Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [531,4,Mod(1,531)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(531, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("531.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 531 = 3^{2} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 531.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: | \(31.3300142130\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 177) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{7} + 34\nu^{6} + 647\nu^{5} - 1483\nu^{4} - 9408\nu^{3} + 16089\nu^{2} + 36118\nu - 20832 ) / 2144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -18\nu^{7} + 11\nu^{6} + 901\nu^{5} - 734\nu^{4} - 12207\nu^{3} + 12861\nu^{2} + 28786\nu - 33792 ) / 2144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} + 8\nu^{6} + 369\nu^{5} - 345\nu^{4} - 5406\nu^{3} + 3695\nu^{2} + 15758\nu - 6352 ) / 536 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -39\nu^{7} + 102\nu^{6} + 1941\nu^{5} - 4449\nu^{4} - 26080\nu^{3} + 48267\nu^{2} + 59042\nu - 62496 ) / 2144 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 57\nu^{7} - 46\nu^{6} - 2775\nu^{5} + 1967\nu^{4} + 36612\nu^{3} - 20593\nu^{2} - 89302\nu + 29824 ) / 2144 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 3\beta_{3} + 23\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{7} + 2\beta_{5} - 8\beta_{4} - 2\beta_{3} + 30\beta_{2} - \beta _1 + 296 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{7} + 30\beta_{6} + 20\beta_{5} - 98\beta_{3} - \beta_{2} + 570\beta _1 + 35 \)
|
| \(\nu^{6}\) | \(=\) |
\( -72\beta_{7} + 27\beta_{6} + 76\beta_{5} - 352\beta_{4} - 73\beta_{3} + 836\beta_{2} - 21\beta _1 + 7300 \)
|
| \(\nu^{7}\) | \(=\) |
\( 438\beta_{7} + 840\beta_{6} + 966\beta_{5} - 8\beta_{4} - 2834\beta_{3} - 48\beta_{2} + 14561\beta _1 + 1554 \)
|
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 |
|
−5.26363 | 0 | 19.7058 | −11.8799 | 0 | −3.36662 | −61.6149 | 0 | 62.5311 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.61734 | 0 | 13.3198 | 3.21787 | 0 | 15.9864 | −24.5634 | 0 | −14.8580 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.17127 | 0 | −3.28558 | −9.58086 | 0 | 14.1591 | 24.5041 | 0 | 20.8027 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.780043 | 0 | −7.39153 | 21.9196 | 0 | 32.1153 | 12.0061 | 0 | −17.0983 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.254436 | 0 | −7.93526 | 10.8225 | 0 | −23.2950 | 4.05451 | 0 | −2.75362 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.67303 | 0 | −5.20096 | 6.76323 | 0 | −19.0526 | −22.0856 | 0 | 11.3151 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.21744 | 0 | 9.78684 | −17.5635 | 0 | 14.8996 | 7.53588 | 0 | −74.0731 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.19624 | 0 | 19.0009 | 8.30102 | 0 | 21.5539 | 57.1634 | 0 | 43.1341 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(59\) | \(-1\) |
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 | 531.4.a.f | 8 | |
| 3.b | odd | 2 | 1 | 177.4.a.c | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 177.4.a.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 531.4.a.f | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} - 49T_{2}^{6} - 89T_{2}^{5} + 648T_{2}^{4} + 1023T_{2}^{3} - 1476T_{2}^{2} - 1940T_{2} - 384 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(531))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots - 384 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 12 T^{7} + \cdots - 85672464 \)
|
| $7$ |
\( T^{8} + \cdots - 3488279296 \)
|
| $11$ |
\( T^{8} + \cdots - 223404635568 \)
|
| $13$ |
\( T^{8} + \cdots - 5466828930044 \)
|
| $17$ |
\( T^{8} + \cdots + 34686030744 \)
|
| $19$ |
\( T^{8} + \cdots + 612675702830144 \)
|
| $23$ |
\( T^{8} + \cdots - 96081715816128 \)
|
| $29$ |
\( T^{8} + \cdots - 238656589193400 \)
|
| $31$ |
\( T^{8} + \cdots + 37\!\cdots\!44 \)
|
| $37$ |
\( T^{8} + \cdots - 17\!\cdots\!08 \)
|
| $41$ |
\( T^{8} + \cdots + 34\!\cdots\!68 \)
|
| $43$ |
\( T^{8} + \cdots - 95\!\cdots\!04 \)
|
| $47$ |
\( T^{8} + \cdots - 61\!\cdots\!48 \)
|
| $53$ |
\( T^{8} + \cdots + 24\!\cdots\!16 \)
|
| $59$ |
\( (T - 59)^{8} \)
|
| $61$ |
\( T^{8} + \cdots + 12\!\cdots\!88 \)
|
| $67$ |
\( T^{8} + \cdots - 74\!\cdots\!36 \)
|
| $71$ |
\( T^{8} + \cdots - 90\!\cdots\!88 \)
|
| $73$ |
\( T^{8} + \cdots + 70\!\cdots\!32 \)
|
| $79$ |
\( T^{8} + \cdots + 47\!\cdots\!16 \)
|
| $83$ |
\( T^{8} + \cdots + 11\!\cdots\!52 \)
|
| $89$ |
\( T^{8} + \cdots - 59\!\cdots\!44 \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!68 \)
|
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