Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,68,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, 68, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 68);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 68 \) |
| 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: | \(255.861316737\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} + \cdots + 20\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{40}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13\cdot 17 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} + \cdots + 20\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 118503 \nu^{4} - 4966651742499 \nu^{3} + \cdots + 74\!\cdots\!12 ) / 12\!\cdots\!76 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2962575 \nu^{4} + 124166293562475 \nu^{3} + \cdots - 45\!\cdots\!84 ) / 12\!\cdots\!76 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18951882417 \nu^{4} + \cdots - 45\!\cdots\!80 ) / 62\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 5 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 25\beta_{2} - 1189792592\beta _1 + 216434076347341359449 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 409640 \beta_{4} - 1180151429 \beta_{3} - 16401249733 \beta_{2} + \cdots - 32\!\cdots\!85 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 51506018070360 \beta_{4} + \cdots + 11\!\cdots\!61 ) / 5184 \)
|
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 |
|
−2.26950e10 | 0 | 3.67490e20 | 1.61198e23 | 0 | 2.93806e28 | −4.99099e30 | 0 | −3.65838e33 | |||||||||||||||||||||||||||||||||
| 1.2 | −8.43209e9 | 0 | −7.64738e19 | −3.83370e23 | 0 | −1.75533e27 | 1.88919e30 | 0 | 3.23261e33 | ||||||||||||||||||||||||||||||||||
| 1.3 | −5.06726e9 | 0 | −1.21897e20 | 8.33551e22 | 0 | −1.15070e28 | 1.36548e30 | 0 | −4.22381e32 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.34339e10 | 0 | 3.28965e19 | 3.01383e23 | 0 | −4.20742e26 | −1.54057e30 | 0 | 4.04876e33 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.72055e10 | 0 | 1.48456e20 | −4.93311e23 | 0 | 1.79351e28 | 1.51842e28 | 0 | −8.48767e33 | ||||||||||||||||||||||||||||||||||
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 | 9.68.a.a | 5 | |
| 3.b | odd | 2 | 1 | 1.68.a.a | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.68.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 9.68.a.a | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 5554901256 T_{2}^{4} + \cdots + 22\!\cdots\!76 \)
acting on \(S_{68}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 22\!\cdots\!76 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 76\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 44\!\cdots\!24 \)
|
| $11$ |
\( T^{5} + \cdots - 13\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots + 65\!\cdots\!68 \)
|
| $17$ |
\( T^{5} + \cdots - 41\!\cdots\!24 \)
|
| $19$ |
\( T^{5} + \cdots + 67\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 13\!\cdots\!68 \)
|
| $29$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 60\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots - 23\!\cdots\!76 \)
|
| $41$ |
\( T^{5} + \cdots + 28\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots - 63\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots - 25\!\cdots\!24 \)
|
| $53$ |
\( T^{5} + \cdots + 34\!\cdots\!32 \)
|
| $59$ |
\( T^{5} + \cdots - 61\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 21\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 42\!\cdots\!24 \)
|
| $71$ |
\( T^{5} + \cdots - 78\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 21\!\cdots\!32 \)
|
| $79$ |
\( T^{5} + \cdots - 93\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 11\!\cdots\!68 \)
|
| $89$ |
\( T^{5} + \cdots - 23\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 51\!\cdots\!24 \)
|
| show more | |
| show less |