Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5577,2,Mod(1,5577)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5577, 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("5577.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5577 = 3 \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5577.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: | \(44.5325692073\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.863825.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 9x^{2} + 8x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 429) |
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 9x^{2} + 8x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + \nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 8\beta_{2} + 10\beta _1 + 14 \)
|
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 |
|
−1.84524 | −1.00000 | 1.40492 | −3.96195 | 1.84524 | 3.60163 | 1.09806 | 1.00000 | 7.31076 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.33178 | −1.00000 | −0.226360 | 3.31232 | 1.33178 | 1.53428 | 2.96502 | 1.00000 | −4.41129 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.847823 | −1.00000 | −1.28120 | 2.90954 | −0.847823 | 4.44226 | −2.78187 | 1.00000 | 2.46677 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.55977 | −1.00000 | 0.432881 | −1.18322 | −1.55977 | −2.91334 | −2.44434 | 1.00000 | −1.84555 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.76943 | −1.00000 | 5.66975 | −3.07670 | −2.76943 | 2.33517 | 10.1631 | 1.00000 | −8.52070 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
| \(13\) | \(1\) |
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 | 5577.2.a.u | 5 | |
| 13.b | even | 2 | 1 | 5577.2.a.o | 5 | ||
| 13.e | even | 6 | 2 | 429.2.i.e | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.2.i.e | ✓ | 10 | 13.e | even | 6 | 2 | |
| 5577.2.a.o | 5 | 13.b | even | 2 | 1 | ||
| 5577.2.a.u | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 6T_{2}^{3} + 9T_{2}^{2} + 8T_{2} - 9 \)
|
|
\( T_{5}^{5} + 2T_{5}^{4} - 21T_{5}^{3} - 34T_{5}^{2} + 108T_{5} + 139 \)
|
|
\( T_{7}^{5} - 9T_{7}^{4} + 16T_{7}^{3} + 57T_{7}^{2} - 207T_{7} + 167 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 9 \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( T^{5} + 2 T^{4} + \cdots + 139 \)
|
| $7$ |
\( T^{5} - 9 T^{4} + \cdots + 167 \)
|
| $11$ |
\( (T - 1)^{5} \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} + 3 T^{4} + \cdots + 1 \)
|
| $19$ |
\( T^{5} + 5 T^{4} + \cdots + 1699 \)
|
| $23$ |
\( T^{5} + 5 T^{4} + \cdots + 137 \)
|
| $29$ |
\( T^{5} - 12 T^{4} + \cdots - 25 \)
|
| $31$ |
\( T^{5} - 18 T^{4} + \cdots + 6225 \)
|
| $37$ |
\( T^{5} - T^{4} + \cdots - 10147 \)
|
| $41$ |
\( T^{5} + 30 T^{4} + \cdots + 653 \)
|
| $43$ |
\( T^{5} + 3 T^{4} + \cdots + 4705 \)
|
| $47$ |
\( T^{5} - 22 T^{4} + \cdots - 59 \)
|
| $53$ |
\( T^{5} - 7 T^{4} + \cdots + 5197 \)
|
| $59$ |
\( T^{5} - 12 T^{4} + \cdots - 5725 \)
|
| $61$ |
\( T^{5} - 18 T^{4} + \cdots - 29371 \)
|
| $67$ |
\( T^{5} - 37 T^{4} + \cdots + 26393 \)
|
| $71$ |
\( T^{5} - 17 T^{4} + \cdots - 51519 \)
|
| $73$ |
\( T^{5} + 2 T^{4} + \cdots - 32777 \)
|
| $79$ |
\( T^{5} + 6 T^{4} + \cdots + 72185 \)
|
| $83$ |
\( T^{5} + 4 T^{4} + \cdots + 879 \)
|
| $89$ |
\( T^{5} + 14 T^{4} + \cdots + 6329 \)
|
| $97$ |
\( T^{5} - 15 T^{4} + \cdots - 269 \)
|
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