Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,86,Mod(1,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 86, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 86);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 86 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.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: | \(411.794619216\) |
| 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: |
\( x^{6} - 3 x^{5} + \cdots - 17\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{65}\cdot 3^{30}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 17^{2} \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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
\( x^{6} - 3 x^{5} + \cdots - 17\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 96\nu - 48 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 55\!\cdots\!13 \nu^{5} + \cdots + 28\!\cdots\!90 ) / 51\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23\!\cdots\!59 \nu^{5} + \cdots - 91\!\cdots\!22 ) / 12\!\cdots\!32 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 39\!\cdots\!99 \nu^{5} + \cdots + 80\!\cdots\!34 ) / 51\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 28\!\cdots\!47 \nu^{5} + \cdots - 40\!\cdots\!18 ) / 51\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 48 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 172\beta_{2} + 2250185580864\beta _1 + 61824811714782289642833408 ) / 9216 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 3813 \beta_{5} + 1321486 \beta_{4} + 183077918961 \beta_{3} + 21305290231923 \beta_{2} + \cdots + 86\!\cdots\!48 ) / 55296 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 630076817944247 \beta_{5} + \cdots + 12\!\cdots\!44 ) / 165888 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 16\!\cdots\!68 \beta_{5} + \cdots + 43\!\cdots\!08 ) / 124416 \)
|
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 |
|
−1.15489e13 | 0 | 9.46906e25 | 4.26442e29 | 0 | 5.79630e35 | −6.46794e38 | 0 | −4.92492e42 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −7.11528e12 | 0 | 1.19416e25 | −1.39389e29 | 0 | −1.69824e35 | 1.90292e38 | 0 | 9.91791e41 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.51077e10 | 0 | −3.86850e25 | −5.79783e28 | 0 | −1.94193e31 | 1.94260e36 | 0 | 1.45570e39 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.74029e12 | 0 | −2.46958e25 | 8.27734e29 | 0 | 1.89091e35 | −2.37065e38 | 0 | 3.09596e42 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 8.02060e12 | 0 | 2.56443e25 | −8.73356e29 | 0 | 1.26096e35 | −1.04599e38 | 0 | −7.00484e42 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.05253e13 | 0 | 7.20957e25 | 7.53919e29 | 0 | −3.48209e35 | 3.51650e38 | 0 | 7.93520e42 | |||||||||||||||||||||||||||||||||||||
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 | 9.86.a.a | 6 | |
| 3.b | odd | 2 | 1 | 1.86.a.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.86.a.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 9.86.a.a | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3596910688800 T_{2}^{5} + \cdots - 65\!\cdots\!64 \)
acting on \(S_{86}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 65\!\cdots\!64 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 15\!\cdots\!44 \)
|
| $11$ |
\( T^{6} + \cdots + 40\!\cdots\!64 \)
|
| $13$ |
\( T^{6} + \cdots + 70\!\cdots\!64 \)
|
| $17$ |
\( T^{6} + \cdots + 45\!\cdots\!96 \)
|
| $19$ |
\( T^{6} + \cdots - 55\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 88\!\cdots\!24 \)
|
| $29$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 22\!\cdots\!36 \)
|
| $37$ |
\( T^{6} + \cdots - 17\!\cdots\!24 \)
|
| $41$ |
\( T^{6} + \cdots + 56\!\cdots\!64 \)
|
| $43$ |
\( T^{6} + \cdots + 22\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 35\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots + 23\!\cdots\!04 \)
|
| $59$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots + 12\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots + 19\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots + 28\!\cdots\!24 \)
|
| $79$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 45\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 34\!\cdots\!16 \)
|
| show more | |
| show less |