Properties

Label 930.2.n.g
Level $930$
Weight $2$
Character orbit 930.n
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} + 31 x^{14} + 20 x^{13} + 474 x^{12} + 463 x^{11} + 6637 x^{10} + 13567 x^{9} + 108596 x^{8} + 172826 x^{7} + 917001 x^{6} + 1583565 x^{5} + 3229491 x^{4} - 1732290 x^{3} + 4672960 x^{2} - 5544800 x + 22848400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + 31 x^{14} + 20 x^{13} + 474 x^{12} + 463 x^{11} + 6637 x^{10} + 13567 x^{9} + 108596 x^{8} + 172826 x^{7} + 917001 x^{6} + 1583565 x^{5} + 3229491 x^{4} - 1732290 x^{3} + 4672960 x^{2} - 5544800 x + 22848400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(15\!\cdots\!70\)\( \nu^{15} + \)\(39\!\cdots\!84\)\( \nu^{14} - \)\(23\!\cdots\!69\)\( \nu^{13} + \)\(13\!\cdots\!94\)\( \nu^{12} - \)\(41\!\cdots\!80\)\( \nu^{11} + \)\(12\!\cdots\!36\)\( \nu^{10} - \)\(41\!\cdots\!08\)\( \nu^{9} + \)\(74\!\cdots\!43\)\( \nu^{8} - \)\(25\!\cdots\!92\)\( \nu^{7} + \)\(33\!\cdots\!54\)\( \nu^{6} - \)\(61\!\cdots\!16\)\( \nu^{5} - \)\(31\!\cdots\!86\)\( \nu^{4} - \)\(37\!\cdots\!45\)\( \nu^{3} - \)\(22\!\cdots\!06\)\( \nu^{2} - \)\(33\!\cdots\!20\)\( \nu - \)\(62\!\cdots\!20\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(32\!\cdots\!22\)\( \nu^{15} + \)\(54\!\cdots\!43\)\( \nu^{14} + \)\(52\!\cdots\!02\)\( \nu^{13} + \)\(16\!\cdots\!00\)\( \nu^{12} + \)\(37\!\cdots\!78\)\( \nu^{11} + \)\(23\!\cdots\!26\)\( \nu^{10} + \)\(65\!\cdots\!99\)\( \nu^{9} + \)\(36\!\cdots\!04\)\( \nu^{8} + \)\(13\!\cdots\!22\)\( \nu^{7} + \)\(60\!\cdots\!62\)\( \nu^{6} + \)\(14\!\cdots\!92\)\( \nu^{5} + \)\(47\!\cdots\!75\)\( \nu^{4} + \)\(15\!\cdots\!02\)\( \nu^{3} + \)\(97\!\cdots\!00\)\( \nu^{2} - \)\(54\!\cdots\!00\)\( \nu + \)\(17\!\cdots\!00\)\(\)\()/ \)\(43\!\cdots\!40\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(26\!\cdots\!25\)\( \nu^{15} + \)\(59\!\cdots\!29\)\( \nu^{14} - \)\(22\!\cdots\!39\)\( \nu^{13} + \)\(16\!\cdots\!64\)\( \nu^{12} - \)\(30\!\cdots\!90\)\( \nu^{11} + \)\(17\!\cdots\!41\)\( \nu^{10} - \)\(32\!\cdots\!53\)\( \nu^{9} + \)\(14\!\cdots\!93\)\( \nu^{8} - \)\(46\!\cdots\!72\)\( \nu^{7} + \)\(25\!\cdots\!14\)\( \nu^{6} - \)\(33\!\cdots\!21\)\( \nu^{5} + \)\(90\!\cdots\!39\)\( \nu^{4} - \)\(31\!\cdots\!95\)\( \nu^{3} - \)\(17\!\cdots\!26\)\( \nu^{2} - \)\(32\!\cdots\!20\)\( \nu - \)\(46\!\cdots\!20\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(57\!\cdots\!19\)\( \nu^{15} - \)\(54\!\cdots\!04\)\( \nu^{14} + \)\(38\!\cdots\!34\)\( \nu^{13} - \)\(16\!\cdots\!90\)\( \nu^{12} + \)\(61\!\cdots\!56\)\( \nu^{11} - \)\(18\!\cdots\!13\)\( \nu^{10} + \)\(71\!\cdots\!58\)\( \nu^{9} - \)\(19\!\cdots\!32\)\( \nu^{8} + \)\(71\!\cdots\!94\)\( \nu^{7} - \)\(26\!\cdots\!46\)\( \nu^{6} + \)\(11\!\cdots\!79\)\( \nu^{5} - \)\(20\!\cdots\!60\)\( \nu^{4} + \)\(17\!\cdots\!54\)\( \nu^{3} - \)\(30\!\cdots\!60\)\( \nu^{2} + \)\(61\!\cdots\!60\)\( \nu - \)\(64\!\cdots\!00\)\(\)\()/ \)\(43\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(18\!\cdots\!65\)\( \nu^{15} + \)\(26\!\cdots\!23\)\( \nu^{14} - \)\(43\!\cdots\!38\)\( \nu^{13} - \)\(24\!\cdots\!22\)\( \nu^{12} - \)\(46\!\cdots\!10\)\( \nu^{11} + \)\(45\!\cdots\!47\)\( \nu^{10} - \)\(65\!\cdots\!91\)\( \nu^{9} - \)\(92\!\cdots\!94\)\( \nu^{8} - \)\(11\!\cdots\!84\)\( \nu^{7} + \)\(27\!\cdots\!68\)\( \nu^{6} - \)\(22\!\cdots\!47\)\( \nu^{5} + \)\(55\!\cdots\!63\)\( \nu^{4} + \)\(54\!\cdots\!10\)\( \nu^{3} + \)\(41\!\cdots\!28\)\( \nu^{2} + \)\(14\!\cdots\!00\)\( \nu - \)\(29\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(23\!\cdots\!76\)\( \nu^{15} + \)\(92\!\cdots\!32\)\( \nu^{14} + \)\(57\!\cdots\!83\)\( \nu^{13} + \)\(41\!\cdots\!48\)\( \nu^{12} + \)\(11\!\cdots\!04\)\( \nu^{11} + \)\(66\!\cdots\!10\)\( \nu^{10} + \)\(18\!\cdots\!76\)\( \nu^{9} + \)\(10\!\cdots\!23\)\( \nu^{8} + \)\(38\!\cdots\!62\)\( \nu^{7} + \)\(16\!\cdots\!94\)\( \nu^{6} + \)\(36\!\cdots\!94\)\( \nu^{5} + \)\(13\!\cdots\!38\)\( \nu^{4} + \)\(20\!\cdots\!11\)\( \nu^{3} + \)\(29\!\cdots\!88\)\( \nu^{2} - \)\(23\!\cdots\!60\)\( \nu + \)\(56\!\cdots\!80\)\(\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(25\!\cdots\!05\)\( \nu^{15} - \)\(12\!\cdots\!14\)\( \nu^{14} + \)\(95\!\cdots\!49\)\( \nu^{13} - \)\(21\!\cdots\!24\)\( \nu^{12} + \)\(11\!\cdots\!80\)\( \nu^{11} - \)\(20\!\cdots\!81\)\( \nu^{10} + \)\(15\!\cdots\!28\)\( \nu^{9} - \)\(76\!\cdots\!73\)\( \nu^{8} + \)\(19\!\cdots\!22\)\( \nu^{7} - \)\(29\!\cdots\!64\)\( \nu^{6} + \)\(16\!\cdots\!11\)\( \nu^{5} + \)\(11\!\cdots\!96\)\( \nu^{4} + \)\(20\!\cdots\!45\)\( \nu^{3} + \)\(40\!\cdots\!06\)\( \nu^{2} + \)\(62\!\cdots\!80\)\( \nu + \)\(22\!\cdots\!20\)\(\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(15\!\cdots\!48\)\( \nu^{15} - \)\(58\!\cdots\!72\)\( \nu^{14} + \)\(68\!\cdots\!67\)\( \nu^{13} - \)\(12\!\cdots\!14\)\( \nu^{12} + \)\(12\!\cdots\!92\)\( \nu^{11} - \)\(14\!\cdots\!82\)\( \nu^{10} + \)\(19\!\cdots\!84\)\( \nu^{9} - \)\(74\!\cdots\!17\)\( \nu^{8} + \)\(25\!\cdots\!80\)\( \nu^{7} - \)\(36\!\cdots\!46\)\( \nu^{6} + \)\(27\!\cdots\!34\)\( \nu^{5} + \)\(11\!\cdots\!66\)\( \nu^{4} + \)\(19\!\cdots\!63\)\( \nu^{3} + \)\(57\!\cdots\!26\)\( \nu^{2} + \)\(74\!\cdots\!80\)\( \nu - \)\(91\!\cdots\!40\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(21\!\cdots\!66\)\( \nu^{15} - \)\(16\!\cdots\!32\)\( \nu^{14} + \)\(59\!\cdots\!57\)\( \nu^{13} - \)\(34\!\cdots\!86\)\( \nu^{12} + \)\(41\!\cdots\!24\)\( \nu^{11} - \)\(51\!\cdots\!56\)\( \nu^{10} - \)\(12\!\cdots\!56\)\( \nu^{9} - \)\(60\!\cdots\!15\)\( \nu^{8} - \)\(74\!\cdots\!36\)\( \nu^{7} - \)\(12\!\cdots\!90\)\( \nu^{6} - \)\(21\!\cdots\!00\)\( \nu^{5} - \)\(95\!\cdots\!66\)\( \nu^{4} - \)\(26\!\cdots\!99\)\( \nu^{3} - \)\(49\!\cdots\!66\)\( \nu^{2} + \)\(49\!\cdots\!00\)\( \nu + \)\(81\!\cdots\!60\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(11\!\cdots\!57\)\( \nu^{15} - \)\(80\!\cdots\!94\)\( \nu^{14} + \)\(34\!\cdots\!84\)\( \nu^{13} + \)\(33\!\cdots\!63\)\( \nu^{12} + \)\(53\!\cdots\!38\)\( \nu^{11} + \)\(89\!\cdots\!13\)\( \nu^{10} + \)\(71\!\cdots\!78\)\( \nu^{9} + \)\(22\!\cdots\!90\)\( \nu^{8} + \)\(12\!\cdots\!43\)\( \nu^{7} + \)\(29\!\cdots\!50\)\( \nu^{6} + \)\(10\!\cdots\!15\)\( \nu^{5} + \)\(27\!\cdots\!28\)\( \nu^{4} + \)\(48\!\cdots\!62\)\( \nu^{3} + \)\(91\!\cdots\!73\)\( \nu^{2} + \)\(92\!\cdots\!90\)\( \nu + \)\(14\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(\)\(39\!\cdots\!31\)\( \nu^{15} - \)\(64\!\cdots\!46\)\( \nu^{14} + \)\(11\!\cdots\!16\)\( \nu^{13} + \)\(21\!\cdots\!10\)\( \nu^{12} + \)\(13\!\cdots\!64\)\( \nu^{11} + \)\(73\!\cdots\!63\)\( \nu^{10} + \)\(17\!\cdots\!72\)\( \nu^{9} + \)\(35\!\cdots\!22\)\( \nu^{8} + \)\(31\!\cdots\!46\)\( \nu^{7} + \)\(28\!\cdots\!46\)\( \nu^{6} + \)\(16\!\cdots\!11\)\( \nu^{5} + \)\(26\!\cdots\!90\)\( \nu^{4} - \)\(35\!\cdots\!04\)\( \nu^{3} - \)\(21\!\cdots\!20\)\( \nu^{2} - \)\(15\!\cdots\!80\)\( \nu + \)\(24\!\cdots\!00\)\(\)\()/ \)\(43\!\cdots\!40\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(36\!\cdots\!58\)\( \nu^{15} - \)\(19\!\cdots\!43\)\( \nu^{14} - \)\(10\!\cdots\!57\)\( \nu^{13} - \)\(24\!\cdots\!06\)\( \nu^{12} - \)\(15\!\cdots\!22\)\( \nu^{11} - \)\(46\!\cdots\!08\)\( \nu^{10} - \)\(22\!\cdots\!39\)\( \nu^{9} - \)\(88\!\cdots\!43\)\( \nu^{8} - \)\(40\!\cdots\!70\)\( \nu^{7} - \)\(11\!\cdots\!84\)\( \nu^{6} - \)\(37\!\cdots\!74\)\( \nu^{5} - \)\(11\!\cdots\!41\)\( \nu^{4} - \)\(15\!\cdots\!93\)\( \nu^{3} - \)\(83\!\cdots\!06\)\( \nu^{2} - \)\(47\!\cdots\!40\)\( \nu + \)\(28\!\cdots\!60\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(42\!\cdots\!33\)\( \nu^{15} - \)\(64\!\cdots\!09\)\( \nu^{14} + \)\(12\!\cdots\!04\)\( \nu^{13} + \)\(22\!\cdots\!39\)\( \nu^{12} + \)\(18\!\cdots\!82\)\( \nu^{11} + \)\(39\!\cdots\!85\)\( \nu^{10} + \)\(28\!\cdots\!43\)\( \nu^{9} + \)\(85\!\cdots\!84\)\( \nu^{8} + \)\(48\!\cdots\!21\)\( \nu^{7} + \)\(11\!\cdots\!82\)\( \nu^{6} + \)\(41\!\cdots\!37\)\( \nu^{5} + \)\(10\!\cdots\!69\)\( \nu^{4} + \)\(18\!\cdots\!38\)\( \nu^{3} + \)\(95\!\cdots\!09\)\( \nu^{2} + \)\(33\!\cdots\!60\)\( \nu + \)\(59\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!20\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(35\!\cdots\!94\)\( \nu^{15} - \)\(67\!\cdots\!59\)\( \nu^{14} + \)\(10\!\cdots\!24\)\( \nu^{13} + \)\(15\!\cdots\!60\)\( \nu^{12} + \)\(16\!\cdots\!06\)\( \nu^{11} + \)\(26\!\cdots\!02\)\( \nu^{10} + \)\(23\!\cdots\!53\)\( \nu^{9} + \)\(62\!\cdots\!98\)\( \nu^{8} + \)\(39\!\cdots\!54\)\( \nu^{7} + \)\(81\!\cdots\!74\)\( \nu^{6} + \)\(32\!\cdots\!24\)\( \nu^{5} + \)\(66\!\cdots\!65\)\( \nu^{4} + \)\(11\!\cdots\!24\)\( \nu^{3} - \)\(10\!\cdots\!60\)\( \nu^{2} - \)\(18\!\cdots\!40\)\( \nu - \)\(24\!\cdots\!20\)\(\)\()/ \)\(12\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{7} + 11 \beta_{6} - \beta_{5}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{14} - 2 \beta_{13} - \beta_{11} + 2 \beta_{10} + 9 \beta_{8} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 11 \beta_{3} - 3 \beta_{2} - 2 \beta_{1} - 6\)
\(\nu^{4}\)\(=\)\(-19 \beta_{15} + 41 \beta_{13} + \beta_{12} + 6 \beta_{11} - 21 \beta_{10} - 19 \beta_{9} + 195 \beta_{8} + 140 \beta_{7} - 208 \beta_{6} + 25 \beta_{5} - 13 \beta_{4} + 25 \beta_{3} + 19 \beta_{2} + \beta_{1} - 159\)
\(\nu^{5}\)\(=\)\(-45 \beta_{15} + 27 \beta_{14} + 76 \beta_{13} - 42 \beta_{12} + 98 \beta_{11} - 27 \beta_{9} - 27 \beta_{8} + 243 \beta_{7} - 270 \beta_{6} + 195 \beta_{5} - 76 \beta_{4} - 15 \beta_{3} + 125 \beta_{2} - 114\)
\(\nu^{6}\)\(=\)\(50 \beta_{15} - 50 \beta_{14} - 394 \beta_{13} + 159 \beta_{12} + 394 \beta_{10} + 394 \beta_{9} - 3524 \beta_{8} + 50 \beta_{7} + 1422 \beta_{6} + 109 \beta_{5} - 266 \beta_{4} - 394 \beta_{3} - 152 \beta_{2} + 18 \beta_{1} + 1078\)
\(\nu^{7}\)\(=\)\(-186 \beta_{15} + 895 \beta_{14} - 1495 \beta_{13} + 3094 \beta_{12} - 2556 \beta_{11} - 186 \beta_{10} - 7303 \beta_{8} - 7489 \beta_{7} + 6875 \beta_{6} + 186 \beta_{5} + 812 \beta_{3} - 1495 \beta_{2} + 812 \beta_{1} + 1495\)
\(\nu^{8}\)\(=\)\(6456 \beta_{14} - 441 \beta_{13} - 6958 \beta_{11} - 6456 \beta_{10} - 7600 \beta_{9} + 23203 \beta_{8} - 45888 \beta_{7} + 441 \beta_{6} - 3961 \beta_{5} + 7399 \beta_{4} + 5305 \beta_{3} - 7399 \beta_{2} - 3961 \beta_{1} - 15804\)
\(\nu^{9}\)\(=\)\(-2670 \beta_{15} + 26144 \beta_{13} - 53791 \beta_{12} + 38690 \beta_{11} - 20425 \beta_{10} - 2670 \beta_{9} + 200625 \beta_{8} + 28062 \beta_{7} - 164605 \beta_{6} - 34675 \beta_{5} + 36020 \beta_{4} - 34675 \beta_{3} + 2670 \beta_{2} - 53791 \beta_{1} - 30732\)
\(\nu^{10}\)\(=\)\(-27884 \beta_{15} - 125831 \beta_{14} + 51094 \beta_{13} - 92134 \beta_{12} + 338467 \beta_{11} + 125831 \beta_{9} + 125831 \beta_{8} + 686648 \beta_{7} - 560817 \beta_{6} - 878 \beta_{5} - 51094 \beta_{4} - 217965 \beta_{3} + 212636 \beta_{2} + 521387\)
\(\nu^{11}\)\(=\)\(359450 \beta_{15} - 359450 \beta_{14} - 525010 \beta_{13} + 352855 \beta_{12} + 525010 \beta_{10} + 525010 \beta_{9} - 4548668 \beta_{8} + 359450 \beta_{7} + 3199820 \beta_{6} - 6595 \beta_{5} - 943324 \beta_{4} - 525010 \beta_{3} + 408482 \beta_{2} + 1000073 \beta_{1} + 3034260\)
\(\nu^{12}\)\(=\)\(2535937 \beta_{15} + 703639 \beta_{14} - 4262215 \beta_{13} + 1894088 \beta_{12} - 7641341 \beta_{11} + 2535937 \beta_{10} - 16550911 \beta_{8} - 14014974 \beta_{7} + 27520344 \beta_{6} - 2535937 \beta_{5} + 2090936 \beta_{3} - 4262215 \beta_{2} + 2090936 \beta_{1} + 4262215\)
\(\nu^{13}\)\(=\)\(5346644 \beta_{14} + 418107 \beta_{13} - 21815283 \beta_{11} - 5346644 \beta_{10} - 12788554 \beta_{9} + 77427864 \beta_{8} - 32508937 \beta_{7} - 418107 \beta_{6} - 7886931 \beta_{5} + 21397176 \beta_{4} + 24567646 \beta_{3} - 21397176 \beta_{2} - 7886931 \beta_{1} - 56030688\)
\(\nu^{14}\)\(=\)\(-52603301 \beta_{15} + 125404283 \beta_{13} - 50673810 \beta_{12} + 100076600 \beta_{11} - 70347306 \beta_{10} - 52603301 \beta_{9} + 674371214 \beta_{8} + 291450406 \beta_{7} - 626897915 \beta_{6} + 48933450 \beta_{5} + 47473299 \beta_{4} + 48933450 \beta_{3} + 52603301 \beta_{2} - 50673810 \beta_{1} - 344053707\)
\(\nu^{15}\)\(=\)\(-157117913 \beta_{15} - 145620408 \beta_{14} + 356746928 \beta_{13} - 180313932 \beta_{12} + 789657437 \beta_{11} + 145620408 \beta_{9} + 145620408 \beta_{8} + 1885602431 \beta_{7} - 1739982023 \beta_{6} + 380897339 \beta_{5} - 356746928 \beta_{4} - 325934340 \beta_{3} + 644037029 \beta_{2} + 77376219\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
−1.17072 3.60310i
−0.530826 1.63371i
1.03459 + 3.18415i
1.47597 + 4.54256i
−2.68002 + 1.94715i
−2.02450 + 1.47089i
1.10063 0.799653i
3.29488 2.39387i
−2.68002 1.94715i
−2.02450 1.47089i
1.10063 + 0.799653i
3.29488 + 2.39387i
−1.17072 + 3.60310i
−0.530826 + 1.63371i
1.03459 3.18415i
1.47597 4.54256i
−0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 1.00000 −1.17072 + 3.60310i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.2 −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 1.00000 −0.530826 + 1.63371i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.3 −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 1.00000 1.03459 3.18415i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.4 −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 1.00000 1.47597 4.54256i 0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
721.1 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 1.00000 −2.68002 1.94715i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.2 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 1.00000 −2.02450 1.47089i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.3 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 1.00000 1.10063 + 0.799653i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.4 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 1.00000 3.29488 + 2.39387i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
841.1 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 1.00000 −2.68002 + 1.94715i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.2 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 1.00000 −2.02450 + 1.47089i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.3 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 1.00000 1.10063 0.799653i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.4 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 1.00000 3.29488 2.39387i −0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
901.1 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 1.00000 −1.17072 3.60310i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.2 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 1.00000 −0.530826 1.63371i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.3 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 1.00000 1.03459 + 3.18415i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.4 −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 1.00000 1.47597 + 4.54256i 0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.g 16
31.d even 5 1 inner 930.2.n.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.g 16 1.a even 1 1 trivial
930.2.n.g 16 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$5$ \( ( -1 + T )^{16} \)
$7$ \( 22848400 - 5544800 T + 4672960 T^{2} - 1732290 T^{3} + 3229491 T^{4} + 1583565 T^{5} + 917001 T^{6} + 172826 T^{7} + 108596 T^{8} + 13567 T^{9} + 6637 T^{10} + 463 T^{11} + 474 T^{12} + 20 T^{13} + 31 T^{14} - T^{15} + T^{16} \)
$11$ \( 249766416 + 228336192 T + 12273120 T^{2} - 122473218 T^{3} + 125414587 T^{4} - 33706925 T^{5} + 16179112 T^{6} - 808330 T^{7} + 993843 T^{8} - 46354 T^{9} + 46975 T^{10} - 3199 T^{11} + 1991 T^{12} - 119 T^{13} + 63 T^{14} + T^{16} \)
$13$ \( 4583881 - 26165161 T + 46092923 T^{2} + 105250801 T^{3} + 90277328 T^{4} + 34499129 T^{5} + 12295400 T^{6} + 2486195 T^{7} + 680221 T^{8} + 85740 T^{9} + 34345 T^{10} + 36 T^{11} + 1763 T^{12} - 111 T^{13} + 68 T^{14} - 4 T^{15} + T^{16} \)
$17$ \( 6497004816 - 548429616 T + 105713568 T^{2} + 155691354 T^{3} + 189615961 T^{4} + 39518372 T^{5} + 29894230 T^{6} + 7291740 T^{7} + 2355216 T^{8} + 292320 T^{9} + 98355 T^{10} + 949 T^{11} + 3004 T^{12} - 236 T^{13} + 65 T^{14} - 3 T^{15} + T^{16} \)
$19$ \( 6400 - 38400 T + 116960 T^{2} - 169180 T^{3} + 172381 T^{4} + 100441 T^{5} + 69759 T^{6} - 221827 T^{7} + 231656 T^{8} + 120933 T^{9} + 208484 T^{10} + 46089 T^{11} + 10164 T^{12} + 1225 T^{13} + 143 T^{14} + 12 T^{15} + T^{16} \)
$23$ \( 2093611536 + 369250920 T + 176720724 T^{2} - 84047358 T^{3} + 201826615 T^{4} - 13815679 T^{5} + 69803405 T^{6} - 10376453 T^{7} + 9215602 T^{8} - 625173 T^{9} + 191978 T^{10} - 11065 T^{11} + 3794 T^{12} + 169 T^{13} + 63 T^{14} + 2 T^{15} + T^{16} \)
$29$ \( 810000 - 9342000 T + 233186400 T^{2} - 301258830 T^{3} + 257027701 T^{4} - 133483840 T^{5} + 43870965 T^{6} - 9188289 T^{7} + 1843755 T^{8} - 487985 T^{9} + 125781 T^{10} - 25180 T^{11} + 5865 T^{12} - 1264 T^{13} + 190 T^{14} - 15 T^{15} + T^{16} \)
$31$ \( 852891037441 - 467714439887 T + 132238048469 T^{2} - 23361387216 T^{3} + 1832265664 T^{4} + 217027435 T^{5} - 82403828 T^{6} + 12804178 T^{7} - 1881257 T^{8} + 413038 T^{9} - 85748 T^{10} + 7285 T^{11} + 1984 T^{12} - 816 T^{13} + 149 T^{14} - 17 T^{15} + T^{16} \)
$37$ \( ( 117899 - 142612 T - 51360 T^{2} + 22650 T^{3} + 7801 T^{4} - 126 T^{5} - 162 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$41$ \( 544644000000 - 1469653200000 T + 1419799536000 T^{2} + 476600834400 T^{3} + 195049097776 T^{4} + 31372312088 T^{5} + 5224933616 T^{6} + 513871258 T^{7} + 103119093 T^{8} + 7595096 T^{9} + 1695711 T^{10} + 120267 T^{11} + 21902 T^{12} + 1639 T^{13} + 177 T^{14} + 7 T^{15} + T^{16} \)
$43$ \( 309830277376 - 264901814592 T + 136705864976 T^{2} - 37310656244 T^{3} + 9455206965 T^{4} - 356185257 T^{5} + 312952455 T^{6} + 12680900 T^{7} + 21558116 T^{8} + 5825 T^{9} + 329965 T^{10} - 7038 T^{11} + 6140 T^{12} + 404 T^{13} + 86 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( 26889440400 - 18797027400 T + 31366652760 T^{2} - 30953179020 T^{3} + 24351124621 T^{4} - 7227232137 T^{5} + 2750271995 T^{6} + 498466592 T^{7} + 80515328 T^{8} + 2941193 T^{9} + 1147695 T^{10} + 43550 T^{11} + 18002 T^{12} + 1046 T^{13} + 174 T^{14} + 16 T^{15} + T^{16} \)
$53$ \( 802242662400 + 68143334400 T + 73097256960 T^{2} + 4752754560 T^{3} + 4947035536 T^{4} + 670311632 T^{5} + 354581128 T^{6} + 52948390 T^{7} + 23259559 T^{8} + 2717819 T^{9} + 546389 T^{10} + 26803 T^{11} + 6666 T^{12} - 147 T^{13} + 89 T^{14} - 9 T^{15} + T^{16} \)
$59$ \( 20322965610000 - 5439338217000 T + 1131946002900 T^{2} - 120012280530 T^{3} + 68233466101 T^{4} + 1050346086 T^{5} + 1607098247 T^{6} + 40930435 T^{7} + 63814329 T^{8} + 1449137 T^{9} + 1416251 T^{10} + 192666 T^{11} + 34341 T^{12} + 2234 T^{13} + 266 T^{14} + 7 T^{15} + T^{16} \)
$61$ \( ( -6144016 + 1936680 T + 468179 T^{2} - 169795 T^{3} - 1328 T^{4} + 3445 T^{5} - 172 T^{6} - 15 T^{7} + T^{8} )^{2} \)
$67$ \( ( 484096 - 217456 T - 156171 T^{2} + 86082 T^{3} - 4504 T^{4} - 3351 T^{5} + 641 T^{6} - 43 T^{7} + T^{8} )^{2} \)
$71$ \( 3033449222400 + 4119630537600 T + 2518597935360 T^{2} + 641231530080 T^{3} + 224035237936 T^{4} + 45678041176 T^{5} + 10307403664 T^{6} + 1353280326 T^{7} + 269248967 T^{8} + 16213995 T^{9} + 4714565 T^{10} - 25207 T^{11} + 36568 T^{12} + 187 T^{13} - 5 T^{14} + 3 T^{15} + T^{16} \)
$73$ \( 280543952896 - 130195648512 T + 52708185344 T^{2} - 13944557424 T^{3} + 28718021393 T^{4} + 10050825581 T^{5} + 3214622999 T^{6} + 544942077 T^{7} + 98608804 T^{8} + 9898861 T^{9} + 1708588 T^{10} + 58121 T^{11} + 7604 T^{12} + 377 T^{13} + 99 T^{14} - 8 T^{15} + T^{16} \)
$79$ \( 8077515625 - 3990450000 T - 1401993125 T^{2} + 344869500 T^{3} + 2657827525 T^{4} - 2535816200 T^{5} + 1263223460 T^{6} - 422898090 T^{7} + 114099981 T^{8} - 24041510 T^{9} + 4430958 T^{10} - 659360 T^{11} + 82824 T^{12} - 8040 T^{13} + 602 T^{14} - 30 T^{15} + T^{16} \)
$83$ \( 10719703296 - 31450923648 T + 32261418432 T^{2} + 16562038656 T^{3} + 10412320336 T^{4} + 1669502592 T^{5} + 409827960 T^{6} - 83090092 T^{7} + 26048369 T^{8} - 5927836 T^{9} + 1763660 T^{10} - 274796 T^{11} + 36006 T^{12} - 3292 T^{13} + 283 T^{14} - 16 T^{15} + T^{16} \)
$89$ \( 1129373798400 + 1719310924800 T + 1078407613440 T^{2} + 33676669440 T^{3} + 225002680336 T^{4} - 8539600712 T^{5} + 14290154448 T^{6} - 379918320 T^{7} + 396069689 T^{8} - 500169 T^{9} + 2420774 T^{10} + 286437 T^{11} + 17031 T^{12} + 837 T^{13} + 344 T^{14} + 29 T^{15} + T^{16} \)
$97$ \( 9201413776 + 65004817080 T + 182200175408 T^{2} + 62724706840 T^{3} + 88337644153 T^{4} + 50020708005 T^{5} + 16701650821 T^{6} + 3031574150 T^{7} + 639032880 T^{8} + 78601125 T^{9} + 10193121 T^{10} + 632255 T^{11} + 24288 T^{12} - 1210 T^{13} + 223 T^{14} + 5 T^{15} + T^{16} \)
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