Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,2,Mod(69,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.69");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 136.c (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: | \(1.08596546749\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.4469724736.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 2x^{5} - 4x^{4} + 4x^{3} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + 2x^{5} - 4x^{4} + 4x^{3} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 4\nu^{3} - 4\nu^{2} + 8 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 2\nu^{5} + 4\nu^{3} - 4\nu^{2} + 8 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 4\nu^{2} - 8\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} + 2\nu^{3} - 4\nu^{2} + 4\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{4} - 2\beta_{3} + \beta_{2} + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{2} + 2\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 2\beta_{5} - 3\beta_{4} + \beta_{2} - 2\beta _1 + 1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{7} + 5\beta_{6} - 2\beta_{5} - \beta_{4} - \beta_{2} + 2\beta _1 + 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
| \(n\) | \(69\) | \(103\) | \(105\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 69.1 |
|
−1.33209 | − | 0.474920i | 2.10562i | 1.54890 | + | 1.26527i | 1.58069i | 1.00000 | − | 2.80487i | −5.01127 | −1.46237 | − | 2.42105i | −1.43364 | 0.750703 | − | 2.10562i | ||||||||||||||||||||||||||||||||
| 69.2 | −1.33209 | + | 0.474920i | − | 2.10562i | 1.54890 | − | 1.26527i | − | 1.58069i | 1.00000 | + | 2.80487i | −5.01127 | −1.46237 | + | 2.42105i | −1.43364 | 0.750703 | + | 2.10562i | |||||||||||||||||||||||||||||||
| 69.3 | −0.733159 | − | 1.20933i | 0.826905i | −0.924955 | + | 1.77326i | 1.12786i | 1.00000 | − | 0.606253i | 1.74755 | 2.82260 | − | 0.181508i | 2.31623 | 1.36396 | − | 0.826905i | |||||||||||||||||||||||||||||||||
| 69.4 | −0.733159 | + | 1.20933i | − | 0.826905i | −0.924955 | − | 1.77326i | − | 1.12786i | 1.00000 | + | 0.606253i | 1.74755 | 2.82260 | + | 0.181508i | 2.31623 | 1.36396 | + | 0.826905i | |||||||||||||||||||||||||||||||
| 69.5 | 0.185533 | − | 1.40199i | 0.713272i | −1.93115 | − | 0.520231i | − | 3.84444i | 1.00000 | + | 0.132335i | −1.15650 | −1.08765 | + | 2.61094i | 2.49124 | −5.38987 | − | 0.713272i | ||||||||||||||||||||||||||||||||
| 69.6 | 0.185533 | + | 1.40199i | − | 0.713272i | −1.93115 | + | 0.520231i | 3.84444i | 1.00000 | − | 0.132335i | −1.15650 | −1.08765 | − | 2.61094i | 2.49124 | −5.38987 | + | 0.713272i | ||||||||||||||||||||||||||||||||
| 69.7 | 1.37971 | − | 0.310478i | 3.22084i | 1.80721 | − | 0.856739i | − | 2.33443i | 1.00000 | + | 4.44384i | −1.57978 | 2.22743 | − | 1.74315i | −7.37384 | −0.724789 | − | 3.22084i | ||||||||||||||||||||||||||||||||
| 69.8 | 1.37971 | + | 0.310478i | − | 3.22084i | 1.80721 | + | 0.856739i | 2.33443i | 1.00000 | − | 4.44384i | −1.57978 | 2.22743 | + | 1.74315i | −7.37384 | −0.724789 | + | 3.22084i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 136.2.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1224.2.f.c | 8 | ||
| 4.b | odd | 2 | 1 | 544.2.c.b | 8 | ||
| 8.b | even | 2 | 1 | inner | 136.2.c.b | ✓ | 8 |
| 8.d | odd | 2 | 1 | 544.2.c.b | 8 | ||
| 12.b | even | 2 | 1 | 4896.2.f.d | 8 | ||
| 16.e | even | 4 | 2 | 4352.2.a.bf | 8 | ||
| 16.f | odd | 4 | 2 | 4352.2.a.bb | 8 | ||
| 24.f | even | 2 | 1 | 4896.2.f.d | 8 | ||
| 24.h | odd | 2 | 1 | 1224.2.f.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.2.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 136.2.c.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 544.2.c.b | 8 | 4.b | odd | 2 | 1 | ||
| 544.2.c.b | 8 | 8.d | odd | 2 | 1 | ||
| 1224.2.f.c | 8 | 3.b | odd | 2 | 1 | ||
| 1224.2.f.c | 8 | 24.h | odd | 2 | 1 | ||
| 4352.2.a.bb | 8 | 16.f | odd | 4 | 2 | ||
| 4352.2.a.bf | 8 | 16.e | even | 4 | 2 | ||
| 4896.2.f.d | 8 | 12.b | even | 2 | 1 | ||
| 4896.2.f.d | 8 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 16T_{3}^{6} + 64T_{3}^{4} + 60T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} + \cdots + 16 \)
|
| $3$ |
\( T^{8} + 16 T^{6} + \cdots + 16 \)
|
| $5$ |
\( T^{8} + 24 T^{6} + \cdots + 256 \)
|
| $7$ |
\( (T^{4} + 6 T^{3} + 2 T^{2} + \cdots - 16)^{2} \)
|
| $11$ |
\( T^{8} + 44 T^{6} + \cdots + 5776 \)
|
| $13$ |
\( T^{8} + 52 T^{6} + \cdots + 256 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} + 80 T^{6} + \cdots + 6400 \)
|
| $23$ |
\( (T^{4} + 8 T^{3} + \cdots + 944)^{2} \)
|
| $29$ |
\( T^{8} + 24 T^{6} + \cdots + 256 \)
|
| $31$ |
\( (T^{4} - 12 T^{3} + \cdots - 160)^{2} \)
|
| $37$ |
\( T^{8} + 160 T^{6} + \cdots + 160000 \)
|
| $41$ |
\( (T^{4} - 56 T^{2} + \cdots - 80)^{2} \)
|
| $43$ |
\( T^{8} + 188 T^{6} + \cdots + 102400 \)
|
| $47$ |
\( (T^{4} - 2 T^{3} + \cdots + 3200)^{2} \)
|
| $53$ |
\( T^{8} + 112 T^{6} + \cdots + 102400 \)
|
| $59$ |
\( T^{8} + 300 T^{6} + \cdots + 10240000 \)
|
| $61$ |
\( T^{8} + 24 T^{6} + \cdots + 256 \)
|
| $67$ |
\( T^{8} + 112 T^{6} + \cdots + 6400 \)
|
| $71$ |
\( (T^{4} + 18 T^{3} + \cdots - 400)^{2} \)
|
| $73$ |
\( (T^{4} - 4 T^{3} + \cdots + 304)^{2} \)
|
| $79$ |
\( (T^{4} - 12 T^{3} + \cdots - 128)^{2} \)
|
| $83$ |
\( T^{8} + 316 T^{6} + \cdots + 36966400 \)
|
| $89$ |
\( (T^{4} - 4 T^{3} + \cdots - 2456)^{2} \)
|
| $97$ |
\( (T^{4} + 4 T^{3} + \cdots - 688)^{2} \)
|
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