Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [539,4,Mod(1,539)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, 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("539.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 539 = 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 539.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: | \(31.8020294931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 44x^{7} + 145x^{6} + 478x^{5} - 1836x^{4} + 149x^{3} + 2126x^{2} - 464x - 216 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 44x^{7} + 145x^{6} + 478x^{5} - 1836x^{4} + 149x^{3} + 2126x^{2} - 464x - 216 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{8} + 116\nu^{6} + 41\nu^{5} - 1375\nu^{4} - 1001\nu^{3} + 6326\nu^{2} + 712\nu - 1176 ) / 448 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{8} - 16\nu^{7} - 116\nu^{6} + 727\nu^{5} + 847\nu^{4} - 8871\nu^{3} + 5658\nu^{2} + 13912\nu - 3240 ) / 896 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} + 8\nu^{7} + 44\nu^{6} - 349\nu^{5} - 365\nu^{4} + 4021\nu^{3} - 2694\nu^{2} - 4520\nu + 2680 ) / 224 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 15 \nu^{8} - 16 \nu^{7} - 708 \nu^{6} + 947 \nu^{5} + 9547 \nu^{4} - 14211 \nu^{3} - 27582 \nu^{2} + \cdots + 11192 ) / 896 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13 \nu^{8} + 32 \nu^{7} + 556 \nu^{6} - 1593 \nu^{5} - 5713 \nu^{4} + 20345 \nu^{3} - 5270 \nu^{2} + \cdots + 6104 ) / 448 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11\nu^{8} - 32\nu^{7} - 500\nu^{6} + 1535\nu^{5} + 5927\nu^{4} - 19135\nu^{3} - 5574\nu^{2} + 18648\nu + 2648 ) / 448 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} - \beta_{6} + \beta_{5} - \beta_{4} + 20\beta _1 - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{8} + \beta_{7} - \beta_{6} + 3\beta_{5} + 3\beta_{4} + 23\beta_{2} - 5\beta _1 + 215 \)
|
| \(\nu^{5}\) | \(=\) |
\( 28\beta_{8} + \beta_{7} - 26\beta_{6} + 35\beta_{5} - 12\beta_{4} + 4\beta_{3} + 433\beta _1 - 306 \)
|
| \(\nu^{6}\) | \(=\) |
\( 61 \beta_{8} + 28 \beta_{7} - 37 \beta_{6} + 107 \beta_{5} + 119 \beta_{4} - 2 \beta_{3} + 513 \beta_{2} + \cdots + 4508 \)
|
| \(\nu^{7}\) | \(=\) |
\( 661 \beta_{8} + 15 \beta_{7} - 598 \beta_{6} + 964 \beta_{5} - 114 \beta_{4} + 164 \beta_{3} + \cdots - 8267 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1491 \beta_{8} + 638 \beta_{7} - 994 \beta_{6} + 2907 \beta_{5} + 3396 \beta_{4} - 172 \beta_{3} + \cdots + 97392 \)
|
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 |
|
−4.72506 | −3.35899 | 14.3261 | −8.56301 | 15.8714 | 0 | −29.8914 | −15.7172 | 40.4607 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.93603 | 2.92998 | 7.49237 | 17.3638 | −11.5325 | 0 | 1.99807 | −18.4152 | −68.3446 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.28945 | 4.68392 | 2.82047 | −15.8206 | −15.4075 | 0 | 17.0378 | −5.06091 | 52.0409 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.32228 | −9.77107 | −6.25159 | 15.2646 | 12.9200 | 0 | 18.8445 | 68.4738 | −20.1841 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.495883 | −3.00425 | −7.75410 | 4.88996 | 1.48976 | 0 | 7.81219 | −17.9745 | −2.42485 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.233052 | 8.99716 | −7.94569 | 18.5555 | 2.09681 | 0 | −3.71617 | 53.9490 | 4.32440 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.02106 | 5.36301 | −6.95743 | −14.7661 | 5.47596 | 0 | −15.2725 | 1.76184 | −15.0771 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.70162 | 8.99759 | 14.1052 | 3.18026 | 42.3033 | 0 | 28.7044 | 53.9567 | 14.9524 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.81296 | 0.162642 | 15.1646 | 11.8954 | 0.782791 | 0 | 34.4830 | −26.9735 | 57.2523 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 539.4.a.k | 9 | |
| 7.b | odd | 2 | 1 | 539.4.a.j | 9 | ||
| 7.d | odd | 6 | 2 | 77.4.e.b | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.e.b | ✓ | 18 | 7.d | odd | 6 | 2 | |
| 539.4.a.j | 9 | 7.b | odd | 2 | 1 | ||
| 539.4.a.k | 9 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):
|
\( T_{2}^{9} + 3T_{2}^{8} - 44T_{2}^{7} - 145T_{2}^{6} + 478T_{2}^{5} + 1836T_{2}^{4} + 149T_{2}^{3} - 2126T_{2}^{2} - 464T_{2} + 216 \)
|
|
\( T_{3}^{9} - 15 T_{3}^{8} - 56 T_{3}^{7} + 1718 T_{3}^{6} - 4899 T_{3}^{5} - 28377 T_{3}^{4} + \cdots + 95551 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 3 T^{8} + \cdots + 216 \)
|
| $3$ |
\( T^{9} - 15 T^{8} + \cdots + 95551 \)
|
| $5$ |
\( T^{9} + \cdots + 1819987272 \)
|
| $7$ |
\( T^{9} \)
|
| $11$ |
\( (T - 11)^{9} \)
|
| $13$ |
\( T^{9} + \cdots - 129605201872125 \)
|
| $17$ |
\( T^{9} + \cdots + 200838998226024 \)
|
| $19$ |
\( T^{9} + \cdots - 12\!\cdots\!96 \)
|
| $23$ |
\( T^{9} + \cdots + 71\!\cdots\!08 \)
|
| $29$ |
\( T^{9} + \cdots + 52\!\cdots\!59 \)
|
| $31$ |
\( T^{9} + \cdots + 25\!\cdots\!68 \)
|
| $37$ |
\( T^{9} + \cdots - 38\!\cdots\!96 \)
|
| $41$ |
\( T^{9} + \cdots + 35\!\cdots\!48 \)
|
| $43$ |
\( T^{9} + \cdots + 91\!\cdots\!32 \)
|
| $47$ |
\( T^{9} + \cdots - 57\!\cdots\!12 \)
|
| $53$ |
\( T^{9} + \cdots + 51\!\cdots\!96 \)
|
| $59$ |
\( T^{9} + \cdots - 35\!\cdots\!57 \)
|
| $61$ |
\( T^{9} + \cdots + 61\!\cdots\!92 \)
|
| $67$ |
\( T^{9} + \cdots + 21\!\cdots\!76 \)
|
| $71$ |
\( T^{9} + \cdots - 37\!\cdots\!56 \)
|
| $73$ |
\( T^{9} + \cdots - 56\!\cdots\!04 \)
|
| $79$ |
\( T^{9} + \cdots + 10\!\cdots\!17 \)
|
| $83$ |
\( T^{9} + \cdots - 30\!\cdots\!84 \)
|
| $89$ |
\( T^{9} + \cdots - 93\!\cdots\!64 \)
|
| $97$ |
\( T^{9} + \cdots + 49\!\cdots\!19 \)
|
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