Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,3,Mod(161,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.161");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.t (of order \(4\), 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: | \(5.66758949869\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | 6.0.20819026944.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 42x^{4} + 441x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 52) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 42x^{4} + 441x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 21\nu ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 21\nu^{2} ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 35\nu^{3} + 27\nu^{2} + 288\nu + 84 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 35\nu^{3} - 27\nu^{2} + 288\nu - 84 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} - 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{2} - 21\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 21\beta_{5} - 21\beta_{4} + 27\beta_{3} + 294 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{5} + 6\beta_{4} - 210\beta_{2} + 447\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
0 | −4.71926 | 0 | −2.77606 | + | 2.77606i | 0 | 5.49532 | + | 5.49532i | 0 | 13.2714 | 0 | ||||||||||||||||||||||||||||||||
| 161.2 | 0 | 0.286838 | 0 | 5.81544 | − | 5.81544i | 0 | −8.10228 | − | 8.10228i | 0 | −8.91772 | 0 | |||||||||||||||||||||||||||||||||
| 161.3 | 0 | 4.43242 | 0 | −6.03938 | + | 6.03938i | 0 | −0.393041 | − | 0.393041i | 0 | 10.6463 | 0 | |||||||||||||||||||||||||||||||||
| 177.1 | 0 | −4.71926 | 0 | −2.77606 | − | 2.77606i | 0 | 5.49532 | − | 5.49532i | 0 | 13.2714 | 0 | |||||||||||||||||||||||||||||||||
| 177.2 | 0 | 0.286838 | 0 | 5.81544 | + | 5.81544i | 0 | −8.10228 | + | 8.10228i | 0 | −8.91772 | 0 | |||||||||||||||||||||||||||||||||
| 177.3 | 0 | 4.43242 | 0 | −6.03938 | − | 6.03938i | 0 | −0.393041 | + | 0.393041i | 0 | 10.6463 | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 208.3.t.d | 6 | |
| 4.b | odd | 2 | 1 | 52.3.g.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 468.3.m.c | 6 | ||
| 13.d | odd | 4 | 1 | inner | 208.3.t.d | 6 | |
| 20.d | odd | 2 | 1 | 1300.3.t.a | 6 | ||
| 20.e | even | 4 | 1 | 1300.3.k.a | 6 | ||
| 20.e | even | 4 | 1 | 1300.3.k.b | 6 | ||
| 52.b | odd | 2 | 1 | 676.3.g.b | 6 | ||
| 52.f | even | 4 | 1 | 52.3.g.a | ✓ | 6 | |
| 52.f | even | 4 | 1 | 676.3.g.b | 6 | ||
| 156.l | odd | 4 | 1 | 468.3.m.c | 6 | ||
| 260.l | odd | 4 | 1 | 1300.3.k.b | 6 | ||
| 260.s | odd | 4 | 1 | 1300.3.k.a | 6 | ||
| 260.u | even | 4 | 1 | 1300.3.t.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.3.g.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 52.3.g.a | ✓ | 6 | 52.f | even | 4 | 1 | |
| 208.3.t.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 208.3.t.d | 6 | 13.d | odd | 4 | 1 | inner | |
| 468.3.m.c | 6 | 12.b | even | 2 | 1 | ||
| 468.3.m.c | 6 | 156.l | odd | 4 | 1 | ||
| 676.3.g.b | 6 | 52.b | odd | 2 | 1 | ||
| 676.3.g.b | 6 | 52.f | even | 4 | 1 | ||
| 1300.3.k.a | 6 | 20.e | even | 4 | 1 | ||
| 1300.3.k.a | 6 | 260.s | odd | 4 | 1 | ||
| 1300.3.k.b | 6 | 20.e | even | 4 | 1 | ||
| 1300.3.k.b | 6 | 260.l | odd | 4 | 1 | ||
| 1300.3.t.a | 6 | 20.d | odd | 2 | 1 | ||
| 1300.3.t.a | 6 | 260.u | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 21T_{3} + 6 \)
acting on \(S_{3}^{\mathrm{new}}(208, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{3} - 21 T + 6)^{2} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 76050 \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 2450 \)
|
| $11$ |
\( T^{6} - 18 T^{5} + \cdots + 596232 \)
|
| $13$ |
\( T^{6} + 30 T^{5} + \cdots + 4826809 \)
|
| $17$ |
\( T^{6} + 1122 T^{4} + \cdots + 14745600 \)
|
| $19$ |
\( T^{6} - 78 T^{5} + \cdots + 29799200 \)
|
| $23$ |
\( T^{6} + 1500 T^{4} + \cdots + 20358144 \)
|
| $29$ |
\( (T^{3} - 30 T^{2} + \cdots + 20316)^{2} \)
|
| $31$ |
\( T^{6} - 6 T^{5} + \cdots + 56180000 \)
|
| $37$ |
\( T^{6} + 54 T^{5} + \cdots + 873842 \)
|
| $41$ |
\( T^{6} + \cdots + 1839089952 \)
|
| $43$ |
\( T^{6} + \cdots + 4493289024 \)
|
| $47$ |
\( T^{6} + \cdots + 7858818450 \)
|
| $53$ |
\( (T^{3} - 114 T^{2} + \cdots - 38076)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 1542123648 \)
|
| $61$ |
\( (T^{3} - 102 T^{2} + \cdots - 2712)^{2} \)
|
| $67$ |
\( T^{6} + 78 T^{5} + \cdots + 30952712 \)
|
| $71$ |
\( T^{6} + \cdots + 2561418738 \)
|
| $73$ |
\( T^{6} + \cdots + 2377740800 \)
|
| $79$ |
\( (T^{3} - 30 T^{2} + \cdots + 37452)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 73972426248 \)
|
| $89$ |
\( T^{6} + \cdots + 2347495200 \)
|
| $97$ |
\( T^{6} + \cdots + 52696863368 \)
|
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