Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4235,2,Mod(1,4235)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, 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("4235.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4235 = 5 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4235.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: | \(33.8166452560\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.173513.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 3x^{2} + 3x - 1 \)
|
| 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
\( x^{5} - 2x^{4} - 5x^{3} + 3x^{2} + 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + \nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 3\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} + 2\beta_{3} + \beta_{2} + 8\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{4} + 9\beta_{3} + 2\beta_{2} + 23\beta _1 + 14 \)
|
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 |
|
−2.40663 | −0.321225 | 3.79186 | −1.00000 | 0.773069 | −1.00000 | −4.31233 | −2.89681 | 2.40663 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.164091 | −3.27813 | −1.97307 | −1.00000 | 0.537912 | −1.00000 | 0.651947 | 7.74611 | 0.164091 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.686507 | 0.347661 | −1.52871 | −1.00000 | 0.238671 | −1.00000 | −2.42248 | −2.87913 | −0.686507 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.65369 | −1.14142 | 0.734678 | −1.00000 | −1.88755 | −1.00000 | −2.09245 | −1.69716 | −1.65369 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.23053 | 2.39311 | 2.97525 | −1.00000 | 5.33790 | −1.00000 | 2.17531 | 2.72699 | −2.23053 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 4235.2.a.be | yes | 5 |
| 11.b | odd | 2 | 1 | 4235.2.a.y | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4235.2.a.y | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 4235.2.a.be | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 12T_{2}^{2} - 4T_{2} - 1 \)
|
|
\( T_{3}^{5} + 2T_{3}^{4} - 7T_{3}^{3} - 9T_{3}^{2} + T_{3} + 1 \)
|
|
\( T_{13}^{5} - 12T_{13}^{4} + 12T_{13}^{3} + 325T_{13}^{2} - 1368T_{13} + 1571 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( (T + 1)^{5} \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} - 12 T^{4} + \cdots + 1571 \)
|
| $17$ |
\( T^{5} - 14 T^{4} + \cdots - 83 \)
|
| $19$ |
\( T^{5} - 9 T^{4} + \cdots - 2259 \)
|
| $23$ |
\( T^{5} - 17 T^{4} + \cdots - 67 \)
|
| $29$ |
\( T^{5} - 3 T^{4} + \cdots - 3197 \)
|
| $31$ |
\( T^{5} - 2 T^{4} + \cdots + 1359 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots + 1051 \)
|
| $41$ |
\( T^{5} + 15 T^{4} + \cdots + 4617 \)
|
| $43$ |
\( T^{5} - 4 T^{4} + \cdots - 10971 \)
|
| $47$ |
\( T^{5} + 2 T^{4} + \cdots + 251 \)
|
| $53$ |
\( T^{5} - 6 T^{4} + \cdots - 22913 \)
|
| $59$ |
\( T^{5} + 6 T^{4} + \cdots + 68011 \)
|
| $61$ |
\( T^{5} - 20 T^{4} + \cdots - 1867 \)
|
| $67$ |
\( T^{5} - 3 T^{4} + \cdots + 2309 \)
|
| $71$ |
\( T^{5} + 6 T^{4} + \cdots + 13 \)
|
| $73$ |
\( T^{5} - 11 T^{4} + \cdots + 1097 \)
|
| $79$ |
\( T^{5} - 19 T^{4} + \cdots + 20143 \)
|
| $83$ |
\( T^{5} - 8 T^{4} + \cdots - 279 \)
|
| $89$ |
\( T^{5} - T^{4} + \cdots - 6047 \)
|
| $97$ |
\( T^{5} + 7 T^{4} + \cdots - 271673 \)
|
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