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: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 67x^{8} - x^{7} + 1529x^{6} + 194x^{5} - 14053x^{4} - 4705x^{3} + 47798x^{2} + 25312x - 25480 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 67x^{8} - x^{7} + 1529x^{6} + 194x^{5} - 14053x^{4} - 4705x^{3} + 47798x^{2} + 25312x - 25480 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6 \nu^{9} - 283 \nu^{8} - 1184 \nu^{7} + 16768 \nu^{6} + 40109 \nu^{5} - 346283 \nu^{4} + \cdots - 6430424 ) / 191184 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18 \nu^{9} - 289 \nu^{8} + 1276 \nu^{7} + 15700 \nu^{6} - 32701 \nu^{5} - 250505 \nu^{4} + \cdots - 655592 ) / 191184 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 98 \nu^{9} + 829 \nu^{8} + 7200 \nu^{7} - 49312 \nu^{6} - 180947 \nu^{5} + 907461 \nu^{4} + \cdots + 4446232 ) / 191184 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 313 \nu^{9} - 64 \nu^{8} - 18142 \nu^{7} + 6247 \nu^{6} + 319823 \nu^{5} - 172699 \nu^{4} + \cdots - 6845664 ) / 191184 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 117 \nu^{9} + 113 \nu^{8} + 7156 \nu^{7} - 6629 \nu^{6} - 137164 \nu^{5} + 100624 \nu^{4} + \cdots - 612920 ) / 47796 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 146 \nu^{9} + 248 \nu^{8} + 9275 \nu^{7} - 15032 \nu^{6} - 190576 \nu^{5} + 264863 \nu^{4} + \cdots + 956088 ) / 47796 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 655 \nu^{9} - 1970 \nu^{8} - 40110 \nu^{7} + 123317 \nu^{6} + 780115 \nu^{5} - 2336127 \nu^{4} + \cdots - 14304416 ) / 191184 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + 20\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} - \beta_{7} - 3\beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} + 31\beta_{2} - 5\beta _1 + 274 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4 \beta_{9} - 40 \beta_{8} + 43 \beta_{7} + 8 \beta_{6} + 24 \beta_{5} + 38 \beta_{4} - 14 \beta_{3} + \cdots - 80 \)
|
| \(\nu^{6}\) | \(=\) |
\( 68 \beta_{9} + 17 \beta_{8} - 43 \beta_{7} - 148 \beta_{6} + 105 \beta_{5} - 113 \beta_{4} - 26 \beta_{3} + \cdots + 6789 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 191 \beta_{9} - 1277 \beta_{8} + 1470 \beta_{7} + 469 \beta_{6} + 519 \beta_{5} + 1340 \beta_{4} + \cdots - 4299 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2885 \beta_{9} + 1083 \beta_{8} - 1613 \beta_{7} - 5507 \beta_{6} + 4011 \beta_{5} - 5966 \beta_{4} + \cdots + 180498 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 7199 \beta_{9} - 38743 \beta_{8} + 46723 \beta_{7} + 19793 \beta_{6} + 10867 \beta_{5} + \cdots - 173931 \)
|
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 |
|
−5.54057 | −7.07442 | 22.6979 | −3.90908 | 39.1963 | 0 | −81.4350 | 23.0474 | 21.6586 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.52997 | 7.18159 | 12.5206 | −4.69123 | −32.5324 | 0 | −20.4782 | 24.5752 | 21.2511 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.83215 | −8.68415 | 0.0210855 | 2.73592 | 24.5948 | 0 | 22.5975 | 48.4144 | −7.74854 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.23599 | 0.897008 | −3.00036 | −4.26737 | −2.00570 | 0 | 24.5967 | −26.1954 | 9.54180 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.34862 | 5.81456 | −6.18124 | 21.0471 | −7.84160 | 0 | 19.1250 | 6.80905 | −28.3844 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.536321 | −3.94501 | −7.71236 | −2.94337 | −2.11579 | 0 | −8.42687 | −11.4369 | −1.57859 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.75655 | 0.571094 | −0.401416 | −20.2159 | 1.57425 | 0 | −23.1589 | −26.6739 | −55.7263 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 3.29788 | −5.36262 | 2.87600 | 8.04041 | −17.6853 | 0 | −16.8983 | 1.75768 | 26.5163 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.62503 | 9.39376 | 13.3909 | −4.01508 | 43.4464 | 0 | 24.9329 | 61.2427 | −18.5699 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.27151 | 1.20818 | 19.7889 | 18.2187 | 6.36896 | 0 | 62.1452 | −25.5403 | 96.0400 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.n | 10 | |
| 7.b | odd | 2 | 1 | 539.4.a.m | 10 | ||
| 7.c | even | 3 | 2 | 77.4.e.c | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.e.c | ✓ | 20 | 7.c | even | 3 | 2 | |
| 539.4.a.m | 10 | 7.b | odd | 2 | 1 | ||
| 539.4.a.n | 10 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(539))\):
|
\( T_{2}^{10} - 67 T_{2}^{8} - T_{2}^{7} + 1529 T_{2}^{6} + 194 T_{2}^{5} - 14053 T_{2}^{4} - 4705 T_{2}^{3} + \cdots - 25480 \)
|
|
\( T_{3}^{10} - 173 T_{3}^{8} - 58 T_{3}^{7} + 9751 T_{3}^{6} + 4754 T_{3}^{5} - 200132 T_{3}^{4} + \cdots + 315541 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 67 T^{8} + \cdots - 25480 \)
|
| $3$ |
\( T^{10} - 173 T^{8} + \cdots + 315541 \)
|
| $5$ |
\( T^{10} + \cdots + 157704916 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T + 11)^{10} \)
|
| $13$ |
\( T^{10} + \cdots + 38048696721917 \)
|
| $17$ |
\( T^{10} + \cdots + 13\!\cdots\!64 \)
|
| $19$ |
\( T^{10} + \cdots - 6123382667604 \)
|
| $23$ |
\( T^{10} + \cdots + 35\!\cdots\!44 \)
|
| $29$ |
\( T^{10} + \cdots + 15\!\cdots\!25 \)
|
| $31$ |
\( T^{10} + \cdots - 35\!\cdots\!68 \)
|
| $37$ |
\( T^{10} + \cdots + 21\!\cdots\!04 \)
|
| $41$ |
\( T^{10} + \cdots + 35\!\cdots\!24 \)
|
| $43$ |
\( T^{10} + \cdots + 91\!\cdots\!20 \)
|
| $47$ |
\( T^{10} + \cdots - 88\!\cdots\!44 \)
|
| $53$ |
\( T^{10} + \cdots + 74\!\cdots\!68 \)
|
| $59$ |
\( T^{10} + \cdots + 49\!\cdots\!45 \)
|
| $61$ |
\( T^{10} + \cdots + 13\!\cdots\!48 \)
|
| $67$ |
\( T^{10} + \cdots + 67\!\cdots\!48 \)
|
| $71$ |
\( T^{10} + \cdots - 14\!\cdots\!96 \)
|
| $73$ |
\( T^{10} + \cdots + 26\!\cdots\!16 \)
|
| $79$ |
\( T^{10} + \cdots + 46\!\cdots\!05 \)
|
| $83$ |
\( T^{10} + \cdots - 30\!\cdots\!56 \)
|
| $89$ |
\( T^{10} + \cdots + 74\!\cdots\!52 \)
|
| $97$ |
\( T^{10} + \cdots + 47\!\cdots\!13 \)
|
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