Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5445,2,Mod(1,5445)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5445, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5445.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5445.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: | \(43.4785439006\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.74043072.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} + 21x^{2} - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 9x^{4} + 21x^{2} - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 9\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 9\nu^{3} - 17\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{4} + 9\beta_{3} + 19\beta_1 \)
|
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 |
|
−2.38098 | 0 | 3.66908 | 1.00000 | 0 | −2.24200 | −3.97405 | 0 | −2.38098 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.57354 | 0 | 0.476024 | 1.00000 | 0 | 0.665985 | 2.39804 | 0 | −1.57354 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.924607 | 0 | −1.14510 | 1.00000 | 0 | 4.64003 | 2.90798 | 0 | −0.924607 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.924607 | 0 | −1.14510 | 1.00000 | 0 | −4.64003 | −2.90798 | 0 | 0.924607 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.57354 | 0 | 0.476024 | 1.00000 | 0 | −0.665985 | −2.39804 | 0 | 1.57354 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.38098 | 0 | 3.66908 | 1.00000 | 0 | 2.24200 | 3.97405 | 0 | 2.38098 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5445.2.a.by | yes | 6 |
| 3.b | odd | 2 | 1 | 5445.2.a.bw | ✓ | 6 | |
| 11.b | odd | 2 | 1 | inner | 5445.2.a.by | yes | 6 |
| 33.d | even | 2 | 1 | 5445.2.a.bw | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5445.2.a.bw | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 5445.2.a.bw | ✓ | 6 | 33.d | even | 2 | 1 | |
| 5445.2.a.by | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 5445.2.a.by | yes | 6 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5445))\):
|
\( T_{2}^{6} - 9T_{2}^{4} + 21T_{2}^{2} - 12 \)
|
|
\( T_{7}^{6} - 27T_{7}^{4} + 120T_{7}^{2} - 48 \)
|
|
\( T_{23}^{3} - 12T_{23}^{2} + 39T_{23} - 24 \)
|
|
\( T_{53}^{3} - 6T_{53}^{2} - 15T_{53} + 84 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{4} + \cdots - 12 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 27 T^{4} + \cdots - 48 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 42 T^{4} + \cdots - 12 \)
|
| $19$ |
\( T^{6} - 27 T^{4} + \cdots - 48 \)
|
| $23$ |
\( (T^{3} - 12 T^{2} + \cdots - 24)^{2} \)
|
| $29$ |
\( (T^{2} - 12)^{3} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} + \cdots - 103)^{2} \)
|
| $37$ |
\( (T^{3} - 15 T^{2} + \cdots - 28)^{2} \)
|
| $41$ |
\( T^{6} - 120 T^{4} + \cdots - 49152 \)
|
| $43$ |
\( T^{6} - 132 T^{4} + \cdots - 192 \)
|
| $47$ |
\( (T^{3} - 6 T^{2} + \cdots + 354)^{2} \)
|
| $53$ |
\( (T^{3} - 6 T^{2} - 15 T + 84)^{2} \)
|
| $59$ |
\( (T^{3} - 18 T^{2} + \cdots + 552)^{2} \)
|
| $61$ |
\( T^{6} - 129 T^{4} + \cdots - 15123 \)
|
| $67$ |
\( (T^{3} + 9 T^{2} - 16)^{2} \)
|
| $71$ |
\( (T^{3} - 18 T^{2} + \cdots + 168)^{2} \)
|
| $73$ |
\( T^{6} - 411 T^{4} + \cdots - 1843968 \)
|
| $79$ |
\( T^{6} - 57 T^{4} + \cdots - 1083 \)
|
| $83$ |
\( T^{6} - 228 T^{4} + \cdots - 69312 \)
|
| $89$ |
\( (T^{3} - 6 T^{2} - 132 T + 24)^{2} \)
|
| $97$ |
\( (T^{3} + 9 T^{2} + \cdots - 1856)^{2} \)
|
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