Properties

Label 930.2.bg.e
Level $930$
Weight $2$
Character orbit 930.bg
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.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 27 x^{14} + 345 x^{12} - 2652 x^{10} + 13244 x^{8} - 43398 x^{6} + 89940 x^{4} - 107433 x^{2} + 58081\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 27 x^{14} + 345 x^{12} - 2652 x^{10} + 13244 x^{8} - 43398 x^{6} + 89940 x^{4} - 107433 x^{2} + 58081\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -481 \nu^{14} + 10935 \nu^{12} - 120641 \nu^{10} + 792237 \nu^{8} - 3303859 \nu^{6} + 8551395 \nu^{4} - 12690614 \nu^{2} + 161849 \nu + 8734563 \)\()/323698\)
\(\beta_{2}\)\(=\)\((\)\( 481 \nu^{14} - 10935 \nu^{12} + 120641 \nu^{10} - 792237 \nu^{8} + 3303859 \nu^{6} - 8551395 \nu^{4} + 12690614 \nu^{2} + 161849 \nu - 8734563 \)\()/323698\)
\(\beta_{3}\)\(=\)\((\)\(-879144720 \nu^{15} - 3144854790 \nu^{14} + 13836683352 \nu^{13} + 78024201152 \nu^{12} - 89278933653 \nu^{11} - 889423402658 \nu^{10} + 95062455881 \nu^{9} + 5888432618929 \nu^{8} + 2063158037274 \nu^{7} - 24026211151887 \nu^{6} - 13843669424702 \nu^{5} + 59633015429809 \nu^{4} + 36965828626322 \nu^{3} - 82401300908132 \nu^{2} - 36427645933254 \nu + 59040737512289\)\()/ 11901469429298 \)
\(\beta_{4}\)\(=\)\((\)\(-879144720 \nu^{15} + 3144854790 \nu^{14} + 13836683352 \nu^{13} - 78024201152 \nu^{12} - 89278933653 \nu^{11} + 889423402658 \nu^{10} + 95062455881 \nu^{9} - 5888432618929 \nu^{8} + 2063158037274 \nu^{7} + 24026211151887 \nu^{6} - 13843669424702 \nu^{5} - 59633015429809 \nu^{4} + 36965828626322 \nu^{3} + 82401300908132 \nu^{2} - 36427645933254 \nu - 59040737512289\)\()/ 11901469429298 \)
\(\beta_{5}\)\(=\)\((\)\(-879144720 \nu^{15} - 10804470307 \nu^{14} + 13836683352 \nu^{13} + 249798214715 \nu^{12} - 89278933653 \nu^{11} - 2731917586661 \nu^{10} + 95062455881 \nu^{9} + 17346636905054 \nu^{8} + 2063158037274 \nu^{7} - 67024837757108 \nu^{6} - 13843669424702 \nu^{5} + 147677522073294 \nu^{4} + 36965828626322 \nu^{3} - 151545537147092 \nu^{2} - 42378380647903 \nu + 44600586148174\)\()/ 11901469429298 \)
\(\beta_{6}\)\(=\)\((\)\(1761178984 \nu^{15} + 11368095320 \nu^{14} - 44779737057 \nu^{13} - 280055462260 \nu^{12} + 534721065978 \nu^{11} + 3239592599263 \nu^{10} - 3768012967205 \nu^{9} - 22120938532531 \nu^{8} + 16738334520669 \nu^{7} + 94843723502693 \nu^{6} - 46192300263954 \nu^{5} - 250479400389093 \nu^{4} + 73208859844642 \nu^{3} + 366581167649995 \nu^{2} - 52214246429899 \nu - 230349536745774\)\()/ 11901469429298 \)
\(\beta_{7}\)\(=\)\((\)\(11502471 \nu^{15} - 100754402 \nu^{14} - 302430222 \nu^{13} + 2300277459 \nu^{12} + 3745368043 \nu^{11} - 25050172064 \nu^{10} - 27330788147 \nu^{9} + 160572232380 \nu^{8} + 125474461758 \nu^{7} - 646303508562 \nu^{6} - 353492024798 \nu^{5} + 1593931115935 \nu^{4} + 543750044164 \nu^{3} - 2207888439990 \nu^{2} - 325676753988 \nu + 1379612638329\)\()/ 49383690578 \)
\(\beta_{8}\)\(=\)\((\)\(13049190 \nu^{15} - 41079768 \nu^{14} - 323751872 \nu^{13} + 888074667 \nu^{12} + 3690553538 \nu^{11} - 9279789799 \nu^{10} - 24433330369 \nu^{9} + 56873654394 \nu^{8} + 99693822207 \nu^{7} - 215754323582 \nu^{6} - 247439898049 \nu^{5} + 481477613042 \nu^{4} + 341914111652 \nu^{3} - 543057264054 \nu^{2} - 244982313329 \nu + 211873877520\)\()/ 49383690578 \)
\(\beta_{9}\)\(=\)\((\)\(158509246 \nu^{15} + 204032769 \nu^{14} - 3519708183 \nu^{13} - 5203789126 \nu^{12} + 37026555855 \nu^{11} + 61795072449 \nu^{10} - 225774923500 \nu^{9} - 430177674564 \nu^{8} + 842415020205 \nu^{7} + 1871226083985 \nu^{6} - 1818260803239 \nu^{5} - 4993951496763 \nu^{4} + 1975431901078 \nu^{3} + 7360174536264 \nu^{2} - 809518665534 \nu - 4537175879612\)\()/ 410395497562 \)
\(\beta_{10}\)\(=\)\((\)\(167082381 \nu^{15} + 204032769 \nu^{14} - 4015307501 \nu^{13} - 5203789126 \nu^{12} + 45907529277 \nu^{11} + 61795072449 \nu^{10} - 312045159727 \nu^{9} - 430177674564 \nu^{8} + 1351068112770 \nu^{7} + 1871226083985 \nu^{6} - 3673028978218 \nu^{5} - 4993951496763 \nu^{4} + 5782517133443 \nu^{3} + 7360174536264 \nu^{2} - 4114840286229 \nu - 4742373628393\)\()/ 410395497562 \)
\(\beta_{11}\)\(=\)\((\)\(-167082381 \nu^{15} + 204032769 \nu^{14} + 4015307501 \nu^{13} - 5203789126 \nu^{12} - 45907529277 \nu^{11} + 61795072449 \nu^{10} + 312045159727 \nu^{9} - 430177674564 \nu^{8} - 1351068112770 \nu^{7} + 1871226083985 \nu^{6} + 3673028978218 \nu^{5} - 4993951496763 \nu^{4} - 5782517133443 \nu^{3} + 7360174536264 \nu^{2} + 4114840286229 \nu - 4742373628393\)\()/ 410395497562 \)
\(\beta_{12}\)\(=\)\((\)\( 36243 \nu^{15} - 862640 \nu^{13} + 9868500 \nu^{11} - 67041955 \nu^{9} + 289073175 \nu^{7} - 776643695 \nu^{5} + 1198809225 \nu^{3} - 835256245 \nu + 39005609 \)\()/78011218\)
\(\beta_{13}\)\(=\)\((\)\(846609 \nu^{15} + 2057746 \nu^{14} - 21592486 \nu^{13} - 48696648 \nu^{12} + 256411089 \nu^{11} + 543806285 \nu^{10} - 1784969604 \nu^{9} - 3575813034 \nu^{8} + 7764423585 \nu^{7} + 14846523620 \nu^{6} - 20721790443 \nu^{5} - 38360465617 \nu^{4} + 30540143304 \nu^{3} + 57407971584 \nu^{2} - 19677898873 \nu - 40266853821\)\()/ 1702885882 \)
\(\beta_{14}\)\(=\)\((\)\(6135024293 \nu^{15} + 13352156499 \nu^{14} - 144306881117 \nu^{13} - 288319318279 \nu^{12} + 1621765377709 \nu^{11} + 2948465510484 \nu^{10} - 10824464034847 \nu^{9} - 17387689074968 \nu^{8} + 45498484447973 \nu^{7} + 62256078466158 \nu^{6} - 117622505367539 \nu^{5} - 126129629458253 \nu^{4} + 168334643403717 \nu^{3} + 119928992729707 \nu^{2} - 106868595929830 \nu - 33115752912402\)\()/ 11901469429298 \)
\(\beta_{15}\)\(=\)\((\)\(-6229064855 \nu^{15} - 4449079069 \nu^{14} + 149688381624 \nu^{13} + 84880417141 \nu^{12} - 1686020685713 \nu^{11} - 788893843317 \nu^{10} + 11169729283807 \nu^{9} + 4060764738779 \nu^{8} - 46468851901548 \nu^{7} - 11418624916134 \nu^{6} + 119958555534177 \nu^{5} + 10381297760156 \nu^{4} - 176885437933337 \nu^{3} + 22259305461325 \nu^{2} + 126156859383031 \nu - 41759097040057\)\()/ 11901469429298 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{2} + \beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{11} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{13} + 4 \beta_{12} - 5 \beta_{10} + 6 \beta_{9} - \beta_{7} - \beta_{6} + 5 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 2 \beta_{1} - 5\)
\(\nu^{4}\)\(=\)\(8 \beta_{15} + \beta_{13} - \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - \beta_{9} + 7 \beta_{8} - \beta_{7} - 6 \beta_{6} - 9 \beta_{5} + \beta_{4} + 6 \beta_{3} - 2 \beta_{2} - 7 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(10 \beta_{15} - 4 \beta_{13} + 21 \beta_{12} - 5 \beta_{11} - 36 \beta_{10} + 51 \beta_{9} - 6 \beta_{8} - 4 \beta_{7} - 10 \beta_{6} + 38 \beta_{4} + 38 \beta_{3} - 5 \beta_{2} - 5 \beta_{1} - 36\)
\(\nu^{6}\)\(=\)\(42 \beta_{15} + 15 \beta_{13} - 4 \beta_{12} - 21 \beta_{11} + 25 \beta_{10} - 4 \beta_{9} + 27 \beta_{8} - 15 \beta_{7} - 34 \beta_{6} - 44 \beta_{5} - 33 \beta_{4} + 69 \beta_{3} - 19 \beta_{2} - 25 \beta_{1} - 28\)
\(\nu^{7}\)\(=\)\(56 \beta_{15} - 26 \beta_{14} + 15 \beta_{13} + 39 \beta_{12} - 77 \beta_{11} - 138 \beta_{10} + 245 \beta_{9} - 71 \beta_{8} - 11 \beta_{7} - 56 \beta_{6} - 13 \beta_{5} + 156 \beta_{4} + 169 \beta_{3} - 93 \beta_{2} - 80 \beta_{1} - 155\)
\(\nu^{8}\)\(=\)\(138 \beta_{15} + 120 \beta_{13} + 67 \beta_{12} + 110 \beta_{11} + 181 \beta_{10} + 67 \beta_{9} + 18 \beta_{8} - 120 \beta_{7} - 272 \beta_{6} - 105 \beta_{5} - 245 \beta_{4} + 484 \beta_{3} - 101 \beta_{2} - 4 \beta_{1} - 438\)
\(\nu^{9}\)\(=\)\(153 \beta_{15} - 402 \beta_{14} + 345 \beta_{13} - 172 \beta_{12} - 614 \beta_{11} - 211 \beta_{10} + 576 \beta_{9} - 498 \beta_{8} - 57 \beta_{7} - 153 \beta_{6} - 201 \beta_{5} + 158 \beta_{4} + 359 \beta_{3} - 661 \beta_{2} - 460 \beta_{1} - 403\)
\(\nu^{10}\)\(=\)\(-33 \beta_{15} + 566 \beta_{13} + 1149 \beta_{12} + 2089 \beta_{11} + 907 \beta_{10} + 1149 \beta_{9} - 599 \beta_{8} - 566 \beta_{7} - 2265 \beta_{6} + 325 \beta_{5} - 499 \beta_{4} + 2472 \beta_{3} - 230 \beta_{2} + 555 \beta_{1} - 2578\)
\(\nu^{11}\)\(=\)\(-413 \beta_{15} - 3342 \beta_{14} + 2841 \beta_{13} - 948 \beta_{12} - 2786 \beta_{11} + 963 \beta_{10} - 1932 \beta_{9} - 2428 \beta_{8} - 501 \beta_{7} + 413 \beta_{6} - 1671 \beta_{5} - 3057 \beta_{4} - 1386 \beta_{3} - 3095 \beta_{2} - 1424 \beta_{1} - 231\)
\(\nu^{12}\)\(=\)\(-4536 \beta_{15} + 702 \beta_{13} + 9525 \beta_{12} + 16543 \beta_{11} + 2482 \beta_{10} + 9525 \beta_{9} - 5238 \beta_{8} - 702 \beta_{7} - 14514 \beta_{6} + 5283 \beta_{5} + 5073 \beta_{4} + 8694 \beta_{3} + 1662 \beta_{2} + 3621 \beta_{1} - 8380\)
\(\nu^{13}\)\(=\)\(-6889 \beta_{15} - 17220 \beta_{14} + 13966 \beta_{13} + 7188 \beta_{12} - 1893 \beta_{11} + 8690 \beta_{10} - 30906 \beta_{9} - 7077 \beta_{8} - 3254 \beta_{7} + 6889 \beta_{6} - 8610 \beta_{5} - 28599 \beta_{4} - 19989 \beta_{3} - 8131 \beta_{2} + 479 \beta_{1} + 3249\)
\(\nu^{14}\)\(=\)\(-40193 \beta_{15} - 13606 \beta_{13} + 48406 \beta_{12} + 79494 \beta_{11} - 9105 \beta_{10} + 48406 \beta_{9} - 26587 \beta_{8} + 13606 \beta_{7} - 56619 \beta_{6} + 34250 \beta_{5} + 55017 \beta_{4} + 7545 \beta_{3} + 24403 \beta_{2} + 9847 \beta_{1} - 374\)
\(\nu^{15}\)\(=\)\(-33940 \beta_{15} - 36120 \beta_{14} + 24745 \beta_{13} + 119836 \beta_{12} + 84770 \beta_{11} + 48950 \beta_{10} - 203780 \beta_{9} + 9195 \beta_{8} - 11375 \beta_{7} + 33940 \beta_{6} - 18060 \beta_{5} - 151395 \beta_{4} - 133335 \beta_{3} + 18000 \beta_{2} + 36060 \beta_{1} + 23912\)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−2.08547 0.994522i
2.08547 0.994522i
2.29670 0.743145i
−2.29670 0.743145i
1.37050 0.406737i
−1.37050 0.406737i
1.37050 + 0.406737i
−1.37050 + 0.406737i
2.29670 + 0.743145i
−2.29670 + 0.743145i
−2.08547 + 0.994522i
2.08547 + 0.994522i
−1.93590 + 0.207912i
1.93590 + 0.207912i
−1.93590 0.207912i
1.93590 0.207912i
0.809017 0.587785i 0.104528 0.994522i 0.309017 0.951057i 0.500000 0.866025i −0.500000 0.866025i −4.44421 + 0.944645i −0.309017 0.951057i −0.978148 0.207912i −0.104528 0.994522i
121.2 0.809017 0.587785i 0.104528 0.994522i 0.309017 0.951057i 0.500000 0.866025i −0.500000 0.866025i 2.15704 0.458493i −0.309017 0.951057i −0.978148 0.207912i −0.104528 0.994522i
361.1 −0.309017 0.951057i −0.669131 0.743145i −0.809017 + 0.587785i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.296127 2.81746i 0.809017 + 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
361.2 −0.309017 0.951057i −0.669131 0.743145i −0.809017 + 0.587785i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.000615948 0.00586036i 0.809017 + 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
391.1 0.809017 + 0.587785i −0.913545 0.406737i 0.309017 + 0.951057i 0.500000 0.866025i −0.500000 0.866025i −1.80375 + 2.00327i −0.309017 + 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
391.2 0.809017 + 0.587785i −0.913545 0.406737i 0.309017 + 0.951057i 0.500000 0.866025i −0.500000 0.866025i 1.16386 1.29260i −0.309017 + 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
421.1 0.809017 0.587785i −0.913545 + 0.406737i 0.309017 0.951057i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.80375 2.00327i −0.309017 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
421.2 0.809017 0.587785i −0.913545 + 0.406737i 0.309017 0.951057i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.16386 + 1.29260i −0.309017 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
541.1 −0.309017 + 0.951057i −0.669131 + 0.743145i −0.809017 0.587785i 0.500000 0.866025i −0.500000 0.866025i −0.296127 + 2.81746i 0.809017 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
541.2 −0.309017 + 0.951057i −0.669131 + 0.743145i −0.809017 0.587785i 0.500000 0.866025i −0.500000 0.866025i 0.000615948 0.00586036i 0.809017 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
661.1 0.809017 + 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i 0.500000 + 0.866025i −0.500000 + 0.866025i −4.44421 0.944645i −0.309017 + 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
661.2 0.809017 + 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i 0.500000 + 0.866025i −0.500000 + 0.866025i 2.15704 + 0.458493i −0.309017 + 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
691.1 −0.309017 + 0.951057i 0.978148 + 0.207912i −0.809017 0.587785i 0.500000 + 0.866025i −0.500000 + 0.866025i −0.731733 + 0.325789i 0.809017 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
691.2 −0.309017 + 0.951057i 0.978148 + 0.207912i −0.809017 0.587785i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.45430 0.647494i 0.809017 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
751.1 −0.309017 0.951057i 0.978148 0.207912i −0.809017 + 0.587785i 0.500000 0.866025i −0.500000 0.866025i −0.731733 0.325789i 0.809017 + 0.587785i 0.913545 0.406737i −0.978148 0.207912i
751.2 −0.309017 0.951057i 0.978148 0.207912i −0.809017 + 0.587785i 0.500000 0.866025i −0.500000 0.866025i 1.45430 + 0.647494i 0.809017 + 0.587785i 0.913545 0.406737i −0.978148 0.207912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bg.e 16
31.g even 15 1 inner 930.2.bg.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.e 16 1.a even 1 1 trivial
930.2.bg.e 16 31.g even 15 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

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$3$ \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{8} \)
$7$ \( 1 - 35 T + 28781 T^{2} + 13800 T^{3} - 36110 T^{4} - 7650 T^{5} + 32714 T^{6} - 15485 T^{7} + 1339 T^{8} - 205 T^{9} + 824 T^{10} - 120 T^{11} - 35 T^{12} - 30 T^{13} - 4 T^{14} + 5 T^{15} + T^{16} \)
$11$ \( 12952801 + 19297838 T + 4906397 T^{2} - 7737749 T^{3} - 5690047 T^{4} - 1002986 T^{5} + 796569 T^{6} + 595013 T^{7} + 224508 T^{8} + 52261 T^{9} + 5901 T^{10} - 373 T^{11} - 232 T^{12} - 13 T^{13} + 8 T^{14} + 4 T^{15} + T^{16} \)
$13$ \( 625 - 2500 T + 13375 T^{2} - 43000 T^{3} + 89825 T^{4} - 59925 T^{5} - 51740 T^{6} + 50480 T^{7} + 16666 T^{8} - 8569 T^{9} + 170 T^{10} + 1244 T^{11} - 81 T^{12} - 36 T^{13} + 10 T^{14} - 4 T^{15} + T^{16} \)
$17$ \( 417344041 + 885270286 T + 375591843 T^{2} - 151858132 T^{3} - 19837017 T^{4} - 2631222 T^{5} + 8903786 T^{6} + 4843854 T^{7} + 1786973 T^{8} + 462212 T^{9} + 112484 T^{10} + 27436 T^{11} + 5608 T^{12} + 1124 T^{13} + 202 T^{14} + 22 T^{15} + T^{16} \)
$19$ \( 1113025 - 822900 T + 5015500 T^{2} - 1811250 T^{3} + 2833905 T^{4} - 393915 T^{5} - 312415 T^{6} - 79290 T^{7} + 115346 T^{8} + 552 T^{9} - 15355 T^{10} + 1407 T^{11} + 1164 T^{12} - 117 T^{13} - 50 T^{14} + 3 T^{15} + T^{16} \)
$23$ \( 841 - 44457 T + 967492 T^{2} - 5947956 T^{3} + 15942666 T^{4} - 8031207 T^{5} + 5104870 T^{6} - 2032758 T^{7} + 801909 T^{8} - 242094 T^{9} + 67300 T^{10} - 14889 T^{11} + 3266 T^{12} - 528 T^{13} + 78 T^{14} - 9 T^{15} + T^{16} \)
$29$ \( 72361 + 5683701 T + 170027863 T^{2} - 63497388 T^{3} + 37619541 T^{4} - 13247964 T^{5} + 5686000 T^{6} - 1043499 T^{7} + 418614 T^{8} - 52383 T^{9} + 19600 T^{10} - 1077 T^{11} + 836 T^{12} + 24 T^{13} + 42 T^{14} + 3 T^{15} + T^{16} \)
$31$ \( 852891037441 + 412689211665 T + 103837930677 T^{2} + 27483984960 T^{3} + 6901472433 T^{4} + 1421924430 T^{5} + 301065924 T^{6} + 60879195 T^{7} + 11016725 T^{8} + 1963845 T^{9} + 313284 T^{10} + 47730 T^{11} + 7473 T^{12} + 960 T^{13} + 117 T^{14} + 15 T^{15} + T^{16} \)
$37$ \( 1579983001 + 939785607 T + 1269425326 T^{2} + 91303733 T^{3} + 401320516 T^{4} + 54418476 T^{5} + 57208369 T^{6} + 7952964 T^{7} + 5620207 T^{8} + 899584 T^{9} + 328159 T^{10} + 47816 T^{11} + 12836 T^{12} + 1733 T^{13} + 256 T^{14} + 17 T^{15} + T^{16} \)
$41$ \( 20475625 - 42648125 T + 446154750 T^{2} - 450039250 T^{3} - 333119175 T^{4} + 529228475 T^{5} + 59865690 T^{6} - 117143970 T^{7} + 45608156 T^{8} - 9674607 T^{9} + 1457860 T^{10} - 180302 T^{11} + 19579 T^{12} - 2423 T^{13} + 325 T^{14} - 28 T^{15} + T^{16} \)
$43$ \( 32041 - 406151 T + 3061263 T^{2} + 8076047 T^{3} + 16335408 T^{4} + 29383257 T^{5} + 32318561 T^{6} + 17952246 T^{7} + 6075308 T^{8} + 1045238 T^{9} + 84464 T^{10} - 19826 T^{11} - 6197 T^{12} + 266 T^{13} + 247 T^{14} + 28 T^{15} + T^{16} \)
$47$ \( 367450561 - 874700639 T + 1195473360 T^{2} - 1296660380 T^{3} + 1456941950 T^{4} - 993503737 T^{5} + 373432473 T^{6} - 53224430 T^{7} + 12538415 T^{8} - 970550 T^{9} + 286443 T^{10} - 28887 T^{11} + 6710 T^{12} - 705 T^{13} + 125 T^{14} - 9 T^{15} + T^{16} \)
$53$ \( 787221435025 + 1259245531300 T + 989688694875 T^{2} + 353802165575 T^{3} + 38833564155 T^{4} - 13177946355 T^{5} - 3981085045 T^{6} + 64526340 T^{7} + 108488516 T^{8} + 3526337 T^{9} - 997210 T^{10} - 34298 T^{11} - 5516 T^{12} - 397 T^{13} + 190 T^{14} - 2 T^{15} + T^{16} \)
$59$ \( 1596721681 - 13901416428 T + 39297649492 T^{2} - 38255103501 T^{3} + 19678570203 T^{4} - 6101155929 T^{5} + 1144584754 T^{6} - 110255193 T^{7} - 1172367 T^{8} + 1592589 T^{9} + 12361 T^{10} - 85182 T^{11} + 22988 T^{12} - 3462 T^{13} + 348 T^{14} - 24 T^{15} + T^{16} \)
$61$ \( ( -8339 + 16485 T + 14969 T^{2} - 8815 T^{3} - 8439 T^{4} - 1630 T^{5} + 9 T^{6} + 20 T^{7} + T^{8} )^{2} \)
$67$ \( 479998166761 - 562568335181 T + 536655909024 T^{2} - 208567024141 T^{3} + 79529870201 T^{4} - 15876306783 T^{5} + 4864276891 T^{6} - 768100148 T^{7} + 192956202 T^{8} - 16536032 T^{9} + 3224971 T^{10} - 149157 T^{11} + 39236 T^{12} - 859 T^{13} + 234 T^{14} + T^{15} + T^{16} \)
$71$ \( 14311278150625 - 7615513051875 T + 673624346625 T^{2} + 377356986875 T^{3} - 87255820425 T^{4} - 80764675 T^{5} + 1597349965 T^{6} - 43791945 T^{7} - 12793574 T^{8} - 6706914 T^{9} + 3251455 T^{10} - 694576 T^{11} + 103209 T^{12} - 10321 T^{13} + 750 T^{14} - 39 T^{15} + T^{16} \)
$73$ \( 708376405801 - 1566848642687 T + 1215812729577 T^{2} - 471947964974 T^{3} + 190762605443 T^{4} - 68819108791 T^{5} + 20525523579 T^{6} - 6227272472 T^{7} + 1811937233 T^{8} - 406496264 T^{9} + 72606531 T^{10} - 9993903 T^{11} + 990983 T^{12} - 67548 T^{13} + 3023 T^{14} - 81 T^{15} + T^{16} \)
$79$ \( 49171383597361 + 94913295379552 T + 43561638775652 T^{2} - 11274043893541 T^{3} + 12650926852793 T^{4} - 4030651837414 T^{5} + 782089982844 T^{6} - 101606472263 T^{7} + 7537641843 T^{8} + 763859 T^{9} - 64086024 T^{10} + 6221998 T^{11} - 166897 T^{12} - 15227 T^{13} + 1628 T^{14} - 64 T^{15} + T^{16} \)
$83$ \( 19217999641 - 119737333025 T + 375553834009 T^{2} - 444697653845 T^{3} + 140860084880 T^{4} + 60027140160 T^{5} + 15168971926 T^{6} + 2348990170 T^{7} + 86892289 T^{8} - 32501240 T^{9} - 6311194 T^{10} - 451040 T^{11} + 9900 T^{12} + 5760 T^{13} + 679 T^{14} + 40 T^{15} + T^{16} \)
$89$ \( 1215209490621025 - 402544001313225 T + 109758607329050 T^{2} - 16669084599150 T^{3} + 2478297239990 T^{4} - 222110283315 T^{5} + 35298347760 T^{6} - 2755186700 T^{7} + 557084366 T^{8} - 38623178 T^{9} + 6514473 T^{10} - 350521 T^{11} + 39965 T^{12} - 101 T^{13} + 78 T^{14} + 7 T^{15} + T^{16} \)
$97$ \( 663205420200625 - 343914146098750 T + 64335812472625 T^{2} + 6508971100000 T^{3} + 1636051676650 T^{4} + 2968115900 T^{5} + 57197276705 T^{6} + 77066200 T^{7} + 954430946 T^{8} + 58701852 T^{9} + 16692383 T^{10} + 1411944 T^{11} + 147975 T^{12} + 7879 T^{13} + 573 T^{14} + 17 T^{15} + T^{16} \)
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