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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2192x^{2} - 12432x + 55224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2192x^{2} - 12432x + 55224 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu^{2} - 2162\nu - 15954 ) / 90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 26\nu^{2} + 4036\nu - 7575 ) / 45 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} + 20\beta_{2} + 16\beta _1 + 4387 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -25\beta_{3} + 260\beta_{2} + 4244\beta _1 + 41881 ) / 4 \)
|
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 | −101.388 | −36.0000 | 49.3291 | −64.0000 | 81.0000 | 405.553 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | 9.00000 | 16.0000 | −7.86228 | −36.0000 | −233.784 | −64.0000 | 81.0000 | 31.4491 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 9.00000 | 16.0000 | 15.7335 | −36.0000 | 69.0882 | −64.0000 | 81.0000 | −62.9340 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 9.00000 | 16.0000 | 83.5169 | −36.0000 | 187.367 | −64.0000 | 81.0000 | −334.068 | |||||||||||||||||||||||||||||||
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.t | 4 | |
| 13.b | even | 2 | 1 | 1014.6.a.u | 4 | ||
| 13.d | odd | 4 | 2 | 78.6.b.b | ✓ | 8 | |
| 39.f | even | 4 | 2 | 234.6.b.d | 8 | ||
| 52.f | even | 4 | 2 | 624.6.c.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.6.b.b | ✓ | 8 | 13.d | odd | 4 | 2 | |
| 234.6.b.d | 8 | 39.f | even | 4 | 2 | ||
| 624.6.c.f | 8 | 52.f | even | 4 | 2 | ||
| 1014.6.a.t | 4 | 1.a | even | 1 | 1 | trivial | |
| 1014.6.a.u | 4 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(1014))\):
|
\( T_{5}^{4} + 10T_{5}^{3} - 8732T_{5}^{2} + 64440T_{5} + 1047456 \)
|
|
\( T_{7}^{4} - 72T_{7}^{3} - 45892T_{7}^{2} + 5345280T_{7} - 149284800 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} + 10 T^{3} + \cdots + 1047456 \)
|
| $7$ |
\( T^{4} - 72 T^{3} + \cdots - 149284800 \)
|
| $11$ |
\( T^{4} + \cdots - 8317015200 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 223697296368 \)
|
| $19$ |
\( T^{4} + \cdots + 837274330368 \)
|
| $23$ |
\( T^{4} + \cdots + 17976140163072 \)
|
| $29$ |
\( T^{4} + \cdots + 12705499286928 \)
|
| $31$ |
\( T^{4} + \cdots + 123917462876160 \)
|
| $37$ |
\( T^{4} + \cdots - 21\!\cdots\!08 \)
|
| $41$ |
\( T^{4} + \cdots - 11\!\cdots\!44 \)
|
| $43$ |
\( T^{4} + \cdots - 53\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 15\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 40\!\cdots\!92 \)
|
| $61$ |
\( T^{4} + \cdots + 53\!\cdots\!60 \)
|
| $67$ |
\( T^{4} + \cdots - 12\!\cdots\!60 \)
|
| $71$ |
\( T^{4} + \cdots + 68\!\cdots\!88 \)
|
| $73$ |
\( T^{4} + \cdots + 55\!\cdots\!92 \)
|
| $79$ |
\( T^{4} + \cdots + 11\!\cdots\!88 \)
|
| $83$ |
\( T^{4} + \cdots - 44\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 14\!\cdots\!20 \)
|
| $97$ |
\( T^{4} + \cdots - 35\!\cdots\!68 \)
|
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