Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,10,Mod(1,81)]
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([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.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: | \(41.7179027293\) |
| 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: |
\( x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{18}\cdot 17 \) |
| Twist minimal: | no (minimal twist has level 9) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 732 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 503 \nu^{7} + 61083 \nu^{6} + 2528604 \nu^{5} - 160366700 \nu^{4} - 8380255800 \nu^{3} + \cdots + 7794440754048 ) / 2707592448 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 15021 \nu^{7} + 418759 \nu^{6} + 58051644 \nu^{5} - 982462380 \nu^{4} + \cdots + 60735117255808 ) / 10830369792 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17033 \nu^{7} + 174427 \nu^{6} + 47937228 \nu^{5} - 340995580 \nu^{4} + \cdots + 21174648020608 ) / 10830369792 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 26447 \nu^{7} - 408067 \nu^{6} + 76309476 \nu^{5} + 1090595804 \nu^{4} + \cdots + 82897833766784 ) / 5415184896 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 18075 \nu^{7} + 103961 \nu^{6} + 50268732 \nu^{5} - 301142724 \nu^{4} + \cdots - 17752715711872 ) / 2707592448 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 732 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} + 1234\beta _1 + 1422 \)
|
| \(\nu^{4}\) | \(=\) |
\( -20\beta_{7} + 4\beta_{6} + 61\beta_{5} + 11\beta_{4} - 15\beta_{3} + 1685\beta_{2} + 5262\beta _1 + 903286 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 60 \beta_{7} + 140 \beta_{6} - 2441 \beta_{5} + 2337 \beta_{4} - 1693 \beta_{3} + 13367 \beta_{2} + \cdots + 3788658 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 47892 \beta_{7} + 1956 \beta_{6} + 163181 \beta_{5} + 33115 \beta_{4} - 40879 \beta_{3} + \cdots + 1270802438 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 258908 \beta_{7} + 333964 \beta_{6} - 4757753 \beta_{5} + 4397969 \beta_{4} - 2584941 \beta_{3} + \cdots + 9953357362 \)
|
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 |
|
−39.7589 | 0 | 1068.77 | 846.447 | 0 | −7043.99 | −22136.7 | 0 | −33653.8 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −25.6540 | 0 | 146.127 | −706.451 | 0 | −4568.42 | 9386.09 | 0 | 18123.3 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −21.4832 | 0 | −50.4713 | 44.8401 | 0 | 9345.55 | 12083.7 | 0 | −963.310 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.67952 | 0 | −509.179 | 1538.61 | 0 | 5656.94 | −1715.10 | 0 | 2584.13 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 7.66432 | 0 | −453.258 | −2210.95 | 0 | −6233.15 | −7398.05 | 0 | −16945.4 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 17.7185 | 0 | −198.054 | 739.877 | 0 | −6027.34 | −12581.1 | 0 | 13109.5 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 34.8660 | 0 | 703.639 | −2006.52 | 0 | 4867.88 | 6681.68 | 0 | −69959.4 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 39.9678 | 0 | 1085.42 | 2207.15 | 0 | 4345.52 | 22918.5 | 0 | 88215.0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
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 | 81.10.a.d | 8 | |
| 3.b | odd | 2 | 1 | 81.10.a.c | 8 | ||
| 9.c | even | 3 | 2 | 27.10.c.a | 16 | ||
| 9.d | odd | 6 | 2 | 9.10.c.a | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.10.c.a | ✓ | 16 | 9.d | odd | 6 | 2 | |
| 27.10.c.a | 16 | 9.c | even | 3 | 2 | ||
| 81.10.a.c | 8 | 3.b | odd | 2 | 1 | ||
| 81.10.a.d | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 15 T_{2}^{7} - 2832 T_{2}^{6} + 36072 T_{2}^{5} + 2299752 T_{2}^{4} - 22804416 T_{2}^{3} + \cdots - 6964475904 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots - 6964475904 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 29\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 13\!\cdots\!64 \)
|
| $11$ |
\( T^{8} + \cdots + 14\!\cdots\!71 \)
|
| $13$ |
\( T^{8} + \cdots + 52\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots - 82\!\cdots\!84 \)
|
| $19$ |
\( T^{8} + \cdots - 88\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 32\!\cdots\!68 \)
|
| $29$ |
\( T^{8} + \cdots + 27\!\cdots\!84 \)
|
| $31$ |
\( T^{8} + \cdots + 26\!\cdots\!60 \)
|
| $37$ |
\( T^{8} + \cdots + 99\!\cdots\!68 \)
|
| $41$ |
\( T^{8} + \cdots - 81\!\cdots\!05 \)
|
| $43$ |
\( T^{8} + \cdots - 12\!\cdots\!11 \)
|
| $47$ |
\( T^{8} + \cdots - 47\!\cdots\!12 \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots - 12\!\cdots\!97 \)
|
| $61$ |
\( T^{8} + \cdots + 87\!\cdots\!48 \)
|
| $67$ |
\( T^{8} + \cdots + 93\!\cdots\!13 \)
|
| $71$ |
\( T^{8} + \cdots - 16\!\cdots\!84 \)
|
| $73$ |
\( T^{8} + \cdots - 10\!\cdots\!04 \)
|
| $79$ |
\( T^{8} + \cdots + 62\!\cdots\!72 \)
|
| $83$ |
\( T^{8} + \cdots - 89\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots - 31\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 44\!\cdots\!49 \)
|
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