Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,3,Mod(127,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.127");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.m (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: | \(9.15533688251\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{3} + \nu^{2} + 3\nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 3\beta_{2} + \beta _1 + 3 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | − | 1.73205i | 0 | −9.58258 | 0 | − | 2.64575i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||
| 127.2 | 0 | − | 1.73205i | 0 | −0.417424 | 0 | 2.64575i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
| 127.3 | 0 | 1.73205i | 0 | −9.58258 | 0 | 2.64575i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
| 127.4 | 0 | 1.73205i | 0 | −0.417424 | 0 | − | 2.64575i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.3.m.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1008.3.m.f | 4 | ||
| 4.b | odd | 2 | 1 | inner | 336.3.m.a | ✓ | 4 |
| 7.b | odd | 2 | 1 | 2352.3.m.k | 4 | ||
| 8.b | even | 2 | 1 | 1344.3.m.c | 4 | ||
| 8.d | odd | 2 | 1 | 1344.3.m.c | 4 | ||
| 12.b | even | 2 | 1 | 1008.3.m.f | 4 | ||
| 28.d | even | 2 | 1 | 2352.3.m.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 336.3.m.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 336.3.m.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 1008.3.m.f | 4 | 3.b | odd | 2 | 1 | ||
| 1008.3.m.f | 4 | 12.b | even | 2 | 1 | ||
| 1344.3.m.c | 4 | 8.b | even | 2 | 1 | ||
| 1344.3.m.c | 4 | 8.d | odd | 2 | 1 | ||
| 2352.3.m.k | 4 | 7.b | odd | 2 | 1 | ||
| 2352.3.m.k | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 10T_{5} + 4 \)
acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 3)^{2} \)
|
| $5$ |
\( (T^{2} + 10 T + 4)^{2} \)
|
| $7$ |
\( (T^{2} + 7)^{2} \)
|
| $11$ |
\( T^{4} + 404 T^{2} + 21904 \)
|
| $13$ |
\( (T^{2} - 16 T - 20)^{2} \)
|
| $17$ |
\( (T^{2} - 10 T + 4)^{2} \)
|
| $19$ |
\( (T^{2} + 448)^{2} \)
|
| $23$ |
\( T^{4} + 1988 T^{2} + 490000 \)
|
| $29$ |
\( (T^{2} + 28 T - 140)^{2} \)
|
| $31$ |
\( (T^{2} + 432)^{2} \)
|
| $37$ |
\( (T^{2} + 104 T + 2620)^{2} \)
|
| $41$ |
\( (T^{2} - 46 T - 500)^{2} \)
|
| $43$ |
\( T^{4} + 1280 T^{2} + 65536 \)
|
| $47$ |
\( T^{4} + 528 T^{2} + 57600 \)
|
| $53$ |
\( (T^{2} - 56 T - 3332)^{2} \)
|
| $59$ |
\( T^{4} + 11984 T^{2} + 21086464 \)
|
| $61$ |
\( (T^{2} - 84 T + 420)^{2} \)
|
| $67$ |
\( T^{4} + 18000 T^{2} + 72182016 \)
|
| $71$ |
\( T^{4} + 6884 T^{2} + 9560464 \)
|
| $73$ |
\( (T^{2} + 16 T - 2036)^{2} \)
|
| $79$ |
\( T^{4} + 7440 T^{2} + 10342656 \)
|
| $83$ |
\( T^{4} + 14432 T^{2} + 14137600 \)
|
| $89$ |
\( (T^{2} - 118 T + 2956)^{2} \)
|
| $97$ |
\( (T^{2} + 152 T + 3676)^{2} \)
|
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