Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4205,2,Mod(1,4205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4205, 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("4205.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4205 = 5 \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4205.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: | \(33.5770940499\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1586009.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 9x^{3} + 4x^{2} + 17x + 7 \)
|
| 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
\( x^{5} - x^{4} - 9x^{3} + 4x^{2} + 17x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 11\nu + 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 8\nu^{2} + 11\nu + 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{4} + 8\beta_{3} + 2\beta_{2} + \beta _1 + 24 \)
|
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 |
|
−2.81969 | 2.21248 | 5.95067 | 1.00000 | −6.23850 | −3.02603 | −11.1397 | 1.89505 | −2.81969 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.03739 | −1.51980 | 2.15097 | 1.00000 | 3.09643 | 2.57663 | −0.307585 | −0.690207 | −2.03739 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.594278 | −3.18209 | −1.64683 | 1.00000 | −1.89105 | −4.07314 | −2.16723 | 7.12569 | 0.594278 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.849666 | 1.37380 | −1.27807 | 1.00000 | 1.16727 | 3.54107 | −2.78526 | −1.11267 | 0.849666 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.41314 | −0.884387 | 3.82326 | 1.00000 | −2.13415 | −2.01854 | 4.39978 | −2.21786 | 2.41314 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 4205.2.a.h | ✓ | 5 |
| 29.b | even | 2 | 1 | 4205.2.a.k | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4205.2.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 4205.2.a.k | yes | 5 | 29.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):
|
\( T_{2}^{5} + T_{2}^{4} - 9T_{2}^{3} - 4T_{2}^{2} + 17T_{2} - 7 \)
|
|
\( T_{3}^{5} + 2T_{3}^{4} - 8T_{3}^{3} - 11T_{3}^{2} + 12T_{3} + 13 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 9 T^{3} + \cdots - 7 \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 13 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} + 3 T^{4} + \cdots + 227 \)
|
| $11$ |
\( T^{5} + 2 T^{4} + \cdots + 67 \)
|
| $13$ |
\( T^{5} - 45 T^{3} + \cdots + 169 \)
|
| $17$ |
\( T^{5} + 19 T^{4} + \cdots + 331 \)
|
| $19$ |
\( T^{5} - T^{4} + \cdots - 17 \)
|
| $23$ |
\( T^{5} - 4 T^{4} + \cdots - 89 \)
|
| $29$ |
\( T^{5} \)
|
| $31$ |
\( T^{5} + 18 T^{4} + \cdots + 211 \)
|
| $37$ |
\( T^{5} + 2 T^{4} + \cdots - 9 \)
|
| $41$ |
\( T^{5} + 2 T^{4} + \cdots - 567 \)
|
| $43$ |
\( T^{5} + 22 T^{4} + \cdots - 1053 \)
|
| $47$ |
\( T^{5} - 2 T^{4} + \cdots - 721 \)
|
| $53$ |
\( T^{5} + 14 T^{4} + \cdots - 6197 \)
|
| $59$ |
\( T^{5} - 20 T^{4} + \cdots + 801 \)
|
| $61$ |
\( T^{5} + 5 T^{4} + \cdots + 38853 \)
|
| $67$ |
\( T^{5} + 21 T^{4} + \cdots + 1267 \)
|
| $71$ |
\( T^{5} + 4 T^{4} + \cdots + 2689 \)
|
| $73$ |
\( T^{5} + 28 T^{4} + \cdots - 65511 \)
|
| $79$ |
\( T^{5} + 3 T^{4} + \cdots + 1053 \)
|
| $83$ |
\( T^{5} - 8 T^{4} + \cdots + 24479 \)
|
| $89$ |
\( T^{5} - 34 T^{4} + \cdots + 448693 \)
|
| $97$ |
\( T^{5} + 9 T^{4} + \cdots + 2273 \)
|
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