Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,3,Mod(29,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.29");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.c (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: | \(2.28883422063\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.116032.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 8x^{2} + 14x + 21 \)
|
| 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 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
\( x^{4} - 2x^{3} + 8x^{2} + 14x + 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 8\nu + 12 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu^{2} - 8\nu - 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{2} - 8\beta _1 - 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(43\) | \(73\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 29.1 |
|
0 | 0.177124 | − | 2.99477i | 0 | − | 3.86775i | 0 | −2.64575 | 0 | −8.93725 | − | 1.06089i | 0 | |||||||||||||||||||||||||
| 29.2 | 0 | 0.177124 | + | 2.99477i | 0 | 3.86775i | 0 | −2.64575 | 0 | −8.93725 | + | 1.06089i | 0 | |||||||||||||||||||||||||||
| 29.3 | 0 | 2.82288 | − | 1.01557i | 0 | 9.43613i | 0 | 2.64575 | 0 | 6.93725 | − | 5.73363i | 0 | |||||||||||||||||||||||||||
| 29.4 | 0 | 2.82288 | + | 1.01557i | 0 | − | 9.43613i | 0 | 2.64575 | 0 | 6.93725 | + | 5.73363i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 84.3.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 84.3.c.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 336.3.d.a | 4 | ||
| 5.b | even | 2 | 1 | 2100.3.g.a | 4 | ||
| 5.c | odd | 4 | 2 | 2100.3.e.a | 8 | ||
| 7.b | odd | 2 | 1 | 588.3.c.h | 4 | ||
| 7.c | even | 3 | 2 | 588.3.p.f | 8 | ||
| 7.d | odd | 6 | 2 | 588.3.p.h | 8 | ||
| 8.b | even | 2 | 1 | 1344.3.d.a | 4 | ||
| 8.d | odd | 2 | 1 | 1344.3.d.g | 4 | ||
| 9.c | even | 3 | 2 | 2268.3.bg.a | 8 | ||
| 9.d | odd | 6 | 2 | 2268.3.bg.a | 8 | ||
| 12.b | even | 2 | 1 | 336.3.d.a | 4 | ||
| 15.d | odd | 2 | 1 | 2100.3.g.a | 4 | ||
| 15.e | even | 4 | 2 | 2100.3.e.a | 8 | ||
| 21.c | even | 2 | 1 | 588.3.c.h | 4 | ||
| 21.g | even | 6 | 2 | 588.3.p.h | 8 | ||
| 21.h | odd | 6 | 2 | 588.3.p.f | 8 | ||
| 24.f | even | 2 | 1 | 1344.3.d.g | 4 | ||
| 24.h | odd | 2 | 1 | 1344.3.d.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.3.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 84.3.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 336.3.d.a | 4 | 4.b | odd | 2 | 1 | ||
| 336.3.d.a | 4 | 12.b | even | 2 | 1 | ||
| 588.3.c.h | 4 | 7.b | odd | 2 | 1 | ||
| 588.3.c.h | 4 | 21.c | even | 2 | 1 | ||
| 588.3.p.f | 8 | 7.c | even | 3 | 2 | ||
| 588.3.p.f | 8 | 21.h | odd | 6 | 2 | ||
| 588.3.p.h | 8 | 7.d | odd | 6 | 2 | ||
| 588.3.p.h | 8 | 21.g | even | 6 | 2 | ||
| 1344.3.d.a | 4 | 8.b | even | 2 | 1 | ||
| 1344.3.d.a | 4 | 24.h | odd | 2 | 1 | ||
| 1344.3.d.g | 4 | 8.d | odd | 2 | 1 | ||
| 1344.3.d.g | 4 | 24.f | even | 2 | 1 | ||
| 2100.3.e.a | 8 | 5.c | odd | 4 | 2 | ||
| 2100.3.e.a | 8 | 15.e | even | 4 | 2 | ||
| 2100.3.g.a | 4 | 5.b | even | 2 | 1 | ||
| 2100.3.g.a | 4 | 15.d | odd | 2 | 1 | ||
| 2268.3.bg.a | 8 | 9.c | even | 3 | 2 | ||
| 2268.3.bg.a | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(84, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 6 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} + 104T^{2} + 1332 \)
|
| $7$ |
\( (T^{2} - 7)^{2} \)
|
| $11$ |
\( T^{4} + 440 T^{2} + 47952 \)
|
| $13$ |
\( (T^{2} - 14 T - 126)^{2} \)
|
| $17$ |
\( T^{4} + 488T^{2} + 5328 \)
|
| $19$ |
\( (T^{2} + 2 T - 62)^{2} \)
|
| $23$ |
\( T^{4} + 416 T^{2} + 21312 \)
|
| $29$ |
\( T^{4} + 3080 T^{2} + 2349648 \)
|
| $31$ |
\( (T^{2} - 20 T + 72)^{2} \)
|
| $37$ |
\( (T^{2} + 56 T + 532)^{2} \)
|
| $41$ |
\( T^{4} + 3960 T^{2} + 3884112 \)
|
| $43$ |
\( (T^{2} - 64 T + 772)^{2} \)
|
| $47$ |
\( T^{4} + 5760 T^{2} + 3068928 \)
|
| $53$ |
\( T^{4} + 10296 T^{2} + 47952 \)
|
| $59$ |
\( T^{4} + 824 T^{2} + 107892 \)
|
| $61$ |
\( (T^{2} - 98 T + 1554)^{2} \)
|
| $67$ |
\( (T^{2} + 204 T + 9956)^{2} \)
|
| $71$ |
\( T^{4} + 5480 T^{2} + 5120208 \)
|
| $73$ |
\( (T^{2} - 10108)^{2} \)
|
| $79$ |
\( (T^{2} + 64 T + 912)^{2} \)
|
| $83$ |
\( T^{4} + 9176 T^{2} + 16411572 \)
|
| $89$ |
\( T^{4} + 3960 T^{2} + 3884112 \)
|
| $97$ |
\( (T^{2} - 12 T - 1756)^{2} \)
|
| show more | |
| show less |