Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(73,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.73");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 585.w (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(4.67124851824\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 20 x^{18} + 228 x^{16} - 1776 x^{14} + 10401 x^{12} - 46976 x^{10} + 166416 x^{8} + \cdots + 1048576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 20 x^{18} + 228 x^{16} - 1776 x^{14} + 10401 x^{12} - 46976 x^{10} + 166416 x^{8} + \cdots + 1048576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{19} + 20 \nu^{17} - 228 \nu^{15} + 1776 \nu^{13} - 10401 \nu^{11} + 46976 \nu^{9} + \cdots + 1048576 \nu ) / 262144 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{18} + 20 \nu^{16} - 228 \nu^{14} + 1776 \nu^{12} - 10401 \nu^{10} + 46976 \nu^{8} + \cdots + 1048576 ) / 65536 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1307 \nu^{18} - 103156 \nu^{16} + 1453108 \nu^{14} - 12833520 \nu^{12} + 74308869 \nu^{10} + \cdots - 4318756864 ) / 77529088 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 423 \nu^{19} + 3730 \nu^{17} - 8612 \nu^{15} - 153944 \nu^{13} + 1894553 \nu^{11} + \cdots - 271482880 \nu ) / 19382272 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3323 \nu^{18} - 68260 \nu^{16} + 752172 \nu^{14} - 5662832 \nu^{12} + 31244635 \nu^{10} + \cdots - 1587675136 ) / 38764544 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3569 \nu^{18} - 34956 \nu^{16} + 161284 \nu^{14} - 31696 \nu^{12} - 4317935 \nu^{10} + \cdots + 597786624 ) / 38764544 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1795 \nu^{18} + 29560 \nu^{16} - 292252 \nu^{14} + 1761088 \nu^{12} - 7467555 \nu^{10} + \cdots - 534233088 ) / 19382272 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 981 \nu^{19} - 12728 \nu^{17} + 120452 \nu^{15} - 828800 \nu^{13} + 4508853 \nu^{11} + \cdots - 272388096 \nu ) / 19382272 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3 \nu^{18} + 56 \nu^{16} - 604 \nu^{14} + 4416 \nu^{12} - 24099 \nu^{10} + 99324 \nu^{8} + \cdots + 901120 ) / 16384 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 15619 \nu^{18} - 253852 \nu^{16} + 2394412 \nu^{14} - 15352400 \nu^{12} + 72211683 \nu^{10} + \cdots - 191823872 ) / 77529088 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 781 \nu^{19} - 11308 \nu^{17} + 101620 \nu^{15} - 595664 \nu^{13} + 2599341 \nu^{11} + \cdots + 71827456 \nu ) / 11075584 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1445 \nu^{19} - 31807 \nu^{17} + 335008 \nu^{15} - 2431996 \nu^{13} + 12757077 \nu^{11} + \cdots - 512094208 \nu ) / 19382272 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 17663 \nu^{18} - 227196 \nu^{16} + 1769564 \nu^{14} - 8609104 \nu^{12} + 28642143 \nu^{10} + \cdots + 1584463872 ) / 77529088 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 4567 \nu^{18} - 81892 \nu^{16} + 830748 \nu^{14} - 5828592 \nu^{12} + 30309815 \nu^{10} + \cdots - 1239416832 ) / 19382272 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 27767 \nu^{19} - 569740 \nu^{17} + 6287100 \nu^{15} - 47403664 \nu^{13} + 262261463 \nu^{11} + \cdots - 14251982848 \nu ) / 310116352 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 28751 \nu^{19} - 436524 \nu^{17} + 3923548 \nu^{15} - 24879120 \nu^{13} + 120011183 \nu^{11} + \cdots - 5355077632 \nu ) / 310116352 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 34509 \nu^{19} + 503060 \nu^{17} - 4923860 \nu^{15} + 31897712 \nu^{13} + \cdots + 711262208 \nu ) / 310116352 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 17741 \nu^{19} - 331820 \nu^{17} + 3509172 \nu^{15} - 25137616 \nu^{13} + 132969197 \nu^{11} + \cdots - 4298375168 \nu ) / 155058176 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 12113 \nu^{19} + 215676 \nu^{17} - 2215684 \nu^{15} + 15495312 \nu^{13} + \cdots + 2595028992 \nu ) / 77529088 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{19} - \beta_{15} - \beta_{11} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{14} + \beta_{13} + \beta_{7} - 2\beta_{6} - 2\beta_{5} + \beta_{2} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{19} - 2\beta_{18} - 2\beta_{16} - \beta_{15} + 2\beta_{12} + \beta_{11} + 2\beta_{8} - 7\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{14} + 5\beta_{13} + 2\beta_{10} + 5\beta_{7} - 8\beta_{6} + 7\beta_{2} - 12 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9 \beta_{19} - 16 \beta_{18} - 8 \beta_{17} - 8 \beta_{16} + 7 \beta_{15} + 16 \beta_{12} + \cdots - 9 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 33 \beta_{14} + 9 \beta_{13} + 20 \beta_{10} + 2 \beta_{9} + 11 \beta_{7} - 8 \beta_{6} + 44 \beta_{5} + \cdots - 78 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 57 \beta_{19} - 30 \beta_{18} - 62 \beta_{17} + 16 \beta_{16} + 23 \beta_{15} + 48 \beta_{12} + \cdots + 37 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 99 \beta_{14} - 27 \beta_{13} + 88 \beta_{10} - 4 \beta_{9} + 3 \beta_{7} + 34 \beta_{6} + 62 \beta_{5} + \cdots - 108 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 193 \beta_{19} + 204 \beta_{18} - 156 \beta_{17} + 204 \beta_{16} - 51 \beta_{15} - 8 \beta_{12} + \cdots + 37 \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 143 \beta_{14} - 169 \beta_{13} + 96 \beta_{10} - 208 \beta_{9} - 25 \beta_{7} + 106 \beta_{6} + \cdots + 12 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 433 \beta_{19} + 1210 \beta_{18} + 296 \beta_{17} + 546 \beta_{16} - 327 \beta_{15} - 530 \beta_{12} + \cdots - 745 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 1145 \beta_{14} + 83 \beta_{13} - 1122 \beta_{10} - 1296 \beta_{9} - 165 \beta_{7} + 136 \beta_{6} + \cdots - 1876 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 135 \beta_{19} + 584 \beta_{18} + 2808 \beta_{17} + 128 \beta_{16} + 1369 \beta_{15} - 2264 \beta_{12} + \cdots - 1023 \beta_1 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 647 \beta_{14} + 2679 \beta_{13} - 7228 \beta_{10} - 1794 \beta_{9} - 3147 \beta_{7} + 1344 \beta_{6} + \cdots - 10674 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 5745 \beta_{19} - 12746 \beta_{18} + 5246 \beta_{17} - 1784 \beta_{16} + 16505 \beta_{15} + \cdots + 18419 \beta_1 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 5485 \beta_{14} - 741 \beta_{13} - 18656 \beta_{10} + 17564 \beta_{9} - 23051 \beta_{7} + \cdots + 11220 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 36241 \beta_{19} - 16852 \beta_{18} - 1004 \beta_{17} - 10572 \beta_{16} + 51139 \beta_{15} + \cdots + 93811 \beta_1 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 102089 \beta_{14} - 81623 \beta_{13} - 1424 \beta_{10} + 101288 \beta_{9} - 61239 \beta_{7} + \cdots + 276396 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 120961 \beta_{19} + 189886 \beta_{18} + 44432 \beta_{17} - 105330 \beta_{16} - 63329 \beta_{15} + \cdots + 61753 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(352\) | \(496\) |
| \(\chi(n)\) | \(1\) | \(\beta_{5}\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
−2.61646 | 0 | 4.84586 | 1.92478 | + | 1.13807i | 0 | − | 3.09894i | −7.44609 | 0 | −5.03612 | − | 2.97772i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | −2.43645 | 0 | 3.93630 | −0.945807 | − | 2.02619i | 0 | 1.25073i | −4.71771 | 0 | 2.30441 | + | 4.93672i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | −2.03024 | 0 | 2.12187 | −1.16108 | + | 1.91099i | 0 | 4.98515i | −0.247424 | 0 | 2.35728 | − | 3.87977i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | −1.33596 | 0 | −0.215211 | 1.24556 | − | 1.85704i | 0 | 0.0928416i | 2.95943 | 0 | −1.66402 | + | 2.48093i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | −1.14507 | 0 | −0.688825 | −0.0335724 | + | 2.23582i | 0 | − | 2.22978i | 3.07888 | 0 | 0.0384426 | − | 2.56016i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 1.14507 | 0 | −0.688825 | 0.0335724 | − | 2.23582i | 0 | − | 2.22978i | −3.07888 | 0 | 0.0384426 | − | 2.56016i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.7 | 1.33596 | 0 | −0.215211 | −1.24556 | + | 1.85704i | 0 | 0.0928416i | −2.95943 | 0 | −1.66402 | + | 2.48093i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.8 | 2.03024 | 0 | 2.12187 | 1.16108 | − | 1.91099i | 0 | 4.98515i | 0.247424 | 0 | 2.35728 | − | 3.87977i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.9 | 2.43645 | 0 | 3.93630 | 0.945807 | + | 2.02619i | 0 | 1.25073i | 4.71771 | 0 | 2.30441 | + | 4.93672i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 73.10 | 2.61646 | 0 | 4.84586 | −1.92478 | − | 1.13807i | 0 | − | 3.09894i | 7.44609 | 0 | −5.03612 | − | 2.97772i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.1 | −2.61646 | 0 | 4.84586 | 1.92478 | − | 1.13807i | 0 | 3.09894i | −7.44609 | 0 | −5.03612 | + | 2.97772i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.2 | −2.43645 | 0 | 3.93630 | −0.945807 | + | 2.02619i | 0 | − | 1.25073i | −4.71771 | 0 | 2.30441 | − | 4.93672i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.3 | −2.03024 | 0 | 2.12187 | −1.16108 | − | 1.91099i | 0 | − | 4.98515i | −0.247424 | 0 | 2.35728 | + | 3.87977i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.4 | −1.33596 | 0 | −0.215211 | 1.24556 | + | 1.85704i | 0 | − | 0.0928416i | 2.95943 | 0 | −1.66402 | − | 2.48093i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.5 | −1.14507 | 0 | −0.688825 | −0.0335724 | − | 2.23582i | 0 | 2.22978i | 3.07888 | 0 | 0.0384426 | + | 2.56016i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.6 | 1.14507 | 0 | −0.688825 | 0.0335724 | + | 2.23582i | 0 | 2.22978i | −3.07888 | 0 | 0.0384426 | + | 2.56016i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.7 | 1.33596 | 0 | −0.215211 | −1.24556 | − | 1.85704i | 0 | − | 0.0928416i | −2.95943 | 0 | −1.66402 | − | 2.48093i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.8 | 2.03024 | 0 | 2.12187 | 1.16108 | + | 1.91099i | 0 | − | 4.98515i | 0.247424 | 0 | 2.35728 | + | 3.87977i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.9 | 2.43645 | 0 | 3.93630 | 0.945807 | − | 2.02619i | 0 | − | 1.25073i | 4.71771 | 0 | 2.30441 | − | 4.93672i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 577.10 | 2.61646 | 0 | 4.84586 | −1.92478 | + | 1.13807i | 0 | 3.09894i | 7.44609 | 0 | −5.03612 | + | 2.97772i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 65.k | even | 4 | 1 | inner |
| 195.j | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 585.2.w.f | yes | 20 |
| 3.b | odd | 2 | 1 | inner | 585.2.w.f | yes | 20 |
| 5.c | odd | 4 | 1 | 585.2.n.f | ✓ | 20 | |
| 13.d | odd | 4 | 1 | 585.2.n.f | ✓ | 20 | |
| 15.e | even | 4 | 1 | 585.2.n.f | ✓ | 20 | |
| 39.f | even | 4 | 1 | 585.2.n.f | ✓ | 20 | |
| 65.k | even | 4 | 1 | inner | 585.2.w.f | yes | 20 |
| 195.j | odd | 4 | 1 | inner | 585.2.w.f | yes | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 585.2.n.f | ✓ | 20 | 5.c | odd | 4 | 1 | |
| 585.2.n.f | ✓ | 20 | 13.d | odd | 4 | 1 | |
| 585.2.n.f | ✓ | 20 | 15.e | even | 4 | 1 | |
| 585.2.n.f | ✓ | 20 | 39.f | even | 4 | 1 | |
| 585.2.w.f | yes | 20 | 1.a | even | 1 | 1 | trivial |
| 585.2.w.f | yes | 20 | 3.b | odd | 2 | 1 | inner |
| 585.2.w.f | yes | 20 | 65.k | even | 4 | 1 | inner |
| 585.2.w.f | yes | 20 | 195.j | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):
|
\( T_{2}^{10} - 20T_{2}^{8} + 148T_{2}^{6} - 496T_{2}^{4} + 737T_{2}^{2} - 392 \)
|
|
\( T_{7}^{10} + 41T_{7}^{8} + 472T_{7}^{6} + 1832T_{7}^{4} + 1872T_{7}^{2} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{10} - 20 T^{8} + \cdots - 392)^{2} \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + 20 T^{18} + \cdots + 9765625 \)
|
| $7$ |
\( (T^{10} + 41 T^{8} + \cdots + 16)^{2} \)
|
| $11$ |
\( T^{20} + \cdots + 623201296 \)
|
| $13$ |
\( (T^{10} + 8 T^{9} + \cdots + 371293)^{2} \)
|
| $17$ |
\( T^{20} + 1793 T^{16} + \cdots + 65536 \)
|
| $19$ |
\( (T^{10} - 144 T^{7} + \cdots + 739328)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 116985856 \)
|
| $29$ |
\( (T^{10} + 130 T^{8} + \cdots + 61952)^{2} \)
|
| $31$ |
\( (T^{10} + 2 T^{9} + \cdots + 59168)^{2} \)
|
| $37$ |
\( (T^{10} + 225 T^{8} + \cdots + 9560464)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 3863599084816 \)
|
| $43$ |
\( (T^{10} - 10 T^{9} + \cdots + 3987488)^{2} \)
|
| $47$ |
\( (T^{10} + 286 T^{8} + \cdots + 242968968)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 19987173376 \)
|
| $59$ |
\( T^{20} + \cdots + 11\!\cdots\!56 \)
|
| $61$ |
\( (T^{5} + 3 T^{4} + \cdots + 11804)^{4} \)
|
| $67$ |
\( (T^{5} + 18 T^{4} + \cdots - 50664)^{4} \)
|
| $71$ |
\( T^{20} + \cdots + 221533456 \)
|
| $73$ |
\( (T^{5} + 20 T^{4} + \cdots + 1824)^{4} \)
|
| $79$ |
\( (T^{10} + 241 T^{8} + \cdots + 6411024)^{2} \)
|
| $83$ |
\( (T^{10} + 284 T^{8} + \cdots + 557568)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 21\!\cdots\!56 \)
|
| $97$ |
\( (T^{5} + 21 T^{4} + \cdots + 7812)^{4} \)
|
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