Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(25.3031870642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 401x^{4} - 1212x^{3} + 17752x^{2} + 15108x - 22632 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{9} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 401x^{4} - 1212x^{3} + 17752x^{2} + 15108x - 22632 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 6641\nu^{5} - 97242\nu^{4} - 2979321\nu^{3} + 31856346\nu^{2} + 298415260\nu - 1031947384 ) / 69200132 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 137781\nu^{5} - 173118\nu^{4} - 53207622\nu^{3} - 105916911\nu^{2} + 2068887384\nu - 436251969 ) / 17300033 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 364490\nu^{5} + 1063461\nu^{4} - 155430825\nu^{3} - 720783666\nu^{2} + 8445696616\nu + 3103806566 ) / 34600066 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 941891 \nu^{5} - 1166304 \nu^{4} + 383121177 \nu^{3} + 1546656714 \nu^{2} - 15736789588 \nu - 20579550384 ) / 69200132 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 491307 \nu^{5} - 1246755 \nu^{4} - 194646384 \nu^{3} - 268379592 \nu^{2} + 10043014260 \nu + 13119275006 ) / 34600066 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 3\beta_{4} + 2\beta_{3} + \beta_{2} - 25\beta _1 - 14 ) / 108 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -9\beta_{5} + 6\beta_{4} + 5\beta_{3} + 18\beta_{2} + 140\beta _1 + 7290 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 67\beta_{5} + 492\beta_{4} + 251\beta_{3} + 454\beta_{2} - 5362\beta _1 + 29884 ) / 54 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2937\beta_{5} + 3552\beta_{4} + 3143\beta_{3} + 6990\beta_{2} + 13250\beta _1 + 2261868 ) / 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7757\beta_{5} + 176550\beta_{4} + 89707\beta_{3} + 197216\beta_{2} - 1758704\beta _1 + 20262596 ) / 54 \)
|
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 |
|
−15.4889 | 0 | 111.906 | −105.528 | 0 | −1522.84 | 249.280 | 0 | 1634.51 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −14.2319 | 0 | 74.5469 | 290.607 | 0 | 1111.88 | 760.739 | 0 | −4135.89 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.07224 | 0 | −126.850 | −95.9733 | 0 | 378.001 | 273.261 | 0 | 102.906 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 6.18840 | 0 | −89.7037 | −335.903 | 0 | −884.050 | −1347.24 | 0 | −2078.70 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 12.1944 | 0 | 20.7039 | 492.052 | 0 | 764.622 | −1308.41 | 0 | 6000.29 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 21.4102 | 0 | 330.397 | −65.2548 | 0 | 236.387 | 4333.37 | 0 | −1397.12 | |||||||||||||||||||||||||||||||||||||
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 | 81.8.a.e | 6 | |
| 3.b | odd | 2 | 1 | 81.8.a.c | 6 | ||
| 9.c | even | 3 | 2 | 9.8.c.a | ✓ | 12 | |
| 9.d | odd | 6 | 2 | 27.8.c.a | 12 | ||
| 36.f | odd | 6 | 2 | 144.8.i.c | 12 | ||
| 36.h | even | 6 | 2 | 432.8.i.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.8.c.a | ✓ | 12 | 9.c | even | 3 | 2 | |
| 27.8.c.a | 12 | 9.d | odd | 6 | 2 | ||
| 81.8.a.c | 6 | 3.b | odd | 2 | 1 | ||
| 81.8.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 144.8.i.c | 12 | 36.f | odd | 6 | 2 | ||
| 432.8.i.c | 12 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 9T_{2}^{5} - 504T_{2}^{4} + 3024T_{2}^{3} + 59184T_{2}^{2} - 296784T_{2} - 381888 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{5} + \cdots - 381888 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 31744053360000 \)
|
| $7$ |
\( T^{6} + \cdots + 10\!\cdots\!48 \)
|
| $11$ |
\( T^{6} + \cdots + 91\!\cdots\!91 \)
|
| $13$ |
\( T^{6} + \cdots - 22\!\cdots\!92 \)
|
| $17$ |
\( T^{6} + \cdots - 16\!\cdots\!76 \)
|
| $19$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 27\!\cdots\!28 \)
|
| $29$ |
\( T^{6} + \cdots + 72\!\cdots\!12 \)
|
| $31$ |
\( T^{6} + \cdots - 15\!\cdots\!16 \)
|
| $37$ |
\( T^{6} + \cdots + 17\!\cdots\!64 \)
|
| $41$ |
\( T^{6} + \cdots + 21\!\cdots\!69 \)
|
| $43$ |
\( T^{6} + \cdots + 52\!\cdots\!63 \)
|
| $47$ |
\( T^{6} + \cdots + 23\!\cdots\!48 \)
|
| $53$ |
\( T^{6} + \cdots + 54\!\cdots\!44 \)
|
| $59$ |
\( T^{6} + \cdots + 86\!\cdots\!83 \)
|
| $61$ |
\( T^{6} + \cdots - 27\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots - 45\!\cdots\!93 \)
|
| $71$ |
\( T^{6} + \cdots + 81\!\cdots\!24 \)
|
| $73$ |
\( T^{6} + \cdots + 34\!\cdots\!16 \)
|
| $79$ |
\( T^{6} + \cdots - 10\!\cdots\!12 \)
|
| $83$ |
\( T^{6} + \cdots + 24\!\cdots\!92 \)
|
| $89$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 91\!\cdots\!63 \)
|
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