Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [79,3,Mod(78,79)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("79.78");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 79 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 79.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: | \(2.15259408845\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 77x^{6} + 2073x^{4} + 23519x^{2} + 95938 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} + 77x^{6} + 2073x^{4} + 23519x^{2} + 95938 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 58\nu^{5} - 943\nu^{3} - 4230\nu ) / 84 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 58\nu^{4} - 943\nu^{2} - 4398 ) / 84 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{6} - 123\nu^{4} - 2271\nu^{2} - 13066 ) / 126 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{7} - 123\nu^{5} - 2271\nu^{3} - 13066\nu ) / 126 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} - 65\nu^{4} - 1286\nu^{2} - 7828 ) / 42 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - 65\nu^{5} - 1286\nu^{3} - 7828\nu ) / 42 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - 3\beta_{4} + 2\beta_{3} - 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 3\beta_{5} + 2\beta_{2} - 24\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -55\beta_{6} + 147\beta_{4} - 86\beta_{3} + 490 \)
|
| \(\nu^{5}\) | \(=\) |
\( -55\beta_{7} + 147\beta_{5} - 86\beta_{2} + 662\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2247\beta_{6} - 5697\beta_{4} + 3018\beta_{3} - 13958 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2247\beta_{7} - 5697\beta_{5} + 3018\beta_{2} - 19994\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/79\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 78.1 |
|
−3.17284 | − | 5.79987i | 6.06689 | 6.39881 | 18.4021i | − | 7.72499i | −6.55790 | −24.6385 | −20.3024 | ||||||||||||||||||||||||||||||||||||||||
| 78.2 | −3.17284 | 5.79987i | 6.06689 | 6.39881 | − | 18.4021i | 7.72499i | −6.55790 | −24.6385 | −20.3024 | ||||||||||||||||||||||||||||||||||||||||||
| 78.3 | −1.76238 | − | 3.67287i | −0.894022 | −5.37838 | 6.47299i | 12.7976i | 8.62512 | −4.48997 | 9.47874 | ||||||||||||||||||||||||||||||||||||||||||
| 78.4 | −1.76238 | 3.67287i | −0.894022 | −5.37838 | − | 6.47299i | − | 12.7976i | 8.62512 | −4.48997 | 9.47874 | |||||||||||||||||||||||||||||||||||||||||
| 78.5 | 0.345246 | − | 3.39431i | −3.88081 | −2.25262 | − | 1.17187i | − | 8.92088i | −2.72082 | −2.52133 | −0.777707 | ||||||||||||||||||||||||||||||||||||||||
| 78.6 | 0.345246 | 3.39431i | −3.88081 | −2.25262 | 1.17187i | 8.92088i | −2.72082 | −2.52133 | −0.777707 | |||||||||||||||||||||||||||||||||||||||||||
| 78.7 | 2.58997 | − | 4.28371i | 2.70794 | 0.232185 | − | 11.0947i | 6.32169i | −3.34640 | −9.35015 | 0.601352 | |||||||||||||||||||||||||||||||||||||||||
| 78.8 | 2.58997 | 4.28371i | 2.70794 | 0.232185 | 11.0947i | − | 6.32169i | −3.34640 | −9.35015 | 0.601352 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 79.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 79.3.b.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 711.3.d.b | 8 | ||
| 4.b | odd | 2 | 1 | 1264.3.e.b | 8 | ||
| 79.b | odd | 2 | 1 | inner | 79.3.b.b | ✓ | 8 |
| 237.b | even | 2 | 1 | 711.3.d.b | 8 | ||
| 316.d | even | 2 | 1 | 1264.3.e.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 79.3.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 79.3.b.b | ✓ | 8 | 79.b | odd | 2 | 1 | inner |
| 711.3.d.b | 8 | 3.b | odd | 2 | 1 | ||
| 711.3.d.b | 8 | 237.b | even | 2 | 1 | ||
| 1264.3.e.b | 8 | 4.b | odd | 2 | 1 | ||
| 1264.3.e.b | 8 | 316.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 8T_{2}^{2} - 12T_{2} + 5 \)
acting on \(S_{3}^{\mathrm{new}}(79, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} - 8 T^{2} + \cdots + 5)^{2} \)
|
| $3$ |
\( T^{8} + 77 T^{6} + \cdots + 95938 \)
|
| $5$ |
\( (T^{4} + T^{3} - 37 T^{2} + \cdots + 18)^{2} \)
|
| $7$ |
\( T^{8} + 343 T^{6} + \cdots + 31083912 \)
|
| $11$ |
\( (T^{4} - 14 T^{3} + \cdots + 1380)^{2} \)
|
| $13$ |
\( (T^{4} + 15 T^{3} + \cdots - 10)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 1989370368 \)
|
| $19$ |
\( (T^{4} + 16 T^{3} + \cdots + 114420)^{2} \)
|
| $23$ |
\( (T^{4} - 30 T^{3} + \cdots + 8000)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 62944921800 \)
|
| $31$ |
\( (T^{4} - 18 T^{3} + \cdots + 295832)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 775978779168 \)
|
| $41$ |
\( T^{8} + \cdots + 77709780000 \)
|
| $43$ |
\( T^{8} + \cdots + 1119020832 \)
|
| $47$ |
\( T^{8} + \cdots + 8534896221312 \)
|
| $53$ |
\( T^{8} + \cdots + 1687514498568 \)
|
| $59$ |
\( T^{8} + \cdots + 11471711998050 \)
|
| $61$ |
\( T^{8} + \cdots + 74679098580000 \)
|
| $67$ |
\( (T^{4} - 66 T^{3} + \cdots - 509800)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 6993880200 \)
|
| $73$ |
\( (T^{4} - 192 T^{3} + \cdots + 782720)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 15\!\cdots\!61 \)
|
| $83$ |
\( (T^{4} - 48 T^{3} + \cdots + 38130280)^{2} \)
|
| $89$ |
\( (T^{4} - 51 T^{3} + \cdots - 160120)^{2} \)
|
| $97$ |
\( (T^{4} - 235 T^{3} + \cdots + 12300170)^{2} \)
|
| show more | |
| show less |