Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [501,4,Mod(1,501)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(501, 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("501.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 501 = 3 \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 501.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: | \(29.5599569129\) |
| Analytic rank: | \(1\) |
| Dimension: | \(19\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{19} - x^{18} - 105 x^{17} + 101 x^{16} + 4534 x^{15} - 4163 x^{14} - 103845 x^{13} + 89794 x^{12} + \cdots - 362016 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{18}\) 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
\( x^{19} - x^{18} - 105 x^{17} + 101 x^{16} + 4534 x^{15} - 4163 x^{14} - 103845 x^{13} + 89794 x^{12} + \cdots - 362016 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 19\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 29\!\cdots\!59 \nu^{18} + \cdots + 65\!\cdots\!56 ) / 54\!\cdots\!48 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 29\!\cdots\!63 \nu^{18} + \cdots + 89\!\cdots\!28 ) / 27\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 15\!\cdots\!93 \nu^{18} + \cdots + 32\!\cdots\!52 ) / 10\!\cdots\!96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\!\cdots\!89 \nu^{18} + \cdots - 57\!\cdots\!12 ) / 36\!\cdots\!32 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 17\!\cdots\!69 \nu^{18} + \cdots + 19\!\cdots\!12 ) / 54\!\cdots\!48 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 44\!\cdots\!03 \nu^{18} + \cdots - 24\!\cdots\!04 ) / 10\!\cdots\!96 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 25\!\cdots\!93 \nu^{18} + \cdots - 35\!\cdots\!60 ) / 54\!\cdots\!48 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 76\!\cdots\!69 \nu^{18} + \cdots + 71\!\cdots\!80 ) / 98\!\cdots\!36 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 90\!\cdots\!09 \nu^{18} + \cdots + 15\!\cdots\!16 ) / 10\!\cdots\!96 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 10\!\cdots\!23 \nu^{18} + \cdots - 13\!\cdots\!44 ) / 12\!\cdots\!92 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 81\!\cdots\!49 \nu^{18} + \cdots + 11\!\cdots\!04 ) / 90\!\cdots\!08 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 33\!\cdots\!03 \nu^{18} + \cdots - 39\!\cdots\!28 ) / 36\!\cdots\!32 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 98\!\cdots\!19 \nu^{18} + \cdots - 13\!\cdots\!24 ) / 98\!\cdots\!36 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 29\!\cdots\!59 \nu^{18} + \cdots - 48\!\cdots\!44 ) / 27\!\cdots\!24 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 15\!\cdots\!73 \nu^{18} + \cdots - 16\!\cdots\!96 ) / 10\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 19\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{18} + 2 \beta_{17} + \beta_{14} - 3 \beta_{13} + \beta_{12} - \beta_{10} + \beta_{6} + \cdots + 205 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{18} + \beta_{17} - 2 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{11} + \cdots + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 66 \beta_{18} + 79 \beta_{17} - 8 \beta_{16} + 6 \beta_{15} + 26 \beta_{14} - 109 \beta_{13} + \cdots + 4287 \)
|
| \(\nu^{7}\) | \(=\) |
\( 119 \beta_{18} + 48 \beta_{17} - 93 \beta_{16} - 86 \beta_{15} - 85 \beta_{14} - 67 \beta_{13} + \cdots + 267 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1738 \beta_{18} + 2356 \beta_{17} - 436 \beta_{16} + 294 \beta_{15} + 545 \beta_{14} - 3031 \beta_{13} + \cdots + 94645 \)
|
| \(\nu^{9}\) | \(=\) |
\( 3493 \beta_{18} + 1723 \beta_{17} - 3154 \beta_{16} - 2740 \beta_{15} - 2629 \beta_{14} - 1655 \beta_{13} + \cdots + 11444 \)
|
| \(\nu^{10}\) | \(=\) |
\( 42785 \beta_{18} + 63928 \beta_{17} - 16536 \beta_{16} + 10044 \beta_{15} + 10388 \beta_{14} + \cdots + 2150203 \)
|
| \(\nu^{11}\) | \(=\) |
\( 92567 \beta_{18} + 55317 \beta_{17} - 95473 \beta_{16} - 77802 \beta_{15} - 71670 \beta_{14} + \cdots + 401715 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1026948 \beta_{18} + 1665629 \beta_{17} - 540041 \beta_{16} + 296962 \beta_{15} + 183242 \beta_{14} + \cdots + 49645316 \)
|
| \(\nu^{13}\) | \(=\) |
\( 2349117 \beta_{18} + 1669162 \beta_{17} - 2737046 \beta_{16} - 2086946 \beta_{15} - 1827602 \beta_{14} + \cdots + 12657152 \)
|
| \(\nu^{14}\) | \(=\) |
\( 24401033 \beta_{18} + 42541311 \beta_{17} - 16280353 \beta_{16} + 8150226 \beta_{15} + 2946758 \beta_{14} + \cdots + 1157468502 \)
|
| \(\nu^{15}\) | \(=\) |
\( 58391698 \beta_{18} + 48354605 \beta_{17} - 76061601 \beta_{16} - 54228426 \beta_{15} - 44728653 \beta_{14} + \cdots + 374428655 \)
|
| \(\nu^{16}\) | \(=\) |
\( 577521988 \beta_{18} + 1074806257 \beta_{17} - 467395062 \beta_{16} + 214117680 \beta_{15} + \cdots + 27158401871 \)
|
| \(\nu^{17}\) | \(=\) |
\( 1435563666 \beta_{18} + 1360781883 \beta_{17} - 2071405499 \beta_{16} - 1382302238 \beta_{15} + \cdots + 10653147827 \)
|
| \(\nu^{18}\) | \(=\) |
\( 13651975105 \beta_{18} + 26979344418 \beta_{17} - 12991265458 \beta_{16} + 5469541150 \beta_{15} + \cdots + 640132568253 \)
|
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 |
|
−4.88200 | −3.00000 | 15.8340 | 9.32800 | 14.6460 | 18.0504 | −38.2455 | 9.00000 | −45.5393 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.85732 | −3.00000 | 15.5936 | −10.9708 | 14.5720 | −29.2321 | −36.8846 | 9.00000 | 53.2888 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.59636 | −3.00000 | 13.1265 | 7.34744 | 13.7891 | −0.0109308 | −23.5634 | 9.00000 | −33.7715 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.73234 | −3.00000 | 5.93034 | −2.13205 | 11.1970 | 18.9972 | 7.72465 | 9.00000 | 7.95752 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.33897 | −3.00000 | 3.14873 | 19.7968 | 10.0169 | 9.05992 | 16.1983 | 9.00000 | −66.1008 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.23872 | −3.00000 | −2.98813 | −21.9733 | 6.71616 | −7.56447 | 24.5994 | 9.00000 | 49.1922 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.19683 | −3.00000 | −3.17393 | 2.22130 | 6.59049 | −25.7563 | 24.5472 | 9.00000 | −4.87982 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.363523 | −3.00000 | −7.86785 | −12.1344 | 1.09057 | −1.08901 | 5.76833 | 9.00000 | 4.41115 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.289740 | −3.00000 | −7.91605 | 17.2255 | 0.869221 | 5.18201 | 4.61152 | 9.00000 | −4.99092 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.188221 | −3.00000 | −7.96457 | −6.05764 | 0.564663 | −0.996265 | 3.00487 | 9.00000 | 1.14018 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.320380 | −3.00000 | −7.89736 | 18.2915 | −0.961139 | −32.1410 | −5.09319 | 9.00000 | 5.86022 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.13862 | −3.00000 | −6.70355 | −7.78898 | −3.41585 | 30.6595 | −16.7417 | 9.00000 | −8.86867 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.85736 | −3.00000 | −4.55020 | −1.16451 | −5.57209 | 10.4266 | −23.3103 | 9.00000 | −2.16293 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 3.25191 | −3.00000 | 2.57491 | 15.1761 | −9.75573 | −5.80377 | −17.6419 | 9.00000 | 49.3514 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 3.27141 | −3.00000 | 2.70211 | 2.49339 | −9.81422 | −3.67685 | −17.3316 | 9.00000 | 8.15690 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 3.41127 | −3.00000 | 3.63673 | −0.160333 | −10.2338 | 23.8286 | −14.8843 | 9.00000 | −0.546937 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 4.60242 | −3.00000 | 13.1822 | −16.1321 | −13.8073 | 11.3925 | 23.8509 | 9.00000 | −74.2468 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 4.83996 | −3.00000 | 15.4252 | 2.88297 | −14.5199 | −33.2643 | 35.9378 | 9.00000 | 13.9534 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 4.99072 | −3.00000 | 16.9072 | −4.24871 | −14.9721 | −16.0617 | 44.4535 | 9.00000 | −21.2041 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(167\) | \( -1 \) |
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 | 501.4.a.b | ✓ | 19 |
| 3.b | odd | 2 | 1 | 1503.4.a.b | 19 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 501.4.a.b | ✓ | 19 | 1.a | even | 1 | 1 | trivial |
| 1503.4.a.b | 19 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{19} - T_{2}^{18} - 105 T_{2}^{17} + 101 T_{2}^{16} + 4534 T_{2}^{15} - 4163 T_{2}^{14} + \cdots - 362016 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(501))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{19} - T^{18} + \cdots - 362016 \)
|
| $3$ |
\( (T + 3)^{19} \)
|
| $5$ |
\( T^{19} + \cdots - 390111740877168 \)
|
| $7$ |
\( T^{19} + \cdots + 34\!\cdots\!76 \)
|
| $11$ |
\( T^{19} + \cdots - 43\!\cdots\!92 \)
|
| $13$ |
\( T^{19} + \cdots - 19\!\cdots\!84 \)
|
| $17$ |
\( T^{19} + \cdots + 63\!\cdots\!56 \)
|
| $19$ |
\( T^{19} + \cdots + 18\!\cdots\!20 \)
|
| $23$ |
\( T^{19} + \cdots + 13\!\cdots\!16 \)
|
| $29$ |
\( T^{19} + \cdots + 34\!\cdots\!20 \)
|
| $31$ |
\( T^{19} + \cdots - 54\!\cdots\!20 \)
|
| $37$ |
\( T^{19} + \cdots + 41\!\cdots\!04 \)
|
| $41$ |
\( T^{19} + \cdots - 14\!\cdots\!60 \)
|
| $43$ |
\( T^{19} + \cdots + 39\!\cdots\!96 \)
|
| $47$ |
\( T^{19} + \cdots - 32\!\cdots\!16 \)
|
| $53$ |
\( T^{19} + \cdots - 14\!\cdots\!80 \)
|
| $59$ |
\( T^{19} + \cdots - 11\!\cdots\!64 \)
|
| $61$ |
\( T^{19} + \cdots + 44\!\cdots\!72 \)
|
| $67$ |
\( T^{19} + \cdots + 14\!\cdots\!92 \)
|
| $71$ |
\( T^{19} + \cdots + 20\!\cdots\!40 \)
|
| $73$ |
\( T^{19} + \cdots - 54\!\cdots\!88 \)
|
| $79$ |
\( T^{19} + \cdots - 30\!\cdots\!24 \)
|
| $83$ |
\( T^{19} + \cdots - 71\!\cdots\!68 \)
|
| $89$ |
\( T^{19} + \cdots + 19\!\cdots\!76 \)
|
| $97$ |
\( T^{19} + \cdots - 54\!\cdots\!56 \)
|
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