Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,8,Mod(28,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), not 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: | \(25.3031870642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 229x^{6} + 13863x^{4} + 85327x^{2} + 40804 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 229x^{6} + 13863x^{4} + 85327x^{2} + 40804 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 330\nu^{5} + 25377\nu^{3} + 117748\nu + 43632 ) / 87264 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 101\nu^{6} + 330\nu^{5} + 11514\nu^{4} + 25377\nu^{3} + 32421\nu^{2} + 248644\nu - 84436 ) / 87264 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 101\nu^{6} + 330\nu^{5} - 11514\nu^{4} + 25377\nu^{3} - 32421\nu^{2} + 248644\nu + 84436 ) / 87264 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 114\nu^{4} + 975\nu^{2} + 75140 ) / 432 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 186\nu^{4} - 8601\nu^{2} - 28444 ) / 48 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 601 \nu^{7} + 909 \nu^{6} + 132882 \nu^{5} + 169074 \nu^{4} + 7855953 \nu^{3} + 7818309 \nu^{2} + \cdots + 25855596 ) / 87264 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 476 \nu^{7} - 101 \nu^{6} - 91632 \nu^{5} - 11514 \nu^{4} - 4487484 \nu^{3} + 98475 \nu^{2} + \cdots + 7589140 ) / 87264 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} - 172 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{7} + 4\beta_{6} + 2\beta_{5} - 2\beta_{4} - 325\beta_{3} - 325\beta_{2} + 150\beta _1 - 75 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{5} - 115\beta_{4} + 133\beta_{3} - 133\beta_{2} + 18560 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 452 \beta_{7} - 464 \beta_{6} - 232 \beta_{5} + 226 \beta_{4} + 36065 \beta_{3} + 36065 \beta_{2} + \cdots + 4209 ) / 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( 76\beta_{5} + 4263\beta_{4} - 5379\beta_{3} + 5379\beta_{2} - 686040 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 15884 \beta_{7} + 17204 \beta_{6} + 8602 \beta_{5} - 7942 \beta_{4} - 1335723 \beta_{3} - 1335723 \beta_{2} + \cdots - 77209 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−7.65402 | − | 13.2572i | 0 | −53.1681 | + | 92.0899i | −26.6928 | + | 46.2333i | 0 | −121.554 | − | 210.538i | −331.630 | 0 | 817.229 | ||||||||||||||||||||||||||||||||||
| 28.2 | 0.874231 | + | 1.51421i | 0 | 62.4714 | − | 108.204i | 180.817 | − | 313.184i | 0 | 758.855 | + | 1314.38i | 442.261 | 0 | 632.303 | |||||||||||||||||||||||||||||||||||
| 28.3 | 3.67973 | + | 6.37349i | 0 | 36.9191 | − | 63.9458i | −234.236 | + | 405.708i | 0 | −655.845 | − | 1135.96i | 1485.42 | 0 | −3447.70 | |||||||||||||||||||||||||||||||||||
| 28.4 | 10.6001 | + | 18.3598i | 0 | −160.722 | + | 278.379i | 176.112 | − | 305.034i | 0 | −381.456 | − | 660.702i | −4101.05 | 0 | 7467.17 | |||||||||||||||||||||||||||||||||||
| 55.1 | −7.65402 | + | 13.2572i | 0 | −53.1681 | − | 92.0899i | −26.6928 | − | 46.2333i | 0 | −121.554 | + | 210.538i | −331.630 | 0 | 817.229 | |||||||||||||||||||||||||||||||||||
| 55.2 | 0.874231 | − | 1.51421i | 0 | 62.4714 | + | 108.204i | 180.817 | + | 313.184i | 0 | 758.855 | − | 1314.38i | 442.261 | 0 | 632.303 | |||||||||||||||||||||||||||||||||||
| 55.3 | 3.67973 | − | 6.37349i | 0 | 36.9191 | + | 63.9458i | −234.236 | − | 405.708i | 0 | −655.845 | + | 1135.96i | 1485.42 | 0 | −3447.70 | |||||||||||||||||||||||||||||||||||
| 55.4 | 10.6001 | − | 18.3598i | 0 | −160.722 | − | 278.379i | 176.112 | + | 305.034i | 0 | −381.456 | + | 660.702i | −4101.05 | 0 | 7467.17 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.8.c.j | 8 | |
| 3.b | odd | 2 | 1 | 81.8.c.i | 8 | ||
| 9.c | even | 3 | 1 | 81.8.a.a | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 81.8.c.j | 8 | |
| 9.d | odd | 6 | 1 | 81.8.a.b | yes | 4 | |
| 9.d | odd | 6 | 1 | 81.8.c.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.8.a.a | ✓ | 4 | 9.c | even | 3 | 1 | |
| 81.8.a.b | yes | 4 | 9.d | odd | 6 | 1 | |
| 81.8.c.i | 8 | 3.b | odd | 2 | 1 | ||
| 81.8.c.i | 8 | 9.d | odd | 6 | 1 | ||
| 81.8.c.j | 8 | 1.a | even | 1 | 1 | trivial | |
| 81.8.c.j | 8 | 9.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 15 T_{2}^{7} + 483 T_{2}^{6} - 1890 T_{2}^{5} + 113940 T_{2}^{4} - 868320 T_{2}^{3} + \cdots + 17438976 \)
acting on \(S_{8}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 15 T^{7} + \cdots + 17438976 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 10\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 13\!\cdots\!56 \)
|
| $11$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
|
| $13$ |
\( T^{8} + \cdots + 50\!\cdots\!01 \)
|
| $17$ |
\( (T^{4} + \cdots + 35\!\cdots\!89)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots - 59\!\cdots\!96)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 27\!\cdots\!36 \)
|
| $29$ |
\( T^{8} + \cdots + 83\!\cdots\!21 \)
|
| $31$ |
\( T^{8} + \cdots + 57\!\cdots\!76 \)
|
| $37$ |
\( (T^{4} + \cdots + 15\!\cdots\!49)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 47\!\cdots\!16 \)
|
| $43$ |
\( T^{8} + \cdots + 29\!\cdots\!56 \)
|
| $47$ |
\( T^{8} + \cdots + 67\!\cdots\!36 \)
|
| $53$ |
\( (T^{4} + \cdots - 94\!\cdots\!04)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $61$ |
\( T^{8} + \cdots + 73\!\cdots\!41 \)
|
| $67$ |
\( T^{8} + \cdots + 67\!\cdots\!76 \)
|
| $71$ |
\( (T^{4} + \cdots - 18\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots + 12\!\cdots\!89)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 16\!\cdots\!36 \)
|
| $83$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots - 34\!\cdots\!79)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
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