Properties

Label 4005.2.a.w
Level 4005
Weight 2
Character orbit 4005.a
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 17
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{5} + ( 1 + \beta_{8} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{5} + ( 1 + \beta_{8} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} ) q^{8} + \beta_{1} q^{10} -\beta_{14} q^{11} -\beta_{9} q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{14} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{10} + \beta_{11} ) q^{16} + ( -1 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{12} ) q^{17} + ( 2 - \beta_{16} ) q^{19} + ( -1 - \beta_{2} ) q^{20} + \beta_{5} q^{22} + ( -1 - \beta_{4} ) q^{23} + q^{25} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{26} + ( 2 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} ) q^{28} + ( \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{16} ) q^{29} + ( 1 - \beta_{6} - \beta_{9} ) q^{31} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{8} - \beta_{11} ) q^{32} + ( 1 + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{14} + \beta_{16} ) q^{34} + ( -1 - \beta_{8} ) q^{35} + ( -\beta_{1} - \beta_{2} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{37} + ( -1 - 2 \beta_{1} - \beta_{6} - \beta_{9} + \beta_{12} + \beta_{16} ) q^{38} + ( 1 + \beta_{1} + \beta_{3} ) q^{40} + ( -\beta_{1} - \beta_{8} - \beta_{15} ) q^{41} + ( 1 - \beta_{1} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{15} + \beta_{16} ) q^{43} + ( 1 - \beta_{1} + \beta_{4} + \beta_{6} + \beta_{9} - 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{44} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{46} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} + ( 4 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{49} -\beta_{1} q^{50} + ( -2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{52} + ( -\beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{16} ) q^{53} + \beta_{14} q^{55} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{56} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{7} + 2 \beta_{9} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{58} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{59} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{61} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} + \beta_{14} - \beta_{16} ) q^{62} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{64} + \beta_{9} q^{65} + ( 3 - \beta_{1} - \beta_{5} + \beta_{15} ) q^{67} + ( 1 - 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - \beta_{15} - \beta_{16} ) q^{68} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{70} + ( 1 + 2 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{16} ) q^{71} + ( 3 - \beta_{2} + \beta_{3} + \beta_{7} - \beta_{9} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} ) q^{73} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{74} + ( 4 + 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 3 \beta_{9} - \beta_{12} + \beta_{14} ) q^{76} + ( \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} + \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{16} ) q^{77} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{79} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{10} - \beta_{11} ) q^{80} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} ) q^{82} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{9} - 2 \beta_{12} + \beta_{14} - \beta_{16} ) q^{83} + ( 1 + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} ) q^{85} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} - \beta_{16} ) q^{86} + ( -2 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{11} - 3 \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} ) q^{88} + q^{89} + ( 2 - 3 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{91} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} - \beta_{16} ) q^{92} + ( 3 + \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{9} + \beta_{11} - \beta_{14} ) q^{94} + ( -2 + \beta_{16} ) q^{95} + ( 3 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} - 3 \beta_{7} + 3 \beta_{9} + \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{16} ) q^{97} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 5q^{2} + 21q^{4} - 17q^{5} + 12q^{7} - 15q^{8} + O(q^{10}) \) \( 17q - 5q^{2} + 21q^{4} - 17q^{5} + 12q^{7} - 15q^{8} + 5q^{10} - 2q^{11} + 8q^{13} - 4q^{14} + 33q^{16} - 10q^{17} + 32q^{19} - 21q^{20} + 8q^{22} - 15q^{23} + 17q^{25} + 15q^{26} + 24q^{28} - q^{29} + 18q^{31} - 25q^{32} + 14q^{34} - 12q^{35} + 12q^{37} - 22q^{38} + 15q^{40} + 7q^{41} + 28q^{43} + 14q^{44} + 4q^{46} - 26q^{47} + 41q^{49} - 5q^{50} + 10q^{52} - 12q^{53} + 2q^{55} - 13q^{56} + 16q^{58} + 23q^{59} + 26q^{61} - 10q^{62} + 59q^{64} - 8q^{65} + 31q^{67} + q^{68} + 4q^{70} + 2q^{71} + 33q^{73} + 10q^{74} + 66q^{76} - 12q^{77} + 33q^{79} - 33q^{80} + 30q^{82} - 13q^{83} + 10q^{85} + 20q^{86} + 12q^{88} + 17q^{89} + 40q^{91} - 16q^{92} + 38q^{94} - 32q^{95} + 45q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} - 2456 x^{9} - 7002 x^{8} + 6279 x^{7} + 7299 x^{6} - 7119 x^{5} - 3066 x^{4} + 3184 x^{3} + 99 x^{2} - 231 x + 24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\(685 \nu^{16} - 1978 \nu^{15} - 23375 \nu^{14} + 80742 \nu^{13} + 247035 \nu^{12} - 1063752 \nu^{11} - 927300 \nu^{10} + 6222061 \nu^{9} + 79227 \nu^{8} - 17370991 \nu^{7} + 6231634 \nu^{6} + 22769593 \nu^{5} - 11572552 \nu^{4} - 11774626 \nu^{3} + 6374744 \nu^{2} + 982243 \nu - 368684\)\()/31022\)
\(\beta_{5}\)\(=\)\((\)\(-1070 \nu^{16} + 1165 \nu^{15} + 40249 \nu^{14} - 75174 \nu^{13} - 484380 \nu^{12} + 1137763 \nu^{11} + 2444697 \nu^{10} - 7082375 \nu^{9} - 4941450 \nu^{8} + 20342129 \nu^{7} + 1670236 \nu^{6} - 26893622 \nu^{5} + 5596139 \nu^{4} + 13977281 \nu^{3} - 4729743 \nu^{2} - 1384065 \nu + 300552\)\()/31022\)
\(\beta_{6}\)\(=\)\((\)\(6613 \nu^{16} - 17420 \nu^{15} - 180375 \nu^{14} + 520530 \nu^{13} + 1766251 \nu^{12} - 5814854 \nu^{11} - 7314342 \nu^{10} + 30806921 \nu^{9} + 9574065 \nu^{8} - 81028001 \nu^{7} + 12506146 \nu^{6} + 103349077 \nu^{5} - 39644890 \nu^{4} - 55867828 \nu^{3} + 25227292 \nu^{2} + 7528791 \nu - 1639268\)\()/62044\)
\(\beta_{7}\)\(=\)\((\)\(-7257 \nu^{16} + 30443 \nu^{15} + 137250 \nu^{14} - 693400 \nu^{13} - 837779 \nu^{12} + 6167935 \nu^{11} + 943861 \nu^{10} - 27194200 \nu^{9} + 8863387 \nu^{8} + 62567552 \nu^{7} - 34862246 \nu^{6} - 72621151 \nu^{5} + 46524231 \nu^{4} + 36780711 \nu^{3} - 21561069 \nu^{2} - 4918520 \nu + 1063920\)\()/62044\)
\(\beta_{8}\)\(=\)\((\)\(6850 \nu^{16} - 35291 \nu^{15} - 94151 \nu^{14} + 714354 \nu^{13} + 174722 \nu^{12} - 5503379 \nu^{11} + 2779047 \nu^{10} + 20604597 \nu^{9} - 16766182 \nu^{8} - 40485931 \nu^{7} + 37622828 \nu^{6} + 41160644 \nu^{5} - 37720701 \nu^{4} - 18925679 \nu^{3} + 14593081 \nu^{2} + 2423683 \nu - 739750\)\()/31022\)
\(\beta_{9}\)\(=\)\((\)\(-14426 \nu^{16} + 61515 \nu^{15} + 278177 \nu^{14} - 1417322 \nu^{13} - 1771986 \nu^{12} + 12781181 \nu^{11} + 2666987 \nu^{10} - 57266037 \nu^{9} + 14936458 \nu^{8} + 134174293 \nu^{7} - 65290780 \nu^{6} - 158712488 \nu^{5} + 91148769 \nu^{4} + 81480431 \nu^{3} - 44547125 \nu^{2} - 10787091 \nu + 2544040\)\()/62044\)
\(\beta_{10}\)\(=\)\((\)\(7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} - 6567131 \nu^{11} + 1851747 \nu^{10} + 26826658 \nu^{9} - 16686955 \nu^{8} - 57856922 \nu^{7} + 43854462 \nu^{6} + 63961259 \nu^{5} - 49324275 \nu^{4} - 30979503 \nu^{3} + 21184979 \nu^{2} + 3902278 \nu - 1263544\)\()/31022\)
\(\beta_{11}\)\(=\)\((\)\(7535 \nu^{16} - 37269 \nu^{15} - 117526 \nu^{14} + 795096 \nu^{13} + 421757 \nu^{12} - 6567131 \nu^{11} + 1851747 \nu^{10} + 26826658 \nu^{9} - 16686955 \nu^{8} - 57856922 \nu^{7} + 43854462 \nu^{6} + 63961259 \nu^{5} - 49293253 \nu^{4} - 30979503 \nu^{3} + 20936803 \nu^{2} + 3871256 \nu - 1015368\)\()/31022\)
\(\beta_{12}\)\(=\)\((\)\(15238 \nu^{16} - 70517 \nu^{15} - 270335 \nu^{14} + 1582686 \nu^{13} + 1432154 \nu^{12} - 13900087 \nu^{11} + 185359 \nu^{10} + 60904047 \nu^{9} - 25130162 \nu^{8} - 141089899 \nu^{7} + 83217368 \nu^{6} + 167347268 \nu^{5} - 106903155 \nu^{4} - 87031309 \nu^{3} + 50859751 \nu^{2} + 11368909 \nu - 3029808\)\()/62044\)
\(\beta_{13}\)\(=\)\((\)\(-5003 \nu^{16} + 23640 \nu^{15} + 78789 \nu^{14} - 491641 \nu^{13} - 318258 \nu^{12} + 3931692 \nu^{11} - 700684 \nu^{10} - 15463187 \nu^{9} + 8035982 \nu^{8} + 32135471 \nu^{7} - 21250233 \nu^{6} - 34474069 \nu^{5} + 23881305 \nu^{4} + 16421556 \nu^{3} - 10374066 \nu^{2} - 1994666 \nu + 580367\)\()/15511\)
\(\beta_{14}\)\(=\)\((\)\(-12523 \nu^{16} + 61545 \nu^{15} + 189010 \nu^{14} - 1274666 \nu^{13} - 638709 \nu^{12} + 10147647 \nu^{11} - 2869597 \nu^{10} - 39770336 \nu^{9} + 23674113 \nu^{8} + 82744596 \nu^{7} - 58289788 \nu^{6} - 89735141 \nu^{5} + 62257615 \nu^{4} + 43991657 \nu^{3} - 25895951 \nu^{2} - 5969520 \nu + 1508748\)\()/31022\)
\(\beta_{15}\)\(=\)\((\)\(-16906 \nu^{16} + 80451 \nu^{15} + 276079 \nu^{14} - 1727532 \nu^{13} - 1200628 \nu^{12} + 14402921 \nu^{11} - 2192535 \nu^{10} - 59629455 \nu^{9} + 30378002 \nu^{8} + 130973989 \nu^{7} - 86449694 \nu^{6} - 148302258 \nu^{5} + 102577437 \nu^{4} + 74063549 \nu^{3} - 46121451 \nu^{2} - 9366361 \nu + 2614408\)\()/31022\)
\(\beta_{16}\)\(=\)\((\)\(-38615 \nu^{16} + 177511 \nu^{15} + 642124 \nu^{14} - 3780588 \nu^{13} - 3038785 \nu^{12} + 31183239 \nu^{11} - 2039975 \nu^{10} - 127378426 \nu^{9} + 54823177 \nu^{8} + 275564722 \nu^{7} - 161135054 \nu^{6} - 306983937 \nu^{5} + 190626927 \nu^{4} + 150613483 \nu^{3} - 84980589 \nu^{2} - 18623890 \nu + 4923384\)\()/62044\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{11} - \beta_{10} + 8 \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{11} - \beta_{8} - \beta_{4} + 9 \beta_{3} + \beta_{2} + 30 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(11 \beta_{11} - 9 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 2 \beta_{3} + 57 \beta_{2} + 11 \beta_{1} + 99\)
\(\nu^{7}\)\(=\)\(-\beta_{16} + \beta_{14} + \beta_{13} + 14 \beta_{11} - \beta_{10} + \beta_{9} - 11 \beta_{8} + 2 \beta_{7} + \beta_{6} - 12 \beta_{4} + 69 \beta_{3} + 15 \beta_{2} + 195 \beta_{1} + 71\)
\(\nu^{8}\)\(=\)\(-2 \beta_{16} - \beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 95 \beta_{11} - 67 \beta_{10} + 12 \beta_{9} - 13 \beta_{8} + 16 \beta_{7} + 14 \beta_{6} + \beta_{5} - 15 \beta_{4} + 33 \beta_{3} + 396 \beta_{2} + 100 \beta_{1} + 654\)
\(\nu^{9}\)\(=\)\(-16 \beta_{16} - \beta_{15} + 15 \beta_{14} + 20 \beta_{13} - \beta_{12} + 141 \beta_{11} - 17 \beta_{10} + 16 \beta_{9} - 89 \beta_{8} + 35 \beta_{7} + 19 \beta_{6} - 110 \beta_{4} + 510 \beta_{3} + 160 \beta_{2} + 1317 \beta_{1} + 556\)
\(\nu^{10}\)\(=\)\(-34 \beta_{16} - 21 \beta_{15} + 18 \beta_{14} + 73 \beta_{13} - 38 \beta_{12} + 758 \beta_{11} - 475 \beta_{10} + 108 \beta_{9} - 119 \beta_{8} + 182 \beta_{7} + 146 \beta_{6} + 16 \beta_{5} - 160 \beta_{4} + 374 \beta_{3} + 2735 \beta_{2} + 863 \beta_{1} + 4461\)
\(\nu^{11}\)\(=\)\(-175 \beta_{16} - 25 \beta_{15} + 157 \beta_{14} + 258 \beta_{13} - 29 \beta_{12} + 1254 \beta_{11} - 192 \beta_{10} + 177 \beta_{9} - 643 \beta_{8} + 419 \beta_{7} + 240 \beta_{6} - 3 \beta_{5} - 920 \beta_{4} + 3738 \beta_{3} + 1489 \beta_{2} + 9078 \beta_{1} + 4378\)
\(\nu^{12}\)\(=\)\(-387 \beta_{16} - 280 \beta_{15} + 209 \beta_{14} + 891 \beta_{13} - 477 \beta_{12} + 5849 \beta_{11} - 3319 \beta_{10} + 891 \beta_{9} - 946 \beta_{8} + 1795 \beta_{7} + 1366 \beta_{6} + 166 \beta_{5} - 1491 \beta_{4} + 3637 \beta_{3} + 18897 \beta_{2} + 7255 \beta_{1} + 30985\)
\(\nu^{13}\)\(=\)\(-1639 \beta_{16} - 379 \beta_{15} + 1418 \beta_{14} + 2756 \beta_{13} - 465 \beta_{12} + 10502 \beta_{11} - 1842 \beta_{10} + 1700 \beta_{9} - 4398 \beta_{8} + 4260 \beta_{7} + 2539 \beta_{6} - 73 \beta_{5} - 7381 \beta_{4} + 27380 \beta_{3} + 12932 \beta_{2} + 63356 \beta_{1} + 34535\)
\(\nu^{14}\)\(=\)\(-3726 \beta_{16} - 3056 \beta_{15} + 2014 \beta_{14} + 9160 \beta_{13} - 4993 \beta_{12} + 44424 \beta_{11} - 23120 \beta_{10} + 7152 \beta_{9} - 6988 \beta_{8} + 16345 \beta_{7} + 12081 \beta_{6} + 1413 \beta_{5} - 12968 \beta_{4} + 32623 \beta_{3} + 130973 \beta_{2} + 59806 \beta_{1} + 217789\)
\(\nu^{15}\)\(=\)\(-14164 \beta_{16} - 4567 \beta_{15} + 11851 \beta_{14} + 26556 \beta_{13} - 5729 \beta_{12} + 85086 \beta_{11} - 16278 \beta_{10} + 15288 \beta_{9} - 29152 \beta_{8} + 39557 \beta_{7} + 24381 \beta_{6} - 1106 \beta_{5} - 57918 \beta_{4} + 200835 \beta_{3} + 107939 \beta_{2} + 446006 \beta_{1} + 271845\)
\(\nu^{16}\)\(=\)\(-32872 \beta_{16} - 29872 \beta_{15} + 17602 \beta_{14} + 85786 \beta_{13} - 47253 \beta_{12} + 334746 \beta_{11} - 161278 \beta_{10} + 57104 \beta_{9} - 49396 \beta_{8} + 141417 \beta_{7} + 103157 \beta_{6} + 10648 \beta_{5} - 108361 \beta_{4} + 278635 \beta_{3} + 911836 \beta_{2} + 485008 \beta_{1} + 1544374\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.74886
2.71108
2.50383
2.12059
1.89951
1.69575
1.08305
0.829580
0.184653
0.159847
−0.320785
−1.02928
−1.30924
−1.58492
−1.59605
−2.53341
−2.56306
−2.74886 0 5.55624 −1.00000 0 4.26511 −9.77563 0 2.74886
1.2 −2.71108 0 5.34994 −1.00000 0 −0.946340 −9.08194 0 2.71108
1.3 −2.50383 0 4.26915 −1.00000 0 −2.88177 −5.68155 0 2.50383
1.4 −2.12059 0 2.49688 −1.00000 0 4.22149 −1.05368 0 2.12059
1.5 −1.89951 0 1.60815 −1.00000 0 2.75204 0.744322 0 1.89951
1.6 −1.69575 0 0.875568 −1.00000 0 −1.30460 1.90676 0 1.69575
1.7 −1.08305 0 −0.826999 −1.00000 0 −2.85554 3.06179 0 1.08305
1.8 −0.829580 0 −1.31180 −1.00000 0 1.06171 2.74740 0 0.829580
1.9 −0.184653 0 −1.96590 −1.00000 0 2.68806 0.732314 0 0.184653
1.10 −0.159847 0 −1.97445 −1.00000 0 −1.46854 0.635302 0 0.159847
1.11 0.320785 0 −1.89710 −1.00000 0 4.94917 −1.25013 0 −0.320785
1.12 1.02928 0 −0.940578 −1.00000 0 −4.61830 −3.02668 0 −1.02928
1.13 1.30924 0 −0.285886 −1.00000 0 4.28141 −2.99278 0 −1.30924
1.14 1.58492 0 0.511971 −1.00000 0 −2.52227 −2.35841 0 −1.58492
1.15 1.59605 0 0.547372 −1.00000 0 1.65189 −2.31847 0 −1.59605
1.16 2.53341 0 4.41816 −1.00000 0 3.58669 6.12619 0 −2.53341
1.17 2.56306 0 4.56927 −1.00000 0 −0.860224 6.58519 0 −2.56306
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

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 4005.2.a.w 17
3.b odd 2 1 4005.2.a.x yes 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4005.2.a.w 17 1.a even 1 1 trivial
4005.2.a.x yes 17 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(89\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{17} + \cdots\)
\(T_{7}^{17} - \cdots\)
\(T_{11}^{17} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 19 T^{2} + 55 T^{3} + 139 T^{4} + 309 T^{5} + 630 T^{6} + 1185 T^{7} + 2104 T^{8} + 3558 T^{9} + 5847 T^{10} + 9393 T^{11} + 14947 T^{12} + 23462 T^{13} + 36422 T^{14} + 55321 T^{15} + 82057 T^{16} + 117804 T^{17} + 164114 T^{18} + 221284 T^{19} + 291376 T^{20} + 375392 T^{21} + 478304 T^{22} + 601152 T^{23} + 748416 T^{24} + 910848 T^{25} + 1077248 T^{26} + 1213440 T^{27} + 1290240 T^{28} + 1265664 T^{29} + 1138688 T^{30} + 901120 T^{31} + 622592 T^{32} + 327680 T^{33} + 131072 T^{34} \)
$3$ \( \)
$5$ \( ( 1 + T )^{17} \)
$7$ \( 1 - 12 T + 111 T^{2} - 764 T^{3} + 4605 T^{4} - 23986 T^{5} + 114013 T^{6} - 492098 T^{7} + 1981418 T^{8} - 7422378 T^{9} + 26304718 T^{10} - 88062636 T^{11} + 281969210 T^{12} - 862666714 T^{13} + 2547360412 T^{14} - 7251180150 T^{15} + 20057796276 T^{16} - 53764090300 T^{17} + 140404573932 T^{18} - 355307827350 T^{19} + 873744621316 T^{20} - 2071262780314 T^{21} + 4739056512470 T^{22} - 10360481062764 T^{23} + 21663066375874 T^{24} - 42788532116778 T^{25} + 79957363274726 T^{26} - 139005505082402 T^{27} + 225440953949659 T^{28} - 331997114803186 T^{29} + 446173892924235 T^{30} - 518162427656636 T^{31} + 526979327603673 T^{32} - 398795166835212 T^{33} + 232630513987207 T^{34} \)
$11$ \( 1 + 2 T + 79 T^{2} + 174 T^{3} + 3380 T^{4} + 8012 T^{5} + 101888 T^{6} + 253738 T^{7} + 2405468 T^{8} + 6125288 T^{9} + 46935436 T^{10} + 119101822 T^{11} + 779897016 T^{12} + 1928251668 T^{13} + 11217890076 T^{14} + 26540972170 T^{15} + 140860529778 T^{16} + 314200764012 T^{17} + 1549465827558 T^{18} + 3211457632570 T^{19} + 14931011691156 T^{20} + 28231532671188 T^{21} + 125603194323816 T^{22} + 210996142884142 T^{23} + 914638867291556 T^{24} + 1313009881482728 T^{25} + 5671967716374388 T^{26} + 6581310243408538 T^{27} + 29069835495213568 T^{28} + 25145088154288652 T^{29} + 116686767046486780 T^{30} + 66076471043483934 T^{31} + 330002605383836429 T^{32} + 91899459727144322 T^{33} + 505447028499293771 T^{34} \)
$13$ \( 1 - 8 T + 119 T^{2} - 698 T^{3} + 6293 T^{4} - 29580 T^{5} + 207927 T^{6} - 815536 T^{7} + 4955708 T^{8} - 16454356 T^{9} + 92109122 T^{10} - 258134796 T^{11} + 1411470574 T^{12} - 3320231152 T^{13} + 18986614136 T^{14} - 38414077962 T^{15} + 242856804484 T^{16} - 465535252992 T^{17} + 3157138458292 T^{18} - 6491979175578 T^{19} + 41713591256792 T^{20} - 94829121932272 T^{21} + 524069143832182 T^{22} - 1245967356545964 T^{23} + 5779710807672074 T^{24} - 13422323683470676 T^{25} + 52552802378771084 T^{26} - 112428563008566064 T^{27} + 372638534250931299 T^{28} - 689157357922987980 T^{29} + 1905993045785048129 T^{30} - 2748288717218103722 T^{31} + 6091121268676800083 T^{32} - 5323332873465438728 T^{33} + 8650415919381337933 T^{34} \)
$17$ \( 1 + 10 T + 189 T^{2} + 1432 T^{3} + 15847 T^{4} + 98408 T^{5} + 817515 T^{6} + 4322504 T^{7} + 29595192 T^{8} + 135937078 T^{9} + 807459670 T^{10} + 3263747640 T^{11} + 17514468522 T^{12} + 63329689330 T^{13} + 321426522742 T^{14} + 1077066935328 T^{15} + 5447632151042 T^{16} + 17937813805916 T^{17} + 92609746567714 T^{18} + 311272344309792 T^{19} + 1579168506231446 T^{20} + 5289358982530930 T^{21} + 24868040732241354 T^{22} + 78778933859087160 T^{23} + 331331929488817910 T^{24} + 948264083366297398 T^{25} + 3509630973801002424 T^{26} + 8714141698666404296 T^{27} + 28017789309934591995 T^{28} + 57334689121306320488 T^{29} + \)\(15\!\cdots\!39\)\( T^{30} + \)\(24\!\cdots\!28\)\( T^{31} + \)\(54\!\cdots\!77\)\( T^{32} + \)\(48\!\cdots\!10\)\( T^{33} + \)\(82\!\cdots\!77\)\( T^{34} \)
$19$ \( 1 - 32 T + 649 T^{2} - 9840 T^{3} + 122670 T^{4} - 1312928 T^{5} + 12423522 T^{6} - 105914584 T^{7} + 824809862 T^{8} - 5926817512 T^{9} + 39600117734 T^{10} - 247488951224 T^{11} + 1453456854138 T^{12} - 8049919891296 T^{13} + 42161437665510 T^{14} - 209243322246720 T^{15} + 985391521687876 T^{16} - 4406958680928496 T^{17} + 18722438912069644 T^{18} - 75536839331065920 T^{19} + 289185300947733090 T^{20} - 1049073610153586016 T^{21} + 3598903063074247662 T^{22} - 11643335748099108344 T^{23} + 35397426103495319426 T^{24} - \)\(10\!\cdots\!92\)\( T^{25} + \)\(26\!\cdots\!98\)\( T^{26} - \)\(64\!\cdots\!84\)\( T^{27} + \)\(14\!\cdots\!18\)\( T^{28} - \)\(29\!\cdots\!08\)\( T^{29} + \)\(51\!\cdots\!30\)\( T^{30} - \)\(78\!\cdots\!40\)\( T^{31} + \)\(98\!\cdots\!51\)\( T^{32} - \)\(92\!\cdots\!92\)\( T^{33} + \)\(54\!\cdots\!39\)\( T^{34} \)
$23$ \( 1 + 15 T + 349 T^{2} + 4080 T^{3} + 55400 T^{4} + 534564 T^{5} + 5450980 T^{6} + 45021280 T^{7} + 378308814 T^{8} + 2740873508 T^{9} + 19867937230 T^{10} + 128408646752 T^{11} + 823950233956 T^{12} + 4804293842108 T^{13} + 27711150712360 T^{14} + 146779037309776 T^{15} + 767862197298948 T^{16} + 3706280807521370 T^{17} + 17660830537875804 T^{18} + 77646110736871504 T^{19} + 337161570717284120 T^{20} + 1344438393069344828 T^{21} + 5303226320671062908 T^{22} + 19009088177219282528 T^{23} + 67646858260102691810 T^{24} + \)\(21\!\cdots\!48\)\( T^{25} + \)\(68\!\cdots\!82\)\( T^{26} + \)\(18\!\cdots\!20\)\( T^{27} + \)\(51\!\cdots\!60\)\( T^{28} + \)\(11\!\cdots\!44\)\( T^{29} + \)\(27\!\cdots\!00\)\( T^{30} + \)\(47\!\cdots\!20\)\( T^{31} + \)\(93\!\cdots\!43\)\( T^{32} + \)\(91\!\cdots\!15\)\( T^{33} + \)\(14\!\cdots\!03\)\( T^{34} \)
$29$ \( 1 + T + 277 T^{2} - 10 T^{3} + 38971 T^{4} - 30631 T^{5} + 3724135 T^{6} - 4860736 T^{7} + 269907750 T^{8} - 442506636 T^{9} + 15658915452 T^{10} - 28574494534 T^{11} + 749448907248 T^{12} - 1416417378440 T^{13} + 30111798773152 T^{14} - 55984885639524 T^{15} + 1025584324573126 T^{16} - 1796861752250804 T^{17} + 29741945412620654 T^{18} - 47083288822839684 T^{19} + 734396660278404128 T^{20} - 1001805099840421640 T^{21} + 15372058204450907952 T^{22} - 16996775734610227414 T^{23} + \)\(27\!\cdots\!68\)\( T^{24} - \)\(22\!\cdots\!96\)\( T^{25} + \)\(39\!\cdots\!50\)\( T^{26} - \)\(20\!\cdots\!36\)\( T^{27} + \)\(45\!\cdots\!15\)\( T^{28} - \)\(10\!\cdots\!71\)\( T^{29} + \)\(39\!\cdots\!19\)\( T^{30} - \)\(29\!\cdots\!10\)\( T^{31} + \)\(23\!\cdots\!73\)\( T^{32} + \)\(25\!\cdots\!21\)\( T^{33} + \)\(72\!\cdots\!09\)\( T^{34} \)
$31$ \( 1 - 18 T + 499 T^{2} - 6682 T^{3} + 108624 T^{4} - 1177976 T^{5} + 14347488 T^{6} - 132034198 T^{7} + 1320417160 T^{8} - 10617407872 T^{9} + 91375214048 T^{10} - 654820383714 T^{11} + 4988056158032 T^{12} - 32300161858568 T^{13} + 221726048831296 T^{14} - 1309311299293806 T^{15} + 8189627555033130 T^{16} - 44307211662986204 T^{17} + 253878454206027030 T^{18} - 1258248158621347566 T^{19} + 6605440720733139136 T^{20} - 29829877779786577928 T^{21} + \)\(14\!\cdots\!32\)\( T^{22} - \)\(58\!\cdots\!34\)\( T^{23} + \)\(25\!\cdots\!28\)\( T^{24} - \)\(90\!\cdots\!52\)\( T^{25} + \)\(34\!\cdots\!60\)\( T^{26} - \)\(10\!\cdots\!98\)\( T^{27} + \)\(36\!\cdots\!28\)\( T^{28} - \)\(92\!\cdots\!36\)\( T^{29} + \)\(26\!\cdots\!84\)\( T^{30} - \)\(50\!\cdots\!22\)\( T^{31} + \)\(11\!\cdots\!49\)\( T^{32} - \)\(13\!\cdots\!58\)\( T^{33} + \)\(22\!\cdots\!11\)\( T^{34} \)
$37$ \( 1 - 12 T + 397 T^{2} - 4042 T^{3} + 75087 T^{4} - 660608 T^{5} + 9018169 T^{6} - 69344954 T^{7} + 772788990 T^{8} - 5234765824 T^{9} + 50506451380 T^{10} - 303764569926 T^{11} + 2649920873188 T^{12} - 14338736065870 T^{13} + 117569063272982 T^{14} - 586731695059110 T^{15} + 4664616261518458 T^{16} - 22220824794657764 T^{17} + 172590801676182946 T^{18} - 803235690535921590 T^{19} + 5955225761966357246 T^{20} - 26873099923946985070 T^{21} + \)\(18\!\cdots\!16\)\( T^{22} - \)\(77\!\cdots\!34\)\( T^{23} + \)\(47\!\cdots\!40\)\( T^{24} - \)\(18\!\cdots\!04\)\( T^{25} + \)\(10\!\cdots\!30\)\( T^{26} - \)\(33\!\cdots\!46\)\( T^{27} + \)\(16\!\cdots\!97\)\( T^{28} - \)\(43\!\cdots\!48\)\( T^{29} + \)\(18\!\cdots\!39\)\( T^{30} - \)\(36\!\cdots\!38\)\( T^{31} + \)\(13\!\cdots\!21\)\( T^{32} - \)\(14\!\cdots\!92\)\( T^{33} + \)\(45\!\cdots\!17\)\( T^{34} \)
$41$ \( 1 - 7 T + 449 T^{2} - 2752 T^{3} + 96431 T^{4} - 523909 T^{5} + 13277743 T^{6} - 64407250 T^{7} + 1324366806 T^{8} - 5759629112 T^{9} + 102595917832 T^{10} - 401547014206 T^{11} + 6476080021990 T^{12} - 22967595824928 T^{13} + 345629831262422 T^{14} - 1124889830723164 T^{15} + 16039592728145046 T^{16} - 48730134350190848 T^{17} + 657623301853946886 T^{18} - 1890939805445638684 T^{19} + 23821153600437386662 T^{20} - 64900936545844370208 T^{21} + \)\(75\!\cdots\!90\)\( T^{22} - \)\(19\!\cdots\!46\)\( T^{23} + \)\(19\!\cdots\!92\)\( T^{24} - \)\(45\!\cdots\!52\)\( T^{25} + \)\(43\!\cdots\!66\)\( T^{26} - \)\(86\!\cdots\!50\)\( T^{27} + \)\(73\!\cdots\!63\)\( T^{28} - \)\(11\!\cdots\!29\)\( T^{29} + \)\(89\!\cdots\!51\)\( T^{30} - \)\(10\!\cdots\!72\)\( T^{31} + \)\(69\!\cdots\!49\)\( T^{32} - \)\(44\!\cdots\!87\)\( T^{33} + \)\(26\!\cdots\!81\)\( T^{34} \)
$43$ \( 1 - 28 T + 841 T^{2} - 15904 T^{3} + 292115 T^{4} - 4292066 T^{5} + 60257593 T^{6} - 734268488 T^{7} + 8538293022 T^{8} - 89489235518 T^{9} + 896888323688 T^{10} - 8267722562308 T^{11} + 73060494204556 T^{12} - 600766728947240 T^{13} + 4745336587342980 T^{14} - 35106921953161868 T^{15} + 249816973801734840 T^{16} - 1669967136500211176 T^{17} + 10742129873474598120 T^{18} - 64912698691396293932 T^{19} + \)\(37\!\cdots\!60\)\( T^{20} - \)\(20\!\cdots\!40\)\( T^{21} + \)\(10\!\cdots\!08\)\( T^{22} - \)\(52\!\cdots\!92\)\( T^{23} + \)\(24\!\cdots\!16\)\( T^{24} - \)\(10\!\cdots\!18\)\( T^{25} + \)\(42\!\cdots\!46\)\( T^{26} - \)\(15\!\cdots\!12\)\( T^{27} + \)\(55\!\cdots\!51\)\( T^{28} - \)\(17\!\cdots\!66\)\( T^{29} + \)\(50\!\cdots\!45\)\( T^{30} - \)\(11\!\cdots\!96\)\( T^{31} + \)\(26\!\cdots\!87\)\( T^{32} - \)\(38\!\cdots\!28\)\( T^{33} + \)\(58\!\cdots\!43\)\( T^{34} \)
$47$ \( 1 + 26 T + 807 T^{2} + 15666 T^{3} + 293205 T^{4} + 4536152 T^{5} + 65133673 T^{6} + 839708870 T^{7} + 10052167162 T^{8} + 111409594520 T^{9} + 1155464450340 T^{10} + 11247324853704 T^{11} + 103177625880472 T^{12} + 894596963517952 T^{13} + 7346317690892662 T^{14} + 57229337503156880 T^{15} + 423478126770338274 T^{16} + 2977412877546770444 T^{17} + 19903471958205898878 T^{18} + \)\(12\!\cdots\!20\)\( T^{19} + \)\(76\!\cdots\!26\)\( T^{20} + \)\(43\!\cdots\!12\)\( T^{21} + \)\(23\!\cdots\!04\)\( T^{22} + \)\(12\!\cdots\!16\)\( T^{23} + \)\(58\!\cdots\!20\)\( T^{24} + \)\(26\!\cdots\!20\)\( T^{25} + \)\(11\!\cdots\!54\)\( T^{26} + \)\(44\!\cdots\!30\)\( T^{27} + \)\(16\!\cdots\!19\)\( T^{28} + \)\(52\!\cdots\!32\)\( T^{29} + \)\(16\!\cdots\!35\)\( T^{30} + \)\(40\!\cdots\!54\)\( T^{31} + \)\(97\!\cdots\!01\)\( T^{32} + \)\(14\!\cdots\!46\)\( T^{33} + \)\(26\!\cdots\!87\)\( T^{34} \)
$53$ \( 1 + 12 T + 699 T^{2} + 7718 T^{3} + 239613 T^{4} + 2428492 T^{5} + 53191169 T^{6} + 494789262 T^{7} + 8537303878 T^{8} + 72902439294 T^{9} + 1049429267600 T^{10} + 8223861909526 T^{11} + 102185401517448 T^{12} + 733920899128816 T^{13} + 8045018467515776 T^{14} + 52817689975366662 T^{15} + 518248862436682984 T^{16} + 3096548522935099796 T^{17} + 27467189709144198152 T^{18} + \)\(14\!\cdots\!58\)\( T^{19} + \)\(11\!\cdots\!52\)\( T^{20} + \)\(57\!\cdots\!96\)\( T^{21} + \)\(42\!\cdots\!64\)\( T^{22} + \)\(18\!\cdots\!54\)\( T^{23} + \)\(12\!\cdots\!00\)\( T^{24} + \)\(45\!\cdots\!34\)\( T^{25} + \)\(28\!\cdots\!74\)\( T^{26} + \)\(86\!\cdots\!38\)\( T^{27} + \)\(49\!\cdots\!93\)\( T^{28} + \)\(11\!\cdots\!72\)\( T^{29} + \)\(62\!\cdots\!49\)\( T^{30} + \)\(10\!\cdots\!42\)\( T^{31} + \)\(51\!\cdots\!43\)\( T^{32} + \)\(46\!\cdots\!52\)\( T^{33} + \)\(20\!\cdots\!13\)\( T^{34} \)
$59$ \( 1 - 23 T + 733 T^{2} - 12504 T^{3} + 236063 T^{4} - 3212595 T^{5} + 45358887 T^{6} - 510421708 T^{7} + 5830371788 T^{8} - 55076568224 T^{9} + 526100289408 T^{10} - 4169232564352 T^{11} + 33833087417970 T^{12} - 221987587494690 T^{13} + 1573186086806432 T^{14} - 8634291575162620 T^{15} + 62230436154361394 T^{16} - 365440139076269452 T^{17} + 3671595733107322246 T^{18} - 30055968973141080220 T^{19} + \)\(32\!\cdots\!28\)\( T^{20} - \)\(26\!\cdots\!90\)\( T^{21} + \)\(24\!\cdots\!30\)\( T^{22} - \)\(17\!\cdots\!32\)\( T^{23} + \)\(13\!\cdots\!52\)\( T^{24} - \)\(80\!\cdots\!04\)\( T^{25} + \)\(50\!\cdots\!32\)\( T^{26} - \)\(26\!\cdots\!08\)\( T^{27} + \)\(13\!\cdots\!33\)\( T^{28} - \)\(57\!\cdots\!95\)\( T^{29} + \)\(24\!\cdots\!77\)\( T^{30} - \)\(77\!\cdots\!44\)\( T^{31} + \)\(26\!\cdots\!67\)\( T^{32} - \)\(49\!\cdots\!43\)\( T^{33} + \)\(12\!\cdots\!19\)\( T^{34} \)
$61$ \( 1 - 26 T + 1043 T^{2} - 20348 T^{3} + 477330 T^{4} - 7564728 T^{5} + 132626666 T^{6} - 1784504004 T^{7} + 25607720218 T^{8} - 300643836032 T^{9} + 3696421602182 T^{10} - 38549330641580 T^{11} + 416981322358510 T^{12} - 3908188136042216 T^{13} + 37798731225533142 T^{14} - 320637551526387668 T^{15} + 2799695048262362968 T^{16} - 21560466918534222860 T^{17} + \)\(17\!\cdots\!48\)\( T^{18} - \)\(11\!\cdots\!28\)\( T^{19} + \)\(85\!\cdots\!02\)\( T^{20} - \)\(54\!\cdots\!56\)\( T^{21} + \)\(35\!\cdots\!10\)\( T^{22} - \)\(19\!\cdots\!80\)\( T^{23} + \)\(11\!\cdots\!22\)\( T^{24} - \)\(57\!\cdots\!92\)\( T^{25} + \)\(29\!\cdots\!38\)\( T^{26} - \)\(12\!\cdots\!04\)\( T^{27} + \)\(57\!\cdots\!26\)\( T^{28} - \)\(20\!\cdots\!88\)\( T^{29} + \)\(77\!\cdots\!30\)\( T^{30} - \)\(20\!\cdots\!68\)\( T^{31} + \)\(62\!\cdots\!43\)\( T^{32} - \)\(95\!\cdots\!86\)\( T^{33} + \)\(22\!\cdots\!21\)\( T^{34} \)
$67$ \( 1 - 31 T + 1295 T^{2} - 29866 T^{3} + 739374 T^{4} - 13697448 T^{5} + 255183014 T^{6} - 3966806290 T^{7} + 60566700648 T^{8} - 811560889924 T^{9} + 10581781605176 T^{10} - 124342473451358 T^{11} + 1416364457565758 T^{12} - 14756241779765080 T^{13} + 148800609577196406 T^{14} - 1383524737566083046 T^{15} + 12443670342641564938 T^{16} - \)\(10\!\cdots\!46\)\( T^{17} + \)\(83\!\cdots\!46\)\( T^{18} - \)\(62\!\cdots\!94\)\( T^{19} + \)\(44\!\cdots\!78\)\( T^{20} - \)\(29\!\cdots\!80\)\( T^{21} + \)\(19\!\cdots\!06\)\( T^{22} - \)\(11\!\cdots\!02\)\( T^{23} + \)\(64\!\cdots\!48\)\( T^{24} - \)\(32\!\cdots\!84\)\( T^{25} + \)\(16\!\cdots\!56\)\( T^{26} - \)\(72\!\cdots\!10\)\( T^{27} + \)\(31\!\cdots\!62\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{29} + \)\(40\!\cdots\!38\)\( T^{30} - \)\(10\!\cdots\!14\)\( T^{31} + \)\(31\!\cdots\!85\)\( T^{32} - \)\(51\!\cdots\!11\)\( T^{33} + \)\(11\!\cdots\!27\)\( T^{34} \)
$71$ \( 1 - 2 T + 545 T^{2} - 2046 T^{3} + 150934 T^{4} - 777752 T^{5} + 28767150 T^{6} - 175768122 T^{7} + 4255604486 T^{8} - 28180002080 T^{9} + 519308265238 T^{10} - 3506347517406 T^{11} + 54139179488374 T^{12} - 357744712977992 T^{13} + 4921493965275326 T^{14} - 31076443488011178 T^{15} + 394426582918463016 T^{16} - 2353419327908305916 T^{17} + 28004287387210874136 T^{18} - \)\(15\!\cdots\!98\)\( T^{19} + \)\(17\!\cdots\!86\)\( T^{20} - \)\(90\!\cdots\!52\)\( T^{21} + \)\(97\!\cdots\!74\)\( T^{22} - \)\(44\!\cdots\!26\)\( T^{23} + \)\(47\!\cdots\!58\)\( T^{24} - \)\(18\!\cdots\!80\)\( T^{25} + \)\(19\!\cdots\!66\)\( T^{26} - \)\(57\!\cdots\!22\)\( T^{27} + \)\(66\!\cdots\!50\)\( T^{28} - \)\(12\!\cdots\!32\)\( T^{29} + \)\(17\!\cdots\!74\)\( T^{30} - \)\(16\!\cdots\!26\)\( T^{31} + \)\(32\!\cdots\!95\)\( T^{32} - \)\(83\!\cdots\!42\)\( T^{33} + \)\(29\!\cdots\!91\)\( T^{34} \)
$73$ \( 1 - 33 T + 1289 T^{2} - 28770 T^{3} + 677538 T^{4} - 11856980 T^{5} + 212943502 T^{6} - 3128588334 T^{7} + 46709496756 T^{8} - 597840744796 T^{9} + 7742309945596 T^{10} - 88217100182922 T^{11} + 1014895374809458 T^{12} - 10429181854194124 T^{13} + 108088538781403166 T^{14} - 1009172259685056806 T^{15} + 9497402766474017934 T^{16} - 80830906277803361350 T^{17} + \)\(69\!\cdots\!82\)\( T^{18} - \)\(53\!\cdots\!74\)\( T^{19} + \)\(42\!\cdots\!22\)\( T^{20} - \)\(29\!\cdots\!84\)\( T^{21} + \)\(21\!\cdots\!94\)\( T^{22} - \)\(13\!\cdots\!58\)\( T^{23} + \)\(85\!\cdots\!12\)\( T^{24} - \)\(48\!\cdots\!76\)\( T^{25} + \)\(27\!\cdots\!28\)\( T^{26} - \)\(13\!\cdots\!66\)\( T^{27} + \)\(66\!\cdots\!54\)\( T^{28} - \)\(27\!\cdots\!80\)\( T^{29} + \)\(11\!\cdots\!54\)\( T^{30} - \)\(35\!\cdots\!30\)\( T^{31} + \)\(11\!\cdots\!73\)\( T^{32} - \)\(21\!\cdots\!13\)\( T^{33} + \)\(47\!\cdots\!53\)\( T^{34} \)
$79$ \( 1 - 33 T + 1065 T^{2} - 22456 T^{3} + 453093 T^{4} - 7380195 T^{5} + 115556793 T^{6} - 1567772582 T^{7} + 20615512790 T^{8} - 242982107518 T^{9} + 2803855212532 T^{10} - 29591533525458 T^{11} + 309421437939350 T^{12} - 3002923498258490 T^{13} + 29262838389737800 T^{14} - 267788772394368092 T^{15} + 2489902131579605596 T^{16} - 21900790885351587120 T^{17} + \)\(19\!\cdots\!84\)\( T^{18} - \)\(16\!\cdots\!72\)\( T^{19} + \)\(14\!\cdots\!00\)\( T^{20} - \)\(11\!\cdots\!90\)\( T^{21} + \)\(95\!\cdots\!50\)\( T^{22} - \)\(71\!\cdots\!18\)\( T^{23} + \)\(53\!\cdots\!88\)\( T^{24} - \)\(36\!\cdots\!98\)\( T^{25} + \)\(24\!\cdots\!10\)\( T^{26} - \)\(14\!\cdots\!82\)\( T^{27} + \)\(86\!\cdots\!47\)\( T^{28} - \)\(43\!\cdots\!95\)\( T^{29} + \)\(21\!\cdots\!27\)\( T^{30} - \)\(82\!\cdots\!36\)\( T^{31} + \)\(31\!\cdots\!35\)\( T^{32} - \)\(75\!\cdots\!93\)\( T^{33} + \)\(18\!\cdots\!59\)\( T^{34} \)
$83$ \( 1 + 13 T + 531 T^{2} + 5020 T^{3} + 138914 T^{4} + 1148780 T^{5} + 26435034 T^{6} + 208330276 T^{7} + 4087943500 T^{8} + 31713279340 T^{9} + 534437566684 T^{10} + 4126832741836 T^{11} + 60834578745426 T^{12} + 468823122766516 T^{13} + 6173634810686442 T^{14} + 46907231445630772 T^{15} + 565077439895866462 T^{16} + 4142730188560919694 T^{17} + 46901427511356916346 T^{18} + \)\(32\!\cdots\!08\)\( T^{19} + \)\(35\!\cdots\!54\)\( T^{20} + \)\(22\!\cdots\!36\)\( T^{21} + \)\(23\!\cdots\!18\)\( T^{22} + \)\(13\!\cdots\!84\)\( T^{23} + \)\(14\!\cdots\!68\)\( T^{24} + \)\(71\!\cdots\!40\)\( T^{25} + \)\(76\!\cdots\!00\)\( T^{26} + \)\(32\!\cdots\!24\)\( T^{27} + \)\(34\!\cdots\!78\)\( T^{28} + \)\(12\!\cdots\!80\)\( T^{29} + \)\(12\!\cdots\!82\)\( T^{30} + \)\(36\!\cdots\!80\)\( T^{31} + \)\(32\!\cdots\!17\)\( T^{32} + \)\(65\!\cdots\!53\)\( T^{33} + \)\(42\!\cdots\!23\)\( T^{34} \)
$89$ \( ( 1 - T )^{17} \)
$97$ \( 1 - 45 T + 1593 T^{2} - 39180 T^{3} + 844530 T^{4} - 15203556 T^{5} + 252728450 T^{6} - 3753208580 T^{7} + 52965948112 T^{8} - 688925759476 T^{9} + 8649158787880 T^{10} - 101786189329420 T^{11} + 1169853811449590 T^{12} - 12761237898693980 T^{13} + 137388122367542326 T^{14} - 1416645831506652900 T^{15} + 14522418446246620814 T^{16} - \)\(14\!\cdots\!82\)\( T^{17} + \)\(14\!\cdots\!58\)\( T^{18} - \)\(13\!\cdots\!00\)\( T^{19} + \)\(12\!\cdots\!98\)\( T^{20} - \)\(11\!\cdots\!80\)\( T^{21} + \)\(10\!\cdots\!30\)\( T^{22} - \)\(84\!\cdots\!80\)\( T^{23} + \)\(69\!\cdots\!40\)\( T^{24} - \)\(53\!\cdots\!36\)\( T^{25} + \)\(40\!\cdots\!04\)\( T^{26} - \)\(27\!\cdots\!20\)\( T^{27} + \)\(18\!\cdots\!50\)\( T^{28} - \)\(10\!\cdots\!96\)\( T^{29} + \)\(56\!\cdots\!10\)\( T^{30} - \)\(25\!\cdots\!20\)\( T^{31} + \)\(10\!\cdots\!49\)\( T^{32} - \)\(27\!\cdots\!45\)\( T^{33} + \)\(59\!\cdots\!37\)\( T^{34} \)
show more
show less