Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,3,Mod(77,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.77");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.g (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: | \(4.25069212402\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 77.1 |
|
0 | −2.50000 | − | 1.65831i | 0 | −3.31662 | 0 | 7.00000i | 0 | 3.50000 | + | 8.29156i | 0 | ||||||||||||||||||||||||||
| 77.2 | 0 | −2.50000 | − | 1.65831i | 0 | 3.31662 | 0 | − | 7.00000i | 0 | 3.50000 | + | 8.29156i | 0 | ||||||||||||||||||||||||||
| 77.3 | 0 | −2.50000 | + | 1.65831i | 0 | −3.31662 | 0 | − | 7.00000i | 0 | 3.50000 | − | 8.29156i | 0 | ||||||||||||||||||||||||||
| 77.4 | 0 | −2.50000 | + | 1.65831i | 0 | 3.31662 | 0 | 7.00000i | 0 | 3.50000 | − | 8.29156i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.3.g.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 156.3.g.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 624.3.l.h | 4 | ||
| 12.b | even | 2 | 1 | 624.3.l.h | 4 | ||
| 13.b | even | 2 | 1 | inner | 156.3.g.b | ✓ | 4 |
| 39.d | odd | 2 | 1 | inner | 156.3.g.b | ✓ | 4 |
| 52.b | odd | 2 | 1 | 624.3.l.h | 4 | ||
| 156.h | even | 2 | 1 | 624.3.l.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.3.g.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 156.3.g.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 156.3.g.b | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 156.3.g.b | ✓ | 4 | 39.d | odd | 2 | 1 | inner |
| 624.3.l.h | 4 | 4.b | odd | 2 | 1 | ||
| 624.3.l.h | 4 | 12.b | even | 2 | 1 | ||
| 624.3.l.h | 4 | 52.b | odd | 2 | 1 | ||
| 624.3.l.h | 4 | 156.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 11 \)
acting on \(S_{3}^{\mathrm{new}}(156, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 5 T + 9)^{2} \)
|
| $5$ |
\( (T^{2} - 11)^{2} \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
|
| $11$ |
\( (T^{2} - 396)^{2} \)
|
| $13$ |
\( (T^{2} + 169)^{2} \)
|
| $17$ |
\( (T^{2} + 539)^{2} \)
|
| $19$ |
\( (T^{2} + 900)^{2} \)
|
| $23$ |
\( (T^{2} + 704)^{2} \)
|
| $29$ |
\( (T^{2} + 1100)^{2} \)
|
| $31$ |
\( (T^{2} + 100)^{2} \)
|
| $37$ |
\( (T^{2} + 169)^{2} \)
|
| $41$ |
\( (T^{2} - 2156)^{2} \)
|
| $43$ |
\( (T - 55)^{4} \)
|
| $47$ |
\( (T^{2} - 2475)^{2} \)
|
| $53$ |
\( (T^{2} + 1584)^{2} \)
|
| $59$ |
\( (T^{2} - 8624)^{2} \)
|
| $61$ |
\( (T - 14)^{4} \)
|
| $67$ |
\( (T^{2} + 256)^{2} \)
|
| $71$ |
\( (T^{2} - 891)^{2} \)
|
| $73$ |
\( (T^{2} + 16)^{2} \)
|
| $79$ |
\( (T - 54)^{4} \)
|
| $83$ |
\( (T^{2} - 1100)^{2} \)
|
| $89$ |
\( (T^{2} - 396)^{2} \)
|
| $97$ |
\( (T^{2} + 6084)^{2} \)
|
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