Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [726,6,Mod(1,726)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(726, 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("726.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 726.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: | \(116.438653184\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8052400.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 288x^{2} + 20131 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 11 \) |
| Twist minimal: | no (minimal twist has level 66) |
| 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
\( x^{4} - 288x^{2} + 20131 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 44\nu + 155 ) / 22 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 5\nu^{2} - 310\nu + 731 ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 143 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 11\beta _1 - 6 ) / 22 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + 143 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 265\beta_{3} + 242\beta_{2} + 1705\beta _1 - 1106 ) / 22 \)
|
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 | −111.486 | −36.0000 | −226.020 | −64.0000 | 81.0000 | 445.944 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | 9.00000 | 16.0000 | −30.6488 | −36.0000 | 1.59188 | −64.0000 | 81.0000 | 122.595 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 9.00000 | 16.0000 | −10.8102 | −36.0000 | 229.182 | −64.0000 | 81.0000 | 43.2409 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 9.00000 | 16.0000 | 2.94494 | −36.0000 | 63.2459 | −64.0000 | 81.0000 | −11.7798 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 726.6.a.bb | 4 | |
| 11.b | odd | 2 | 1 | 726.6.a.be | 4 | ||
| 11.d | odd | 10 | 2 | 66.6.e.a | ✓ | 8 | |
| 33.f | even | 10 | 2 | 198.6.f.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 66.6.e.a | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 198.6.f.c | 8 | 33.f | even | 10 | 2 | ||
| 726.6.a.bb | 4 | 1.a | even | 1 | 1 | trivial | |
| 726.6.a.be | 4 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(726))\):
|
\( T_{5}^{4} + 150T_{5}^{3} + 4503T_{5}^{2} + 22350T_{5} - 108779 \)
|
|
\( T_{7}^{4} - 68T_{7}^{3} - 51494T_{7}^{2} + 3358260T_{7} - 5215195 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} + 150 T^{3} + \cdots - 108779 \)
|
| $7$ |
\( T^{4} - 68 T^{3} + \cdots - 5215195 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + \cdots + 7071504001 \)
|
| $17$ |
\( T^{4} + \cdots - 216904750819 \)
|
| $19$ |
\( T^{4} + \cdots + 3791385857941 \)
|
| $23$ |
\( T^{4} + \cdots - 3587654924095 \)
|
| $29$ |
\( T^{4} + \cdots - 12882507946544 \)
|
| $31$ |
\( T^{4} + \cdots + 5230193354341 \)
|
| $37$ |
\( T^{4} + \cdots + 168866357471505 \)
|
| $41$ |
\( T^{4} + \cdots + 358733822777605 \)
|
| $43$ |
\( T^{4} + \cdots - 878847087788595 \)
|
| $47$ |
\( T^{4} + \cdots + 637450817857505 \)
|
| $53$ |
\( T^{4} + \cdots + 28\!\cdots\!89 \)
|
| $59$ |
\( T^{4} + \cdots + 67\!\cdots\!61 \)
|
| $61$ |
\( T^{4} + \cdots + 44\!\cdots\!81 \)
|
| $67$ |
\( T^{4} + \cdots - 20\!\cdots\!51 \)
|
| $71$ |
\( T^{4} + \cdots + 28\!\cdots\!81 \)
|
| $73$ |
\( T^{4} + \cdots + 607575386880720 \)
|
| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!45 \)
|
| $83$ |
\( T^{4} + \cdots + 28\!\cdots\!01 \)
|
| $89$ |
\( T^{4} + \cdots - 84\!\cdots\!99 \)
|
| $97$ |
\( T^{4} + \cdots - 22\!\cdots\!39 \)
|
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