Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [529,4,Mod(1,529)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(529, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("529.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 529 = 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 529.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: | \(31.2120103930\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 28x^{3} - 8x^{2} + 151x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{5} - 28x^{3} - 8x^{2} + 151x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 19\nu^{2} + 3\nu + 28 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 27\nu^{2} - 11\nu - 116 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 17\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 18\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 20\beta_{3} + 28\beta_{2} + 34\beta _1 + 186 \)
|
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 |
|
−4.18688 | −4.69621 | 9.52992 | 0.353159 | 19.6624 | 32.9954 | −6.40560 | −4.94564 | −1.47863 | |||||||||||||||||||||||||||||||||
| 1.2 | −3.05318 | 7.41773 | 1.32189 | −4.68644 | −22.6476 | −18.4238 | 20.3894 | 28.0227 | 14.3085 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.0264564 | −4.46196 | −7.99930 | 13.4877 | 0.118048 | −26.4212 | 0.423284 | −7.09089 | −0.356837 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.46142 | 3.78046 | −1.94143 | −3.30045 | 9.30529 | 16.6457 | −24.4700 | −12.7081 | −8.12378 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.80509 | −9.04002 | 15.0889 | −9.85398 | −43.4381 | 9.20386 | 34.0629 | 54.7219 | −47.3493 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \( -1 \) |
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 | 529.4.a.h | ✓ | 5 |
| 23.b | odd | 2 | 1 | 529.4.a.i | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 529.4.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 529.4.a.i | yes | 5 | 23.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(529))\):
|
\( T_{2}^{5} - 28T_{2}^{3} - 8T_{2}^{2} + 151T_{2} + 4 \)
|
|
\( T_{3}^{5} + 7T_{3}^{4} - 72T_{3}^{3} - 462T_{3}^{2} + 788T_{3} + 5312 \)
|
|
\( T_{5}^{5} + 4T_{5}^{4} - 148T_{5}^{3} - 1066T_{5}^{2} - 1661T_{5} + 726 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 28 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{5} + 7 T^{4} + \cdots + 5312 \)
|
| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 726 \)
|
| $7$ |
\( T^{5} - 14 T^{4} + \cdots - 2460688 \)
|
| $11$ |
\( T^{5} - 54 T^{4} + \cdots - 68489152 \)
|
| $13$ |
\( T^{5} + 30 T^{4} + \cdots + 9900468 \)
|
| $17$ |
\( T^{5} + \cdots + 1832603296 \)
|
| $19$ |
\( T^{5} + \cdots - 1028652424 \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 2151398052 \)
|
| $31$ |
\( T^{5} + \cdots - 513947655360 \)
|
| $37$ |
\( T^{5} + \cdots - 106646547328 \)
|
| $41$ |
\( T^{5} + \cdots - 42621420230 \)
|
| $43$ |
\( T^{5} + \cdots - 84288484512 \)
|
| $47$ |
\( T^{5} + \cdots - 47734727136 \)
|
| $53$ |
\( T^{5} + \cdots + 6225734313576 \)
|
| $59$ |
\( T^{5} + \cdots - 57188888376 \)
|
| $61$ |
\( T^{5} + \cdots + 4541927640324 \)
|
| $67$ |
\( T^{5} + \cdots + 1218238739456 \)
|
| $71$ |
\( T^{5} + \cdots + 59857805265200 \)
|
| $73$ |
\( T^{5} + \cdots - 336797251627 \)
|
| $79$ |
\( T^{5} + \cdots - 15527969777664 \)
|
| $83$ |
\( T^{5} + \cdots + 48647467969344 \)
|
| $89$ |
\( T^{5} + \cdots + 224496464777930 \)
|
| $97$ |
\( T^{5} + \cdots + 238007597477798 \)
|
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