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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 28x^{6} + 265x^{4} + 1008x^{2} + 1296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{18} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 28x^{6} + 265x^{4} + 1008x^{2} + 1296 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 19\nu^{7} + 460\nu^{5} + 3235\nu^{3} + 6336\nu ) / 216 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -10\nu^{7} - 127\nu^{5} + 446\nu^{3} + 4761\nu ) / 54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 10\nu^{4} + 131\nu^{2} + 870 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{7} - 160\nu^{5} - 1063\nu^{3} - 1836\nu ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{6} + 292\nu^{4} + 1861\nu^{2} + 3312 ) / 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 4\nu^{6} + 121\nu^{4} + 1015\nu^{2} + 2139 ) / 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 109\nu^{7} + 2494\nu^{5} + 16879\nu^{3} + 36306\nu ) / 54 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{7} + 19\beta_{4} - \beta_{2} + 15\beta_1 ) / 243 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{5} + 5\beta_{3} - 564 ) / 81 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{7} - 188\beta_{4} + 26\beta_{2} - 408\beta_1 ) / 243 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 22\beta_{6} - 38\beta_{5} - 71\beta_{3} + 5097 ) / 81 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -10\beta_{7} + 2173\beta_{4} - 301\beta_{2} + 6801\beta_1 ) / 243 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -117\beta_{6} + 214\beta_{5} + 293\beta_{3} - 18128 ) / 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 767\beta_{7} - 26936\beta_{4} + 3194\beta_{2} - 97428\beta_1 ) / 243 \)
|
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 |
|
− | 7.16404i | 0 | −35.3234 | 39.7307i | 0 | 14.6404 | 138.434i | 0 | 284.632 | |||||||||||||||||||||||||||||||||||||||||
| 80.2 | − | 5.33705i | 0 | −12.4841 | − | 39.7818i | 0 | 65.7210 | − | 18.7643i | 0 | −212.317 | ||||||||||||||||||||||||||||||||||||||||
| 80.3 | − | 3.78414i | 0 | 1.68029 | 1.50784i | 0 | −78.7017 | − | 66.9047i | 0 | 5.70588 | |||||||||||||||||||||||||||||||||||||||||
| 80.4 | − | 1.36848i | 0 | 14.1273 | 13.1381i | 0 | 24.3403 | − | 41.2286i | 0 | 17.9792 | |||||||||||||||||||||||||||||||||||||||||
| 80.5 | 1.36848i | 0 | 14.1273 | − | 13.1381i | 0 | 24.3403 | 41.2286i | 0 | 17.9792 | ||||||||||||||||||||||||||||||||||||||||||
| 80.6 | 3.78414i | 0 | 1.68029 | − | 1.50784i | 0 | −78.7017 | 66.9047i | 0 | 5.70588 | ||||||||||||||||||||||||||||||||||||||||||
| 80.7 | 5.33705i | 0 | −12.4841 | 39.7818i | 0 | 65.7210 | 18.7643i | 0 | −212.317 | |||||||||||||||||||||||||||||||||||||||||||
| 80.8 | 7.16404i | 0 | −35.3234 | − | 39.7307i | 0 | 14.6404 | − | 138.434i | 0 | 284.632 | |||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 81.5.b.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | 1296.5.e.d | 8 | ||
| 9.c | even | 3 | 2 | 81.5.d.e | 16 | ||
| 9.d | odd | 6 | 2 | 81.5.d.e | 16 | ||
| 12.b | even | 2 | 1 | 1296.5.e.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 81.5.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 81.5.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 81.5.d.e | 16 | 9.c | even | 3 | 2 | ||
| 81.5.d.e | 16 | 9.d | odd | 6 | 2 | ||
| 1296.5.e.d | 8 | 4.b | odd | 2 | 1 | ||
| 1296.5.e.d | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 96T_{2}^{6} + 2781T_{2}^{4} + 25812T_{2}^{2} + 39204 \)
acting on \(S_{5}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 96 T^{6} + \cdots + 39204 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 3336 T^{6} + \cdots + 980378721 \)
|
| $7$ |
\( (T^{4} - 26 T^{3} + \cdots - 1843184)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 21\!\cdots\!84 \)
|
| $13$ |
\( (T^{4} + 130 T^{3} + \cdots + 99328537)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 16\!\cdots\!89 \)
|
| $19$ |
\( (T^{4} + 154 T^{3} + \cdots + 22562369872)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 17\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 10\!\cdots\!09 \)
|
| $31$ |
\( (T^{4} - 1472 T^{3} + \cdots - 54220173056)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 3469511755343)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 40\!\cdots\!56 \)
|
| $43$ |
\( (T^{4} + \cdots + 4089330165328)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 12\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 18\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + \cdots + 17\!\cdots\!24 \)
|
| $61$ |
\( (T^{4} + \cdots + 5808975410077)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 61141889718512)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 25\!\cdots\!76 \)
|
| $73$ |
\( (T^{4} + \cdots - 214837668024959)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 495439676622736)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 91\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 57\!\cdots\!61 \)
|
| $97$ |
\( (T^{4} + \cdots - 152531203430144)^{2} \)
|
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