Properties

Label 8030.2.a.bf
Level 8030
Weight 2
Character orbit 8030.a
Self dual yes
Analytic conductor 64.120
Analytic rank 1
Dimension 15
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 4 x^{14} - 23 x^{13} + 99 x^{12} + 191 x^{11} - 922 x^{10} - 702 x^{9} + 4108 x^{8} + 957 x^{7} - 8875 x^{6} + 479 x^{5} + 7698 x^{4} - 1731 x^{3} - 626 x^{2} + 46 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} -\beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} -\beta_{3} q^{7} + q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} -\beta_{3} q^{7} + q^{8} + ( 1 + \beta_{2} ) q^{9} - q^{10} - q^{11} -\beta_{1} q^{12} -\beta_{8} q^{13} -\beta_{3} q^{14} + \beta_{1} q^{15} + q^{16} + ( \beta_{1} + \beta_{5} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + ( -1 - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{19} - q^{20} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{21} - q^{22} + ( -1 + \beta_{1} + \beta_{3} - \beta_{11} - \beta_{13} ) q^{23} -\beta_{1} q^{24} + q^{25} -\beta_{8} q^{26} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{27} -\beta_{3} q^{28} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} - \beta_{14} ) q^{29} + \beta_{1} q^{30} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{31} + q^{32} + \beta_{1} q^{33} + ( \beta_{1} + \beta_{5} ) q^{34} + \beta_{3} q^{35} + ( 1 + \beta_{2} ) q^{36} + ( -2 + \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{37} + ( -1 - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{38} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{39} - q^{40} + ( 1 - \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{41} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{42} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} ) q^{43} - q^{44} + ( -1 - \beta_{2} ) q^{45} + ( -1 + \beta_{1} + \beta_{3} - \beta_{11} - \beta_{13} ) q^{46} + ( -1 + \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{47} -\beta_{1} q^{48} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{49} + q^{50} + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{51} -\beta_{8} q^{52} + ( -3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + \beta_{13} + \beta_{14} ) q^{53} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{54} + q^{55} -\beta_{3} q^{56} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{57} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} - \beta_{14} ) q^{58} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{59} + \beta_{1} q^{60} + ( -5 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{14} ) q^{61} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{11} - \beta_{12} ) q^{62} + ( \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{63} + q^{64} + \beta_{8} q^{65} + \beta_{1} q^{66} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{67} + ( \beta_{1} + \beta_{5} ) q^{68} + ( -2 + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} ) q^{69} + \beta_{3} q^{70} + ( -\beta_{1} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{71} + ( 1 + \beta_{2} ) q^{72} + q^{73} + ( -2 + \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{74} -\beta_{1} q^{75} + ( -1 - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{76} + \beta_{3} q^{77} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{78} + ( -2 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{79} - q^{80} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} ) q^{81} + ( 1 - \beta_{1} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{82} + ( -4 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{13} + 3 \beta_{14} ) q^{83} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{84} + ( -\beta_{1} - \beta_{5} ) q^{85} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{12} ) q^{86} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{87} - q^{88} + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} + 2 \beta_{12} + \beta_{14} ) q^{89} + ( -1 - \beta_{2} ) q^{90} + ( -2 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{8} + 2 \beta_{9} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{91} + ( -1 + \beta_{1} + \beta_{3} - \beta_{11} - \beta_{13} ) q^{92} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{93} + ( -1 + \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{13} - \beta_{14} ) q^{94} + ( 1 + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{12} ) q^{95} -\beta_{1} q^{96} + ( -1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{12} + 2 \beta_{13} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{98} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 4q^{3} + 15q^{4} - 15q^{5} - 4q^{6} - 6q^{7} + 15q^{8} + 17q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 4q^{3} + 15q^{4} - 15q^{5} - 4q^{6} - 6q^{7} + 15q^{8} + 17q^{9} - 15q^{10} - 15q^{11} - 4q^{12} - 6q^{13} - 6q^{14} + 4q^{15} + 15q^{16} + 2q^{17} + 17q^{18} - 8q^{19} - 15q^{20} - 17q^{21} - 15q^{22} - 4q^{23} - 4q^{24} + 15q^{25} - 6q^{26} - 19q^{27} - 6q^{28} - 13q^{29} + 4q^{30} - 20q^{31} + 15q^{32} + 4q^{33} + 2q^{34} + 6q^{35} + 17q^{36} - 15q^{37} - 8q^{38} - 11q^{39} - 15q^{40} + 2q^{41} - 17q^{42} - 26q^{43} - 15q^{44} - 17q^{45} - 4q^{46} - 14q^{47} - 4q^{48} + 11q^{49} + 15q^{50} - 39q^{51} - 6q^{52} - 21q^{53} - 19q^{54} + 15q^{55} - 6q^{56} + q^{57} - 13q^{58} - 14q^{59} + 4q^{60} - 45q^{61} - 20q^{62} - 17q^{63} + 15q^{64} + 6q^{65} + 4q^{66} - 10q^{67} + 2q^{68} - 23q^{69} + 6q^{70} - 9q^{71} + 17q^{72} + 15q^{73} - 15q^{74} - 4q^{75} - 8q^{76} + 6q^{77} - 11q^{78} - 26q^{79} - 15q^{80} + 15q^{81} + 2q^{82} - 30q^{83} - 17q^{84} - 2q^{85} - 26q^{86} - 14q^{87} - 15q^{88} + 10q^{89} - 17q^{90} - 17q^{91} - 4q^{92} - 8q^{93} - 14q^{94} + 8q^{95} - 4q^{96} - 27q^{97} + 11q^{98} - 17q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15} - 4 x^{14} - 23 x^{13} + 99 x^{12} + 191 x^{11} - 922 x^{10} - 702 x^{9} + 4108 x^{8} + 957 x^{7} - 8875 x^{6} + 479 x^{5} + 7698 x^{4} - 1731 x^{3} - 626 x^{2} + 46 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(366035 \nu^{14} - 1741279 \nu^{13} - 10239084 \nu^{12} + 56953737 \nu^{11} + 90005878 \nu^{10} - 692885868 \nu^{9} - 160464288 \nu^{8} + 3898870430 \nu^{7} - 1431593699 \nu^{6} - 10202941704 \nu^{5} + 6932586289 \nu^{4} + 9832804595 \nu^{3} - 8728032188 \nu^{2} + 521031646 \nu + 261843832\)\()/85835268\)
\(\beta_{4}\)\(=\)\((\)\(221319 \nu^{14} - 1050335 \nu^{13} - 1195170 \nu^{12} + 13572861 \nu^{11} - 36046464 \nu^{10} + 17357120 \nu^{9} + 359392846 \nu^{8} - 782126404 \nu^{7} - 979688875 \nu^{6} + 3235047750 \nu^{5} + 29989029 \nu^{4} - 3986272579 \nu^{3} + 2073107780 \nu^{2} + 49446534 \nu - 70626556\)\()/28611756\)
\(\beta_{5}\)\(=\)\((\)\(960341 \nu^{14} - 2079073 \nu^{13} - 24085422 \nu^{12} + 48433869 \nu^{11} + 218289340 \nu^{10} - 426649374 \nu^{9} - 810364518 \nu^{8} + 1857649106 \nu^{7} + 448142761 \nu^{6} - 4071404154 \nu^{5} + 4203159601 \nu^{4} + 3189119675 \nu^{3} - 7486177226 \nu^{2} + 1214106778 \nu + 498906736\)\()/85835268\)
\(\beta_{6}\)\(=\)\((\)\(1401323 \nu^{14} - 3706999 \nu^{13} - 32508126 \nu^{12} + 80618787 \nu^{11} + 269290000 \nu^{10} - 618823506 \nu^{9} - 949617366 \nu^{8} + 2084627546 \nu^{7} + 1045234987 \nu^{6} - 2989722630 \nu^{5} + 1208212627 \nu^{4} + 1426926773 \nu^{3} - 2405458214 \nu^{2} - 487849418 \nu + 246176608\)\()/85835268\)
\(\beta_{7}\)\(=\)\((\)\(-3305617 \nu^{14} + 10474127 \nu^{13} + 80134722 \nu^{12} - 252534357 \nu^{11} - 718017836 \nu^{10} + 2266987038 \nu^{9} + 2954517636 \nu^{8} - 9607202860 \nu^{7} - 5141897405 \nu^{6} + 19320093438 \nu^{5} + 943404775 \nu^{4} - 14598872695 \nu^{3} + 4690178716 \nu^{2} - 529393910 \nu + 60835336\)\()/85835268\)
\(\beta_{8}\)\(=\)\((\)\(-1280809 \nu^{14} + 4426957 \nu^{13} + 33773790 \nu^{12} - 119493219 \nu^{11} - 333339020 \nu^{10} + 1227637336 \nu^{9} + 1518438146 \nu^{8} - 6030088596 \nu^{7} - 2987617231 \nu^{6} + 14236309866 \nu^{5} + 1004705713 \nu^{4} - 13138428591 \nu^{3} + 2615461640 \nu^{2} + 577080874 \nu - 18158464\)\()/28611756\)
\(\beta_{9}\)\(=\)\((\)\(2319073 \nu^{14} - 9327080 \nu^{13} - 53426214 \nu^{12} + 228472740 \nu^{11} + 453199208 \nu^{10} - 2100319149 \nu^{9} - 1804697133 \nu^{8} + 9216665638 \nu^{7} + 3306920312 \nu^{6} - 19523419710 \nu^{5} - 1610019898 \nu^{4} + 16363093441 \nu^{3} - 1640072710 \nu^{2} - 831757504 \nu - 66381772\)\()/42917634\)
\(\beta_{10}\)\(=\)\((\)\(-5483185 \nu^{14} + 25183331 \nu^{13} + 106053138 \nu^{12} - 580526073 \nu^{11} - 625755272 \nu^{10} + 4911689490 \nu^{9} + 780653028 \nu^{8} - 19433892424 \nu^{7} + 4280176927 \nu^{6} + 36275272134 \nu^{5} - 14021936813 \nu^{4} - 25646611771 \nu^{3} + 11525790424 \nu^{2} - 141037466 \nu - 214919840\)\()/85835268\)
\(\beta_{11}\)\(=\)\((\)\(1064697 \nu^{14} - 3945448 \nu^{13} - 25941384 \nu^{12} + 100332648 \nu^{11} + 233803794 \nu^{10} - 966740297 \nu^{9} - 965567359 \nu^{8} + 4476683578 \nu^{7} + 1656705742 \nu^{6} - 10082589522 \nu^{5} - 83605584 \nu^{4} + 9118578187 \nu^{3} - 2014418084 \nu^{2} - 730702446 \nu + 23240302\)\()/14305878\)
\(\beta_{12}\)\(=\)\((\)\(7527395 \nu^{14} - 29036557 \nu^{13} - 178480998 \nu^{12} + 733439763 \nu^{11} + 1531273792 \nu^{10} - 6991802706 \nu^{9} - 5708071680 \nu^{8} + 31839705608 \nu^{7} + 6904612423 \nu^{6} - 69771551034 \nu^{5} + 9165309703 \nu^{4} + 59543491481 \nu^{3} - 22043262632 \nu^{2} - 1450048838 \nu + 771923284\)\()/85835268\)
\(\beta_{13}\)\(=\)\((\)\(-438707 \nu^{14} + 1765566 \nu^{13} + 10104105 \nu^{12} - 43697817 \nu^{11} - 84343819 \nu^{10} + 406761742 \nu^{9} + 315281564 \nu^{8} - 1809931594 \nu^{7} - 460844871 \nu^{6} + 3901113249 \nu^{5} - 112742041 \nu^{4} - 3368581534 \nu^{3} + 677370717 \nu^{2} + 249964550 \nu - 15724668\)\()/4768626\)
\(\beta_{14}\)\(=\)\((\)\(-3990373 \nu^{14} + 16003502 \nu^{13} + 91665171 \nu^{12} - 395775153 \nu^{11} - 759666443 \nu^{10} + 3679695978 \nu^{9} + 2782401546 \nu^{8} - 16337319754 \nu^{7} - 3770851553 \nu^{6} + 35037808089 \nu^{5} - 1839177515 \nu^{4} - 29893392208 \nu^{3} + 6483494065 \nu^{2} + 2067898252 \nu - 1774460\)\()/42917634\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{12} - \beta_{11} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} + 10 \beta_{2} + 2 \beta_{1} + 28\)
\(\nu^{5}\)\(=\)\(-14 \beta_{14} - 3 \beta_{13} + 13 \beta_{12} - 4 \beta_{11} + 13 \beta_{10} - 16 \beta_{9} + 10 \beta_{8} + \beta_{7} - 28 \beta_{6} + 17 \beta_{5} + 13 \beta_{4} - 17 \beta_{3} + 18 \beta_{2} + 44 \beta_{1} + 30\)
\(\nu^{6}\)\(=\)\(-7 \beta_{14} - 5 \beta_{13} + 13 \beta_{12} - 23 \beta_{11} - 9 \beta_{9} - 20 \beta_{8} + 31 \beta_{7} + 10 \beta_{6} + 24 \beta_{5} - 2 \beta_{4} - 40 \beta_{3} + 109 \beta_{2} + 28 \beta_{1} + 240\)
\(\nu^{7}\)\(=\)\(-163 \beta_{14} - 60 \beta_{13} + 141 \beta_{12} - 79 \beta_{11} + 137 \beta_{10} - 208 \beta_{9} + 79 \beta_{8} + 26 \beta_{7} - 315 \beta_{6} + 222 \beta_{5} + 133 \beta_{4} - 235 \beta_{3} + 248 \beta_{2} + 373 \beta_{1} + 359\)
\(\nu^{8}\)\(=\)\(-147 \beta_{14} - 110 \beta_{13} + 151 \beta_{12} - 348 \beta_{11} - 2 \beta_{10} - 213 \beta_{9} - 286 \beta_{8} + 385 \beta_{7} + 57 \beta_{6} + 382 \beta_{5} - 44 \beta_{4} - 607 \beta_{3} + 1224 \beta_{2} + 318 \beta_{1} + 2305\)
\(\nu^{9}\)\(=\)\(-1806 \beta_{14} - 849 \beta_{13} + 1479 \beta_{12} - 1140 \beta_{11} + 1356 \beta_{10} - 2511 \beta_{9} + 527 \beta_{8} + 443 \beta_{7} - 3345 \beta_{6} + 2652 \beta_{5} + 1267 \beta_{4} - 2999 \beta_{3} + 3110 \beta_{2} + 3438 \beta_{1} + 4104\)
\(\nu^{10}\)\(=\)\(-2195 \beta_{14} - 1700 \beta_{13} + 1761 \beta_{12} - 4531 \beta_{11} - 57 \beta_{10} - 3498 \beta_{9} - 3640 \beta_{8} + 4476 \beta_{7} - 56 \beta_{6} + 5203 \beta_{5} - 664 \beta_{4} - 8221 \beta_{3} + 13832 \beta_{2} + 3466 \beta_{1} + 23478\)
\(\nu^{11}\)\(=\)\(-19682 \beta_{14} - 10588 \beta_{13} + 15478 \beta_{12} - 14625 \beta_{11} + 13103 \beta_{10} - 29304 \beta_{9} + 2483 \beta_{8} + 6410 \beta_{7} - 34927 \beta_{6} + 30511 \beta_{5} + 11740 \beta_{4} - 36688 \beta_{3} + 37382 \beta_{2} + 33217 \beta_{1} + 46607\)
\(\nu^{12}\)\(=\)\(-28912 \beta_{14} - 22917 \beta_{13} + 20741 \beta_{12} - 55160 \beta_{11} - 1004 \beta_{10} - 49590 \beta_{9} - 43920 \beta_{8} + 50750 \beta_{7} - 7934 \beta_{6} + 65839 \beta_{5} - 8623 \beta_{4} - 104759 \beta_{3} + 156247 \beta_{2} + 37881 \beta_{1} + 246608\)
\(\nu^{13}\)\(=\)\(-213402 \beta_{14} - 124866 \beta_{13} + 162850 \beta_{12} - 177555 \beta_{11} + 125423 \beta_{10} - 336065 \beta_{9} - 3879 \beta_{8} + 85351 \beta_{7} - 363634 \beta_{6} + 345042 \beta_{5} + 107373 \beta_{4} - 437482 \beta_{3} + 439621 \beta_{2} + 330209 \beta_{1} + 530114\)
\(\nu^{14}\)\(=\)\(-359398 \beta_{14} - 289003 \beta_{13} + 244941 \beta_{12} - 649452 \beta_{11} - 14003 \beta_{10} - 651509 \beta_{9} - 514992 \beta_{8} + 569310 \beta_{7} - 161754 \beta_{6} + 801163 \beta_{5} - 103760 \beta_{4} - 1286815 \beta_{3} + 1761997 \beta_{2} + 419582 \beta_{1} + 2635917\)

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
3.35557
2.96469
2.44568
2.39402
1.82644
1.49088
0.440451
0.142263
−0.0958822
−0.205398
−1.67023
−1.82374
−1.99698
−2.16715
−3.10061
1.00000 −3.35557 1.00000 −1.00000 −3.35557 3.99638 1.00000 8.25982 −1.00000
1.2 1.00000 −2.96469 1.00000 −1.00000 −2.96469 −3.63894 1.00000 5.78937 −1.00000
1.3 1.00000 −2.44568 1.00000 −1.00000 −2.44568 −0.745785 1.00000 2.98137 −1.00000
1.4 1.00000 −2.39402 1.00000 −1.00000 −2.39402 −0.390362 1.00000 2.73133 −1.00000
1.5 1.00000 −1.82644 1.00000 −1.00000 −1.82644 3.00453 1.00000 0.335869 −1.00000
1.6 1.00000 −1.49088 1.00000 −1.00000 −1.49088 −3.98101 1.00000 −0.777287 −1.00000
1.7 1.00000 −0.440451 1.00000 −1.00000 −0.440451 3.12770 1.00000 −2.80600 −1.00000
1.8 1.00000 −0.142263 1.00000 −1.00000 −0.142263 −2.21205 1.00000 −2.97976 −1.00000
1.9 1.00000 0.0958822 1.00000 −1.00000 0.0958822 −1.44050 1.00000 −2.99081 −1.00000
1.10 1.00000 0.205398 1.00000 −1.00000 0.205398 3.29352 1.00000 −2.95781 −1.00000
1.11 1.00000 1.67023 1.00000 −1.00000 1.67023 1.44867 1.00000 −0.210343 −1.00000
1.12 1.00000 1.82374 1.00000 −1.00000 1.82374 −3.76421 1.00000 0.326027 −1.00000
1.13 1.00000 1.99698 1.00000 −1.00000 1.99698 0.970755 1.00000 0.987937 −1.00000
1.14 1.00000 2.16715 1.00000 −1.00000 2.16715 −3.23897 1.00000 1.69652 −1.00000
1.15 1.00000 3.10061 1.00000 −1.00000 3.10061 −2.42975 1.00000 6.61377 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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 8030.2.a.bf 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.bf 15 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(1\)
\(73\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\(T_{3}^{15} + \cdots\)
\(T_{7}^{15} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{15} \)
$3$ \( 1 + 4 T + 22 T^{2} + 69 T^{3} + 239 T^{4} + 634 T^{5} + 1740 T^{6} + 4058 T^{7} + 9507 T^{8} + 19957 T^{9} + 41717 T^{10} + 80655 T^{11} + 155223 T^{12} + 281903 T^{13} + 511309 T^{14} + 884920 T^{15} + 1533927 T^{16} + 2537127 T^{17} + 4191021 T^{18} + 6533055 T^{19} + 10137231 T^{20} + 14548653 T^{21} + 20791809 T^{22} + 26624538 T^{23} + 34248420 T^{24} + 37437066 T^{25} + 42338133 T^{26} + 36669429 T^{27} + 35075106 T^{28} + 19131876 T^{29} + 14348907 T^{30} \)
$5$ \( ( 1 + T )^{15} \)
$7$ \( 1 + 6 T + 65 T^{2} + 298 T^{3} + 1990 T^{4} + 7736 T^{5} + 40337 T^{6} + 138744 T^{7} + 612649 T^{8} + 1902404 T^{9} + 7396193 T^{10} + 20964196 T^{11} + 73348256 T^{12} + 190889318 T^{13} + 608670725 T^{14} + 1456630564 T^{15} + 4260695075 T^{16} + 9353576582 T^{17} + 25158451808 T^{18} + 50335034596 T^{19} + 124307815751 T^{20} + 223815928196 T^{21} + 504542795407 T^{22} + 799831549944 T^{23} + 1627743445559 T^{24} + 2185228526264 T^{25} + 3934880218570 T^{26} + 4124703585898 T^{27} + 6297785676455 T^{28} + 4069338437094 T^{29} + 4747561509943 T^{30} \)
$11$ \( ( 1 + T )^{15} \)
$13$ \( 1 + 6 T + 102 T^{2} + 527 T^{3} + 5012 T^{4} + 22396 T^{5} + 156661 T^{6} + 612478 T^{7} + 3523179 T^{8} + 12223617 T^{9} + 62012681 T^{10} + 194918505 T^{11} + 922628577 T^{12} + 2710398369 T^{13} + 12503218383 T^{14} + 35609399084 T^{15} + 162541838979 T^{16} + 458057324361 T^{17} + 2027014983669 T^{18} + 5567067421305 T^{19} + 23024874366533 T^{20} + 59001064548153 T^{21} + 221074257375543 T^{22} + 499617120536638 T^{23} + 1661311476273553 T^{24} + 3087478783450204 T^{25} + 8982307894913444 T^{26} + 12278090859547487 T^{27} + 30893260872409806 T^{28} + 23624258314195734 T^{29} + 51185893014090757 T^{30} \)
$17$ \( 1 - 2 T + 134 T^{2} - 191 T^{3} + 9115 T^{4} - 8981 T^{5} + 419768 T^{6} - 278562 T^{7} + 14696259 T^{8} - 6482502 T^{9} + 414775793 T^{10} - 123836564 T^{11} + 9746111691 T^{12} - 2122369251 T^{13} + 194030275531 T^{14} - 35675225982 T^{15} + 3298514684027 T^{16} - 613364713539 T^{17} + 47882646737883 T^{18} - 10342953661844 T^{19} + 588922313121601 T^{20} - 156471839317638 T^{21} + 6030443416124307 T^{22} - 1943180944279842 T^{23} + 49779395741392696 T^{24} - 18105641219932469 T^{25} + 312388334844074795 T^{26} - 111280847310884351 T^{27} + 1327213456409395558 T^{28} - 336755653118801858 T^{29} + 2862423051509815793 T^{30} \)
$19$ \( 1 + 8 T + 190 T^{2} + 1109 T^{3} + 15492 T^{4} + 69142 T^{5} + 766745 T^{6} + 2698505 T^{7} + 27481104 T^{8} + 78840098 T^{9} + 797846440 T^{10} + 1946993949 T^{11} + 19896122088 T^{12} + 43134997596 T^{13} + 433278304638 T^{14} + 863944699522 T^{15} + 8232287788122 T^{16} + 15571734132156 T^{17} + 136467501401592 T^{18} + 253734198427629 T^{19} + 1975546772237560 T^{20} + 3709101868536338 T^{21} + 24564582222119856 T^{22} + 45830229783953705 T^{23} + 247419178833559355 T^{24} + 423914183196876742 T^{25} + 1804667090851208748 T^{26} + 2454566245244372549 T^{27} + 7990066857828841210 T^{28} + 6392053486263072968 T^{29} + 15181127029874798299 T^{30} \)
$23$ \( 1 + 4 T + 137 T^{2} + 540 T^{3} + 9816 T^{4} + 40658 T^{5} + 488997 T^{6} + 2196138 T^{7} + 19241973 T^{8} + 93213272 T^{9} + 639191689 T^{10} + 3236350054 T^{11} + 18566533668 T^{12} + 94292235386 T^{13} + 478738702187 T^{14} + 2341976521496 T^{15} + 11010990150301 T^{16} + 49880592519194 T^{17} + 225899015138556 T^{18} + 905663435461414 T^{19} + 4114056953153327 T^{20} + 13798909587118808 T^{21} + 65515559320886931 T^{22} + 171981730593044778 T^{23} + 880758247997422611 T^{24} + 1684319092924541042 T^{25} + 9352780583683107432 T^{26} + 11833897193290973340 T^{27} + 69052981585296031471 T^{28} + 46371345298154999236 T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \)
$29$ \( 1 + 13 T + 236 T^{2} + 2677 T^{3} + 30613 T^{4} + 293419 T^{5} + 2669100 T^{6} + 22210577 T^{7} + 173777969 T^{8} + 1279335506 T^{9} + 8918989221 T^{10} + 59014792217 T^{11} + 372754028883 T^{12} + 2239875051690 T^{13} + 12922514506077 T^{14} + 70951951570914 T^{15} + 374752920676233 T^{16} + 1883734918471290 T^{17} + 9091098010427487 T^{18} + 41740041254031977 T^{19} + 182938716841324929 T^{20} + 760978594352135426 T^{21} + 2997648470479236421 T^{22} + 11110761474044088497 T^{23} + 38721023324191947900 T^{24} + \)\(12\!\cdots\!19\)\( T^{25} + \)\(37\!\cdots\!77\)\( T^{26} + \)\(94\!\cdots\!57\)\( T^{27} + \)\(24\!\cdots\!04\)\( T^{28} + \)\(38\!\cdots\!53\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \)
$31$ \( 1 + 20 T + 373 T^{2} + 4530 T^{3} + 52938 T^{4} + 501474 T^{5} + 4678912 T^{6} + 38103560 T^{7} + 307308078 T^{8} + 2231593258 T^{9} + 16070511204 T^{10} + 106023046606 T^{11} + 694663742879 T^{12} + 4208933059168 T^{13} + 25331963929151 T^{14} + 141521653820144 T^{15} + 785290881803681 T^{16} + 4044784669860448 T^{17} + 20694727564108289 T^{18} + 97914510024619726 T^{19} + 460085091906507804 T^{20} + 1980547230969782698 T^{21} + 8454848563207088658 T^{22} + 32498184818595389960 T^{23} + \)\(12\!\cdots\!52\)\( T^{24} + \)\(41\!\cdots\!74\)\( T^{25} + \)\(13\!\cdots\!78\)\( T^{26} + \)\(35\!\cdots\!30\)\( T^{27} + \)\(91\!\cdots\!43\)\( T^{28} + \)\(15\!\cdots\!20\)\( T^{29} + \)\(23\!\cdots\!51\)\( T^{30} \)
$37$ \( 1 + 15 T + 381 T^{2} + 4274 T^{3} + 66123 T^{4} + 615088 T^{5} + 7366499 T^{6} + 59809861 T^{7} + 605192511 T^{8} + 4406079724 T^{9} + 39226207747 T^{10} + 259633232653 T^{11} + 2077386047516 T^{12} + 12582722068635 T^{13} + 91571260529114 T^{14} + 508637710739164 T^{15} + 3388136639577218 T^{16} + 17225746511961315 T^{17} + 105225835464827948 T^{18} + 486594478942179133 T^{19} + 2720100463281034879 T^{20} + 11304795108026231116 T^{21} + 57452061096063750963 T^{22} + \)\(21\!\cdots\!81\)\( T^{23} + \)\(95\!\cdots\!23\)\( T^{24} + \)\(29\!\cdots\!12\)\( T^{25} + \)\(11\!\cdots\!99\)\( T^{26} + \)\(28\!\cdots\!94\)\( T^{27} + \)\(92\!\cdots\!57\)\( T^{28} + \)\(13\!\cdots\!35\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \)
$41$ \( 1 - 2 T + 336 T^{2} - 1146 T^{3} + 56059 T^{4} - 265772 T^{5} + 6238575 T^{6} - 36715578 T^{7} + 523265499 T^{8} - 3525881144 T^{9} + 35270131085 T^{10} - 254770670622 T^{11} + 1980717479384 T^{12} - 14465041570394 T^{13} + 94587232548861 T^{14} - 659584811782556 T^{15} + 3878076534503301 T^{16} - 24315734879832314 T^{17} + 136513029396624664 T^{18} - 719921024987493342 T^{19} + 4086263396280108085 T^{20} - 16748302975376331704 T^{21} + \)\(10\!\cdots\!19\)\( T^{22} - \)\(29\!\cdots\!38\)\( T^{23} + \)\(20\!\cdots\!75\)\( T^{24} - \)\(35\!\cdots\!72\)\( T^{25} + \)\(30\!\cdots\!19\)\( T^{26} - \)\(25\!\cdots\!26\)\( T^{27} + \)\(31\!\cdots\!56\)\( T^{28} - \)\(75\!\cdots\!22\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \)
$43$ \( 1 + 26 T + 662 T^{2} + 10939 T^{3} + 170447 T^{4} + 2145348 T^{5} + 25549590 T^{6} + 265370391 T^{7} + 2630070773 T^{8} + 23614493736 T^{9} + 203973480694 T^{10} + 1631413077953 T^{11} + 12630719621537 T^{12} + 91685343497130 T^{13} + 646565169288236 T^{14} + 4299119842361434 T^{15} + 27802302279394148 T^{16} + 169526200126193370 T^{17} + 1004230624949542259 T^{18} + 5577476662318794353 T^{19} + 29985823810115499442 T^{20} + \)\(14\!\cdots\!64\)\( T^{21} + \)\(71\!\cdots\!11\)\( T^{22} + \)\(31\!\cdots\!91\)\( T^{23} + \)\(12\!\cdots\!70\)\( T^{24} + \)\(46\!\cdots\!52\)\( T^{25} + \)\(15\!\cdots\!29\)\( T^{26} + \)\(43\!\cdots\!39\)\( T^{27} + \)\(11\!\cdots\!66\)\( T^{28} + \)\(19\!\cdots\!74\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \)
$47$ \( 1 + 14 T + 341 T^{2} + 3791 T^{3} + 53146 T^{4} + 488477 T^{5} + 5125283 T^{6} + 39629280 T^{7} + 343206647 T^{8} + 2239947173 T^{9} + 16972506164 T^{10} + 93403473945 T^{11} + 664672526391 T^{12} + 3183342965904 T^{13} + 24510227623949 T^{14} + 121448379717360 T^{15} + 1151980698325603 T^{16} + 7032004611681936 T^{17} + 69008295707492793 T^{18} + 455779157143411545 T^{19} + 3892559544990123148 T^{20} + 24144872903351814917 T^{21} + \)\(17\!\cdots\!61\)\( T^{22} + \)\(94\!\cdots\!80\)\( T^{23} + \)\(57\!\cdots\!61\)\( T^{24} + \)\(25\!\cdots\!73\)\( T^{25} + \)\(13\!\cdots\!38\)\( T^{26} + \)\(44\!\cdots\!31\)\( T^{27} + \)\(18\!\cdots\!07\)\( T^{28} + \)\(35\!\cdots\!66\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \)
$53$ \( 1 + 21 T + 586 T^{2} + 8297 T^{3} + 138081 T^{4} + 1521197 T^{5} + 19302475 T^{6} + 177125274 T^{7} + 1886114775 T^{8} + 15002925123 T^{9} + 141921087687 T^{10} + 1011548435111 T^{11} + 8899553490522 T^{12} + 58937201208539 T^{13} + 500364028930445 T^{14} + 3190309073782988 T^{15} + 26519293533313585 T^{16} + 165554598194786051 T^{17} + 1324938825008443794 T^{18} + 7981603707823078391 T^{19} + 59350759232361194691 T^{20} + \)\(33\!\cdots\!67\)\( T^{21} + \)\(22\!\cdots\!75\)\( T^{22} + \)\(11\!\cdots\!14\)\( T^{23} + \)\(63\!\cdots\!75\)\( T^{24} + \)\(26\!\cdots\!53\)\( T^{25} + \)\(12\!\cdots\!57\)\( T^{26} + \)\(40\!\cdots\!77\)\( T^{27} + \)\(15\!\cdots\!78\)\( T^{28} + \)\(28\!\cdots\!49\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \)
$59$ \( 1 + 14 T + 580 T^{2} + 7015 T^{3} + 157193 T^{4} + 1674474 T^{5} + 26534005 T^{6} + 252679170 T^{7} + 3143624231 T^{8} + 27182704930 T^{9} + 281832364853 T^{10} + 2258246071105 T^{11} + 20484714925532 T^{12} + 156030120721062 T^{13} + 1301761725738091 T^{14} + 9603701502260684 T^{15} + 76803941818547369 T^{16} + 543140850230016822 T^{17} + 4207130266690836628 T^{18} + 27363982870410953905 T^{19} + \)\(20\!\cdots\!47\)\( T^{20} + \)\(11\!\cdots\!30\)\( T^{21} + \)\(78\!\cdots\!89\)\( T^{22} + \)\(37\!\cdots\!70\)\( T^{23} + \)\(22\!\cdots\!95\)\( T^{24} + \)\(85\!\cdots\!74\)\( T^{25} + \)\(47\!\cdots\!87\)\( T^{26} + \)\(12\!\cdots\!15\)\( T^{27} + \)\(60\!\cdots\!20\)\( T^{28} + \)\(86\!\cdots\!54\)\( T^{29} + \)\(36\!\cdots\!99\)\( T^{30} \)
$61$ \( 1 + 45 T + 1375 T^{2} + 31296 T^{3} + 592536 T^{4} + 9612794 T^{5} + 138287927 T^{6} + 1789352730 T^{7} + 21180426956 T^{8} + 231405858336 T^{9} + 2359353134895 T^{10} + 22602506609164 T^{11} + 205183544649314 T^{12} + 1774331201208089 T^{13} + 14704268867126264 T^{14} + 117055614485664292 T^{15} + 896960400894702104 T^{16} + 6602286399695299169 T^{17} + 46572766148045941034 T^{18} + \)\(31\!\cdots\!24\)\( T^{19} + \)\(19\!\cdots\!95\)\( T^{20} + \)\(11\!\cdots\!96\)\( T^{21} + \)\(66\!\cdots\!76\)\( T^{22} + \)\(34\!\cdots\!30\)\( T^{23} + \)\(16\!\cdots\!07\)\( T^{24} + \)\(68\!\cdots\!94\)\( T^{25} + \)\(25\!\cdots\!96\)\( T^{26} + \)\(83\!\cdots\!16\)\( T^{27} + \)\(22\!\cdots\!75\)\( T^{28} + \)\(44\!\cdots\!45\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \)
$67$ \( 1 + 10 T + 552 T^{2} + 3735 T^{3} + 142378 T^{4} + 708984 T^{5} + 24973659 T^{6} + 98649624 T^{7} + 3394474435 T^{8} + 11113558175 T^{9} + 374639930087 T^{10} + 1051685212347 T^{11} + 34550654075625 T^{12} + 86061193529599 T^{13} + 2706670402205361 T^{14} + 6157519150190172 T^{15} + 181346916947759187 T^{16} + 386328697754369911 T^{17} + 10391558371747201875 T^{18} + 21192635967915090987 T^{19} + \)\(50\!\cdots\!09\)\( T^{20} + \)\(10\!\cdots\!75\)\( T^{21} + \)\(20\!\cdots\!05\)\( T^{22} + \)\(40\!\cdots\!84\)\( T^{23} + \)\(67\!\cdots\!73\)\( T^{24} + \)\(12\!\cdots\!16\)\( T^{25} + \)\(17\!\cdots\!74\)\( T^{26} + \)\(30\!\cdots\!35\)\( T^{27} + \)\(30\!\cdots\!24\)\( T^{28} + \)\(36\!\cdots\!90\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \)
$71$ \( 1 + 9 T + 652 T^{2} + 5397 T^{3} + 203974 T^{4} + 1563427 T^{5} + 40982801 T^{6} + 290848769 T^{7} + 5984092616 T^{8} + 39173493754 T^{9} + 683599945217 T^{10} + 4117553631003 T^{11} + 64431015625764 T^{12} + 358530807073962 T^{13} + 5211913721888649 T^{14} + 27080096501483998 T^{15} + 370045874254094079 T^{16} + 1807353798459842442 T^{17} + 23060569233632819004 T^{18} + \)\(10\!\cdots\!43\)\( T^{19} + \)\(12\!\cdots\!67\)\( T^{20} + \)\(50\!\cdots\!34\)\( T^{21} + \)\(54\!\cdots\!56\)\( T^{22} + \)\(18\!\cdots\!09\)\( T^{23} + \)\(18\!\cdots\!31\)\( T^{24} + \)\(50\!\cdots\!27\)\( T^{25} + \)\(47\!\cdots\!54\)\( T^{26} + \)\(88\!\cdots\!77\)\( T^{27} + \)\(75\!\cdots\!72\)\( T^{28} + \)\(74\!\cdots\!29\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \)
$73$ \( ( 1 - T )^{15} \)
$79$ \( 1 + 26 T + 1012 T^{2} + 19874 T^{3} + 455261 T^{4} + 7245592 T^{5} + 124598652 T^{6} + 1676481960 T^{7} + 23698872918 T^{8} + 277547393180 T^{9} + 3385543083415 T^{10} + 35307215598638 T^{11} + 383627882439550 T^{12} + 3629088838207090 T^{13} + 35886579923223179 T^{14} + 311703087878366736 T^{15} + 2835039813934631141 T^{16} + 22649143439250448690 T^{17} + \)\(18\!\cdots\!50\)\( T^{18} + \)\(13\!\cdots\!78\)\( T^{19} + \)\(10\!\cdots\!85\)\( T^{20} + \)\(67\!\cdots\!80\)\( T^{21} + \)\(45\!\cdots\!62\)\( T^{22} + \)\(25\!\cdots\!60\)\( T^{23} + \)\(14\!\cdots\!88\)\( T^{24} + \)\(68\!\cdots\!92\)\( T^{25} + \)\(34\!\cdots\!19\)\( T^{26} + \)\(11\!\cdots\!34\)\( T^{27} + \)\(47\!\cdots\!68\)\( T^{28} + \)\(95\!\cdots\!06\)\( T^{29} + \)\(29\!\cdots\!99\)\( T^{30} \)
$83$ \( 1 + 30 T + 821 T^{2} + 14657 T^{3} + 250804 T^{4} + 3538169 T^{5} + 49628309 T^{6} + 625997345 T^{7} + 7833578857 T^{8} + 90579055489 T^{9} + 1024864746295 T^{10} + 10899887563123 T^{11} + 113220306919074 T^{12} + 1121409374133760 T^{13} + 10830251092365263 T^{14} + 100065551358223254 T^{15} + 898910840666316829 T^{16} + 7725389178407472640 T^{17} + 64737899632336565238 T^{18} + \)\(51\!\cdots\!83\)\( T^{19} + \)\(40\!\cdots\!85\)\( T^{20} + \)\(29\!\cdots\!41\)\( T^{21} + \)\(21\!\cdots\!39\)\( T^{22} + \)\(14\!\cdots\!45\)\( T^{23} + \)\(92\!\cdots\!27\)\( T^{24} + \)\(54\!\cdots\!81\)\( T^{25} + \)\(32\!\cdots\!68\)\( T^{26} + \)\(15\!\cdots\!77\)\( T^{27} + \)\(72\!\cdots\!23\)\( T^{28} + \)\(22\!\cdots\!70\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \)
$89$ \( 1 - 10 T + 532 T^{2} - 3641 T^{3} + 127591 T^{4} - 600472 T^{5} + 19838201 T^{6} - 64716173 T^{7} + 2409715473 T^{8} - 6146671149 T^{9} + 258498620556 T^{10} - 684275825807 T^{11} + 26266693191977 T^{12} - 82473836849208 T^{13} + 2531657672238726 T^{14} - 8355736850814146 T^{15} + 225317532829246614 T^{16} - 653275261682576568 T^{17} + 18517204432854833713 T^{18} - 42932998773256813487 T^{19} + \)\(14\!\cdots\!44\)\( T^{20} - \)\(30\!\cdots\!89\)\( T^{21} + \)\(10\!\cdots\!17\)\( T^{22} - \)\(25\!\cdots\!13\)\( T^{23} + \)\(69\!\cdots\!09\)\( T^{24} - \)\(18\!\cdots\!72\)\( T^{25} + \)\(35\!\cdots\!99\)\( T^{26} - \)\(89\!\cdots\!61\)\( T^{27} + \)\(11\!\cdots\!08\)\( T^{28} - \)\(19\!\cdots\!10\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \)
$97$ \( 1 + 27 T + 1143 T^{2} + 24206 T^{3} + 619512 T^{4} + 11016095 T^{5} + 214831058 T^{6} + 3311736267 T^{7} + 53522994013 T^{8} + 728845721405 T^{9} + 10162691163729 T^{10} + 123596651232022 T^{11} + 1519819029048788 T^{12} + 16606406030195273 T^{13} + 182302384725858172 T^{14} + 1792986990106848930 T^{15} + 17683331318408242684 T^{16} + \)\(15\!\cdots\!57\)\( T^{17} + \)\(13\!\cdots\!24\)\( T^{18} + \)\(10\!\cdots\!82\)\( T^{19} + \)\(87\!\cdots\!53\)\( T^{20} + \)\(60\!\cdots\!45\)\( T^{21} + \)\(43\!\cdots\!69\)\( T^{22} + \)\(25\!\cdots\!87\)\( T^{23} + \)\(16\!\cdots\!86\)\( T^{24} + \)\(81\!\cdots\!55\)\( T^{25} + \)\(44\!\cdots\!36\)\( T^{26} + \)\(16\!\cdots\!46\)\( T^{27} + \)\(76\!\cdots\!11\)\( T^{28} + \)\(17\!\cdots\!63\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \)
show more
show less