Properties

Label 380.2.u.a
Level $380$
Weight $2$
Character orbit 380.u
Analytic conductor $3.034$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 3 x^{17} + 3 x^{15} + 36 x^{14} + 72 x^{13} - 134 x^{12} - 741 x^{11} + 486 x^{10} + 5700 x^{9} + 11637 x^{8} + 13878 x^{7} + 13978 x^{6} + 10005 x^{5} + 5949 x^{4} + 2649 x^{3} + 819 x^{2} + 135 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{7} ) q^{3} -\beta_{4} q^{5} + ( \beta_{4} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{17} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{11} - \beta_{13} - \beta_{17} ) q^{9} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{7} ) q^{3} -\beta_{4} q^{5} + ( \beta_{4} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{15} - \beta_{17} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{11} - \beta_{13} - \beta_{17} ) q^{9} + ( -2 + 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{17} ) q^{11} + ( 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{15} ) q^{13} -\beta_{1} q^{15} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{16} ) q^{17} + ( -\beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{14} - \beta_{15} ) q^{19} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} ) q^{21} + ( -1 + \beta_{1} + \beta_{4} + \beta_{8} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{23} + \beta_{9} q^{25} + ( -1 + \beta_{2} - 3 \beta_{4} + 2 \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} - \beta_{17} ) q^{27} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{29} + ( \beta_{2} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{31} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + \beta_{13} - 2 \beta_{16} - 2 \beta_{17} ) q^{33} + ( 1 - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{17} ) q^{35} + ( 4 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{37} + ( -4 - \beta_{1} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 3 \beta_{11} - 2 \beta_{12} ) q^{39} + ( -2 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + 2 \beta_{17} ) q^{41} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + 3 \beta_{9} + \beta_{11} + 5 \beta_{12} - 4 \beta_{13} + \beta_{16} + \beta_{17} ) q^{43} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{16} ) q^{45} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{11} + 3 \beta_{12} + \beta_{14} + \beta_{15} - \beta_{17} ) q^{47} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{9} + \beta_{10} - 4 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} ) q^{49} + ( -2 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{51} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{53} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{17} ) q^{55} + ( 5 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{57} + ( 2 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{15} - \beta_{16} + \beta_{17} ) q^{59} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{10} + \beta_{11} - \beta_{12} + 4 \beta_{13} + 2 \beta_{17} ) q^{61} + ( -3 - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - 6 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{63} + ( 2 \beta_{2} - \beta_{4} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{65} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{11} + 2 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} ) q^{67} + ( 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{13} - \beta_{16} ) q^{69} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{16} + \beta_{17} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + \beta_{10} + 5 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{16} + 3 \beta_{17} ) q^{73} -\beta_{5} q^{75} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{77} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{16} + \beta_{17} ) q^{79} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{81} + ( \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + 7 \beta_{13} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{83} + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{16} ) q^{85} + ( 3 - 6 \beta_{1} - \beta_{2} - 6 \beta_{3} + \beta_{4} + 3 \beta_{6} + 3 \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + 4 \beta_{13} - 2 \beta_{14} - \beta_{15} + \beta_{16} + 2 \beta_{17} ) q^{87} + ( -2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{89} + ( -1 + 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} - \beta_{10} + 4 \beta_{12} - 5 \beta_{13} + \beta_{15} - \beta_{16} - \beta_{17} ) q^{91} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{93} + ( 1 - \beta_{2} + \beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} ) q^{95} + ( 3 + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{9} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 3 \beta_{15} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{10} - 2 \beta_{11} + 3 \beta_{13} + \beta_{14} - \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 3q^{3} - 9q^{9} + O(q^{10}) \) \( 18q + 3q^{3} - 9q^{9} + 9q^{13} - 3q^{15} + 12q^{17} + 18q^{19} + 15q^{21} - 9q^{23} - 33q^{29} - 18q^{31} + 9q^{33} + 12q^{35} + 60q^{37} - 84q^{39} - 18q^{41} + 18q^{43} + 6q^{45} - 15q^{47} - 15q^{49} - 9q^{51} + 24q^{53} + 3q^{55} + 66q^{57} + 48q^{59} - 18q^{61} - 54q^{63} + 3q^{65} - 36q^{67} - 9q^{69} + 18q^{73} - 6q^{75} + 9q^{79} - 9q^{81} - 30q^{83} - 12q^{85} - 9q^{87} + 6q^{89} + 30q^{91} + 21q^{95} + 21q^{97} - 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 3 x^{17} + 3 x^{15} + 36 x^{14} + 72 x^{13} - 134 x^{12} - 741 x^{11} + 486 x^{10} + 5700 x^{9} + 11637 x^{8} + 13878 x^{7} + 13978 x^{6} + 10005 x^{5} + 5949 x^{4} + 2649 x^{3} + 819 x^{2} + 135 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(23\!\cdots\!01\)\( \nu^{17} + \)\(38\!\cdots\!21\)\( \nu^{16} + \)\(75\!\cdots\!17\)\( \nu^{15} - \)\(35\!\cdots\!36\)\( \nu^{14} - \)\(97\!\cdots\!47\)\( \nu^{13} - \)\(28\!\cdots\!19\)\( \nu^{12} + \)\(36\!\cdots\!60\)\( \nu^{11} + \)\(20\!\cdots\!28\)\( \nu^{10} + \)\(14\!\cdots\!39\)\( \nu^{9} - \)\(13\!\cdots\!08\)\( \nu^{8} - \)\(46\!\cdots\!27\)\( \nu^{7} - \)\(75\!\cdots\!17\)\( \nu^{6} - \)\(83\!\cdots\!72\)\( \nu^{5} - \)\(68\!\cdots\!82\)\( \nu^{4} - \)\(45\!\cdots\!97\)\( \nu^{3} - \)\(20\!\cdots\!07\)\( \nu^{2} - \)\(72\!\cdots\!43\)\( \nu - \)\(93\!\cdots\!61\)\(\)\()/ \)\(12\!\cdots\!51\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(80\!\cdots\!23\)\( \nu^{17} - \)\(29\!\cdots\!98\)\( \nu^{16} + \)\(18\!\cdots\!13\)\( \nu^{15} + \)\(18\!\cdots\!59\)\( \nu^{14} + \)\(27\!\cdots\!69\)\( \nu^{13} + \)\(38\!\cdots\!66\)\( \nu^{12} - \)\(14\!\cdots\!56\)\( \nu^{11} - \)\(51\!\cdots\!75\)\( \nu^{10} + \)\(78\!\cdots\!31\)\( \nu^{9} + \)\(42\!\cdots\!69\)\( \nu^{8} + \)\(62\!\cdots\!00\)\( \nu^{7} + \)\(56\!\cdots\!66\)\( \nu^{6} + \)\(54\!\cdots\!81\)\( \nu^{5} + \)\(25\!\cdots\!22\)\( \nu^{4} + \)\(15\!\cdots\!60\)\( \nu^{3} + \)\(44\!\cdots\!11\)\( \nu^{2} + \)\(17\!\cdots\!95\)\( \nu + \)\(42\!\cdots\!45\)\(\)\()/ \)\(12\!\cdots\!51\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(50\!\cdots\!40\)\( \nu^{17} + \)\(96\!\cdots\!97\)\( \nu^{16} + \)\(19\!\cdots\!94\)\( \nu^{15} - \)\(23\!\cdots\!03\)\( \nu^{14} - \)\(19\!\cdots\!06\)\( \nu^{13} - \)\(55\!\cdots\!78\)\( \nu^{12} + \)\(38\!\cdots\!86\)\( \nu^{11} + \)\(46\!\cdots\!17\)\( \nu^{10} + \)\(12\!\cdots\!22\)\( \nu^{9} - \)\(33\!\cdots\!90\)\( \nu^{8} - \)\(88\!\cdots\!10\)\( \nu^{7} - \)\(11\!\cdots\!43\)\( \nu^{6} - \)\(11\!\cdots\!78\)\( \nu^{5} - \)\(93\!\cdots\!51\)\( \nu^{4} - \)\(52\!\cdots\!20\)\( \nu^{3} - \)\(25\!\cdots\!76\)\( \nu^{2} - \)\(76\!\cdots\!36\)\( \nu - \)\(11\!\cdots\!00\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(18\!\cdots\!41\)\( \nu^{17} - \)\(64\!\cdots\!98\)\( \nu^{16} + \)\(29\!\cdots\!61\)\( \nu^{15} + \)\(51\!\cdots\!22\)\( \nu^{14} + \)\(63\!\cdots\!66\)\( \nu^{13} + \)\(98\!\cdots\!58\)\( \nu^{12} - \)\(30\!\cdots\!59\)\( \nu^{11} - \)\(12\!\cdots\!54\)\( \nu^{10} + \)\(15\!\cdots\!30\)\( \nu^{9} + \)\(98\!\cdots\!10\)\( \nu^{8} + \)\(16\!\cdots\!41\)\( \nu^{7} + \)\(15\!\cdots\!86\)\( \nu^{6} + \)\(14\!\cdots\!17\)\( \nu^{5} + \)\(75\!\cdots\!20\)\( \nu^{4} + \)\(41\!\cdots\!72\)\( \nu^{3} + \)\(11\!\cdots\!92\)\( \nu^{2} + \)\(14\!\cdots\!00\)\( \nu - \)\(15\!\cdots\!20\)\(\)\()/ \)\(12\!\cdots\!51\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-30103266206878174 \nu^{17} + 89982817832438707 \nu^{16} + 6253022931308790 \nu^{15} - 108503893341585330 \nu^{14} - 1078183524913991538 \nu^{13} - 2161022396547272742 \nu^{12} + 4191654100641764381 \nu^{11} + 22642405411909503703 \nu^{10} - 15290748700288928310 \nu^{9} - 175348207542851549889 \nu^{8} - 347746927977772126974 \nu^{7} - 392225643626669668809 \nu^{6} - 378292079701243787365 \nu^{5} - 263844419857316268116 \nu^{4} - 145783314394989192354 \nu^{3} - 62431169231720649660 \nu^{2} - 15738989858455126452 \nu - 1709886947115762396\)\()/ 1682094761273482869 \)
\(\beta_{7}\)\(=\)\((\)\(\)\(35\!\cdots\!66\)\( \nu^{17} - \)\(11\!\cdots\!00\)\( \nu^{16} + \)\(32\!\cdots\!74\)\( \nu^{15} + \)\(97\!\cdots\!80\)\( \nu^{14} + \)\(12\!\cdots\!28\)\( \nu^{13} + \)\(21\!\cdots\!66\)\( \nu^{12} - \)\(53\!\cdots\!68\)\( \nu^{11} - \)\(24\!\cdots\!16\)\( \nu^{10} + \)\(24\!\cdots\!23\)\( \nu^{9} + \)\(19\!\cdots\!35\)\( \nu^{8} + \)\(35\!\cdots\!76\)\( \nu^{7} + \)\(39\!\cdots\!75\)\( \nu^{6} + \)\(38\!\cdots\!78\)\( \nu^{5} + \)\(24\!\cdots\!77\)\( \nu^{4} + \)\(14\!\cdots\!31\)\( \nu^{3} + \)\(56\!\cdots\!96\)\( \nu^{2} + \)\(14\!\cdots\!40\)\( \nu + \)\(13\!\cdots\!58\)\(\)\()/ \)\(12\!\cdots\!51\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(47\!\cdots\!25\)\( \nu^{17} - \)\(17\!\cdots\!02\)\( \nu^{16} + \)\(98\!\cdots\!46\)\( \nu^{15} + \)\(12\!\cdots\!61\)\( \nu^{14} + \)\(16\!\cdots\!94\)\( \nu^{13} + \)\(23\!\cdots\!31\)\( \nu^{12} - \)\(82\!\cdots\!91\)\( \nu^{11} - \)\(30\!\cdots\!67\)\( \nu^{10} + \)\(44\!\cdots\!92\)\( \nu^{9} + \)\(24\!\cdots\!93\)\( \nu^{8} + \)\(37\!\cdots\!77\)\( \nu^{7} + \)\(34\!\cdots\!34\)\( \nu^{6} + \)\(32\!\cdots\!97\)\( \nu^{5} + \)\(15\!\cdots\!99\)\( \nu^{4} + \)\(92\!\cdots\!13\)\( \nu^{3} + \)\(26\!\cdots\!77\)\( \nu^{2} + \)\(31\!\cdots\!97\)\( \nu + \)\(29\!\cdots\!10\)\(\)\()/ \)\(12\!\cdots\!51\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(14\!\cdots\!15\)\( \nu^{17} - \)\(44\!\cdots\!14\)\( \nu^{16} + \)\(89\!\cdots\!94\)\( \nu^{15} + \)\(36\!\cdots\!06\)\( \nu^{14} + \)\(50\!\cdots\!63\)\( \nu^{13} + \)\(93\!\cdots\!73\)\( \nu^{12} - \)\(20\!\cdots\!08\)\( \nu^{11} - \)\(10\!\cdots\!47\)\( \nu^{10} + \)\(84\!\cdots\!15\)\( \nu^{9} + \)\(78\!\cdots\!07\)\( \nu^{8} + \)\(15\!\cdots\!48\)\( \nu^{7} + \)\(17\!\cdots\!70\)\( \nu^{6} + \)\(18\!\cdots\!72\)\( \nu^{5} + \)\(12\!\cdots\!32\)\( \nu^{4} + \)\(76\!\cdots\!69\)\( \nu^{3} + \)\(32\!\cdots\!55\)\( \nu^{2} + \)\(10\!\cdots\!52\)\( \nu + \)\(13\!\cdots\!40\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(24\!\cdots\!75\)\( \nu^{17} - \)\(78\!\cdots\!28\)\( \nu^{16} + \)\(25\!\cdots\!60\)\( \nu^{15} + \)\(47\!\cdots\!25\)\( \nu^{14} + \)\(86\!\cdots\!82\)\( \nu^{13} + \)\(15\!\cdots\!91\)\( \nu^{12} - \)\(34\!\cdots\!90\)\( \nu^{11} - \)\(16\!\cdots\!01\)\( \nu^{10} + \)\(14\!\cdots\!71\)\( \nu^{9} + \)\(12\!\cdots\!39\)\( \nu^{8} + \)\(25\!\cdots\!89\)\( \nu^{7} + \)\(30\!\cdots\!93\)\( \nu^{6} + \)\(31\!\cdots\!72\)\( \nu^{5} + \)\(21\!\cdots\!40\)\( \nu^{4} + \)\(13\!\cdots\!06\)\( \nu^{3} + \)\(58\!\cdots\!97\)\( \nu^{2} + \)\(20\!\cdots\!00\)\( \nu + \)\(29\!\cdots\!70\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(26\!\cdots\!47\)\( \nu^{17} - \)\(84\!\cdots\!67\)\( \nu^{16} + \)\(20\!\cdots\!57\)\( \nu^{15} + \)\(71\!\cdots\!09\)\( \nu^{14} + \)\(92\!\cdots\!18\)\( \nu^{13} + \)\(16\!\cdots\!45\)\( \nu^{12} - \)\(38\!\cdots\!29\)\( \nu^{11} - \)\(18\!\cdots\!70\)\( \nu^{10} + \)\(16\!\cdots\!88\)\( \nu^{9} + \)\(14\!\cdots\!27\)\( \nu^{8} + \)\(26\!\cdots\!85\)\( \nu^{7} + \)\(30\!\cdots\!62\)\( \nu^{6} + \)\(29\!\cdots\!57\)\( \nu^{5} + \)\(19\!\cdots\!68\)\( \nu^{4} + \)\(11\!\cdots\!97\)\( \nu^{3} + \)\(43\!\cdots\!96\)\( \nu^{2} + \)\(11\!\cdots\!96\)\( \nu + \)\(90\!\cdots\!16\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(31\!\cdots\!71\)\( \nu^{17} - \)\(10\!\cdots\!42\)\( \nu^{16} + \)\(25\!\cdots\!06\)\( \nu^{15} + \)\(90\!\cdots\!30\)\( \nu^{14} + \)\(11\!\cdots\!93\)\( \nu^{13} + \)\(19\!\cdots\!35\)\( \nu^{12} - \)\(47\!\cdots\!14\)\( \nu^{11} - \)\(22\!\cdots\!75\)\( \nu^{10} + \)\(21\!\cdots\!29\)\( \nu^{9} + \)\(17\!\cdots\!24\)\( \nu^{8} + \)\(32\!\cdots\!29\)\( \nu^{7} + \)\(34\!\cdots\!10\)\( \nu^{6} + \)\(33\!\cdots\!11\)\( \nu^{5} + \)\(21\!\cdots\!64\)\( \nu^{4} + \)\(12\!\cdots\!14\)\( \nu^{3} + \)\(45\!\cdots\!66\)\( \nu^{2} + \)\(10\!\cdots\!94\)\( \nu + \)\(56\!\cdots\!50\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{13}\)\(=\)\((\)\(189987438568418044 \nu^{17} - 600065581912132306 \nu^{16} + 89982817832438707 \nu^{15} + 576215338636562922 \nu^{14} + 6731043895121464254 \nu^{13} + 12600912052012107630 \nu^{12} - 27619339164715290638 \nu^{11} - 136589037878556006223 \nu^{10} + 114976300556160673087 \nu^{9} + 1067637651139693922490 \nu^{8} + 2035535615077829228139 \nu^{7} + 2288898744474733487658 \nu^{6} + 2263418772682677750223 \nu^{5} + 1522532243175778742855 \nu^{4} + 866390852186202675640 \nu^{3} + 357493410372750206202 \nu^{2} + 93168542955813728376 \nu + 8227219587007826619\)\()/ 1682094761273482869 \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(80\!\cdots\!05\)\( \nu^{17} + \)\(26\!\cdots\!25\)\( \nu^{16} - \)\(73\!\cdots\!50\)\( \nu^{15} - \)\(21\!\cdots\!88\)\( \nu^{14} - \)\(28\!\cdots\!00\)\( \nu^{13} - \)\(50\!\cdots\!30\)\( \nu^{12} + \)\(12\!\cdots\!84\)\( \nu^{11} + \)\(56\!\cdots\!85\)\( \nu^{10} - \)\(54\!\cdots\!00\)\( \nu^{9} - \)\(44\!\cdots\!53\)\( \nu^{8} - \)\(81\!\cdots\!18\)\( \nu^{7} - \)\(89\!\cdots\!75\)\( \nu^{6} - \)\(87\!\cdots\!01\)\( \nu^{5} - \)\(56\!\cdots\!46\)\( \nu^{4} - \)\(32\!\cdots\!88\)\( \nu^{3} - \)\(12\!\cdots\!32\)\( \nu^{2} - \)\(32\!\cdots\!82\)\( \nu - \)\(19\!\cdots\!59\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(97\!\cdots\!98\)\( \nu^{17} - \)\(29\!\cdots\!96\)\( \nu^{16} - \)\(18\!\cdots\!01\)\( \nu^{15} + \)\(30\!\cdots\!84\)\( \nu^{14} + \)\(35\!\cdots\!09\)\( \nu^{13} + \)\(69\!\cdots\!15\)\( \nu^{12} - \)\(13\!\cdots\!63\)\( \nu^{11} - \)\(72\!\cdots\!49\)\( \nu^{10} + \)\(49\!\cdots\!02\)\( \nu^{9} + \)\(56\!\cdots\!14\)\( \nu^{8} + \)\(11\!\cdots\!63\)\( \nu^{7} + \)\(13\!\cdots\!25\)\( \nu^{6} + \)\(13\!\cdots\!64\)\( \nu^{5} + \)\(91\!\cdots\!93\)\( \nu^{4} + \)\(52\!\cdots\!98\)\( \nu^{3} + \)\(22\!\cdots\!67\)\( \nu^{2} + \)\(63\!\cdots\!63\)\( \nu + \)\(64\!\cdots\!72\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{16}\)\(=\)\((\)\(\)\(12\!\cdots\!10\)\( \nu^{17} - \)\(37\!\cdots\!47\)\( \nu^{16} + \)\(39\!\cdots\!46\)\( \nu^{15} + \)\(38\!\cdots\!51\)\( \nu^{14} + \)\(43\!\cdots\!99\)\( \nu^{13} + \)\(81\!\cdots\!69\)\( \nu^{12} - \)\(17\!\cdots\!03\)\( \nu^{11} - \)\(87\!\cdots\!27\)\( \nu^{10} + \)\(70\!\cdots\!22\)\( \nu^{9} + \)\(68\!\cdots\!57\)\( \nu^{8} + \)\(13\!\cdots\!55\)\( \nu^{7} + \)\(14\!\cdots\!41\)\( \nu^{6} + \)\(14\!\cdots\!19\)\( \nu^{5} + \)\(98\!\cdots\!32\)\( \nu^{4} + \)\(55\!\cdots\!59\)\( \nu^{3} + \)\(23\!\cdots\!50\)\( \nu^{2} + \)\(59\!\cdots\!03\)\( \nu + \)\(65\!\cdots\!98\)\(\)\()/ \)\(36\!\cdots\!53\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(60\!\cdots\!28\)\( \nu^{17} + \)\(19\!\cdots\!27\)\( \nu^{16} - \)\(49\!\cdots\!38\)\( \nu^{15} - \)\(16\!\cdots\!26\)\( \nu^{14} - \)\(21\!\cdots\!42\)\( \nu^{13} - \)\(38\!\cdots\!34\)\( \nu^{12} + \)\(90\!\cdots\!51\)\( \nu^{11} + \)\(42\!\cdots\!45\)\( \nu^{10} - \)\(39\!\cdots\!54\)\( \nu^{9} - \)\(33\!\cdots\!24\)\( \nu^{8} - \)\(62\!\cdots\!23\)\( \nu^{7} - \)\(69\!\cdots\!85\)\( \nu^{6} - \)\(68\!\cdots\!52\)\( \nu^{5} - \)\(44\!\cdots\!85\)\( \nu^{4} - \)\(26\!\cdots\!45\)\( \nu^{3} - \)\(10\!\cdots\!21\)\( \nu^{2} - \)\(26\!\cdots\!87\)\( \nu - \)\(24\!\cdots\!27\)\(\)\()/ \)\(12\!\cdots\!51\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{14} + \beta_{13} - \beta_{12} - 3 \beta_{11} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + \beta_{10} - 4 \beta_{9} - \beta_{8} + 7 \beta_{5} - \beta_{4} - 7 \beta_{2} + 1\)
\(\nu^{4}\)\(=\)\(-7 \beta_{17} - 8 \beta_{13} - 21 \beta_{12} - 8 \beta_{11} - 21 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 12 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} + 7 \beta_{1} - 9\)
\(\nu^{5}\)\(=\)\(-12 \beta_{17} + 4 \beta_{16} + 7 \beta_{15} - 46 \beta_{13} - 48 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} - 27 \beta_{9} + 8 \beta_{8} + 55 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - \beta_{1} - 48\)
\(\nu^{6}\)\(=\)\(-5 \beta_{17} + 55 \beta_{16} + 4 \beta_{15} + 5 \beta_{14} - 189 \beta_{13} - 81 \beta_{12} + 67 \beta_{11} - \beta_{10} - 14 \beta_{9} + 119 \beta_{7} - 59 \beta_{6} + 69 \beta_{4} - 60 \beta_{3} - 46 \beta_{2} - 60 \beta_{1} - 185\)
\(\nu^{7}\)\(=\)\(119 \beta_{16} + 59 \beta_{15} + 119 \beta_{14} - 550 \beta_{13} + 119 \beta_{12} + 199 \beta_{11} + 46 \beta_{10} + 93 \beta_{9} - 60 \beta_{8} + 94 \beta_{7} - 469 \beta_{6} - 94 \beta_{5} + 411 \beta_{4} - 23 \beta_{3} + 23 \beta_{2} - 411\)
\(\nu^{8}\)\(=\)\(94 \beta_{17} + 94 \beta_{16} + 469 \beta_{15} + 469 \beta_{14} - 1234 \beta_{13} + 1328 \beta_{12} + 94 \beta_{11} + 71 \beta_{10} + 694 \beta_{9} - 23 \beta_{8} - 1138 \beta_{6} - 1138 \beta_{5} + 1750 \beta_{4} + 304 \beta_{3} + 457 \beta_{2} + 457 \beta_{1} - 634\)
\(\nu^{9}\)\(=\)\(1138 \beta_{17} + 1138 \beta_{15} + 681 \beta_{14} + 5515 \beta_{12} - 727 \beta_{11} + 681 \beta_{10} + 1865 \beta_{9} + 304 \beta_{8} - 1248 \beta_{7} - 1248 \beta_{6} - 4189 \beta_{5} + 4834 \beta_{4} + 906 \beta_{3} + 342 \beta_{1} + 423\)
\(\nu^{10}\)\(=\)\(4189 \beta_{17} - 1248 \beta_{16} + 1248 \beta_{15} + 906 \beta_{14} + 11563 \beta_{13} + 16578 \beta_{12} - 2659 \beta_{11} + 4189 \beta_{10} + 906 \beta_{8} - 10842 \beta_{7} - 7246 \beta_{5} + 7374 \beta_{4} + 7246 \beta_{3} - 3596 \beta_{2} - 2023 \beta_{1} + 5942\)
\(\nu^{11}\)\(=\)\(7246 \beta_{17} - 10842 \beta_{16} + 2023 \beta_{14} + 54370 \beta_{13} + 31546 \beta_{12} - 4748 \beta_{11} + 10842 \beta_{10} - 17601 \beta_{9} + 7246 \beta_{8} - 38476 \beta_{7} + 14438 \beta_{6} - 10177 \beta_{5} - 10842 \beta_{4} + 38476 \beta_{3} - 4261 \beta_{2} - 10177 \beta_{1} + 15578\)
\(\nu^{12}\)\(=\)\(10177 \beta_{17} - 38476 \beta_{16} - 14438 \beta_{15} - 4261 \beta_{14} + 174807 \beta_{13} + 13855 \beta_{12} + 28293 \beta_{11} + 14438 \beta_{10} - 67935 \beta_{9} + 38476 \beta_{8} - 74249 \beta_{7} + 103688 \beta_{6} - 13456 \beta_{5} - 124527 \beta_{4} + 103688 \beta_{3} + 13456 \beta_{2} - 74249 \beta_{1} + 4261\)
\(\nu^{13}\)\(=\)\(13456 \beta_{17} - 74249 \beta_{16} - 103688 \beta_{15} - 29439 \beta_{14} + 407806 \beta_{13} - 154758 \beta_{12} + 304118 \beta_{11} - 125319 \beta_{9} + 103688 \beta_{8} - 107318 \beta_{7} + 359866 \beta_{6} + 48563 \beta_{5} - 532720 \beta_{4} + 155881 \beta_{3} + 107318 \beta_{2} - 359866 \beta_{1} - 138370\)
\(\nu^{14}\)\(=\)\(-48563 \beta_{17} - 107318 \beta_{16} - 359866 \beta_{15} + 490832 \beta_{13} - 673789 \beta_{12} + 1399960 \beta_{11} - 155881 \beta_{10} + 179529 \beta_{9} + 155881 \beta_{8} - 88369 \beta_{7} + 745254 \beta_{6} + 251020 \beta_{5} - 1555841 \beta_{4} + 745254 \beta_{2} - 996274 \beta_{1} - 673789\)
\(\nu^{15}\)\(=\)\(-251020 \beta_{17} - 88369 \beta_{16} - 745254 \beta_{15} + 251020 \beta_{14} - 789064 \beta_{13} - 1363369 \beta_{12} + 4210017 \beta_{11} - 996274 \beta_{10} + 2846648 \beta_{9} + 526665 \beta_{7} + 1092384 \beta_{6} - 2935017 \beta_{4} - 1619049 \beta_{3} + 3407251 \beta_{2} - 1619049 \beta_{1} - 1534318\)
\(\nu^{16}\)\(=\)\(526665 \beta_{16} - 1092384 \beta_{15} + 526665 \beta_{14} - 5580452 \beta_{13} + 526665 \beta_{12} + 8419308 \beta_{11} - 3407251 \beta_{10} + 14020626 \beta_{9} - 1619049 \beta_{8} + 2221838 \beta_{7} + 560842 \beta_{6} - 2221838 \beta_{5} - 1219807 \beta_{4} - 9609816 \beta_{3} + 9609816 \beta_{2} + 1219807\)
\(\nu^{17}\)\(=\)\(2221838 \beta_{17} + 2221838 \beta_{16} - 560842 \beta_{15} - 560842 \beta_{14} - 12855684 \beta_{13} + 15077522 \beta_{12} + 2221838 \beta_{11} - 7387978 \beta_{10} + 43431387 \beta_{9} - 9609816 \beta_{8} - 5546275 \beta_{6} - 5546275 \beta_{5} + 15212498 \beta_{4} - 32531806 \beta_{3} + 16431857 \beta_{2} + 16431857 \beta_{1} + 28353865\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(\beta_{12}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−1.37827 1.15651i
−0.344606 0.289159i
2.39653 + 2.01093i
−1.37827 + 1.15651i
−0.344606 + 0.289159i
2.39653 2.01093i
0.128174 0.726910i
−0.124000 + 0.703239i
−0.443867 + 2.51729i
2.91971 1.06269i
−0.168248 + 0.0612373i
−1.48541 + 0.540646i
0.128174 + 0.726910i
−0.124000 0.703239i
−0.443867 2.51729i
2.91971 + 1.06269i
−0.168248 0.0612373i
−1.48541 0.540646i
0 −0.312429 1.77187i 0 −0.766044 + 0.642788i 0 −0.336777 + 0.583316i 0 −0.222848 + 0.0811099i 0
61.2 0 −0.0781159 0.443017i 0 −0.766044 + 0.642788i 0 −1.06981 + 1.85297i 0 2.62892 0.956847i 0
61.3 0 0.543249 + 3.08092i 0 −0.766044 + 0.642788i 0 0.0481505 0.0833990i 0 −6.37785 + 2.32135i 0
81.1 0 −0.312429 + 1.77187i 0 −0.766044 0.642788i 0 −0.336777 0.583316i 0 −0.222848 0.0811099i 0
81.2 0 −0.0781159 + 0.443017i 0 −0.766044 0.642788i 0 −1.06981 1.85297i 0 2.62892 + 0.956847i 0
81.3 0 0.543249 3.08092i 0 −0.766044 0.642788i 0 0.0481505 + 0.0833990i 0 −6.37785 2.32135i 0
101.1 0 −0.693610 0.252453i 0 −0.173648 0.984808i 0 −2.32856 + 4.03318i 0 −1.88077 1.57815i 0
101.2 0 0.671023 + 0.244232i 0 −0.173648 0.984808i 0 1.15961 2.00851i 0 −1.90751 1.60059i 0
101.3 0 2.40197 + 0.874246i 0 −0.173648 0.984808i 0 −0.118043 + 0.204456i 0 2.70703 + 2.27147i 0
161.1 0 −2.38017 + 1.99720i 0 0.939693 + 0.342020i 0 2.42255 4.19598i 0 1.15545 6.55290i 0
161.2 0 0.137157 0.115088i 0 0.939693 + 0.342020i 0 −1.55670 + 2.69628i 0 −0.515378 + 2.92285i 0
161.3 0 1.21092 1.01608i 0 0.939693 + 0.342020i 0 1.77958 3.08232i 0 −0.0870410 + 0.493634i 0
301.1 0 −0.693610 + 0.252453i 0 −0.173648 + 0.984808i 0 −2.32856 4.03318i 0 −1.88077 + 1.57815i 0
301.2 0 0.671023 0.244232i 0 −0.173648 + 0.984808i 0 1.15961 + 2.00851i 0 −1.90751 + 1.60059i 0
301.3 0 2.40197 0.874246i 0 −0.173648 + 0.984808i 0 −0.118043 0.204456i 0 2.70703 2.27147i 0
321.1 0 −2.38017 1.99720i 0 0.939693 0.342020i 0 2.42255 + 4.19598i 0 1.15545 + 6.55290i 0
321.2 0 0.137157 + 0.115088i 0 0.939693 0.342020i 0 −1.55670 2.69628i 0 −0.515378 2.92285i 0
321.3 0 1.21092 + 1.01608i 0 0.939693 0.342020i 0 1.77958 + 3.08232i 0 −0.0870410 0.493634i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.u.a 18
19.e even 9 1 inner 380.2.u.a 18
19.e even 9 1 7220.2.a.v 9
19.f odd 18 1 7220.2.a.x 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.u.a 18 1.a even 1 1 trivial
380.2.u.a 18 19.e even 9 1 inner
7220.2.a.v 9 19.e even 9 1
7220.2.a.x 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \)
$3$ \( 9 - 81 T + 333 T^{2} - 321 T^{3} + 864 T^{4} - 696 T^{5} - 2681 T^{6} + 2961 T^{7} + 2952 T^{8} - 4587 T^{9} + 3537 T^{10} - 2082 T^{11} + 1135 T^{12} - 297 T^{13} + 54 T^{14} - 24 T^{15} + 9 T^{16} - 3 T^{17} + T^{18} \)
$5$ \( ( 1 - T^{3} + T^{6} )^{3} \)
$7$ \( 361 - 1653 T + 27006 T^{2} + 138287 T^{3} + 929244 T^{4} + 1375698 T^{5} + 1958534 T^{6} + 574365 T^{7} + 607725 T^{8} + 114305 T^{9} + 135885 T^{10} + 12939 T^{11} + 14237 T^{12} + 624 T^{13} + 1092 T^{14} + 20 T^{15} + 39 T^{16} + T^{18} \)
$11$ \( 531441 - 3011499 T + 14348907 T^{2} - 23733324 T^{3} + 37654308 T^{4} + 19015722 T^{5} + 29224809 T^{6} + 6576633 T^{7} + 5393259 T^{8} + 576852 T^{9} + 678159 T^{10} + 33150 T^{11} + 49594 T^{12} - 651 T^{13} + 2586 T^{14} - 28 T^{15} + 60 T^{16} + T^{18} \)
$13$ \( 199402641 + 242485812 T + 318064887 T^{2} + 194978367 T^{3} + 16157070 T^{4} - 7776 T^{5} + 25141006 T^{6} + 14289723 T^{7} + 2890386 T^{8} + 324925 T^{9} + 23364 T^{10} - 18378 T^{11} - 33 T^{12} + 720 T^{13} + 180 T^{14} - 65 T^{15} + 36 T^{16} - 9 T^{17} + T^{18} \)
$17$ \( 6561 + 1351566 T + 78428007 T^{2} - 53832762 T^{3} - 29663496 T^{4} - 1825092 T^{5} + 41510565 T^{6} - 19127772 T^{7} + 4433085 T^{8} - 1578156 T^{9} + 598572 T^{10} - 207255 T^{11} + 82384 T^{12} - 20205 T^{13} + 4971 T^{14} - 835 T^{15} + 126 T^{16} - 12 T^{17} + T^{18} \)
$19$ \( 322687697779 - 305704134738 T + 115309454331 T^{2} - 16042645421 T^{3} - 3974138895 T^{4} + 2653465881 T^{5} - 627193819 T^{6} + 35565720 T^{7} + 25351605 T^{8} - 9355997 T^{9} + 1334295 T^{10} + 98520 T^{11} - 91441 T^{12} + 20361 T^{13} - 1605 T^{14} - 341 T^{15} + 129 T^{16} - 18 T^{17} + T^{18} \)
$23$ \( 998001 - 7390602 T + 22051035 T^{2} + 37396944 T^{3} + 49444074 T^{4} + 33979293 T^{5} + 18398962 T^{6} + 3008349 T^{7} - 331038 T^{8} - 1111894 T^{9} - 212940 T^{10} + 89955 T^{11} + 45273 T^{12} + 10998 T^{13} + 1602 T^{14} - 61 T^{15} + 9 T^{17} + T^{18} \)
$29$ \( 387420489 + 1937102445 T + 3348078300 T^{2} + 2604946635 T^{3} + 3030099435 T^{4} + 1769248008 T^{5} + 1012124727 T^{6} + 495517662 T^{7} + 179399691 T^{8} + 53991999 T^{9} + 14238459 T^{10} + 2829096 T^{11} + 410626 T^{12} + 69870 T^{13} + 18570 T^{14} + 3880 T^{15} + 483 T^{16} + 33 T^{17} + T^{18} \)
$31$ \( 13082689 + 39465087 T + 84511188 T^{2} + 101006179 T^{3} + 97374624 T^{4} + 66254175 T^{5} + 42409844 T^{6} + 22039371 T^{7} + 11410299 T^{8} + 4677016 T^{9} + 1819128 T^{10} + 564957 T^{11} + 174419 T^{12} + 43959 T^{13} + 10152 T^{14} + 1690 T^{15} + 225 T^{16} + 18 T^{17} + T^{18} \)
$37$ \( ( -1864711 + 2459718 T - 981039 T^{2} + 20789 T^{3} + 74589 T^{4} - 15909 T^{5} + 226 T^{6} + 267 T^{7} - 30 T^{8} + T^{9} )^{2} \)
$41$ \( 15485980059729 - 8783559404028 T + 3992785127874 T^{2} - 425852224227 T^{3} + 631304779851 T^{4} + 163908523863 T^{5} + 27101527974 T^{6} + 7938374544 T^{7} + 674897814 T^{8} - 167833416 T^{9} - 13337559 T^{10} + 689628 T^{11} + 529402 T^{12} + 60330 T^{13} + 75 T^{14} - 859 T^{15} + 39 T^{16} + 18 T^{17} + T^{18} \)
$43$ \( 213364449 + 1219056399 T + 54881483418 T^{2} + 32013886446 T^{3} + 803377179342 T^{4} - 917237781144 T^{5} + 434851621398 T^{6} + 26919260982 T^{7} - 9016418871 T^{8} + 119916570 T^{9} + 48387447 T^{10} - 5099547 T^{11} + 2547988 T^{12} - 265989 T^{13} - 15564 T^{14} + 3602 T^{15} - 30 T^{16} - 18 T^{17} + T^{18} \)
$47$ \( 130045445640009 - 179958688344261 T + 66525997842135 T^{2} - 1583823950019 T^{3} + 5561944470882 T^{4} + 811622311416 T^{5} + 225297571743 T^{6} + 25878153537 T^{7} + 4037264694 T^{8} + 372713481 T^{9} + 124418511 T^{10} + 20752362 T^{11} + 2055249 T^{12} + 35397 T^{13} - 13572 T^{14} - 1824 T^{15} - 9 T^{16} + 15 T^{17} + T^{18} \)
$53$ \( 1556065809 + 1677483675 T + 1141036632 T^{2} + 867908520 T^{3} + 460714149 T^{4} + 158692608 T^{5} + 75192147 T^{6} + 28770768 T^{7} + 8976483 T^{8} + 4198461 T^{9} + 1506294 T^{10} + 310899 T^{11} + 61258 T^{12} - 8433 T^{13} - 3243 T^{14} - 292 T^{15} + 225 T^{16} - 24 T^{17} + T^{18} \)
$59$ \( 191850201 - 3574472166 T + 26395493490 T^{2} - 82277997426 T^{3} + 35284697145 T^{4} + 231199512732 T^{5} + 489374808867 T^{6} - 397263915666 T^{7} + 140510486322 T^{8} - 32139657504 T^{9} + 5471687331 T^{10} - 729912123 T^{11} + 78402493 T^{12} - 6910323 T^{13} + 501945 T^{14} - 29629 T^{15} + 1389 T^{16} - 48 T^{17} + T^{18} \)
$61$ \( 1917895384161 - 2453998052952 T + 923702612067 T^{2} - 476846048859 T^{3} + 394578824094 T^{4} + 125946090816 T^{5} - 5430620597 T^{6} - 9379503003 T^{7} - 589650561 T^{8} + 258044884 T^{9} + 57734451 T^{10} + 3906459 T^{11} + 243915 T^{12} + 57996 T^{13} + 15387 T^{14} + 2710 T^{15} + 294 T^{16} + 18 T^{17} + T^{18} \)
$67$ \( 8000542361529 + 9594938348784 T + 5483911823568 T^{2} - 410232162927 T^{3} + 1576396010331 T^{4} + 55562474559 T^{5} + 150920381278 T^{6} + 62440358178 T^{7} + 17115773571 T^{8} + 4282268604 T^{9} + 645487137 T^{10} + 53889972 T^{11} + 5783086 T^{12} - 100668 T^{13} - 74340 T^{14} - 3243 T^{15} + 351 T^{16} + 36 T^{17} + T^{18} \)
$71$ \( 77975557962129 - 45260971520823 T + 13158492505074 T^{2} + 2870713777149 T^{3} + 1393180389090 T^{4} + 456931747809 T^{5} + 80270827956 T^{6} + 12192008625 T^{7} + 1341879696 T^{8} - 48961827 T^{9} - 16679520 T^{10} - 738558 T^{11} - 61146 T^{12} + 29025 T^{13} + 11862 T^{14} + 540 T^{15} - 81 T^{16} + T^{18} \)
$73$ \( 18281242781649 + 4619749048389 T + 5929670417301 T^{2} + 3372504905502 T^{3} - 66574281609 T^{4} - 307082624616 T^{5} - 29619969569 T^{6} - 7540633971 T^{7} + 7909649424 T^{8} + 1394603224 T^{9} + 107666946 T^{10} - 14341167 T^{11} + 119541 T^{12} + 41787 T^{13} + 18252 T^{14} - 884 T^{15} + 81 T^{16} - 18 T^{17} + T^{18} \)
$79$ \( 400527093527929 - 464463797719392 T + 194555267459565 T^{2} - 23591455974487 T^{3} - 5099277560556 T^{4} + 1496620764804 T^{5} + 3997246799 T^{6} - 12254366697 T^{7} + 12422251125 T^{8} - 3847121434 T^{9} + 544406913 T^{10} - 65816004 T^{11} + 8010821 T^{12} - 448428 T^{13} + 24753 T^{14} - 1459 T^{15} + 237 T^{16} - 9 T^{17} + T^{18} \)
$83$ \( 2004181007481 + 13236737748129 T + 87492664439262 T^{2} + 2640915559449 T^{3} + 10358289668214 T^{4} + 1583572901391 T^{5} + 900190256725 T^{6} + 128275228986 T^{7} + 40743651921 T^{8} + 6089618079 T^{9} + 1312052094 T^{10} + 152005605 T^{11} + 19839787 T^{12} + 1648857 T^{13} + 165567 T^{14} + 11160 T^{15} + 801 T^{16} + 30 T^{17} + T^{18} \)
$89$ \( 51059270662929 - 61285941863982 T + 41570041091607 T^{2} - 19726905894237 T^{3} + 7788272037597 T^{4} - 2108143362240 T^{5} + 288072941113 T^{6} + 14487726129 T^{7} - 10454641674 T^{8} + 1076680870 T^{9} + 95151402 T^{10} - 47745702 T^{11} + 8144598 T^{12} - 682938 T^{13} + 31689 T^{14} - 1172 T^{15} + 33 T^{16} - 6 T^{17} + T^{18} \)
$97$ \( 3020598812169 - 4993061114313 T + 6003707481783 T^{2} - 4602149009151 T^{3} + 1437262403445 T^{4} + 51134589816 T^{5} + 12720397951 T^{6} - 28934451225 T^{7} + 8151651675 T^{8} - 837258990 T^{9} + 31127139 T^{10} + 3019470 T^{11} - 972008 T^{12} + 102114 T^{13} + 6327 T^{14} - 2418 T^{15} + 297 T^{16} - 21 T^{17} + T^{18} \)
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