Properties

Label 252.9.z.e
Level $252$
Weight $9$
Character orbit 252.z
Analytic conductor $102.659$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.z (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} - 516815190850700874 x^{14} + 3617886719518590784 x^{13} + 122992744079100130844521 x^{12} - 738003497470953185965030 x^{11} - 17716220051138913180459215577 x^{10} + 88587865373982244785457265884 x^{9} + 1507689855516745584812805692686751 x^{8} - 6031290957377412523625955897396396 x^{7} - 69448792452425943361067315312590506770 x^{6} + 208367487247705803791441930629487279968 x^{5} + 1331037342615679684155036722672299751825731 x^{4} - 2662421971413455508046552715438433455340210 x^{3} - 47007135303470257351659397734299414285784783 x^{2} + 48338381018096768126520835100261119832719644 x + 515596507492955532983572496163412311048918663\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{26}\cdot 7^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{4} ) q^{5} + ( -577 - 430 \beta_{3} + \beta_{7} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{4} ) q^{5} + ( -577 - 430 \beta_{3} + \beta_{7} ) q^{7} + ( -2 \beta_{1} - \beta_{4} - \beta_{11} ) q^{11} + ( -875 - 1750 \beta_{3} - 4 \beta_{5} - 3 \beta_{6} + \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 4 \beta_{12} - \beta_{14} - \beta_{15} ) q^{13} + ( 9 \beta_{4} + 2 \beta_{10} + \beta_{11} - \beta_{17} ) q^{17} + ( -3565 + 3577 \beta_{3} + 15 \beta_{5} + 19 \beta_{6} - 19 \beta_{7} + 2 \beta_{9} + 4 \beta_{12} - 10 \beta_{14} ) q^{19} + ( -41 \beta_{1} + 41 \beta_{4} - \beta_{10} - \beta_{13} ) q^{23} + ( 23 - 134842 \beta_{3} - 37 \beta_{5} - 37 \beta_{6} + 6 \beta_{7} - \beta_{8} - 2 \beta_{9} - 6 \beta_{12} - 48 \beta_{14} - 24 \beta_{15} ) q^{25} + ( -29 \beta_{1} + \beta_{2} - 58 \beta_{4} - 9 \beta_{10} - 9 \beta_{11} + 2 \beta_{17} + \beta_{18} - \beta_{19} ) q^{29} + ( -315576 - 157781 \beta_{3} + \beta_{5} - \beta_{6} - 138 \beta_{7} - 9 \beta_{8} - 138 \beta_{12} - 23 \beta_{15} ) q^{31} + ( 154 \beta_{1} + \beta_{2} + 356 \beta_{4} + 9 \beta_{10} + 20 \beta_{11} - \beta_{13} + 3 \beta_{16} - 8 \beta_{17} + \beta_{18} ) q^{35} + ( 680173 + 680168 \beta_{3} + 126 \beta_{5} + 189 \beta_{6} + 189 \beta_{7} + 110 \beta_{8} + 55 \beta_{9} + 315 \beta_{12} + 60 \beta_{14} + 120 \beta_{15} ) q^{37} + ( -733 \beta_{1} + 10 \beta_{2} + 30 \beta_{10} - 30 \beta_{11} - 8 \beta_{13} - 4 \beta_{16} - \beta_{18} - \beta_{19} ) q^{41} + ( -29966 - 226 \beta_{5} - 354 \beta_{6} + 580 \beta_{7} - 97 \beta_{8} + 97 \beta_{9} + 226 \beta_{12} + 95 \beta_{14} - 95 \beta_{15} ) q^{43} + ( 14 \beta_{1} - 47 \beta_{2} + 14 \beta_{4} - 11 \beta_{10} - 22 \beta_{11} - 3 \beta_{13} - 6 \beta_{16} - 48 \beta_{17} - \beta_{18} + 2 \beta_{19} ) q^{47} + ( -727895 + 469448 \beta_{3} - 91 \beta_{5} + 98 \beta_{6} - 392 \beta_{7} - 343 \beta_{9} - 392 \beta_{12} + 686 \beta_{14} + 343 \beta_{15} ) q^{49} + ( 312 \beta_{1} + 69 \beta_{2} + 156 \beta_{4} - 144 \beta_{11} + 20 \beta_{13} + 20 \beta_{16} + 34 \beta_{17} + \beta_{19} ) q^{53} + ( -403043 - 806086 \beta_{3} + 1421 \beta_{5} + 648 \beta_{6} - 773 \beta_{7} - 920 \beta_{8} - 920 \beta_{9} + 1421 \beta_{12} - 156 \beta_{14} - 156 \beta_{15} ) q^{55} + ( -\beta_{2} - 3137 \beta_{4} + 368 \beta_{10} + 184 \beta_{11} + 17 \beta_{13} - 17 \beta_{16} + 44 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{59} + ( 1085615 - 1085135 \beta_{3} - 2327 \beta_{5} - 1389 \beta_{6} + 1389 \beta_{7} + 187 \beta_{9} + 938 \beta_{12} - 293 \beta_{14} ) q^{61} + ( 1630 \beta_{1} - 126 \beta_{2} - 1630 \beta_{4} - 1209 \beta_{10} + 8 \beta_{13} + 125 \beta_{17} - \beta_{18} ) q^{65} + ( 1548 + 2601397 \beta_{3} + 2490 \beta_{5} + 2490 \beta_{6} - 3025 \beta_{7} + 864 \beta_{8} + 1728 \beta_{9} + 3025 \beta_{12} - 1368 \beta_{14} - 684 \beta_{15} ) q^{67} + ( 1826 \beta_{1} + 76 \beta_{2} + 3652 \beta_{4} + 343 \beta_{10} + 343 \beta_{11} - 13 \beta_{16} + 152 \beta_{17} - 15 \beta_{18} + 15 \beta_{19} ) q^{71} + ( -3526803 - 1763581 \beta_{3} - 1413 \beta_{5} + 1413 \beta_{6} + 585 \beta_{7} + 1088 \beta_{8} + 585 \beta_{12} + 1447 \beta_{15} ) q^{73} + ( 9976 \beta_{1} + 42 \beta_{2} + 4797 \beta_{4} + 807 \beta_{10} + 1896 \beta_{11} + 12 \beta_{13} - 215 \beta_{17} - 16 \beta_{18} + \beta_{19} ) q^{77} + ( -12993035 - 12988829 \beta_{3} + 898 \beta_{5} - 2141 \beta_{6} - 2141 \beta_{7} + 4810 \beta_{8} + 2405 \beta_{9} - 1243 \beta_{12} - 1801 \beta_{14} - 3602 \beta_{15} ) q^{79} + ( -598 \beta_{1} + 56 \beta_{2} + 660 \beta_{10} - 660 \beta_{11} + 58 \beta_{13} + 29 \beta_{16} + 16 \beta_{18} + 16 \beta_{19} ) q^{83} + ( -4189291 - 11125 \beta_{5} - 5009 \beta_{6} + 16134 \beta_{7} - 4155 \beta_{8} + 4155 \beta_{9} + 11125 \beta_{12} - 779 \beta_{14} + 779 \beta_{15} ) q^{85} + ( 4396 \beta_{1} - 318 \beta_{2} + 4396 \beta_{4} - 2198 \beta_{10} - 4396 \beta_{11} + 20 \beta_{13} + 40 \beta_{16} - 302 \beta_{17} + 16 \beta_{18} - 32 \beta_{19} ) q^{89} + ( -1686010 - 7592602 \beta_{3} + 2046 \beta_{5} + 6370 \beta_{6} - 3788 \beta_{7} + 980 \beta_{8} - 7007 \beta_{9} + 16660 \beta_{12} - 1813 \beta_{14} - 1176 \beta_{15} ) q^{91} + ( -61828 \beta_{1} + 255 \beta_{2} - 30914 \beta_{4} - 1657 \beta_{11} - 187 \beta_{13} - 187 \beta_{16} + 136 \beta_{17} - 17 \beta_{19} ) q^{95} + ( 5988500 + 11977000 \beta_{3} + 32318 \beta_{5} + 14311 \beta_{6} - 18007 \beta_{7} - 10506 \beta_{8} - 10506 \beta_{9} + 32318 \beta_{12} + 6653 \beta_{14} + 6653 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 7238 q^{7} + O(q^{10}) \) \( 20 q - 7238 q^{7} - 107256 q^{19} + 1348832 q^{25} - 4734426 q^{31} + 6802748 q^{37} - 596128 q^{43} - 19249258 q^{49} + 32573772 q^{61} - 25999304 q^{67} - 52899612 q^{73} - 129945530 q^{79} - 83755728 q^{85} + 42306852 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 10 x^{19} - 1751639 x^{18} + 15765036 x^{17} + 1288400424989 x^{16} - 10307560742020 x^{15} - 516815190850700874 x^{14} + 3617886719518590784 x^{13} + 122992744079100130844521 x^{12} - 738003497470953185965030 x^{11} - 17716220051138913180459215577 x^{10} + 88587865373982244785457265884 x^{9} + 1507689855516745584812805692686751 x^{8} - 6031290957377412523625955897396396 x^{7} - 69448792452425943361067315312590506770 x^{6} + 208367487247705803791441930629487279968 x^{5} + 1331037342615679684155036722672299751825731 x^{4} - 2662421971413455508046552715438433455340210 x^{3} - 47007135303470257351659397734299414285784783 x^{2} + 48338381018096768126520835100261119832719644 x + 515596507492955532983572496163412311048918663\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(12\!\cdots\!75\)\( \nu^{18} + \)\(10\!\cdots\!75\)\( \nu^{17} + \)\(19\!\cdots\!47\)\( \nu^{16} - \)\(15\!\cdots\!76\)\( \nu^{15} - \)\(12\!\cdots\!54\)\( \nu^{14} + \)\(90\!\cdots\!58\)\( \nu^{13} + \)\(45\!\cdots\!02\)\( \nu^{12} - \)\(27\!\cdots\!24\)\( \nu^{11} - \)\(91\!\cdots\!73\)\( \nu^{10} + \)\(45\!\cdots\!49\)\( \nu^{9} + \)\(10\!\cdots\!57\)\( \nu^{8} - \)\(41\!\cdots\!72\)\( \nu^{7} - \)\(62\!\cdots\!42\)\( \nu^{6} + \)\(18\!\cdots\!46\)\( \nu^{5} + \)\(15\!\cdots\!86\)\( \nu^{4} - \)\(30\!\cdots\!72\)\( \nu^{3} - \)\(28\!\cdots\!83\)\( \nu^{2} + \)\(28\!\cdots\!51\)\( \nu + \)\(51\!\cdots\!39\)\(\)\()/ \)\(27\!\cdots\!72\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(14\!\cdots\!49\)\( \nu^{18} - \)\(13\!\cdots\!41\)\( \nu^{17} - \)\(26\!\cdots\!01\)\( \nu^{16} + \)\(20\!\cdots\!04\)\( \nu^{15} + \)\(19\!\cdots\!66\)\( \nu^{14} - \)\(13\!\cdots\!18\)\( \nu^{13} - \)\(76\!\cdots\!10\)\( \nu^{12} + \)\(46\!\cdots\!24\)\( \nu^{11} + \)\(18\!\cdots\!63\)\( \nu^{10} - \)\(91\!\cdots\!31\)\( \nu^{9} - \)\(26\!\cdots\!67\)\( \nu^{8} + \)\(10\!\cdots\!80\)\( \nu^{7} + \)\(22\!\cdots\!46\)\( \nu^{6} - \)\(66\!\cdots\!38\)\( \nu^{5} - \)\(10\!\cdots\!14\)\( \nu^{4} + \)\(20\!\cdots\!56\)\( \nu^{3} + \)\(18\!\cdots\!89\)\( \nu^{2} - \)\(18\!\cdots\!57\)\( \nu - \)\(34\!\cdots\!13\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(41\!\cdots\!46\)\( \nu^{19} - \)\(39\!\cdots\!37\)\( \nu^{18} - \)\(72\!\cdots\!75\)\( \nu^{17} + \)\(62\!\cdots\!31\)\( \nu^{16} + \)\(53\!\cdots\!12\)\( \nu^{15} - \)\(40\!\cdots\!34\)\( \nu^{14} - \)\(21\!\cdots\!26\)\( \nu^{13} + \)\(13\!\cdots\!86\)\( \nu^{12} + \)\(51\!\cdots\!74\)\( \nu^{11} - \)\(28\!\cdots\!23\)\( \nu^{10} - \)\(73\!\cdots\!01\)\( \nu^{9} + \)\(33\!\cdots\!81\)\( \nu^{8} + \)\(62\!\cdots\!64\)\( \nu^{7} - \)\(21\!\cdots\!74\)\( \nu^{6} - \)\(28\!\cdots\!22\)\( \nu^{5} + \)\(71\!\cdots\!62\)\( \nu^{4} + \)\(54\!\cdots\!02\)\( \nu^{3} - \)\(82\!\cdots\!49\)\( \nu^{2} - \)\(88\!\cdots\!15\)\( \nu - \)\(63\!\cdots\!61\)\(\)\()/ \)\(21\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(20\!\cdots\!23\)\( \nu^{19} + \)\(16\!\cdots\!31\)\( \nu^{18} + \)\(36\!\cdots\!75\)\( \nu^{17} - \)\(25\!\cdots\!48\)\( \nu^{16} - \)\(26\!\cdots\!46\)\( \nu^{15} + \)\(17\!\cdots\!82\)\( \nu^{14} + \)\(10\!\cdots\!58\)\( \nu^{13} - \)\(60\!\cdots\!88\)\( \nu^{12} - \)\(25\!\cdots\!97\)\( \nu^{11} + \)\(12\!\cdots\!29\)\( \nu^{10} + \)\(36\!\cdots\!73\)\( \nu^{9} - \)\(13\!\cdots\!48\)\( \nu^{8} - \)\(30\!\cdots\!62\)\( \nu^{7} + \)\(82\!\cdots\!82\)\( \nu^{6} + \)\(14\!\cdots\!26\)\( \nu^{5} - \)\(20\!\cdots\!16\)\( \nu^{4} - \)\(27\!\cdots\!31\)\( \nu^{3} + \)\(41\!\cdots\!67\)\( \nu^{2} + \)\(11\!\cdots\!95\)\( \nu - \)\(74\!\cdots\!32\)\(\)\()/ \)\(72\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(43\!\cdots\!25\)\( \nu^{19} + \)\(41\!\cdots\!08\)\( \nu^{18} + \)\(76\!\cdots\!38\)\( \nu^{17} - \)\(65\!\cdots\!59\)\( \nu^{16} - \)\(56\!\cdots\!66\)\( \nu^{15} + \)\(42\!\cdots\!28\)\( \nu^{14} + \)\(22\!\cdots\!72\)\( \nu^{13} - \)\(14\!\cdots\!26\)\( \nu^{12} - \)\(53\!\cdots\!67\)\( \nu^{11} + \)\(29\!\cdots\!92\)\( \nu^{10} + \)\(77\!\cdots\!02\)\( \nu^{9} - \)\(34\!\cdots\!65\)\( \nu^{8} - \)\(65\!\cdots\!22\)\( \nu^{7} + \)\(23\!\cdots\!72\)\( \nu^{6} + \)\(30\!\cdots\!76\)\( \nu^{5} - \)\(75\!\cdots\!74\)\( \nu^{4} - \)\(58\!\cdots\!97\)\( \nu^{3} + \)\(87\!\cdots\!72\)\( \nu^{2} + \)\(94\!\cdots\!46\)\( \nu - \)\(74\!\cdots\!27\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(18\!\cdots\!80\)\( \nu^{19} - \)\(17\!\cdots\!89\)\( \nu^{18} - \)\(31\!\cdots\!47\)\( \nu^{17} + \)\(27\!\cdots\!85\)\( \nu^{16} + \)\(23\!\cdots\!60\)\( \nu^{15} - \)\(17\!\cdots\!34\)\( \nu^{14} - \)\(94\!\cdots\!26\)\( \nu^{13} + \)\(61\!\cdots\!30\)\( \nu^{12} + \)\(22\!\cdots\!80\)\( \nu^{11} - \)\(12\!\cdots\!47\)\( \nu^{10} - \)\(32\!\cdots\!53\)\( \nu^{9} + \)\(14\!\cdots\!19\)\( \nu^{8} + \)\(27\!\cdots\!80\)\( \nu^{7} - \)\(96\!\cdots\!18\)\( \nu^{6} - \)\(12\!\cdots\!66\)\( \nu^{5} + \)\(31\!\cdots\!58\)\( \nu^{4} + \)\(24\!\cdots\!60\)\( \nu^{3} - \)\(36\!\cdots\!89\)\( \nu^{2} - \)\(39\!\cdots\!67\)\( \nu + \)\(10\!\cdots\!73\)\(\)\()/ \)\(51\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(18\!\cdots\!80\)\( \nu^{19} - \)\(17\!\cdots\!31\)\( \nu^{18} - \)\(31\!\cdots\!69\)\( \nu^{17} + \)\(26\!\cdots\!11\)\( \nu^{16} + \)\(23\!\cdots\!84\)\( \nu^{15} - \)\(17\!\cdots\!06\)\( \nu^{14} - \)\(94\!\cdots\!54\)\( \nu^{13} + \)\(60\!\cdots\!94\)\( \nu^{12} + \)\(22\!\cdots\!68\)\( \nu^{11} - \)\(12\!\cdots\!25\)\( \nu^{10} - \)\(32\!\cdots\!91\)\( \nu^{9} + \)\(14\!\cdots\!09\)\( \nu^{8} + \)\(27\!\cdots\!88\)\( \nu^{7} - \)\(96\!\cdots\!86\)\( \nu^{6} - \)\(12\!\cdots\!50\)\( \nu^{5} + \)\(31\!\cdots\!54\)\( \nu^{4} + \)\(24\!\cdots\!68\)\( \nu^{3} - \)\(36\!\cdots\!27\)\( \nu^{2} - \)\(39\!\cdots\!01\)\( \nu - \)\(66\!\cdots\!89\)\(\)\()/ \)\(51\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(19\!\cdots\!15\)\( \nu^{19} + \)\(19\!\cdots\!61\)\( \nu^{18} + \)\(34\!\cdots\!85\)\( \nu^{17} - \)\(30\!\cdots\!02\)\( \nu^{16} - \)\(25\!\cdots\!58\)\( \nu^{15} + \)\(19\!\cdots\!06\)\( \nu^{14} + \)\(10\!\cdots\!74\)\( \nu^{13} - \)\(69\!\cdots\!48\)\( \nu^{12} - \)\(24\!\cdots\!01\)\( \nu^{11} + \)\(13\!\cdots\!67\)\( \nu^{10} + \)\(34\!\cdots\!35\)\( \nu^{9} - \)\(16\!\cdots\!82\)\( \nu^{8} - \)\(29\!\cdots\!66\)\( \nu^{7} + \)\(10\!\cdots\!90\)\( \nu^{6} + \)\(13\!\cdots\!26\)\( \nu^{5} - \)\(35\!\cdots\!76\)\( \nu^{4} - \)\(26\!\cdots\!31\)\( \nu^{3} + \)\(39\!\cdots\!73\)\( \nu^{2} + \)\(42\!\cdots\!09\)\( \nu - \)\(17\!\cdots\!70\)\(\)\()/ \)\(51\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(19\!\cdots\!15\)\( \nu^{19} + \)\(18\!\cdots\!24\)\( \nu^{18} + \)\(34\!\cdots\!18\)\( \nu^{17} - \)\(28\!\cdots\!41\)\( \nu^{16} - \)\(25\!\cdots\!94\)\( \nu^{15} + \)\(18\!\cdots\!64\)\( \nu^{14} + \)\(10\!\cdots\!16\)\( \nu^{13} - \)\(63\!\cdots\!94\)\( \nu^{12} - \)\(24\!\cdots\!33\)\( \nu^{11} + \)\(12\!\cdots\!84\)\( \nu^{10} + \)\(34\!\cdots\!42\)\( \nu^{9} - \)\(15\!\cdots\!67\)\( \nu^{8} - \)\(29\!\cdots\!78\)\( \nu^{7} + \)\(10\!\cdots\!92\)\( \nu^{6} + \)\(13\!\cdots\!52\)\( \nu^{5} - \)\(33\!\cdots\!70\)\( \nu^{4} - \)\(26\!\cdots\!43\)\( \nu^{3} + \)\(39\!\cdots\!80\)\( \nu^{2} + \)\(42\!\cdots\!10\)\( \nu + \)\(13\!\cdots\!23\)\(\)\()/ \)\(51\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(13\!\cdots\!09\)\( \nu^{19} - \)\(82\!\cdots\!74\)\( \nu^{18} - \)\(24\!\cdots\!04\)\( \nu^{17} + \)\(14\!\cdots\!21\)\( \nu^{16} + \)\(17\!\cdots\!78\)\( \nu^{15} - \)\(10\!\cdots\!00\)\( \nu^{14} - \)\(71\!\cdots\!84\)\( \nu^{13} + \)\(39\!\cdots\!54\)\( \nu^{12} + \)\(16\!\cdots\!15\)\( \nu^{11} - \)\(89\!\cdots\!62\)\( \nu^{10} - \)\(24\!\cdots\!80\)\( \nu^{9} + \)\(12\!\cdots\!99\)\( \nu^{8} + \)\(20\!\cdots\!26\)\( \nu^{7} - \)\(98\!\cdots\!80\)\( \nu^{6} - \)\(95\!\cdots\!08\)\( \nu^{5} + \)\(42\!\cdots\!94\)\( \nu^{4} + \)\(18\!\cdots\!57\)\( \nu^{3} - \)\(75\!\cdots\!10\)\( \nu^{2} - \)\(29\!\cdots\!84\)\( \nu + \)\(13\!\cdots\!41\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(13\!\cdots\!09\)\( \nu^{19} + \)\(56\!\cdots\!03\)\( \nu^{18} - \)\(24\!\cdots\!97\)\( \nu^{17} - \)\(10\!\cdots\!52\)\( \nu^{16} + \)\(17\!\cdots\!70\)\( \nu^{15} + \)\(74\!\cdots\!18\)\( \nu^{14} - \)\(71\!\cdots\!98\)\( \nu^{13} - \)\(29\!\cdots\!76\)\( \nu^{12} + \)\(16\!\cdots\!67\)\( \nu^{11} + \)\(70\!\cdots\!37\)\( \nu^{10} - \)\(24\!\cdots\!43\)\( \nu^{9} - \)\(10\!\cdots\!92\)\( \nu^{8} + \)\(20\!\cdots\!66\)\( \nu^{7} + \)\(83\!\cdots\!78\)\( \nu^{6} - \)\(96\!\cdots\!82\)\( \nu^{5} - \)\(37\!\cdots\!28\)\( \nu^{4} + \)\(18\!\cdots\!45\)\( \nu^{3} + \)\(70\!\cdots\!87\)\( \nu^{2} - \)\(17\!\cdots\!45\)\( \nu - \)\(12\!\cdots\!08\)\(\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(15\!\cdots\!05\)\( \nu^{19} + \)\(14\!\cdots\!01\)\( \nu^{18} + \)\(26\!\cdots\!89\)\( \nu^{17} - \)\(22\!\cdots\!96\)\( \nu^{16} - \)\(19\!\cdots\!74\)\( \nu^{15} + \)\(14\!\cdots\!26\)\( \nu^{14} + \)\(79\!\cdots\!34\)\( \nu^{13} - \)\(51\!\cdots\!84\)\( \nu^{12} - \)\(18\!\cdots\!23\)\( \nu^{11} + \)\(10\!\cdots\!55\)\( \nu^{10} + \)\(27\!\cdots\!71\)\( \nu^{9} - \)\(12\!\cdots\!84\)\( \nu^{8} - \)\(23\!\cdots\!18\)\( \nu^{7} + \)\(81\!\cdots\!66\)\( \nu^{6} + \)\(10\!\cdots\!90\)\( \nu^{5} - \)\(26\!\cdots\!64\)\( \nu^{4} - \)\(20\!\cdots\!73\)\( \nu^{3} + \)\(30\!\cdots\!57\)\( \nu^{2} + \)\(33\!\cdots\!41\)\( \nu + \)\(45\!\cdots\!84\)\(\)\()/ \)\(31\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(19\!\cdots\!79\)\( \nu^{19} + \)\(21\!\cdots\!44\)\( \nu^{18} + \)\(33\!\cdots\!74\)\( \nu^{17} - \)\(37\!\cdots\!01\)\( \nu^{16} - \)\(24\!\cdots\!18\)\( \nu^{15} + \)\(26\!\cdots\!00\)\( \nu^{14} + \)\(98\!\cdots\!04\)\( \nu^{13} - \)\(10\!\cdots\!74\)\( \nu^{12} - \)\(23\!\cdots\!65\)\( \nu^{11} + \)\(24\!\cdots\!72\)\( \nu^{10} + \)\(33\!\cdots\!30\)\( \nu^{9} - \)\(32\!\cdots\!19\)\( \nu^{8} - \)\(28\!\cdots\!06\)\( \nu^{7} + \)\(23\!\cdots\!80\)\( \nu^{6} + \)\(13\!\cdots\!48\)\( \nu^{5} - \)\(86\!\cdots\!14\)\( \nu^{4} - \)\(25\!\cdots\!67\)\( \nu^{3} + \)\(10\!\cdots\!60\)\( \nu^{2} + \)\(40\!\cdots\!54\)\( \nu - \)\(19\!\cdots\!21\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(69\!\cdots\!55\)\( \nu^{19} + \)\(65\!\cdots\!83\)\( \nu^{18} + \)\(12\!\cdots\!31\)\( \nu^{17} - \)\(10\!\cdots\!72\)\( \nu^{16} - \)\(89\!\cdots\!98\)\( \nu^{15} + \)\(67\!\cdots\!38\)\( \nu^{14} + \)\(35\!\cdots\!22\)\( \nu^{13} - \)\(23\!\cdots\!48\)\( \nu^{12} - \)\(85\!\cdots\!61\)\( \nu^{11} + \)\(46\!\cdots\!53\)\( \nu^{10} + \)\(12\!\cdots\!89\)\( \nu^{9} - \)\(55\!\cdots\!64\)\( \nu^{8} - \)\(10\!\cdots\!26\)\( \nu^{7} + \)\(36\!\cdots\!14\)\( \nu^{6} + \)\(47\!\cdots\!34\)\( \nu^{5} - \)\(12\!\cdots\!40\)\( \nu^{4} - \)\(91\!\cdots\!31\)\( \nu^{3} + \)\(13\!\cdots\!35\)\( \nu^{2} + \)\(14\!\cdots\!95\)\( \nu + \)\(47\!\cdots\!16\)\(\)\()/ \)\(49\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(69\!\cdots\!55\)\( \nu^{19} + \)\(65\!\cdots\!62\)\( \nu^{18} + \)\(12\!\cdots\!20\)\( \nu^{17} - \)\(10\!\cdots\!59\)\( \nu^{16} - \)\(89\!\cdots\!86\)\( \nu^{15} + \)\(66\!\cdots\!52\)\( \nu^{14} + \)\(35\!\cdots\!08\)\( \nu^{13} - \)\(23\!\cdots\!66\)\( \nu^{12} - \)\(85\!\cdots\!17\)\( \nu^{11} + \)\(46\!\cdots\!14\)\( \nu^{10} + \)\(12\!\cdots\!20\)\( \nu^{9} - \)\(54\!\cdots\!69\)\( \nu^{8} - \)\(10\!\cdots\!22\)\( \nu^{7} + \)\(36\!\cdots\!80\)\( \nu^{6} + \)\(47\!\cdots\!92\)\( \nu^{5} - \)\(11\!\cdots\!42\)\( \nu^{4} - \)\(91\!\cdots\!27\)\( \nu^{3} + \)\(13\!\cdots\!66\)\( \nu^{2} + \)\(14\!\cdots\!28\)\( \nu - \)\(62\!\cdots\!15\)\(\)\()/ \)\(49\!\cdots\!00\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(38\!\cdots\!58\)\( \nu^{19} - \)\(36\!\cdots\!01\)\( \nu^{18} - \)\(66\!\cdots\!31\)\( \nu^{17} + \)\(56\!\cdots\!39\)\( \nu^{16} + \)\(49\!\cdots\!88\)\( \nu^{15} - \)\(36\!\cdots\!42\)\( \nu^{14} - \)\(19\!\cdots\!42\)\( \nu^{13} + \)\(12\!\cdots\!18\)\( \nu^{12} + \)\(46\!\cdots\!42\)\( \nu^{11} - \)\(25\!\cdots\!75\)\( \nu^{10} - \)\(67\!\cdots\!13\)\( \nu^{9} + \)\(30\!\cdots\!17\)\( \nu^{8} + \)\(57\!\cdots\!52\)\( \nu^{7} - \)\(20\!\cdots\!62\)\( \nu^{6} - \)\(26\!\cdots\!90\)\( \nu^{5} + \)\(66\!\cdots\!46\)\( \nu^{4} + \)\(51\!\cdots\!62\)\( \nu^{3} - \)\(76\!\cdots\!13\)\( \nu^{2} - \)\(28\!\cdots\!99\)\( \nu + \)\(14\!\cdots\!23\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{17}\)\(=\)\((\)\(\)\(27\!\cdots\!01\)\( \nu^{19} - \)\(65\!\cdots\!85\)\( \nu^{18} - \)\(48\!\cdots\!65\)\( \nu^{17} + \)\(10\!\cdots\!20\)\( \nu^{16} + \)\(35\!\cdots\!38\)\( \nu^{15} - \)\(76\!\cdots\!66\)\( \nu^{14} - \)\(14\!\cdots\!58\)\( \nu^{13} + \)\(29\!\cdots\!16\)\( \nu^{12} + \)\(34\!\cdots\!11\)\( \nu^{11} - \)\(66\!\cdots\!31\)\( \nu^{10} - \)\(49\!\cdots\!39\)\( \nu^{9} + \)\(90\!\cdots\!28\)\( \nu^{8} + \)\(41\!\cdots\!34\)\( \nu^{7} - \)\(72\!\cdots\!66\)\( \nu^{6} - \)\(19\!\cdots\!74\)\( \nu^{5} + \)\(31\!\cdots\!80\)\( \nu^{4} + \)\(37\!\cdots\!17\)\( \nu^{3} - \)\(54\!\cdots\!29\)\( \nu^{2} - \)\(15\!\cdots\!69\)\( \nu + \)\(10\!\cdots\!12\)\(\)\()/ \)\(56\!\cdots\!00\)\( \)
\(\beta_{18}\)\(=\)\((\)\(\)\(33\!\cdots\!09\)\( \nu^{19} - \)\(73\!\cdots\!41\)\( \nu^{18} - \)\(59\!\cdots\!01\)\( \nu^{17} + \)\(13\!\cdots\!04\)\( \nu^{16} + \)\(43\!\cdots\!46\)\( \nu^{15} - \)\(10\!\cdots\!78\)\( \nu^{14} - \)\(17\!\cdots\!90\)\( \nu^{13} + \)\(44\!\cdots\!84\)\( \nu^{12} + \)\(41\!\cdots\!23\)\( \nu^{11} - \)\(11\!\cdots\!91\)\( \nu^{10} - \)\(59\!\cdots\!07\)\( \nu^{9} + \)\(19\!\cdots\!60\)\( \nu^{8} + \)\(50\!\cdots\!86\)\( \nu^{7} - \)\(18\!\cdots\!98\)\( \nu^{6} - \)\(23\!\cdots\!54\)\( \nu^{5} + \)\(99\!\cdots\!56\)\( \nu^{4} + \)\(44\!\cdots\!09\)\( \nu^{3} - \)\(21\!\cdots\!97\)\( \nu^{2} - \)\(39\!\cdots\!53\)\( \nu + \)\(39\!\cdots\!20\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(33\!\cdots\!09\)\( \nu^{19} - \)\(98\!\cdots\!70\)\( \nu^{18} + \)\(59\!\cdots\!00\)\( \nu^{17} + \)\(33\!\cdots\!35\)\( \nu^{16} - \)\(43\!\cdots\!02\)\( \nu^{15} - \)\(38\!\cdots\!96\)\( \nu^{14} + \)\(17\!\cdots\!92\)\( \nu^{13} + \)\(22\!\cdots\!06\)\( \nu^{12} - \)\(41\!\cdots\!59\)\( \nu^{11} - \)\(72\!\cdots\!66\)\( \nu^{10} + \)\(59\!\cdots\!16\)\( \nu^{9} + \)\(13\!\cdots\!53\)\( \nu^{8} - \)\(50\!\cdots\!06\)\( \nu^{7} - \)\(15\!\cdots\!96\)\( \nu^{6} + \)\(23\!\cdots\!36\)\( \nu^{5} + \)\(87\!\cdots\!90\)\( \nu^{4} - \)\(45\!\cdots\!93\)\( \nu^{3} - \)\(20\!\cdots\!74\)\( \nu^{2} + \)\(46\!\cdots\!76\)\( \nu + \)\(37\!\cdots\!87\)\(\)\()/ \)\(18\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{4} + 3 \beta_{3} + \beta_{1} + 3\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(24 \beta_{15} - 24 \beta_{14} + 31 \beta_{12} - \beta_{9} + \beta_{8} + 37 \beta_{7} - 6 \beta_{6} - 31 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{1} + 525513\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(312 \beta_{19} - 312 \beta_{18} - 82 \beta_{17} + 72 \beta_{16} + 432 \beta_{15} + 216 \beta_{14} + 333 \beta_{12} + 12207 \beta_{11} + 12207 \beta_{10} + 9 \beta_{9} + 18 \beta_{8} + 279 \beta_{7} + 279 \beta_{6} + 54 \beta_{5} + 1688879 \beta_{4} + 4729203 \beta_{3} - 41 \beta_{2} + 844426 \beta_{1} + 4729401\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-1248 \beta_{18} - 164 \beta_{17} + 25204541 \beta_{15} - 25203245 \beta_{14} - 288 \beta_{13} + 61012892 \beta_{12} + 48828 \beta_{10} - 3420598 \beta_{9} + 3420652 \beta_{8} + 40115653 \beta_{7} + 20897689 \beta_{6} - 61012118 \beta_{5} + 3377752 \beta_{4} + 9458397 \beta_{3} - 1084 \beta_{2} - 3377788 \beta_{1} + 442327961807\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(140042625 \beta_{19} - 140045745 \beta_{18} + 1780888 \beta_{17} - 11773292 \beta_{16} + 252040370 \beta_{15} + 126021265 \beta_{14} - 720 \beta_{13} + 200577875 \beta_{12} + 5309934535 \beta_{11} + 5310056605 \beta_{10} + 17103200 \beta_{9} + 34206310 \beta_{8} + 305062810 \beta_{7} + 305064100 \beta_{6} - 104484185 \beta_{5} + 531463073003 \beta_{4} + 2211421976391 \beta_{3} + 887939 \beta_{2} + 265718869864 \beta_{1} + 2211515128385\)\()/9\)
\(\nu^{6}\)\(=\)\((\)\(-6240 \beta_{19} - 840268230 \beta_{18} + 5343074 \beta_{17} - 1440 \beta_{16} + 8522005327209 \beta_{15} - 8520871145544 \beta_{14} + 70635432 \beta_{13} + 25164989114637 \beta_{12} - 244140 \beta_{11} + 31860095490 \beta_{10} - 1345383708042 \beta_{9} + 1345537636437 \beta_{8} + 16158490285845 \beta_{7} + 9007727474772 \beta_{6} - 25164700835502 \beta_{5} + 1594380774632 \beta_{4} + 6634242283185 \beta_{3} - 845615084 \beta_{2} - 1594431441218 \beta_{1} + 139028330770172811\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(52072729741221 \beta_{19} - 52075670690946 \beta_{18} + 28432828051446 \beta_{17} - 16679167623216 \beta_{16} + 119301899602949 \beta_{15} + 59652273014929 \beta_{14} + 247226532 \beta_{13} + 113105102375447 \beta_{12} + 1852958477116866 \beta_{11} + 1853069987878326 \beta_{10} + 9418524011183 \beta_{9} + 18836688855985 \beta_{8} + 176155713950272 \beta_{7} + 176156386603393 \beta_{6} - 63047744733155 \beta_{5} + 174442122316256613 \beta_{4} + 973097851837729389 \beta_{3} + 14213445031317 \beta_{2} + 87212690526061653 \beta_{1} + 973132603378742228\)\()/9\)
\(\nu^{8}\)\(=\)\((\)\(-7842544248 \beta_{19} - 416597523053208 \beta_{18} + 113731287271056 \beta_{17} + 659268064 \beta_{16} + 2877769307876710303 \beta_{15} - 2877053496479083537 \beta_{14} + 133435318781856 \beta_{13} + 9454857725975145904 \beta_{12} - 297362486288 \beta_{11} + 14824262543274688 \beta_{10} - 417128560187269028 \beta_{9} + 417241580320405316 \beta_{8} + 6409228057973845097 \beta_{7} + 3046586490260333489 \beta_{6} - 9454657497889877560 \beta_{5} + 697761048829292920 \beta_{4} + 3892360447575665613 \beta_{3} - 530336802435552 \beta_{2} - 697805692294892800 \beta_{1} + 45624936241950913519630\)\()/9\)
\(\nu^{9}\)\(=\)\((\)\(18503031228866217159 \beta_{19} - 18504905935365720465 \beta_{18} + 21793504891175213672 \beta_{17} - 9373129931013290876 \beta_{16} + 51794836885285963704 \beta_{15} + 25898492162915139735 \beta_{14} + 600457451156136 \beta_{13} + 57678668881733224863 \beta_{12} + 605409228217469709649 \beta_{11} + 605475938067981577975 \beta_{10} + 3754948184628707412 \beta_{9} + 7509557307996006192 \beta_{8} + 85094632065248735904 \beta_{7} + 85095232751118914928 \beta_{6} - 27413092709857989207 \beta_{5} + 58172699664182175123885 \beta_{4} + 410580142326494583319710 \beta_{3} + 10894110052394242891 \beta_{2} + 29081639794339633835343 \beta_{1} + 410590606943598970644558\)\()/9\)
\(\nu^{10}\)\(=\)\((\)\(-6249045192696180 \beta_{19} - 185042810449630394790 \beta_{18} + 108966671471258937778 \beta_{17} + 2001526814979216 \beta_{16} + 974802994838814022258677 \beta_{15} - 974414533562150559677052 \beta_{14} + 93737303896511274536 \beta_{13} + 3450711810563854148978748 \beta_{12} - 222367060462404060 \beta_{11} + 6054537018971881644490 \beta_{10} - 114156299340710813287923 \beta_{9} + 114212621020524015832278 \beta_{8} + 2481094561247469001818006 \beta_{7} + 970179798119899977065082 \beta_{6} - 3450560484184703394362973 \beta_{5} + 290858265124205304456198 \beta_{4} + 2052871518975555747796776 \beta_{3} - 294020848877823113464 \beta_{2} - 290889665509812485032764 \beta_{1} + 15218323881856267442431339410\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(6488045496180850901924637 \beta_{19} - 6489063248823295200557322 \beta_{18} + 11823804462157139785711024 \beta_{17} - 4298878934382940546864072 \beta_{16} + 21442342413291405573533841 \beta_{15} + 10721883389589910932514461 \beta_{14} + 515549667240229245140 \beta_{13} + 27288972306514588401499350 \beta_{12} + 193010659734526106944348427 \beta_{11} + 193043960299643548428189087 \beta_{10} + 1256201158909304760572970 \beta_{9} + 2512195803749398412320587 \beta_{8} + 37958280304711483827649209 \beta_{7} + 37958835170304226997606238 \beta_{6} - 10667245308725566463264637 \beta_{5} + 19530565736083753065605890586 \beta_{4} + 167382633622627462010261231856 \beta_{3} + 5909985482285100850577822 \beta_{2} + 9762883157827734977727104125 \beta_{1} + 167384570848620440891365298949\)\()/9\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(40\!\cdots\!08\)\( \beta_{19} - \)\(77\!\cdots\!04\)\( \beta_{18} + \)\(70\!\cdots\!76\)\( \beta_{17} + \)\(20\!\cdots\!24\)\( \beta_{16} + \)\(33\!\cdots\!58\)\( \beta_{15} - \)\(33\!\cdots\!28\)\( \beta_{14} + \)\(51\!\cdots\!04\)\( \beta_{13} + \)\(12\!\cdots\!74\)\( \beta_{12} - \)\(13\!\cdots\!48\)\( \beta_{11} + \)\(23\!\cdots\!64\)\( \beta_{10} - \)\(27\!\cdots\!44\)\( \beta_{9} + \)\(27\!\cdots\!34\)\( \beta_{8} + \)\(94\!\cdots\!30\)\( \beta_{7} + \)\(30\!\cdots\!04\)\( \beta_{6} - \)\(12\!\cdots\!44\)\( \beta_{5} + \)\(11\!\cdots\!92\)\( \beta_{4} + \)\(10\!\cdots\!50\)\( \beta_{3} - \)\(14\!\cdots\!08\)\( \beta_{2} - \)\(11\!\cdots\!84\)\( \beta_{1} + \)\(51\!\cdots\!73\)\(\)\()/9\)
\(\nu^{13}\)\(=\)\((\)\(\)\(22\!\cdots\!10\)\( \beta_{19} - \)\(22\!\cdots\!24\)\( \beta_{18} + \)\(55\!\cdots\!44\)\( \beta_{17} - \)\(18\!\cdots\!52\)\( \beta_{16} + \)\(86\!\cdots\!34\)\( \beta_{15} + \)\(43\!\cdots\!06\)\( \beta_{14} + \)\(33\!\cdots\!40\)\( \beta_{13} + \)\(12\!\cdots\!62\)\( \beta_{12} + \)\(60\!\cdots\!20\)\( \beta_{11} + \)\(60\!\cdots\!44\)\( \beta_{10} + \)\(36\!\cdots\!90\)\( \beta_{9} + \)\(72\!\cdots\!94\)\( \beta_{8} + \)\(16\!\cdots\!16\)\( \beta_{7} + \)\(16\!\cdots\!38\)\( \beta_{6} - \)\(39\!\cdots\!42\)\( \beta_{5} + \)\(65\!\cdots\!12\)\( \beta_{4} + \)\(66\!\cdots\!77\)\( \beta_{3} + \)\(27\!\cdots\!94\)\( \beta_{2} + \)\(32\!\cdots\!27\)\( \beta_{1} + \)\(66\!\cdots\!95\)\(\)\()/9\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(23\!\cdots\!36\)\( \beta_{19} - \)\(31\!\cdots\!24\)\( \beta_{18} + \)\(38\!\cdots\!88\)\( \beta_{17} + \)\(15\!\cdots\!20\)\( \beta_{16} + \)\(11\!\cdots\!30\)\( \beta_{15} - \)\(11\!\cdots\!50\)\( \beta_{14} + \)\(25\!\cdots\!88\)\( \beta_{13} + \)\(44\!\cdots\!87\)\( \beta_{12} - \)\(70\!\cdots\!76\)\( \beta_{11} + \)\(85\!\cdots\!24\)\( \beta_{10} - \)\(56\!\cdots\!85\)\( \beta_{9} + \)\(56\!\cdots\!89\)\( \beta_{8} + \)\(35\!\cdots\!87\)\( \beta_{7} + \)\(93\!\cdots\!72\)\( \beta_{6} - \)\(44\!\cdots\!87\)\( \beta_{5} + \)\(46\!\cdots\!90\)\( \beta_{4} + \)\(46\!\cdots\!97\)\( \beta_{3} - \)\(70\!\cdots\!80\)\( \beta_{2} - \)\(46\!\cdots\!42\)\( \beta_{1} + \)\(17\!\cdots\!37\)\(\)\()/9\)
\(\nu^{15}\)\(=\)\((\)\(\)\(78\!\cdots\!86\)\( \beta_{19} - \)\(79\!\cdots\!06\)\( \beta_{18} + \)\(23\!\cdots\!58\)\( \beta_{17} - \)\(72\!\cdots\!28\)\( \beta_{16} + \)\(33\!\cdots\!84\)\( \beta_{15} + \)\(16\!\cdots\!78\)\( \beta_{14} + \)\(19\!\cdots\!04\)\( \beta_{13} + \)\(52\!\cdots\!01\)\( \beta_{12} + \)\(19\!\cdots\!91\)\( \beta_{11} + \)\(19\!\cdots\!31\)\( \beta_{10} + \)\(84\!\cdots\!27\)\( \beta_{9} + \)\(16\!\cdots\!26\)\( \beta_{8} + \)\(66\!\cdots\!29\)\( \beta_{7} + \)\(66\!\cdots\!17\)\( \beta_{6} - \)\(14\!\cdots\!28\)\( \beta_{5} + \)\(22\!\cdots\!83\)\( \beta_{4} + \)\(25\!\cdots\!97\)\( \beta_{3} + \)\(11\!\cdots\!07\)\( \beta_{2} + \)\(11\!\cdots\!52\)\( \beta_{1} + \)\(25\!\cdots\!43\)\(\)\()/9\)
\(\nu^{16}\)\(=\)\((\)\(-\)\(12\!\cdots\!64\)\( \beta_{19} - \)\(12\!\cdots\!76\)\( \beta_{18} + \)\(19\!\cdots\!24\)\( \beta_{17} + \)\(10\!\cdots\!60\)\( \beta_{16} + \)\(38\!\cdots\!77\)\( \beta_{15} - \)\(38\!\cdots\!23\)\( \beta_{14} + \)\(11\!\cdots\!68\)\( \beta_{13} + \)\(15\!\cdots\!20\)\( \beta_{12} - \)\(34\!\cdots\!24\)\( \beta_{11} + \)\(30\!\cdots\!56\)\( \beta_{10} - \)\(61\!\cdots\!76\)\( \beta_{9} + \)\(61\!\cdots\!88\)\( \beta_{8} + \)\(13\!\cdots\!75\)\( \beta_{7} + \)\(28\!\cdots\!23\)\( \beta_{6} - \)\(15\!\cdots\!84\)\( \beta_{5} + \)\(17\!\cdots\!08\)\( \beta_{4} + \)\(20\!\cdots\!47\)\( \beta_{3} - \)\(31\!\cdots\!12\)\( \beta_{2} - \)\(17\!\cdots\!20\)\( \beta_{1} + \)\(58\!\cdots\!57\)\(\)\()/9\)
\(\nu^{17}\)\(=\)\((\)\(\)\(27\!\cdots\!33\)\( \beta_{19} - \)\(27\!\cdots\!31\)\( \beta_{18} + \)\(97\!\cdots\!36\)\( \beta_{17} - \)\(28\!\cdots\!08\)\( \beta_{16} + \)\(13\!\cdots\!04\)\( \beta_{15} + \)\(65\!\cdots\!81\)\( \beta_{14} + \)\(99\!\cdots\!04\)\( \beta_{13} + \)\(22\!\cdots\!57\)\( \beta_{12} + \)\(59\!\cdots\!43\)\( \beta_{11} + \)\(59\!\cdots\!81\)\( \beta_{10} + \)\(10\!\cdots\!16\)\( \beta_{9} + \)\(21\!\cdots\!20\)\( \beta_{8} + \)\(27\!\cdots\!64\)\( \beta_{7} + \)\(27\!\cdots\!36\)\( \beta_{6} - \)\(48\!\cdots\!97\)\( \beta_{5} + \)\(75\!\cdots\!25\)\( \beta_{4} + \)\(98\!\cdots\!53\)\( \beta_{3} + \)\(48\!\cdots\!05\)\( \beta_{2} + \)\(37\!\cdots\!34\)\( \beta_{1} + \)\(98\!\cdots\!93\)\(\)\()/9\)
\(\nu^{18}\)\(=\)\((\)\(-\)\(64\!\cdots\!24\)\( \beta_{19} - \)\(49\!\cdots\!86\)\( \beta_{18} + \)\(87\!\cdots\!38\)\( \beta_{17} + \)\(59\!\cdots\!28\)\( \beta_{16} + \)\(13\!\cdots\!43\)\( \beta_{15} - \)\(13\!\cdots\!88\)\( \beta_{14} + \)\(51\!\cdots\!16\)\( \beta_{13} + \)\(56\!\cdots\!35\)\( \beta_{12} - \)\(15\!\cdots\!24\)\( \beta_{11} + \)\(10\!\cdots\!66\)\( \beta_{10} + \)\(21\!\cdots\!90\)\( \beta_{9} - \)\(21\!\cdots\!65\)\( \beta_{8} + \)\(48\!\cdots\!03\)\( \beta_{7} + \)\(87\!\cdots\!72\)\( \beta_{6} - \)\(56\!\cdots\!10\)\( \beta_{5} + \)\(67\!\cdots\!52\)\( \beta_{4} + \)\(89\!\cdots\!55\)\( \beta_{3} - \)\(13\!\cdots\!04\)\( \beta_{2} - \)\(67\!\cdots\!86\)\( \beta_{1} + \)\(19\!\cdots\!67\)\(\)\()/9\)
\(\nu^{19}\)\(=\)\((\)\(\)\(95\!\cdots\!99\)\( \beta_{19} - \)\(95\!\cdots\!50\)\( \beta_{18} + \)\(38\!\cdots\!70\)\( \beta_{17} - \)\(10\!\cdots\!52\)\( \beta_{16} + \)\(49\!\cdots\!87\)\( \beta_{15} + \)\(24\!\cdots\!55\)\( \beta_{14} + \)\(48\!\cdots\!68\)\( \beta_{13} + \)\(91\!\cdots\!49\)\( \beta_{12} + \)\(18\!\cdots\!94\)\( \beta_{11} + \)\(18\!\cdots\!50\)\( \beta_{10} - \)\(40\!\cdots\!27\)\( \beta_{9} - \)\(81\!\cdots\!77\)\( \beta_{8} + \)\(10\!\cdots\!28\)\( \beta_{7} + \)\(10\!\cdots\!75\)\( \beta_{6} - \)\(16\!\cdots\!61\)\( \beta_{5} + \)\(25\!\cdots\!19\)\( \beta_{4} + \)\(37\!\cdots\!17\)\( \beta_{3} + \)\(19\!\cdots\!19\)\( \beta_{2} + \)\(12\!\cdots\!57\)\( \beta_{1} + \)\(37\!\cdots\!74\)\(\)\()/9\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
590.770 + 0.866025i
564.961 + 0.866025i
357.003 + 0.866025i
286.330 + 0.866025i
4.87946 + 0.866025i
−3.87946 + 0.866025i
−285.330 + 0.866025i
−356.003 + 0.866025i
−563.961 + 0.866025i
−589.770 + 0.866025i
590.770 0.866025i
564.961 0.866025i
357.003 0.866025i
286.330 0.866025i
4.87946 0.866025i
−3.87946 0.866025i
−285.330 0.866025i
−356.003 0.866025i
−563.961 0.866025i
−589.770 0.866025i
0 0 0 −885.405 + 511.189i 0 765.093 + 2275.84i 0 0 0
73.2 0 0 0 −846.691 + 488.837i 0 −1709.42 1686.02i 0 0 0
73.3 0 0 0 −534.755 + 308.741i 0 −750.736 2280.61i 0 0 0
73.4 0 0 0 −428.745 + 247.536i 0 1933.79 1423.12i 0 0 0
73.5 0 0 0 −6.56919 + 3.79272i 0 −2048.22 + 1252.83i 0 0 0
73.6 0 0 0 6.56919 3.79272i 0 −2048.22 + 1252.83i 0 0 0
73.7 0 0 0 428.745 247.536i 0 1933.79 1423.12i 0 0 0
73.8 0 0 0 534.755 308.741i 0 −750.736 2280.61i 0 0 0
73.9 0 0 0 846.691 488.837i 0 −1709.42 1686.02i 0 0 0
73.10 0 0 0 885.405 511.189i 0 765.093 + 2275.84i 0 0 0
145.1 0 0 0 −885.405 511.189i 0 765.093 2275.84i 0 0 0
145.2 0 0 0 −846.691 488.837i 0 −1709.42 + 1686.02i 0 0 0
145.3 0 0 0 −534.755 308.741i 0 −750.736 + 2280.61i 0 0 0
145.4 0 0 0 −428.745 247.536i 0 1933.79 + 1423.12i 0 0 0
145.5 0 0 0 −6.56919 3.79272i 0 −2048.22 1252.83i 0 0 0
145.6 0 0 0 6.56919 + 3.79272i 0 −2048.22 1252.83i 0 0 0
145.7 0 0 0 428.745 + 247.536i 0 1933.79 + 1423.12i 0 0 0
145.8 0 0 0 534.755 + 308.741i 0 −750.736 + 2280.61i 0 0 0
145.9 0 0 0 846.691 + 488.837i 0 −1709.42 + 1686.02i 0 0 0
145.10 0 0 0 885.405 + 511.189i 0 765.093 2275.84i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.z.e 20
3.b odd 2 1 inner 252.9.z.e 20
7.d odd 6 1 inner 252.9.z.e 20
21.g even 6 1 inner 252.9.z.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.z.e 20 1.a even 1 1 trivial
252.9.z.e 20 3.b odd 2 1 inner
252.9.z.e 20 7.d odd 6 1 inner
252.9.z.e 20 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(45\!\cdots\!38\)\( T_{5}^{16} - \)\(45\!\cdots\!33\)\( T_{5}^{14} + \)\(32\!\cdots\!34\)\( T_{5}^{12} - \)\(14\!\cdots\!45\)\( T_{5}^{10} + \)\(44\!\cdots\!25\)\( T_{5}^{8} - \)\(75\!\cdots\!00\)\( T_{5}^{6} + \)\(87\!\cdots\!00\)\( T_{5}^{4} - \)\(50\!\cdots\!00\)\( T_{5}^{2} + \)\(28\!\cdots\!00\)\( \)">\(T_{5}^{20} - \cdots\) acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( \)\(28\!\cdots\!00\)\( - \)\(50\!\cdots\!00\)\( T^{2} + \)\(87\!\cdots\!00\)\( T^{4} - \)\(75\!\cdots\!00\)\( T^{6} + \)\(44\!\cdots\!25\)\( T^{8} - \)\(14\!\cdots\!45\)\( T^{10} + \)\(32\!\cdots\!34\)\( T^{12} - 4538708430919915833 T^{14} + 4557812756238 T^{16} - 2627541 T^{18} + T^{20} \)
$7$ \( ( \)\(63\!\cdots\!01\)\( + \)\(39\!\cdots\!19\)\( T + \)\(21\!\cdots\!95\)\( T^{2} + \)\(70\!\cdots\!86\)\( T^{3} + \)\(38\!\cdots\!29\)\( T^{4} + 129916718471073765 T^{5} + 66473263967229 T^{6} + 21337173186 T^{7} + 11360895 T^{8} + 3619 T^{9} + T^{10} )^{2} \)
$11$ \( \)\(76\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T^{2} + \)\(75\!\cdots\!00\)\( T^{4} + \)\(17\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!01\)\( T^{8} + \)\(29\!\cdots\!07\)\( T^{10} + \)\(15\!\cdots\!06\)\( T^{12} + \)\(45\!\cdots\!23\)\( T^{14} + 984480158423293686 T^{16} + 1209688767 T^{18} + T^{20} \)
$13$ \( ( \)\(19\!\cdots\!00\)\( + \)\(72\!\cdots\!04\)\( T^{2} + \)\(46\!\cdots\!24\)\( T^{4} + 9686523009231857025 T^{6} + 5740976634 T^{8} + T^{10} )^{2} \)
$17$ \( \)\(52\!\cdots\!00\)\( - \)\(40\!\cdots\!00\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!24\)\( T^{8} - \)\(20\!\cdots\!96\)\( T^{10} + \)\(23\!\cdots\!56\)\( T^{12} - \)\(19\!\cdots\!76\)\( T^{14} + \)\(11\!\cdots\!44\)\( T^{16} - 43075543896 T^{18} + T^{20} \)
$19$ \( ( \)\(42\!\cdots\!00\)\( - \)\(15\!\cdots\!00\)\( T + \)\(19\!\cdots\!04\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} - \)\(37\!\cdots\!88\)\( T^{4} + \)\(38\!\cdots\!20\)\( T^{5} + \)\(79\!\cdots\!29\)\( T^{6} - 1653071447654532 T^{7} - 29866128591 T^{8} + 53628 T^{9} + T^{10} )^{2} \)
$23$ \( \)\(87\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{4} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(39\!\cdots\!00\)\( T^{8} + \)\(58\!\cdots\!40\)\( T^{10} + \)\(60\!\cdots\!56\)\( T^{12} + \)\(39\!\cdots\!28\)\( T^{14} + \)\(19\!\cdots\!88\)\( T^{16} + 540169907508 T^{18} + T^{20} \)
$29$ \( ( -\)\(14\!\cdots\!00\)\( + \)\(26\!\cdots\!60\)\( T^{2} - \)\(81\!\cdots\!29\)\( T^{4} + \)\(98\!\cdots\!19\)\( T^{6} - 5171113544751 T^{8} + T^{10} )^{2} \)
$31$ \( ( \)\(27\!\cdots\!07\)\( - \)\(25\!\cdots\!95\)\( T - \)\(10\!\cdots\!61\)\( T^{2} + \)\(95\!\cdots\!10\)\( T^{3} + \)\(31\!\cdots\!99\)\( T^{4} - \)\(50\!\cdots\!73\)\( T^{5} - \)\(25\!\cdots\!23\)\( T^{6} + 286842812692529286 T^{7} + 1989072345345 T^{8} + 2367213 T^{9} + T^{10} )^{2} \)
$37$ \( ( \)\(15\!\cdots\!96\)\( - \)\(40\!\cdots\!24\)\( T + \)\(18\!\cdots\!96\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!56\)\( T^{4} - \)\(36\!\cdots\!04\)\( T^{5} + \)\(55\!\cdots\!01\)\( T^{6} - 16797007861750404510 T^{7} + 14862322970331 T^{8} - 3401374 T^{9} + T^{10} )^{2} \)
$41$ \( ( \)\(85\!\cdots\!00\)\( + \)\(17\!\cdots\!40\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{4} + \)\(61\!\cdots\!72\)\( T^{6} + 47701796739156 T^{8} + T^{10} )^{2} \)
$43$ \( ( \)\(18\!\cdots\!00\)\( - \)\(59\!\cdots\!40\)\( T + 61130048163516606068 T^{2} - 22420995393101 T^{3} + 149032 T^{4} + T^{5} )^{4} \)
$47$ \( \)\(38\!\cdots\!00\)\( - \)\(80\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{4} - \)\(53\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!24\)\( T^{8} - \)\(13\!\cdots\!88\)\( T^{10} + \)\(11\!\cdots\!32\)\( T^{12} - \)\(41\!\cdots\!52\)\( T^{14} + \)\(10\!\cdots\!28\)\( T^{16} - 120593971151388 T^{18} + T^{20} \)
$53$ \( \)\(76\!\cdots\!00\)\( + \)\(86\!\cdots\!00\)\( T^{2} + \)\(72\!\cdots\!00\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(67\!\cdots\!25\)\( T^{8} + \)\(37\!\cdots\!75\)\( T^{10} + \)\(15\!\cdots\!50\)\( T^{12} + \)\(15\!\cdots\!35\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} + 396373925510199 T^{18} + T^{20} \)
$59$ \( \)\(34\!\cdots\!00\)\( - \)\(18\!\cdots\!00\)\( T^{2} + \)\(60\!\cdots\!00\)\( T^{4} - \)\(12\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!89\)\( T^{8} - \)\(18\!\cdots\!49\)\( T^{10} + \)\(14\!\cdots\!46\)\( T^{12} - \)\(78\!\cdots\!09\)\( T^{14} + \)\(30\!\cdots\!94\)\( T^{16} - 689407494920109 T^{18} + T^{20} \)
$61$ \( ( \)\(25\!\cdots\!00\)\( - \)\(24\!\cdots\!40\)\( T + \)\(80\!\cdots\!64\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(32\!\cdots\!04\)\( T^{4} + \)\(15\!\cdots\!04\)\( T^{5} + \)\(83\!\cdots\!56\)\( T^{6} + \)\(14\!\cdots\!32\)\( T^{7} + 2330597999520 T^{8} - 16286886 T^{9} + T^{10} )^{2} \)
$67$ \( ( \)\(54\!\cdots\!00\)\( - \)\(53\!\cdots\!00\)\( T + \)\(68\!\cdots\!00\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!00\)\( T^{4} - \)\(56\!\cdots\!80\)\( T^{5} + \)\(49\!\cdots\!09\)\( T^{6} - \)\(10\!\cdots\!16\)\( T^{7} + 881210578909677 T^{8} + 12999652 T^{9} + T^{10} )^{2} \)
$71$ \( ( -\)\(15\!\cdots\!00\)\( + \)\(40\!\cdots\!00\)\( T^{2} - \)\(37\!\cdots\!40\)\( T^{4} + \)\(14\!\cdots\!16\)\( T^{6} - 2210194026495252 T^{8} + T^{10} )^{2} \)
$73$ \( ( \)\(49\!\cdots\!52\)\( + \)\(45\!\cdots\!56\)\( T - \)\(20\!\cdots\!24\)\( T^{2} - \)\(32\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!88\)\( T^{4} + \)\(20\!\cdots\!84\)\( T^{5} + \)\(45\!\cdots\!29\)\( T^{6} - \)\(23\!\cdots\!30\)\( T^{7} - 670940250687993 T^{8} + 26449806 T^{9} + T^{10} )^{2} \)
$79$ \( ( \)\(24\!\cdots\!25\)\( - \)\(16\!\cdots\!15\)\( T + \)\(92\!\cdots\!51\)\( T^{2} - \)\(11\!\cdots\!10\)\( T^{3} + \)\(13\!\cdots\!77\)\( T^{4} + \)\(74\!\cdots\!55\)\( T^{5} + \)\(98\!\cdots\!53\)\( T^{6} + \)\(81\!\cdots\!70\)\( T^{7} + 6223661480453127 T^{8} + 64972765 T^{9} + T^{10} )^{2} \)
$83$ \( ( \)\(12\!\cdots\!00\)\( + \)\(25\!\cdots\!60\)\( T^{2} + \)\(99\!\cdots\!43\)\( T^{4} + \)\(13\!\cdots\!27\)\( T^{6} + 6856065632807181 T^{8} + T^{10} )^{2} \)
$89$ \( \)\(34\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{4} - \)\(90\!\cdots\!20\)\( T^{6} + \)\(47\!\cdots\!84\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{10} + \)\(33\!\cdots\!16\)\( T^{12} - \)\(46\!\cdots\!64\)\( T^{14} + \)\(46\!\cdots\!44\)\( T^{16} - 26478777167636724 T^{18} + T^{20} \)
$97$ \( ( \)\(45\!\cdots\!00\)\( + \)\(65\!\cdots\!24\)\( T^{2} + \)\(16\!\cdots\!79\)\( T^{4} + \)\(15\!\cdots\!15\)\( T^{6} + 65277242250659649 T^{8} + T^{10} )^{2} \)
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