Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,2,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("605.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.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: | \(4.83094932229\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2525.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 3 \)
|
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 |
|
−1.87603 | −1.77748 | 1.51949 | 1.00000 | 3.33461 | 4.25800 | 0.901454 | 0.159450 | −1.87603 | ||||||||||||||||||||||||||||||
| 1.2 | 0.0935099 | 1.46673 | −1.99126 | 1.00000 | 0.137154 | 4.52452 | −0.373222 | −0.848698 | 0.0935099 | |||||||||||||||||||||||||||||||
| 1.3 | 2.25800 | 0.777484 | 3.09855 | 1.00000 | 1.75556 | 0.123970 | 2.48051 | −2.39552 | 2.25800 | |||||||||||||||||||||||||||||||
| 1.4 | 2.52452 | −2.46673 | 4.37322 | 1.00000 | −6.22732 | 2.09351 | 5.99126 | 3.08477 | 2.52452 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
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 | 605.2.a.l | 4 | |
| 3.b | odd | 2 | 1 | 5445.2.a.bg | 4 | ||
| 4.b | odd | 2 | 1 | 9680.2.a.cs | 4 | ||
| 5.b | even | 2 | 1 | 3025.2.a.v | 4 | ||
| 11.b | odd | 2 | 1 | 605.2.a.i | 4 | ||
| 11.c | even | 5 | 2 | 55.2.g.a | ✓ | 8 | |
| 11.c | even | 5 | 2 | 605.2.g.j | 8 | ||
| 11.d | odd | 10 | 2 | 605.2.g.g | 8 | ||
| 11.d | odd | 10 | 2 | 605.2.g.n | 8 | ||
| 33.d | even | 2 | 1 | 5445.2.a.bu | 4 | ||
| 33.h | odd | 10 | 2 | 495.2.n.f | 8 | ||
| 44.c | even | 2 | 1 | 9680.2.a.cv | 4 | ||
| 44.h | odd | 10 | 2 | 880.2.bo.e | 8 | ||
| 55.d | odd | 2 | 1 | 3025.2.a.be | 4 | ||
| 55.j | even | 10 | 2 | 275.2.h.b | 8 | ||
| 55.k | odd | 20 | 4 | 275.2.z.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.2.g.a | ✓ | 8 | 11.c | even | 5 | 2 | |
| 275.2.h.b | 8 | 55.j | even | 10 | 2 | ||
| 275.2.z.b | 16 | 55.k | odd | 20 | 4 | ||
| 495.2.n.f | 8 | 33.h | odd | 10 | 2 | ||
| 605.2.a.i | 4 | 11.b | odd | 2 | 1 | ||
| 605.2.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 605.2.g.g | 8 | 11.d | odd | 10 | 2 | ||
| 605.2.g.j | 8 | 11.c | even | 5 | 2 | ||
| 605.2.g.n | 8 | 11.d | odd | 10 | 2 | ||
| 880.2.bo.e | 8 | 44.h | odd | 10 | 2 | ||
| 3025.2.a.v | 4 | 5.b | even | 2 | 1 | ||
| 3025.2.a.be | 4 | 55.d | odd | 2 | 1 | ||
| 5445.2.a.bg | 4 | 3.b | odd | 2 | 1 | ||
| 5445.2.a.bu | 4 | 33.d | even | 2 | 1 | ||
| 9680.2.a.cs | 4 | 4.b | odd | 2 | 1 | ||
| 9680.2.a.cv | 4 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):
|
\( T_{2}^{4} - 3T_{2}^{3} - 3T_{2}^{2} + 11T_{2} - 1 \)
|
|
\( T_{3}^{4} + 2T_{3}^{3} - 4T_{3}^{2} - 5T_{3} + 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots - 1 \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 5 \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( T^{4} - 11 T^{3} + \cdots + 5 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 7 T^{3} + \cdots - 11 \)
|
| $17$ |
\( T^{4} - 3 T^{3} + \cdots - 11 \)
|
| $19$ |
\( T^{4} - 12 T^{3} + \cdots + 25 \)
|
| $23$ |
\( T^{4} + 9 T^{3} + \cdots - 1669 \)
|
| $29$ |
\( T^{4} + 8 T^{3} + \cdots - 55 \)
|
| $31$ |
\( T^{4} - 3 T^{3} + \cdots + 101 \)
|
| $37$ |
\( T^{4} + 3 T^{3} + \cdots + 151 \)
|
| $41$ |
\( T^{4} + 7 T^{3} + \cdots - 499 \)
|
| $43$ |
\( T^{4} - 21 T^{3} + \cdots - 59 \)
|
| $47$ |
\( T^{4} + 3 T^{3} + \cdots + 71 \)
|
| $53$ |
\( T^{4} + 11 T^{3} + \cdots - 1 \)
|
| $59$ |
\( T^{4} + 7 T^{3} + \cdots - 3025 \)
|
| $61$ |
\( T^{4} - 4 T^{3} + \cdots + 55 \)
|
| $67$ |
\( T^{4} + T^{3} + \cdots - 101 \)
|
| $71$ |
\( T^{4} + 15 T^{3} + \cdots - 7799 \)
|
| $73$ |
\( T^{4} + 9 T^{3} + \cdots - 389 \)
|
| $79$ |
\( T^{4} - 6 T^{3} + \cdots - 2155 \)
|
| $83$ |
\( T^{4} - 15 T^{3} + \cdots - 29 \)
|
| $89$ |
\( T^{4} - 150 T^{2} + \cdots + 725 \)
|
| $97$ |
\( T^{4} - 6 T^{3} + \cdots - 25 \)
|
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