Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,90,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, 90, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 90);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 90 \) |
| 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: | \(451.461862736\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3 x^{6} + \cdots + 56\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{83}\cdot 3^{43}\cdot 5^{9}\cdot 7^{5}\cdot 11^{2}\cdot 13^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3 x^{6} + \cdots + 56\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 48\nu - 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2304\nu^{2} + 228081245760096\nu - 921949945033343757008509810 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 33\!\cdots\!39 \nu^{6} + \cdots + 29\!\cdots\!16 ) / 35\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25\!\cdots\!71 \nu^{6} + \cdots + 60\!\cdots\!60 ) / 17\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\!\cdots\!91 \nu^{6} + \cdots + 19\!\cdots\!00 ) / 44\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 19\!\cdots\!91 \nu^{6} + \cdots - 23\!\cdots\!40 ) / 17\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 21 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 4751692620002\beta _1 + 921949945033243971463489768 ) / 2304 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 42 \beta_{6} - 316 \beta_{5} + 10264134 \beta_{4} + 1609670278248 \beta_{3} - 535021439735 \beta_{2} + \cdots - 27\!\cdots\!22 ) / 6912 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 42626265665746 \beta_{6} - 194107592272340 \beta_{5} + \cdots + 58\!\cdots\!86 ) / 20736 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 43\!\cdots\!82 \beta_{6} + \cdots - 39\!\cdots\!70 ) / 62208 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 19\!\cdots\!78 \beta_{6} + \cdots + 15\!\cdots\!10 ) / 62208 \)
|
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 |
|
−4.42254e13 | 0 | 1.33691e27 | −3.67006e30 | 0 | −4.35255e37 | −3.17513e40 | 0 | 1.62310e44 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.92430e13 | 0 | −2.48679e26 | 3.02831e30 | 0 | 1.70526e37 | 1.66961e40 | 0 | −5.82737e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79872e13 | 0 | −2.95431e26 | 4.75717e30 | 0 | 3.90338e37 | 1.64475e40 | 0 | −8.55680e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.03857e13 | 0 | −5.11108e26 | −2.06848e31 | 0 | −1.95248e37 | −1.17366e40 | 0 | −2.14826e44 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.24646e13 | 0 | −1.14310e26 | 1.81181e31 | 0 | −2.76134e37 | −1.64729e40 | 0 | 4.07017e44 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.19029e13 | 0 | 3.98828e26 | −1.00222e30 | 0 | 1.60489e37 | −7.02319e39 | 0 | −3.19739e43 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.81096e13 | 0 | 1.69556e27 | −1.07790e31 | 0 | 5.70296e37 | 5.17945e40 | 0 | −5.18572e44 | ||||||||||||||||||||||||||||||||||||||||
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 | 9.90.a.b | 7 | |
| 3.b | odd | 2 | 1 | 1.90.a.a | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.90.a.a | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 9.90.a.b | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 31407330351408 T_{2}^{6} + \cdots + 54\!\cdots\!12 \)
acting on \(S_{90}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 54\!\cdots\!12 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + \cdots - 21\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots + 14\!\cdots\!48 \)
|
| $11$ |
\( T^{7} + \cdots + 39\!\cdots\!48 \)
|
| $13$ |
\( T^{7} + \cdots + 39\!\cdots\!36 \)
|
| $17$ |
\( T^{7} + \cdots + 15\!\cdots\!32 \)
|
| $19$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots + 11\!\cdots\!64 \)
|
| $29$ |
\( T^{7} + \cdots + 50\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!88 \)
|
| $37$ |
\( T^{7} + \cdots + 13\!\cdots\!08 \)
|
| $41$ |
\( T^{7} + \cdots - 32\!\cdots\!92 \)
|
| $43$ |
\( T^{7} + \cdots + 45\!\cdots\!36 \)
|
| $47$ |
\( T^{7} + \cdots - 12\!\cdots\!28 \)
|
| $53$ |
\( T^{7} + \cdots + 35\!\cdots\!64 \)
|
| $59$ |
\( T^{7} + \cdots - 20\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 77\!\cdots\!52 \)
|
| $67$ |
\( T^{7} + \cdots - 40\!\cdots\!32 \)
|
| $71$ |
\( T^{7} + \cdots + 57\!\cdots\!68 \)
|
| $73$ |
\( T^{7} + \cdots - 21\!\cdots\!64 \)
|
| $79$ |
\( T^{7} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 58\!\cdots\!36 \)
|
| $89$ |
\( T^{7} + \cdots - 26\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 61\!\cdots\!72 \)
|
| show more | |
| show less |