Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4508,2,Mod(1,4508)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4508, 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("4508.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4508.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: | \(35.9965612312\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.6963152.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 644) |
| 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
\( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 7\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 9\nu + 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 7\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 7\beta_{2} + 12\beta _1 + 29 \)
|
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 |
|
0 | −3.25688 | 0 | 3.04771 | 0 | 0 | 0 | 7.60729 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.18229 | 0 | −4.04332 | 0 | 0 | 0 | 1.76237 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.43585 | 0 | −2.34236 | 0 | 0 | 0 | −0.938323 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.76321 | 0 | 3.11657 | 0 | 0 | 0 | 0.108911 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.11181 | 0 | −1.77860 | 0 | 0 | 0 | 1.45975 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 4508.2.a.f | 5 | |
| 7.b | odd | 2 | 1 | 644.2.a.d | ✓ | 5 | |
| 21.c | even | 2 | 1 | 5796.2.a.t | 5 | ||
| 28.d | even | 2 | 1 | 2576.2.a.bb | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 644.2.a.d | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 2576.2.a.bb | 5 | 28.d | even | 2 | 1 | ||
| 4508.2.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 5796.2.a.t | 5 | 21.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4508))\):
|
\( T_{3}^{5} + 3T_{3}^{4} - 8T_{3}^{3} - 22T_{3}^{2} + 16T_{3} + 38 \)
|
|
\( T_{5}^{5} + 2T_{5}^{4} - 20T_{5}^{3} - 34T_{5}^{2} + 94T_{5} + 160 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 3 T^{4} + \cdots + 38 \)
|
| $5$ |
\( T^{5} + 2 T^{4} + \cdots + 160 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots - 360 \)
|
| $13$ |
\( T^{5} + 13 T^{4} + \cdots - 1424 \)
|
| $17$ |
\( T^{5} + 4 T^{4} + \cdots - 256 \)
|
| $19$ |
\( T^{5} + 12 T^{4} + \cdots + 1024 \)
|
| $23$ |
\( (T + 1)^{5} \)
|
| $29$ |
\( T^{5} - 13 T^{4} + \cdots - 796 \)
|
| $31$ |
\( T^{5} - 3 T^{4} + \cdots - 20810 \)
|
| $37$ |
\( T^{5} + 4 T^{4} + \cdots + 1152 \)
|
| $41$ |
\( T^{5} + T^{4} + \cdots + 2032 \)
|
| $43$ |
\( T^{5} + 8 T^{4} + \cdots - 1984 \)
|
| $47$ |
\( T^{5} + 5 T^{4} + \cdots + 1198 \)
|
| $53$ |
\( T^{5} + 8 T^{4} + \cdots + 28224 \)
|
| $59$ |
\( T^{5} + 12 T^{4} + \cdots + 216 \)
|
| $61$ |
\( T^{5} + 20 T^{4} + \cdots - 292 \)
|
| $67$ |
\( T^{5} + 12 T^{4} + \cdots - 9008 \)
|
| $71$ |
\( T^{5} - 9 T^{4} + \cdots - 19840 \)
|
| $73$ |
\( T^{5} - 9 T^{4} + \cdots - 1296 \)
|
| $79$ |
\( T^{5} + 8 T^{4} + \cdots - 6320 \)
|
| $83$ |
\( T^{5} - 28 T^{4} + \cdots + 4096 \)
|
| $89$ |
\( T^{5} + 32 T^{4} + \cdots - 200 \)
|
| $97$ |
\( T^{5} + 4 T^{4} + \cdots - 27700 \)
|
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