Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,9,Mod(26,27)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.26");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.b (of order \(2\), degree \(1\), 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: | \(10.9992224717\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.6171673600.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{21} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 295\nu^{5} - 669\nu^{4} + 458\nu^{3} + 13712\nu^{2} + 58285\nu + 20675 ) / 1085 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -792\nu^{5} + 4239\nu^{4} - 13464\nu^{3} - 15840\nu^{2} - 7920\nu + 142110 ) / 1085 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5994\nu^{5} - 50058\nu^{4} + 101898\nu^{3} + 119880\nu^{2} + 59940\nu - 4754700 ) / 1085 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1653\nu^{5} - 3987\nu^{4} + 6618\nu^{3} + 62355\nu^{2} + 377835\nu + 133725 ) / 217 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -46063\nu^{5} + 107679\nu^{4} - 126212\nu^{3} - 1708319\nu^{2} - 9792715\nu - 3469625 ) / 1085 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 9\beta_{4} + \beta_{3} + 18\beta_{2} - 68\beta _1 + 972 ) / 2916 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10\beta_{5} + 27\beta_{4} + 805\beta_1 ) / 2187 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 35\beta_{5} + 171\beta_{4} - 23\beta_{3} - 306\beta_{2} + 320\beta _1 - 60264 ) / 2916 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -44\beta_{3} - 333\beta_{2} - 149202 ) / 729 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2615\beta_{5} - 11151\beta_{4} - 1683\beta_{3} - 18306\beta_{2} - 78680\beta _1 - 4968864 ) / 8748 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
− | 29.0422i | 0 | −587.450 | − | 1141.55i | 0 | −618.310 | 9626.03i | 0 | −33153.0 | ||||||||||||||||||||||||||||||||||
| 26.2 | − | 15.9629i | 0 | 1.18662 | 230.735i | 0 | 3842.93 | − | 4105.44i | 0 | 3683.19 | |||||||||||||||||||||||||||||||||||
| 26.3 | − | 7.92067i | 0 | 193.263 | 799.282i | 0 | −4073.62 | − | 3558.46i | 0 | 6330.85 | |||||||||||||||||||||||||||||||||||
| 26.4 | 7.92067i | 0 | 193.263 | − | 799.282i | 0 | −4073.62 | 3558.46i | 0 | 6330.85 | ||||||||||||||||||||||||||||||||||||
| 26.5 | 15.9629i | 0 | 1.18662 | − | 230.735i | 0 | 3842.93 | 4105.44i | 0 | 3683.19 | ||||||||||||||||||||||||||||||||||||
| 26.6 | 29.0422i | 0 | −587.450 | 1141.55i | 0 | −618.310 | − | 9626.03i | 0 | −33153.0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 27.9.b.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 27.9.b.d | ✓ | 6 |
| 4.b | odd | 2 | 1 | 432.9.e.k | 6 | ||
| 9.c | even | 3 | 2 | 81.9.d.f | 12 | ||
| 9.d | odd | 6 | 2 | 81.9.d.f | 12 | ||
| 12.b | even | 2 | 1 | 432.9.e.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.9.b.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 27.9.b.d | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 81.9.d.f | 12 | 9.c | even | 3 | 2 | ||
| 81.9.d.f | 12 | 9.d | odd | 6 | 2 | ||
| 432.9.e.k | 6 | 4.b | odd | 2 | 1 | ||
| 432.9.e.k | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 1161T_{2}^{4} + 283824T_{2}^{2} + 13483584 \)
acting on \(S_{9}^{\mathrm{new}}(27, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 1161 T^{4} + \cdots + 13483584 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 44\!\cdots\!25 \)
|
| $7$ |
\( (T^{3} + 849 T^{2} + \cdots - 9679397525)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 69\!\cdots\!25 \)
|
| $13$ |
\( (T^{3} + \cdots + 3314333225800)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 16\!\cdots\!76 \)
|
| $19$ |
\( (T^{3} + \cdots + 54072465923584)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 83\!\cdots\!01)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 22\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 19\!\cdots\!96 \)
|
| $53$ |
\( T^{6} + \cdots + 28\!\cdots\!89 \)
|
| $59$ |
\( T^{6} + \cdots + 58\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots - 30\!\cdots\!04)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 81\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 43\!\cdots\!75)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 45\!\cdots\!80)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 14\!\cdots\!49 \)
|
| $89$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots + 69\!\cdots\!25)^{2} \)
|
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