Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,5,Mod(80,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.80");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.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: | \(8.37296700979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.39400128.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{9} \) |
| Twist minimal: | no (minimal twist has level 9) |
| 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
\( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 11\nu^{4} - 121\nu^{3} + 98\nu^{2} - 2176\nu - 220 ) / 1098 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} + 55\nu^{4} - 56\nu^{3} + 490\nu^{2} + 100\nu + 3841 ) / 183 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{5} - 143\nu^{4} + 475\nu^{3} - 1274\nu^{2} - 260\nu - 5924 ) / 122 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1168\nu^{5} + 1319\nu^{4} - 12862\nu^{3} - 15649\nu^{2} - 110596\nu - 11557 ) / 549 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2971\nu^{5} - 3035\nu^{4} + 33385\nu^{3} + 34948\nu^{2} + 298528\nu + 31054 ) / 1098 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} - 3\beta_{2} - 27\beta _1 + 9 ) / 54 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{5} - 9\beta_{4} + 9\beta_{2} - 43\beta _1 - 189 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{3} + 39\beta_{2} - 333 ) / 27 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 79\beta_{5} + 99\beta_{4} + 12\beta_{3} + 135\beta_{2} + 799\beta _1 - 2241 ) / 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 183\beta_{5} + 207\beta_{4} - 112\beta_{3} - 543\beta_{2} + 4035\beta _1 + 5949 ) / 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 80.1 |
|
− | 6.55250i | 0 | −26.9353 | − | 11.7830i | 0 | −53.2728 | 71.6534i | 0 | −77.2081 | ||||||||||||||||||||||||||||||||||
| 80.2 | − | 4.46526i | 0 | −3.93857 | 16.0226i | 0 | 72.4837 | − | 53.8574i | 0 | 71.5451 | |||||||||||||||||||||||||||||||||||
| 80.3 | − | 0.355188i | 0 | 15.8738 | − | 34.7338i | 0 | −31.2109 | − | 11.3212i | 0 | −12.3370 | ||||||||||||||||||||||||||||||||||
| 80.4 | 0.355188i | 0 | 15.8738 | 34.7338i | 0 | −31.2109 | 11.3212i | 0 | −12.3370 | |||||||||||||||||||||||||||||||||||||
| 80.5 | 4.46526i | 0 | −3.93857 | − | 16.0226i | 0 | 72.4837 | 53.8574i | 0 | 71.5451 | ||||||||||||||||||||||||||||||||||||
| 80.6 | 6.55250i | 0 | −26.9353 | 11.7830i | 0 | −53.2728 | − | 71.6534i | 0 | −77.2081 | ||||||||||||||||||||||||||||||||||||
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 | 81.5.b.a | 6 | |
| 3.b | odd | 2 | 1 | inner | 81.5.b.a | 6 | |
| 4.b | odd | 2 | 1 | 1296.5.e.c | 6 | ||
| 9.c | even | 3 | 1 | 9.5.d.a | ✓ | 6 | |
| 9.c | even | 3 | 1 | 27.5.d.a | 6 | ||
| 9.d | odd | 6 | 1 | 9.5.d.a | ✓ | 6 | |
| 9.d | odd | 6 | 1 | 27.5.d.a | 6 | ||
| 12.b | even | 2 | 1 | 1296.5.e.c | 6 | ||
| 36.f | odd | 6 | 1 | 144.5.q.a | 6 | ||
| 36.f | odd | 6 | 1 | 432.5.q.a | 6 | ||
| 36.h | even | 6 | 1 | 144.5.q.a | 6 | ||
| 36.h | even | 6 | 1 | 432.5.q.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.5.d.a | ✓ | 6 | 9.c | even | 3 | 1 | |
| 9.5.d.a | ✓ | 6 | 9.d | odd | 6 | 1 | |
| 27.5.d.a | 6 | 9.c | even | 3 | 1 | ||
| 27.5.d.a | 6 | 9.d | odd | 6 | 1 | ||
| 81.5.b.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 81.5.b.a | 6 | 3.b | odd | 2 | 1 | inner | |
| 144.5.q.a | 6 | 36.f | odd | 6 | 1 | ||
| 144.5.q.a | 6 | 36.h | even | 6 | 1 | ||
| 432.5.q.a | 6 | 36.f | odd | 6 | 1 | ||
| 432.5.q.a | 6 | 36.h | even | 6 | 1 | ||
| 1296.5.e.c | 6 | 4.b | odd | 2 | 1 | ||
| 1296.5.e.c | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 63T_{2}^{4} + 864T_{2}^{2} + 108 \)
acting on \(S_{5}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 63 T^{4} + \cdots + 108 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 1602 T^{4} + \cdots + 43001388 \)
|
| $7$ |
\( (T^{3} + 12 T^{2} + \cdots - 120518)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 481294471563 \)
|
| $13$ |
\( (T^{3} - 6 T^{2} + \cdots - 841652)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 47166451632 \)
|
| $19$ |
\( (T^{3} + 129 T^{2} + \cdots - 1195028)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10049071819968 \)
|
| $29$ |
\( T^{6} + \cdots + 92\!\cdots\!28 \)
|
| $31$ |
\( (T^{3} + 1290 T^{2} + \cdots - 252529712)^{2} \)
|
| $37$ |
\( (T^{3} - 6 T^{2} + \cdots + 1276743376)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 32\!\cdots\!63 \)
|
| $43$ |
\( (T^{3} - 285 T^{2} + \cdots + 4656095011)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 65\!\cdots\!28 \)
|
| $53$ |
\( T^{6} + \cdots + 34\!\cdots\!52 \)
|
| $59$ |
\( T^{6} + \cdots + 74\!\cdots\!87 \)
|
| $61$ |
\( (T^{3} + 3630 T^{2} + \cdots + 48924712)^{2} \)
|
| $67$ |
\( (T^{3} - 5055 T^{2} + \cdots + 108250248169)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 26\!\cdots\!12 \)
|
| $73$ |
\( (T^{3} + 7311 T^{2} + \cdots - 15741832472)^{2} \)
|
| $79$ |
\( (T^{3} + 4764 T^{2} + \cdots - 1639793258)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!28 \)
|
| $89$ |
\( T^{6} + \cdots + 53\!\cdots\!92 \)
|
| $97$ |
\( (T^{3} - 28959 T^{2} + \cdots - 722818034117)^{2} \)
|
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