Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,50,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, 50, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 50);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 50 \) |
| 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: | \(136.859584589\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 13737637404392x^{2} + 1893846760763995968x + 12173460045273257880973824 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{11}\cdot 5^{2}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 13737637404392x^{2} + 1893846760763995968x + 12173460045273257880973824 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{3} + 4054077\nu^{2} - 322226356091280\nu + 10503479357158835616 ) / 23120 \)
|
| \(\beta_{3}\) | \(=\) |
\( 144\nu^{2} + 29777208\nu - 989109900560562 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 3 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2481434\beta _1 + 989109893116260 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -450453\beta_{3} + 369920\beta_{2} + 430752910844642\beta _1 - 613601899493015670516 ) / 432 \)
|
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 |
|
−4.03827e7 | 0 | 1.06782e15 | 4.14868e16 | 0 | −1.21820e20 | −2.03878e22 | 0 | −1.67535e24 | ||||||||||||||||||||||||||||||
| 1.2 | −1.11973e7 | 0 | −4.37571e14 | −6.40494e16 | 0 | 2.87552e20 | 1.12031e22 | 0 | 7.17179e23 | |||||||||||||||||||||||||||||||
| 1.3 | 1.23257e7 | 0 | −4.11028e14 | 2.45495e17 | 0 | −2.23658e20 | −1.20049e22 | 0 | 3.02589e24 | |||||||||||||||||||||||||||||||
| 1.4 | 4.53619e7 | 0 | 1.49475e15 | 8.38698e16 | 0 | 5.82632e20 | 4.22681e22 | 0 | 3.80449e24 | |||||||||||||||||||||||||||||||
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 | 9.50.a.b | 4 | |
| 3.b | odd | 2 | 1 | 3.50.a.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.50.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 9.50.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 6107508 T_{2}^{3} + \cdots + 25\!\cdots\!84 \)
acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 25\!\cdots\!84 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots - 54\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots - 91\!\cdots\!56 \)
|
| $13$ |
\( T^{4} + \cdots - 29\!\cdots\!76 \)
|
| $17$ |
\( T^{4} + \cdots - 14\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots + 56\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 43\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 99\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 35\!\cdots\!44 \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 88\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots + 44\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots - 12\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots - 15\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 99\!\cdots\!56 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 28\!\cdots\!56 \)
|
| show more | |
| show less |