Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,6,Mod(1,1014)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1014.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.a (trivial) |
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: | yes |
| Analytic conductor: | \(162.629193290\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 727x - 6617 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 78) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 727x - 6617 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 18\nu - 478 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 484 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 484 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.00000 | −9.00000 | 16.0000 | −37.1158 | 36.0000 | 176.407 | −64.0000 | 81.0000 | 148.463 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | −9.00000 | 16.0000 | −9.73803 | 36.0000 | −105.184 | −64.0000 | 81.0000 | 38.9521 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −9.00000 | 16.0000 | 86.8538 | 36.0000 | 98.7774 | −64.0000 | 81.0000 | −347.415 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.6.a.o | 3 | |
| 13.b | even | 2 | 1 | 1014.6.a.q | 3 | ||
| 13.d | odd | 4 | 2 | 78.6.b.a | ✓ | 6 | |
| 39.f | even | 4 | 2 | 234.6.b.c | 6 | ||
| 52.f | even | 4 | 2 | 624.6.c.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.6.b.a | ✓ | 6 | 13.d | odd | 4 | 2 | |
| 234.6.b.c | 6 | 39.f | even | 4 | 2 | ||
| 624.6.c.d | 6 | 52.f | even | 4 | 2 | ||
| 1014.6.a.o | 3 | 1.a | even | 1 | 1 | trivial | |
| 1014.6.a.q | 3 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1014))\):
|
\( T_{5}^{3} - 40T_{5}^{2} - 3708T_{5} - 31392 \)
|
|
\( T_{7}^{3} - 170T_{7}^{2} - 11520T_{7} + 1832832 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( T^{3} - 40 T^{2} + \cdots - 31392 \)
|
| $7$ |
\( T^{3} - 170 T^{2} + \cdots + 1832832 \)
|
| $11$ |
\( T^{3} - 162 T^{2} + \cdots - 29513160 \)
|
| $13$ |
\( T^{3} \)
|
| $17$ |
\( T^{3} + \cdots + 1783218312 \)
|
| $19$ |
\( T^{3} - 66 T^{2} + \cdots - 14973984 \)
|
| $23$ |
\( T^{3} + \cdots - 2602266624 \)
|
| $29$ |
\( T^{3} + \cdots - 2546571960 \)
|
| $31$ |
\( T^{3} + \cdots - 4280450400 \)
|
| $37$ |
\( T^{3} + \cdots + 39144102912 \)
|
| $41$ |
\( T^{3} + \cdots + 20206815120 \)
|
| $43$ |
\( T^{3} + \cdots + 1729964364544 \)
|
| $47$ |
\( T^{3} + \cdots + 4517894971368 \)
|
| $53$ |
\( T^{3} + \cdots - 26960052479832 \)
|
| $59$ |
\( T^{3} + \cdots - 10593394872552 \)
|
| $61$ |
\( T^{3} + \cdots + 17380383329800 \)
|
| $67$ |
\( T^{3} + \cdots - 30813396636672 \)
|
| $71$ |
\( T^{3} + \cdots - 47139250910760 \)
|
| $73$ |
\( T^{3} + \cdots - 3323872221696 \)
|
| $79$ |
\( T^{3} + \cdots - 35835411156480 \)
|
| $83$ |
\( T^{3} + \cdots - 1086071695416 \)
|
| $89$ |
\( T^{3} + \cdots + 59175760270896 \)
|
| $97$ |
\( T^{3} + \cdots - 17578655448192 \)
|
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