Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [78,6,Mod(55,78)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(78, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("78.55");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 78 = 2 \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 78.e (of order \(3\), degree \(2\), 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: | \(12.5099379454\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{649})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 163x^{2} + 162x + 26244 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 163x^{2} + 162x + 26244 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 163\nu^{2} - 163\nu + 26244 ) / 26406 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 325 ) / 163 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 162\beta_{2} + \beta _1 - 163 \)
|
| \(\nu^{3}\) | \(=\) |
\( 163\beta_{3} - 325 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(67\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
2.00000 | + | 3.46410i | −4.50000 | − | 7.79423i | −8.00000 | + | 13.8564i | −0.237739 | 18.0000 | − | 31.1769i | 22.0943 | − | 38.2685i | −64.0000 | −40.5000 | + | 70.1481i | −0.475478 | − | 0.823553i | ||||||||||||||||
| 55.2 | 2.00000 | + | 3.46410i | −4.50000 | − | 7.79423i | −8.00000 | + | 13.8564i | 25.2377 | 18.0000 | − | 31.1769i | −41.5943 | + | 72.0435i | −64.0000 | −40.5000 | + | 70.1481i | 50.4755 | + | 87.4261i | |||||||||||||||||
| 61.1 | 2.00000 | − | 3.46410i | −4.50000 | + | 7.79423i | −8.00000 | − | 13.8564i | −0.237739 | 18.0000 | + | 31.1769i | 22.0943 | + | 38.2685i | −64.0000 | −40.5000 | − | 70.1481i | −0.475478 | + | 0.823553i | |||||||||||||||||
| 61.2 | 2.00000 | − | 3.46410i | −4.50000 | + | 7.79423i | −8.00000 | − | 13.8564i | 25.2377 | 18.0000 | + | 31.1769i | −41.5943 | − | 72.0435i | −64.0000 | −40.5000 | − | 70.1481i | 50.4755 | − | 87.4261i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 78.6.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 234.6.h.a | 4 | ||
| 13.c | even | 3 | 1 | inner | 78.6.e.a | ✓ | 4 |
| 13.c | even | 3 | 1 | 1014.6.a.j | 2 | ||
| 13.e | even | 6 | 1 | 1014.6.a.m | 2 | ||
| 39.i | odd | 6 | 1 | 234.6.h.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.6.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 78.6.e.a | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 234.6.h.a | 4 | 3.b | odd | 2 | 1 | ||
| 234.6.h.a | 4 | 39.i | odd | 6 | 1 | ||
| 1014.6.a.j | 2 | 13.c | even | 3 | 1 | ||
| 1014.6.a.m | 2 | 13.e | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 25T_{5} - 6 \)
acting on \(S_{6}^{\mathrm{new}}(78, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( (T^{2} + 9 T + 81)^{2} \)
|
| $5$ |
\( (T^{2} - 25 T - 6)^{2} \)
|
| $7$ |
\( T^{4} + 39 T^{3} + \cdots + 13512976 \)
|
| $11$ |
\( T^{4} + 362 T^{3} + \cdots + 273439296 \)
|
| $13$ |
\( T^{4} + \cdots + 137858491849 \)
|
| $17$ |
\( T^{4} + \cdots + 16519347360000 \)
|
| $19$ |
\( T^{4} + \cdots + 3158439840000 \)
|
| $23$ |
\( T^{4} + \cdots + 220554215424 \)
|
| $29$ |
\( T^{4} + \cdots + 327676104900 \)
|
| $31$ |
\( (T^{2} - 12117 T + 34329920)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 174462623064900 \)
|
| $41$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $47$ |
\( (T^{2} + 3946 T - 98395512)^{2} \)
|
| $53$ |
\( (T^{2} + 5229 T - 541881072)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 12\!\cdots\!25 \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 75\!\cdots\!24 \)
|
| $73$ |
\( (T^{2} - 7492 T - 649324013)^{2} \)
|
| $79$ |
\( (T^{2} - 68935 T - 2297644700)^{2} \)
|
| $83$ |
\( (T^{2} - 96696 T + 733730688)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
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