Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,3,Mod(90,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("91.90");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 91.b (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: | \(2.47957040568\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 28x^{4} + 235x^{2} + 608 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 28x^{4} + 235x^{2} + 608 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 21\nu^{2} + 92 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 21\nu^{3} + 21\nu^{2} + 92\nu + 92 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 25\nu^{3} + 21\nu^{2} + 136\nu + 92 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} - 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} - 21\beta_{2} + 97 \)
|
| \(\nu^{5}\) | \(=\) |
\( -21\beta_{5} + 25\beta_{4} - 2\beta_{3} + 139\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(66\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 90.1 |
|
− | 3.88732i | − | 3.88732i | −11.1113 | 5.61812 | −15.1113 | 6.35783 | − | 2.92882i | 27.6437i | −6.11126 | − | 21.8394i | |||||||||||||||||||||||||||||||
| 90.2 | − | 2.75361i | − | 2.75361i | −3.58235 | −8.28632 | −7.58235 | 2.01669 | + | 6.70321i | − | 1.15003i | 1.41765 | 22.8173i | ||||||||||||||||||||||||||||||||
| 90.3 | − | 2.30356i | − | 2.30356i | −1.30639 | −1.33180 | −5.30639 | −4.87452 | − | 5.02385i | − | 6.20490i | 3.69361 | 3.06788i | ||||||||||||||||||||||||||||||||
| 90.4 | 2.30356i | 2.30356i | −1.30639 | −1.33180 | −5.30639 | −4.87452 | + | 5.02385i | 6.20490i | 3.69361 | − | 3.06788i | ||||||||||||||||||||||||||||||||||
| 90.5 | 2.75361i | 2.75361i | −3.58235 | −8.28632 | −7.58235 | 2.01669 | − | 6.70321i | 1.15003i | 1.41765 | − | 22.8173i | ||||||||||||||||||||||||||||||||||
| 90.6 | 3.88732i | 3.88732i | −11.1113 | 5.61812 | −15.1113 | 6.35783 | + | 2.92882i | − | 27.6437i | −6.11126 | 21.8394i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 91.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 91.3.b.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 819.3.d.g | 6 | ||
| 7.b | odd | 2 | 1 | 91.3.b.e | yes | 6 | |
| 13.b | even | 2 | 1 | 91.3.b.e | yes | 6 | |
| 21.c | even | 2 | 1 | 819.3.d.f | 6 | ||
| 39.d | odd | 2 | 1 | 819.3.d.f | 6 | ||
| 91.b | odd | 2 | 1 | inner | 91.3.b.d | ✓ | 6 |
| 273.g | even | 2 | 1 | 819.3.d.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.3.b.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 91.3.b.d | ✓ | 6 | 91.b | odd | 2 | 1 | inner |
| 91.3.b.e | yes | 6 | 7.b | odd | 2 | 1 | |
| 91.3.b.e | yes | 6 | 13.b | even | 2 | 1 | |
| 819.3.d.f | 6 | 21.c | even | 2 | 1 | ||
| 819.3.d.f | 6 | 39.d | odd | 2 | 1 | ||
| 819.3.d.g | 6 | 3.b | odd | 2 | 1 | ||
| 819.3.d.g | 6 | 273.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(91, [\chi])\):
|
\( T_{2}^{6} + 28T_{2}^{4} + 235T_{2}^{2} + 608 \)
|
|
\( T_{5}^{3} + 4T_{5}^{2} - 43T_{5} - 62 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 28 T^{4} + \cdots + 608 \)
|
| $3$ |
\( T^{6} + 28 T^{4} + \cdots + 608 \)
|
| $5$ |
\( (T^{3} + 4 T^{2} - 43 T - 62)^{2} \)
|
| $7$ |
\( T^{6} - 7 T^{5} + \cdots + 117649 \)
|
| $11$ |
\( T^{6} + 325 T^{4} + \cdots + 60800 \)
|
| $13$ |
\( T^{6} - 24 T^{5} + \cdots + 4826809 \)
|
| $17$ |
\( T^{6} + 655 T^{4} + \cdots + 877952 \)
|
| $19$ |
\( (T^{3} - 13 T^{2} + \cdots - 3032)^{2} \)
|
| $23$ |
\( (T^{3} + 20 T^{2} + \cdots - 18160)^{2} \)
|
| $29$ |
\( (T^{3} - 35 T^{2} + \cdots + 14324)^{2} \)
|
| $31$ |
\( (T^{3} - 66 T^{2} + \cdots + 6898)^{2} \)
|
| $37$ |
\( T^{6} + 2884 T^{4} + \cdots + 112942688 \)
|
| $41$ |
\( (T^{3} - 83 T^{2} + \cdots - 13988)^{2} \)
|
| $43$ |
\( (T^{3} - 10 T^{2} + \cdots + 35650)^{2} \)
|
| $47$ |
\( (T^{3} - 119 T^{2} + \cdots - 54745)^{2} \)
|
| $53$ |
\( (T^{3} + 95 T^{2} + \cdots + 16000)^{2} \)
|
| $59$ |
\( (T^{3} + 6 T^{2} + \cdots - 4096)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 7967475200 \)
|
| $67$ |
\( T^{6} + \cdots + 13729554432 \)
|
| $71$ |
\( T^{6} + \cdots + 1615395200 \)
|
| $73$ |
\( (T^{3} + 222 T^{2} + \cdots + 252760)^{2} \)
|
| $79$ |
\( (T^{3} - 80 T^{2} + \cdots + 27002)^{2} \)
|
| $83$ |
\( (T^{3} - 5 T^{2} + \cdots + 71480)^{2} \)
|
| $89$ |
\( (T^{3} + 197 T^{2} + \cdots + 177868)^{2} \)
|
| $97$ |
\( (T^{3} - 50 T^{2} + \cdots - 49750)^{2} \)
|
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