Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,2,Mod(165,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.165");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.p (of order \(6\), degree \(2\), not 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: | \(3.13013575923\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 56) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 165.1 |
|
−1.22474 | + | 0.707107i | 1.22474 | + | 0.707107i | 1.00000 | − | 1.73205i | 1.22474 | − | 0.707107i | −2.00000 | 0 | 2.82843i | −0.500000 | − | 0.866025i | −1.00000 | + | 1.73205i | ||||||||||||||||||
| 165.2 | 1.22474 | − | 0.707107i | −1.22474 | − | 0.707107i | 1.00000 | − | 1.73205i | −1.22474 | + | 0.707107i | −2.00000 | 0 | − | 2.82843i | −0.500000 | − | 0.866025i | −1.00000 | + | 1.73205i | ||||||||||||||||||
| 373.1 | −1.22474 | − | 0.707107i | 1.22474 | − | 0.707107i | 1.00000 | + | 1.73205i | 1.22474 | + | 0.707107i | −2.00000 | 0 | − | 2.82843i | −0.500000 | + | 0.866025i | −1.00000 | − | 1.73205i | ||||||||||||||||||
| 373.2 | 1.22474 | + | 0.707107i | −1.22474 | + | 0.707107i | 1.00000 | + | 1.73205i | −1.22474 | − | 0.707107i | −2.00000 | 0 | 2.82843i | −0.500000 | + | 0.866025i | −1.00000 | − | 1.73205i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 56.p | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.2.p.a | 4 | |
| 4.b | odd | 2 | 1 | 1568.2.t.c | 4 | ||
| 7.b | odd | 2 | 1 | 392.2.p.b | 4 | ||
| 7.c | even | 3 | 1 | 56.2.b.a | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 392.2.p.a | 4 | |
| 7.d | odd | 6 | 1 | 392.2.b.b | 2 | ||
| 7.d | odd | 6 | 1 | 392.2.p.b | 4 | ||
| 8.b | even | 2 | 1 | inner | 392.2.p.a | 4 | |
| 8.d | odd | 2 | 1 | 1568.2.t.c | 4 | ||
| 21.h | odd | 6 | 1 | 504.2.c.a | 2 | ||
| 28.d | even | 2 | 1 | 1568.2.t.b | 4 | ||
| 28.f | even | 6 | 1 | 1568.2.b.a | 2 | ||
| 28.f | even | 6 | 1 | 1568.2.t.b | 4 | ||
| 28.g | odd | 6 | 1 | 224.2.b.a | 2 | ||
| 28.g | odd | 6 | 1 | 1568.2.t.c | 4 | ||
| 56.e | even | 2 | 1 | 1568.2.t.b | 4 | ||
| 56.h | odd | 2 | 1 | 392.2.p.b | 4 | ||
| 56.j | odd | 6 | 1 | 392.2.b.b | 2 | ||
| 56.j | odd | 6 | 1 | 392.2.p.b | 4 | ||
| 56.k | odd | 6 | 1 | 224.2.b.a | 2 | ||
| 56.k | odd | 6 | 1 | 1568.2.t.c | 4 | ||
| 56.m | even | 6 | 1 | 1568.2.b.a | 2 | ||
| 56.m | even | 6 | 1 | 1568.2.t.b | 4 | ||
| 56.p | even | 6 | 1 | 56.2.b.a | ✓ | 2 | |
| 56.p | even | 6 | 1 | inner | 392.2.p.a | 4 | |
| 84.n | even | 6 | 1 | 2016.2.c.a | 2 | ||
| 112.u | odd | 12 | 2 | 1792.2.a.p | 2 | ||
| 112.w | even | 12 | 2 | 1792.2.a.n | 2 | ||
| 168.s | odd | 6 | 1 | 504.2.c.a | 2 | ||
| 168.v | even | 6 | 1 | 2016.2.c.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.2.b.a | ✓ | 2 | 7.c | even | 3 | 1 | |
| 56.2.b.a | ✓ | 2 | 56.p | even | 6 | 1 | |
| 224.2.b.a | 2 | 28.g | odd | 6 | 1 | ||
| 224.2.b.a | 2 | 56.k | odd | 6 | 1 | ||
| 392.2.b.b | 2 | 7.d | odd | 6 | 1 | ||
| 392.2.b.b | 2 | 56.j | odd | 6 | 1 | ||
| 392.2.p.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 392.2.p.a | 4 | 7.c | even | 3 | 1 | inner | |
| 392.2.p.a | 4 | 8.b | even | 2 | 1 | inner | |
| 392.2.p.a | 4 | 56.p | even | 6 | 1 | inner | |
| 392.2.p.b | 4 | 7.b | odd | 2 | 1 | ||
| 392.2.p.b | 4 | 7.d | odd | 6 | 1 | ||
| 392.2.p.b | 4 | 56.h | odd | 2 | 1 | ||
| 392.2.p.b | 4 | 56.j | odd | 6 | 1 | ||
| 504.2.c.a | 2 | 21.h | odd | 6 | 1 | ||
| 504.2.c.a | 2 | 168.s | odd | 6 | 1 | ||
| 1568.2.b.a | 2 | 28.f | even | 6 | 1 | ||
| 1568.2.b.a | 2 | 56.m | even | 6 | 1 | ||
| 1568.2.t.b | 4 | 28.d | even | 2 | 1 | ||
| 1568.2.t.b | 4 | 28.f | even | 6 | 1 | ||
| 1568.2.t.b | 4 | 56.e | even | 2 | 1 | ||
| 1568.2.t.b | 4 | 56.m | even | 6 | 1 | ||
| 1568.2.t.c | 4 | 4.b | odd | 2 | 1 | ||
| 1568.2.t.c | 4 | 8.d | odd | 2 | 1 | ||
| 1568.2.t.c | 4 | 28.g | odd | 6 | 1 | ||
| 1568.2.t.c | 4 | 56.k | odd | 6 | 1 | ||
| 1792.2.a.n | 2 | 112.w | even | 12 | 2 | ||
| 1792.2.a.p | 2 | 112.u | odd | 12 | 2 | ||
| 2016.2.c.a | 2 | 84.n | even | 6 | 1 | ||
| 2016.2.c.a | 2 | 168.v | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):
|
\( T_{3}^{4} - 2T_{3}^{2} + 4 \)
|
|
\( T_{17}^{2} - 6T_{17} + 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $3$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $5$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 8T^{2} + 64 \)
|
| $13$ |
\( (T^{2} + 18)^{2} \)
|
| $17$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $19$ |
\( T^{4} - 18T^{2} + 324 \)
|
| $23$ |
\( (T^{2} - 6 T + 36)^{2} \)
|
| $29$ |
\( (T^{2} + 8)^{2} \)
|
| $31$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $37$ |
\( T^{4} - 72T^{2} + 5184 \)
|
| $41$ |
\( (T - 6)^{4} \)
|
| $43$ |
\( (T^{2} + 72)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} - 32T^{2} + 1024 \)
|
| $59$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $61$ |
\( T^{4} - 162 T^{2} + 26244 \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $79$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $83$ |
\( (T^{2} + 242)^{2} \)
|
| $89$ |
\( (T^{2} + 6 T + 36)^{2} \)
|
| $97$ |
\( (T + 10)^{4} \)
|
| show more | |
| show less |