Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [138,3,Mod(91,138)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(138, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("138.91");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 138.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: | \(3.76022764817\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1358954496.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} + 4\nu^{2} + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{6} - 6\nu^{4} - 7\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} + 7\nu^{5} + 13\nu^{3} + 5\nu \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} + 7\nu^{5} + 15\nu^{3} + 11\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 3\beta_{3} + 3\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} - 2\beta_{4} + \beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -5\beta_{7} + 5\beta_{6} + 11\beta_{3} - 9\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 17\beta_{5} + 15\beta_{4} - 12\beta _1 - 40 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 22\beta_{7} - 20\beta_{6} - 43\beta_{3} + 29\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−1.41421 | −1.73205 | 2.00000 | − | 1.69484i | 2.44949 | 4.18388i | −2.82843 | 3.00000 | 2.39686i | |||||||||||||||||||||||||||||||||||||||||
| 91.2 | −1.41421 | −1.73205 | 2.00000 | 1.69484i | 2.44949 | − | 4.18388i | −2.82843 | 3.00000 | − | 2.39686i | |||||||||||||||||||||||||||||||||||||||||
| 91.3 | −1.41421 | 1.73205 | 2.00000 | − | 0.918288i | −2.44949 | − | 10.4925i | −2.82843 | 3.00000 | 1.29866i | |||||||||||||||||||||||||||||||||||||||||
| 91.4 | −1.41421 | 1.73205 | 2.00000 | 0.918288i | −2.44949 | 10.4925i | −2.82843 | 3.00000 | − | 1.29866i | ||||||||||||||||||||||||||||||||||||||||||
| 91.5 | 1.41421 | −1.73205 | 2.00000 | − | 6.00464i | −2.44949 | − | 4.69017i | 2.82843 | 3.00000 | − | 8.49185i | ||||||||||||||||||||||||||||||||||||||||
| 91.6 | 1.41421 | −1.73205 | 2.00000 | 6.00464i | −2.44949 | 4.69017i | 2.82843 | 3.00000 | 8.49185i | |||||||||||||||||||||||||||||||||||||||||||
| 91.7 | 1.41421 | 1.73205 | 2.00000 | − | 4.92225i | 2.44949 | − | 5.13851i | 2.82843 | 3.00000 | − | 6.96111i | ||||||||||||||||||||||||||||||||||||||||
| 91.8 | 1.41421 | 1.73205 | 2.00000 | 4.92225i | 2.44949 | 5.13851i | 2.82843 | 3.00000 | 6.96111i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 138.3.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 414.3.b.c | 8 | ||
| 4.b | odd | 2 | 1 | 1104.3.c.c | 8 | ||
| 23.b | odd | 2 | 1 | inner | 138.3.b.a | ✓ | 8 |
| 69.c | even | 2 | 1 | 414.3.b.c | 8 | ||
| 92.b | even | 2 | 1 | 1104.3.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.3.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 138.3.b.a | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
| 414.3.b.c | 8 | 3.b | odd | 2 | 1 | ||
| 414.3.b.c | 8 | 69.c | even | 2 | 1 | ||
| 1104.3.c.c | 8 | 4.b | odd | 2 | 1 | ||
| 1104.3.c.c | 8 | 92.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(138, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{4} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( T^{8} + 64 T^{6} + \cdots + 2116 \)
|
| $7$ |
\( T^{8} + 176 T^{6} + \cdots + 1119364 \)
|
| $11$ |
\( T^{8} + 512 T^{6} + \cdots + 2262016 \)
|
| $13$ |
\( (T^{4} - 8 T^{3} + \cdots - 956)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 1138792516 \)
|
| $19$ |
\( T^{8} + \cdots + 14491825924 \)
|
| $23$ |
\( T^{8} + \cdots + 78310985281 \)
|
| $29$ |
\( (T^{4} + 72 T^{3} + \cdots - 6128)^{2} \)
|
| $31$ |
\( (T^{4} + 64 T^{3} + \cdots + 1600)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 2955978735616 \)
|
| $41$ |
\( (T^{4} + 8 T^{3} + \cdots + 1208464)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 26854687876 \)
|
| $47$ |
\( (T^{4} + 56 T^{3} + \cdots + 256036)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 842259883497796 \)
|
| $59$ |
\( (T^{4} - 40 T^{3} + \cdots + 6720292)^{2} \)
|
| $61$ |
\( T^{8} + 24704 T^{6} + \cdots + 138674176 \)
|
| $67$ |
\( T^{8} + \cdots + 15376496004 \)
|
| $71$ |
\( (T^{4} - 16 T^{3} + \cdots - 812912)^{2} \)
|
| $73$ |
\( (T^{4} - 32 T^{3} + \cdots + 590848)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 7772598291844 \)
|
| $83$ |
\( T^{8} + \cdots + 5440313672704 \)
|
| $89$ |
\( T^{8} + \cdots + 69\!\cdots\!16 \)
|
| $97$ |
\( T^{8} + \cdots + 380951699424256 \)
|
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