.d.i | 8 | 31.h | odd | 30 | 4 | ||
| 961.2.d.l | 8 | 31.g | even | 15 | 4 | ||
| 961.2.g.o | 16 | 31.d | even | 5 | 4 | ||
| 961.2.g.o | 16 | 31.g | even | 15 | 4 | ||
| 961.2.g.r | 16 | 31.f | odd | 10 | 4 | ||
| 961.2.g.r | 16 | 31.h | odd | 30 | 4 | ||
| 8649.2.a.k | 2 | 93.g | even | 6 | 1 | ||
| 8649.2.a.l | 2 | 93.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(31, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T - 1)^{2} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} + T + 1)^{2} \)
|
| $7$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $11$ |
\( T^{4} - 2 T^{3} + \cdots + 289 \)
|
| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 49 \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 1 \)
|
| $19$ |
\( T^{4} + 6 T^{3} + \cdots + 49 \)
|
| $23$ |
\( (T + 4)^{4} \)
|
| $29$ |
\( (T^{2} + 8 T + 8)^{2} \)
|
| $31$ |
\( (T^{2} + 10 T + 31)^{2} \)
|
| $37$ |
\( (T^{2} + T + 1)^{2} \)
|
| $41$ |
\( T^{4} + 2 T^{3} + \cdots + 5041 \)
|
| $43$ |
\( T^{4} - 2 T^{3} + \cdots + 9409 \)
|
| $47$ |
\( (T^{2} - 8 T - 16)^{2} \)
|
| $53$ |
\( T^{4} + 6 T^{3} + \cdots + 1 \)
|
| $59$ |
\( T^{4} - 6 T^{3} + \cdots + 1681 \)
|
| $61$ |
\( (T^{2} - 8)^{2} \)
|
| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 289 \)
|
| $71$ |
\( T^{4} - 14 T^{3} + \cdots + 1 \)
|
| $73$ |
\( T^{4} + 2 T^{3} + \cdots + 49 \)
|
| $79$ |
\( T^{4} - 22 T^{3} + \cdots + 10609 \)
|
| $83$ |
\( T^{4} - 6 T^{3} + \cdots + 1681 \)
|
| $89$ |
\( (T^{2} + 8 T - 56)^{2} \)
|
| $97$ |
\( (T^{2} - 16 T + 56)^{2} \)
|
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