Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,22,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.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: | \(125.764804929\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 125326x + 2416960 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 125326x + 2416960 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{2} + 246\nu + 166985 ) / 105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{2} + 1782\nu - 501900 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 45\beta _1 + 135 ) / 360 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 41\beta_{2} - 4455\beta _1 + 10024635 ) / 120 \)
|
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 |
|
−999.805 | 0 | −1.09754e6 | −9.76562e6 | 0 | 9.82313e7 | 3.19407e9 | 0 | 9.76372e9 | |||||||||||||||||||||||||||
| 1.2 | 904.285 | 0 | −1.27942e6 | −9.76562e6 | 0 | −5.27665e8 | −3.05338e9 | 0 | −8.83091e9 | ||||||||||||||||||||||||||||
| 1.3 | 2395.52 | 0 | 3.64136e6 | −9.76562e6 | 0 | 8.95100e8 | 3.69919e9 | 0 | −2.33937e10 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 45.22.a.e | 3 | |
| 3.b | odd | 2 | 1 | 15.22.a.b | ✓ | 3 | |
| 15.d | odd | 2 | 1 | 75.22.a.g | 3 | ||
| 15.e | even | 4 | 2 | 75.22.b.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 45.22.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
| 75.22.a.g | 3 | 15.d | odd | 2 | 1 | ||
| 75.22.b.g | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 2300T_{2}^{2} - 1132928T_{2} + 2165809152 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 2165809152 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T + 9765625)^{3} \)
|
| $7$ |
\( T^{3} + \cdots + 46\!\cdots\!00 \)
|
| $11$ |
\( T^{3} + \cdots + 24\!\cdots\!32 \)
|
| $13$ |
\( T^{3} + \cdots - 16\!\cdots\!04 \)
|
| $17$ |
\( T^{3} + \cdots - 41\!\cdots\!92 \)
|
| $19$ |
\( T^{3} + \cdots - 10\!\cdots\!20 \)
|
| $23$ |
\( T^{3} + \cdots + 96\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots + 56\!\cdots\!40 \)
|
| $31$ |
\( T^{3} + \cdots - 49\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 45\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots - 16\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 36\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots - 40\!\cdots\!72 \)
|
| $53$ |
\( T^{3} + \cdots + 45\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots - 34\!\cdots\!60 \)
|
| $61$ |
\( T^{3} + \cdots - 87\!\cdots\!28 \)
|
| $67$ |
\( T^{3} + \cdots + 23\!\cdots\!32 \)
|
| $71$ |
\( T^{3} + \cdots + 51\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots + 62\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 82\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 24\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 27\!\cdots\!68 \)
|
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