Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1205,2,Mod(1,1205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1205, 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("1205.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1205 = 5 \cdot 241 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1205.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: | \(9.62197344356\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.38569.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 5x^{3} + 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 5x^{3} + 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 5\nu^{2} - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 5\nu^{2} - 4\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{3} + 5\beta_{2} + 7 \)
|
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 |
|
−2.54794 | 1.35541 | 4.49198 | 1.00000 | −3.45349 | −1.88303 | −6.34942 | −1.16287 | −2.54794 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.717838 | 0.928169 | −1.48471 | 1.00000 | −0.666275 | −1.52425 | 2.50146 | −2.13850 | −0.717838 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.526087 | −2.87779 | −1.72323 | 1.00000 | 1.51397 | −4.16547 | 1.95874 | 5.28166 | −0.526087 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.442330 | −2.59927 | −1.80434 | 1.00000 | −1.14974 | 1.77251 | −1.68277 | 3.75621 | 0.442330 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.34953 | −1.80652 | 3.52030 | 1.00000 | −4.24447 | −4.19975 | 3.57200 | 0.263500 | 2.34953 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(241\) | \(-1\) |
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 | 1205.2.a.a | ✓ | 5 |
| 5.b | even | 2 | 1 | 6025.2.a.e | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1205.2.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 6025.2.a.e | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + T_{2}^{4} - 6T_{2}^{3} - 5T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1205))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 6 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{5} + 5 T^{4} + \cdots + 17 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} + 10 T^{4} + \cdots - 89 \)
|
| $11$ |
\( T^{5} + 3 T^{4} + \cdots - 1 \)
|
| $13$ |
\( T^{5} + T^{4} - 17 T^{3} + \cdots + 1 \)
|
| $17$ |
\( T^{5} + 5 T^{4} + \cdots + 361 \)
|
| $19$ |
\( T^{5} - 3 T^{4} + \cdots - 91 \)
|
| $23$ |
\( T^{5} + 8 T^{4} + \cdots + 259 \)
|
| $29$ |
\( T^{5} - 9 T^{4} + \cdots + 479 \)
|
| $31$ |
\( T^{5} + 16 T^{4} + \cdots - 13 \)
|
| $37$ |
\( T^{5} - 7 T^{4} + \cdots - 3757 \)
|
| $41$ |
\( T^{5} - 9 T^{4} + \cdots + 11 \)
|
| $43$ |
\( T^{5} + 32 T^{4} + \cdots - 2953 \)
|
| $47$ |
\( T^{5} + 7 T^{4} + \cdots + 593 \)
|
| $53$ |
\( T^{5} + 32 T^{4} + \cdots + 5687 \)
|
| $59$ |
\( T^{5} + 8 T^{4} + \cdots - 421 \)
|
| $61$ |
\( T^{5} + 12 T^{4} + \cdots + 20479 \)
|
| $67$ |
\( T^{5} + 5 T^{4} + \cdots + 23093 \)
|
| $71$ |
\( T^{5} + 11 T^{4} + \cdots - 289 \)
|
| $73$ |
\( T^{5} + 29 T^{4} + \cdots + 2251 \)
|
| $79$ |
\( T^{5} - 16 T^{4} + \cdots + 1 \)
|
| $83$ |
\( T^{5} + 10 T^{4} + \cdots + 9649 \)
|
| $89$ |
\( T^{5} - 9 T^{4} + \cdots + 20647 \)
|
| $97$ |
\( T^{5} + 43 T^{4} + \cdots + 2219 \)
|
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