Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,2,Mod(23,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.23");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.r (of order \(6\), 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: | \(1.24566627153\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−1.41421 | 1.50000 | − | 0.866025i | 2.00000 | 2.82843 | −2.12132 | + | 1.22474i | −1.50000 | + | 2.59808i | −2.82843 | 1.50000 | − | 2.59808i | −4.00000 | ||||||||||||||||||||||
| 23.2 | 1.41421 | 1.50000 | − | 0.866025i | 2.00000 | −2.82843 | 2.12132 | − | 1.22474i | −1.50000 | + | 2.59808i | 2.82843 | 1.50000 | − | 2.59808i | −4.00000 | |||||||||||||||||||||||
| 95.1 | −1.41421 | 1.50000 | + | 0.866025i | 2.00000 | 2.82843 | −2.12132 | − | 1.22474i | −1.50000 | − | 2.59808i | −2.82843 | 1.50000 | + | 2.59808i | −4.00000 | |||||||||||||||||||||||
| 95.2 | 1.41421 | 1.50000 | + | 0.866025i | 2.00000 | −2.82843 | 2.12132 | + | 1.22474i | −1.50000 | − | 2.59808i | 2.82843 | 1.50000 | + | 2.59808i | −4.00000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 52.i | odd | 6 | 1 | inner |
| 156.r | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.2.r.b | yes | 4 |
| 3.b | odd | 2 | 1 | inner | 156.2.r.b | yes | 4 |
| 4.b | odd | 2 | 1 | 156.2.r.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 156.2.r.a | ✓ | 4 | |
| 13.e | even | 6 | 1 | 156.2.r.a | ✓ | 4 | |
| 39.h | odd | 6 | 1 | 156.2.r.a | ✓ | 4 | |
| 52.i | odd | 6 | 1 | inner | 156.2.r.b | yes | 4 |
| 156.r | even | 6 | 1 | inner | 156.2.r.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.2.r.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 156.2.r.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 156.2.r.a | ✓ | 4 | 13.e | even | 6 | 1 | |
| 156.2.r.a | ✓ | 4 | 39.h | odd | 6 | 1 | |
| 156.2.r.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 156.2.r.b | yes | 4 | 3.b | odd | 2 | 1 | inner |
| 156.2.r.b | yes | 4 | 52.i | odd | 6 | 1 | inner |
| 156.2.r.b | yes | 4 | 156.r | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\):
|
\( T_{5}^{2} - 8 \)
|
|
\( T_{7}^{2} + 3T_{7} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{2} \)
|
| $5$ |
\( (T^{2} - 8)^{2} \)
|
| $7$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 7 T + 13)^{2} \)
|
| $17$ |
\( T^{4} - 24T^{2} + 576 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 72T^{2} + 5184 \)
|
| $29$ |
\( T^{4} - 24T^{2} + 576 \)
|
| $31$ |
\( (T - 3)^{4} \)
|
| $37$ |
\( (T^{2} + 12 T + 48)^{2} \)
|
| $41$ |
\( T^{4} + 32T^{2} + 1024 \)
|
| $43$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
| $47$ |
\( (T^{2} + 96)^{2} \)
|
| $53$ |
\( (T^{2} + 96)^{2} \)
|
| $59$ |
\( T^{4} - 24T^{2} + 576 \)
|
| $61$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $67$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $71$ |
\( T^{4} - 24T^{2} + 576 \)
|
| $73$ |
\( (T^{2} + 3)^{2} \)
|
| $79$ |
\( (T^{2} + 27)^{2} \)
|
| $83$ |
\( (T^{2} + 24)^{2} \)
|
| $89$ |
\( T^{4} + 8T^{2} + 64 \)
|
| $97$ |
\( (T^{2} - 9 T + 27)^{2} \)
|
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