Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4851,2,Mod(1,4851)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4851.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4851.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: | \(38.7354300205\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.672323328.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 11x^{4} - 2x^{3} + 33x^{2} + 10x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 693) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 11x^{4} - 2x^{3} + 33x^{2} + 10x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 9\nu^{3} + 7\nu^{2} + 18\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 9\nu^{3} - 15\nu^{2} - 18\nu + 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{2} + 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 9\beta_{3} + \beta_{2} + 27\beta _1 + 10 \)
|
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 |
|
−2.60998 | 0 | 4.81199 | 3.81199 | 0 | 0 | −7.33922 | 0 | −9.94920 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.07171 | 0 | 2.29197 | 1.29197 | 0 | 0 | −0.604878 | 0 | −2.67659 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.726544 | 0 | −1.47213 | −2.47213 | 0 | 0 | 2.52266 | 0 | 1.79611 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.34104 | 0 | −0.201605 | −1.20161 | 0 | 0 | −2.95245 | 0 | −1.61140 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.64709 | 0 | 0.712889 | −0.287111 | 0 | 0 | −2.11998 | 0 | −0.472896 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.42010 | 0 | 3.85689 | 2.85689 | 0 | 0 | 4.49387 | 0 | 6.91397 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 4851.2.a.cd | 6 | |
| 3.b | odd | 2 | 1 | 4851.2.a.cc | 6 | ||
| 7.b | odd | 2 | 1 | 4851.2.a.cb | 6 | ||
| 7.c | even | 3 | 2 | 693.2.i.k | ✓ | 12 | |
| 21.c | even | 2 | 1 | 4851.2.a.ce | 6 | ||
| 21.h | odd | 6 | 2 | 693.2.i.l | yes | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 693.2.i.k | ✓ | 12 | 7.c | even | 3 | 2 | |
| 693.2.i.l | yes | 12 | 21.h | odd | 6 | 2 | |
| 4851.2.a.cb | 6 | 7.b | odd | 2 | 1 | ||
| 4851.2.a.cc | 6 | 3.b | odd | 2 | 1 | ||
| 4851.2.a.cd | 6 | 1.a | even | 1 | 1 | trivial | |
| 4851.2.a.ce | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):
|
\( T_{2}^{6} - 11T_{2}^{4} + 2T_{2}^{3} + 33T_{2}^{2} - 10T_{2} - 21 \)
|
|
\( T_{5}^{6} - 4T_{5}^{5} - 8T_{5}^{4} + 32T_{5}^{3} + 16T_{5}^{2} - 40T_{5} - 12 \)
|
|
\( T_{13}^{6} + 2T_{13}^{5} - 42T_{13}^{4} - 80T_{13}^{3} + 340T_{13}^{2} + 360T_{13} - 356 \)
|
|
\( T_{17}^{6} - 8T_{17}^{5} - 31T_{17}^{4} + 278T_{17}^{3} - 187T_{17}^{2} - 722T_{17} + 15 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 11 T^{4} + \cdots - 21 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 4 T^{5} + \cdots - 12 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} + 2 T^{5} + \cdots - 356 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 15 \)
|
| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 1791 \)
|
| $23$ |
\( T^{6} + 2 T^{5} + \cdots - 615 \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots - 11853 \)
|
| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 100 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots - 55503 \)
|
| $41$ |
\( T^{6} - 30 T^{5} + \cdots + 2988 \)
|
| $43$ |
\( T^{6} + 12 T^{5} + \cdots - 44001 \)
|
| $47$ |
\( T^{6} - 14 T^{5} + \cdots + 30273 \)
|
| $53$ |
\( T^{6} - 16 T^{5} + \cdots + 2916 \)
|
| $59$ |
\( T^{6} - 34 T^{5} + \cdots - 1239 \)
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| $61$ |
\( T^{6} + 2 T^{5} + \cdots + 492 \)
|
| $67$ |
\( T^{6} + 20 T^{5} + \cdots - 75676 \)
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| $71$ |
\( T^{6} + 6 T^{5} + \cdots + 561573 \)
|
| $73$ |
\( T^{6} - 26 T^{5} + \cdots + 41900 \)
|
| $79$ |
\( T^{6} - 2 T^{5} + \cdots - 1028 \)
|
| $83$ |
\( T^{6} - 8 T^{5} + \cdots + 17280 \)
|
| $89$ |
\( T^{6} - 20 T^{5} + \cdots - 60336 \)
|
| $97$ |
\( T^{6} + 14 T^{5} + \cdots + 1909 \)
|
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