Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,4,Mod(337,1014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1014.337");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.b (of order \(2\), degree \(1\), not 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: | \(59.8279367458\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{673})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 337x^{2} + 28224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 78) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 337x^{2} + 28224 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 169\nu ) / 168 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 169 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 169 \)
|
| \(\nu^{3}\) | \(=\) |
\( 168\beta_{2} - 169\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 2.00000i | 3.00000 | −4.00000 | − | 19.4711i | − | 6.00000i | − | 8.47112i | 8.00000i | 9.00000 | −38.9422 | ||||||||||||||||||||||||||
| 337.2 | − | 2.00000i | 3.00000 | −4.00000 | 6.47112i | − | 6.00000i | 17.4711i | 8.00000i | 9.00000 | 12.9422 | |||||||||||||||||||||||||||||
| 337.3 | 2.00000i | 3.00000 | −4.00000 | − | 6.47112i | 6.00000i | − | 17.4711i | − | 8.00000i | 9.00000 | 12.9422 | ||||||||||||||||||||||||||||
| 337.4 | 2.00000i | 3.00000 | −4.00000 | 19.4711i | 6.00000i | 8.47112i | − | 8.00000i | 9.00000 | −38.9422 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.4.b.j | 4 | |
| 13.b | even | 2 | 1 | inner | 1014.4.b.j | 4 | |
| 13.d | odd | 4 | 1 | 1014.4.a.m | 2 | ||
| 13.d | odd | 4 | 1 | 1014.4.a.s | 2 | ||
| 13.f | odd | 12 | 2 | 78.4.e.b | ✓ | 4 | |
| 39.k | even | 12 | 2 | 234.4.h.g | 4 | ||
| 52.l | even | 12 | 2 | 624.4.q.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.4.e.b | ✓ | 4 | 13.f | odd | 12 | 2 | |
| 234.4.h.g | 4 | 39.k | even | 12 | 2 | ||
| 624.4.q.f | 4 | 52.l | even | 12 | 2 | ||
| 1014.4.a.m | 2 | 13.d | odd | 4 | 1 | ||
| 1014.4.a.s | 2 | 13.d | odd | 4 | 1 | ||
| 1014.4.b.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 1014.4.b.j | 4 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1014, [\chi])\):
|
\( T_{5}^{4} + 421T_{5}^{2} + 15876 \)
|
|
\( T_{7}^{4} + 377T_{7}^{2} + 21904 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 421 T^{2} + 15876 \)
|
| $7$ |
\( T^{4} + 377 T^{2} + 21904 \)
|
| $11$ |
\( T^{4} + 2068 T^{2} + 97344 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 99 T + 936)^{2} \)
|
| $19$ |
\( T^{4} + 21664 T^{2} + 114575616 \)
|
| $23$ |
\( (T^{2} - 14 T - 624)^{2} \)
|
| $29$ |
\( (T^{2} - 121 T - 16698)^{2} \)
|
| $31$ |
\( T^{4} + 15257 T^{2} + 614656 \)
|
| $37$ |
\( T^{4} + \cdots + 1418426244 \)
|
| $41$ |
\( T^{4} + \cdots + 2160018576 \)
|
| $43$ |
\( (T^{2} + 645 T + 83648)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 6584348736 \)
|
| $53$ |
\( (T^{2} + 729 T + 119232)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 25787221056 \)
|
| $61$ |
\( (T^{2} + 846 T + 176237)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 5515141696 \)
|
| $71$ |
\( T^{4} + \cdots + 44089920576 \)
|
| $73$ |
\( T^{4} + \cdots + 136795679881 \)
|
| $79$ |
\( (T^{2} + 599 T + 51844)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 5267275776 \)
|
| $89$ |
\( T^{4} + \cdots + 70471135296 \)
|
| $97$ |
\( T^{4} + \cdots + 101729102500 \)
|
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