Properties

Label 930.2.bg.d
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: \(x^{16} + 21 x^{14} + 167 x^{12} + 653 x^{10} + 1350 x^{8} + 1472 x^{6} + 777 x^{4} + 149 x^{2} + 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 21 x^{14} + 167 x^{12} + 653 x^{10} + 1350 x^{8} + 1472 x^{6} + 777 x^{4} + 149 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(831 \nu^{15} - 200 \nu^{14} + 16820 \nu^{13} - 3799 \nu^{12} + 125756 \nu^{11} - 27571 \nu^{10} + 444458 \nu^{9} - 108351 \nu^{8} + 785743 \nu^{7} - 272115 \nu^{6} + 709604 \nu^{5} - 427777 \nu^{4} + 363860 \nu^{3} - 317523 \nu^{2} + 96303 \nu - 39009\)\()/37642\)
\(\beta_{2}\)\(=\)\((\)\(831 \nu^{15} + 200 \nu^{14} + 16820 \nu^{13} + 3799 \nu^{12} + 125756 \nu^{11} + 27571 \nu^{10} + 444458 \nu^{9} + 108351 \nu^{8} + 785743 \nu^{7} + 272115 \nu^{6} + 709604 \nu^{5} + 427777 \nu^{4} + 363860 \nu^{3} + 317523 \nu^{2} + 96303 \nu + 39009\)\()/37642\)
\(\beta_{3}\)\(=\)\((\)\(-3182 \nu^{15} - 4831 \nu^{14} - 54984 \nu^{13} - 92800 \nu^{12} - 299191 \nu^{11} - 639534 \nu^{10} - 421263 \nu^{9} - 1990385 \nu^{8} + 1159795 \nu^{7} - 2840263 \nu^{6} + 3840543 \nu^{5} - 1717923 \nu^{4} + 3329765 \nu^{3} - 371643 \nu^{2} + 779461 \nu - 48359\)\()/37642\)
\(\beta_{4}\)\(=\)\((\)\( 88 \nu^{15} + 529 \nu^{14} + 1740 \nu^{13} + 10382 \nu^{12} + 12405 \nu^{11} + 74046 \nu^{10} + 39051 \nu^{9} + 243052 \nu^{8} + 46842 \nu^{7} + 375106 \nu^{6} - 12102 \nu^{5} + 244291 \nu^{4} - 58629 \nu^{3} + 42548 \nu^{2} - 24516 \nu - 2484 \)\()/3422\)
\(\beta_{5}\)\(=\)\((\)\(88 \nu^{15} - 529 \nu^{14} + 1740 \nu^{13} - 10382 \nu^{12} + 12405 \nu^{11} - 74046 \nu^{10} + 39051 \nu^{9} - 243052 \nu^{8} + 46842 \nu^{7} - 375106 \nu^{6} - 12102 \nu^{5} - 244291 \nu^{4} - 58629 \nu^{3} - 42548 \nu^{2} - 24516 \nu + 2484\)\()/3422\)
\(\beta_{6}\)\(=\)\((\)\( -262 \nu^{15} - 5100 \nu^{13} - 35800 \nu^{11} - 113585 \nu^{9} - 160570 \nu^{7} - 74527 \nu^{5} + 19115 \nu^{3} + 13610 \nu - 649 \)\()/1298\)
\(\beta_{7}\)\(=\)\((\)\(419 \nu^{15} + 8649 \nu^{14} + 11629 \nu^{13} + 170781 \nu^{12} + 126552 \nu^{11} + 1230044 \nu^{10} + 689145 \nu^{9} + 4100400 \nu^{8} + 1992478 \nu^{7} + 6485029 \nu^{6} + 2975631 \nu^{5} + 4401064 \nu^{4} + 2039426 \nu^{3} + 887173 \nu^{2} + 462755 \nu + 9899\)\()/37642\)
\(\beta_{8}\)\(=\)\((\)\(419 \nu^{15} - 8649 \nu^{14} + 11629 \nu^{13} - 170781 \nu^{12} + 126552 \nu^{11} - 1230044 \nu^{10} + 689145 \nu^{9} - 4100400 \nu^{8} + 1992478 \nu^{7} - 6485029 \nu^{6} + 2975631 \nu^{5} - 4401064 \nu^{4} + 2039426 \nu^{3} - 887173 \nu^{2} + 462755 \nu - 9899\)\()/37642\)
\(\beta_{9}\)\(=\)\((\)\(4768 \nu^{15} + 10016 \nu^{14} + 91321 \nu^{13} + 199288 \nu^{12} + 622662 \nu^{11} + 1454534 \nu^{10} + 1867137 \nu^{9} + 4965480 \nu^{8} + 2297667 \nu^{7} + 8222128 \nu^{6} + 447420 \nu^{5} + 6141173 \nu^{4} - 973480 \nu^{3} + 1540376 \nu^{2} - 450734 \nu - 9836\)\()/37642\)
\(\beta_{10}\)\(=\)\((\)\( -402 \nu^{14} - 7954 \nu^{12} - 57501 \nu^{10} - 193130 \nu^{8} - 311137 \nu^{6} - 222689 \nu^{4} - 52648 \nu^{2} + 649 \nu - 262 \)\()/1298\)
\(\beta_{11}\)\(=\)\((\)\( -402 \nu^{14} - 7954 \nu^{12} - 57501 \nu^{10} - 193130 \nu^{8} - 311137 \nu^{6} - 222689 \nu^{4} - 52648 \nu^{2} - 649 \nu - 262 \)\()/1298\)
\(\beta_{12}\)\(=\)\((\)\(-6019 \nu^{15} + 7817 \nu^{14} - 118001 \nu^{13} + 155730 \nu^{12} - 842077 \nu^{11} + 1137181 \nu^{10} - 2781923 \nu^{9} + 3874759 \nu^{8} - 4398281 \nu^{7} + 6376893 \nu^{6} - 3114978 \nu^{5} + 4709137 \nu^{4} - 785551 \nu^{3} + 1167568 \nu^{2} - 30506 \nu + 10235\)\()/37642\)
\(\beta_{13}\)\(=\)\((\)\(4768 \nu^{15} - 10016 \nu^{14} + 91321 \nu^{13} - 199288 \nu^{12} + 622662 \nu^{11} - 1454534 \nu^{10} + 1867137 \nu^{9} - 4965480 \nu^{8} + 2297667 \nu^{7} - 8222128 \nu^{6} + 447420 \nu^{5} - 6141173 \nu^{4} - 973480 \nu^{3} - 1540376 \nu^{2} - 450734 \nu + 9836\)\()/37642\)
\(\beta_{14}\)\(=\)\((\)\(-2982 \nu^{15} - 13836 \nu^{14} - 51185 \nu^{13} - 273731 \nu^{12} - 271620 \nu^{11} - 1979258 \nu^{10} - 312912 \nu^{9} - 6656682 \nu^{8} + 1431910 \nu^{7} - 10775174 \nu^{6} + 4268320 \nu^{5} - 7824115 \nu^{4} + 3647288 \nu^{3} - 1953119 \nu^{2} + 799649 \nu - 59562\)\()/37642\)
\(\beta_{15}\)\(=\)\((\)\(-831 \nu^{15} + 15834 \nu^{14} - 16820 \nu^{13} + 315259 \nu^{12} - 125756 \nu^{11} + 2301933 \nu^{10} - 444458 \nu^{9} + 7857869 \nu^{8} - 785743 \nu^{7} + 13025901 \nu^{6} - 709604 \nu^{5} + 9846051 \nu^{4} - 363860 \nu^{3} + 2652659 \nu^{2} - 96303 \nu + 97121\)\()/37642\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{11} + \beta_{10}\)
\(\nu^{2}\)\(=\)\(\beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} + 2 \beta_{7} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + 5 \beta_{11} - 5 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 2 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - 8 \beta_{12} + 2 \beta_{11} - 14 \beta_{10} - 9 \beta_{9} - 2 \beta_{8} - 14 \beta_{7} - 16 \beta_{5} + 8 \beta_{4} + 16 \beta_{3} + 2 \beta_{2} - 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(8 \beta_{15} + 4 \beta_{14} + 9 \beta_{13} - 12 \beta_{12} - 30 \beta_{11} + 30 \beta_{10} + 9 \beta_{9} - 5 \beta_{8} - 5 \beta_{7} + 24 \beta_{6} + 6 \beta_{5} + 10 \beta_{4} + 17 \beta_{2} + 21 \beta_{1} + 8\)
\(\nu^{6}\)\(=\)\(-23 \beta_{15} + 62 \beta_{14} - 70 \beta_{13} + 62 \beta_{12} - 26 \beta_{11} + 98 \beta_{10} + 70 \beta_{9} + 19 \beta_{8} + 105 \beta_{7} + 130 \beta_{5} - 68 \beta_{4} - 124 \beta_{3} - 21 \beta_{2} + 60 \beta_{1} - 50\)
\(\nu^{7}\)\(=\)\(-60 \beta_{15} - 9 \beta_{14} - 77 \beta_{13} + 111 \beta_{12} + 208 \beta_{11} - 208 \beta_{10} - 77 \beta_{9} + 20 \beta_{8} + 20 \beta_{7} - 228 \beta_{6} - 79 \beta_{5} - 88 \beta_{4} - 132 \beta_{2} - 183 \beta_{1} - 63\)
\(\nu^{8}\)\(=\)\(217 \beta_{15} - 490 \beta_{14} + 551 \beta_{13} - 490 \beta_{12} + 253 \beta_{11} - 727 \beta_{10} - 551 \beta_{9} - 154 \beta_{8} - 826 \beta_{7} - 1065 \beta_{5} + 575 \beta_{4} + 980 \beta_{3} + 188 \beta_{2} - 461 \beta_{1} + 368\)
\(\nu^{9}\)\(=\)\(461 \beta_{15} - 31 \beta_{14} + 650 \beta_{13} - 953 \beta_{12} - 1567 \beta_{11} + 1567 \beta_{10} + 650 \beta_{9} - 66 \beta_{8} - 66 \beta_{7} + 2000 \beta_{6} + 774 \beta_{5} + 743 \beta_{4} + 1041 \beta_{2} + 1533 \beta_{1} + 508\)
\(\nu^{10}\)\(=\)\(-1911 \beta_{15} + 3940 \beta_{14} - 4427 \beta_{13} + 3940 \beta_{12} - 2235 \beta_{11} + 5645 \beta_{10} + 4427 \beta_{9} + 1231 \beta_{8} + 6649 \beta_{7} + 8744 \beta_{5} - 4804 \beta_{4} - 7880 \beta_{3} - 1603 \beta_{2} + 3632 \beta_{1} - 2862\)
\(\nu^{11}\)\(=\)\(-3632 \beta_{15} + 698 \beta_{14} - 5431 \beta_{13} + 7962 \beta_{12} + 12330 \beta_{11} - 12330 \beta_{10} - 5431 \beta_{9} + 113 \beta_{8} + 113 \beta_{7} - 16954 \beta_{6} - 6873 \beta_{5} - 6175 \beta_{4} - 8367 \beta_{2} - 12697 \beta_{1} - 4147\)
\(\nu^{12}\)\(=\)\(16295 \beta_{15} - 32013 \beta_{14} + 36004 \beta_{13} - 32013 \beta_{12} + 18982 \beta_{11} - 45044 \beta_{10} - 36004 \beta_{9} - 9922 \beta_{8} - 54104 \beta_{7} - 71827 \beta_{5} + 39814 \beta_{4} + 64026 \beta_{3} + 13393 \beta_{2} - 29111 \beta_{1} + 22795\)
\(\nu^{13}\)\(=\)\(29111 \beta_{15} - 7594 \beta_{14} + 45064 \beta_{13} - 65816 \beta_{12} - 99171 \beta_{11} + 99171 \beta_{10} + 45064 \beta_{9} + 916 \beta_{8} + 916 \beta_{7} + 141448 \beta_{6} + 58589 \beta_{5} + 50995 \beta_{4} + 68017 \beta_{2} + 104722 \beta_{1} + 34019\)
\(\nu^{14}\)\(=\)\(-136628 \beta_{15} + 261566 \beta_{14} - 294541 \beta_{13} + 261566 \beta_{12} - 158424 \beta_{11} + 364708 \beta_{10} + 294541 \beta_{9} + 80626 \beta_{8} + 442506 \beta_{7} + 590091 \beta_{5} - 328525 \beta_{4} - 523132 \beta_{3} - 110880 \beta_{2} + 235818 \beta_{1} - 183918\)
\(\nu^{15}\)\(=\)\(-235818 \beta_{15} + 70191 \beta_{14} - 372339 \beta_{13} + 541827 \beta_{12} + 806355 \beta_{11} - 806355 \beta_{10} - 372339 \beta_{9} - 15420 \beta_{8} - 15420 \beta_{7} - 1171068 \beta_{6} - 490181 \beta_{5} - 419990 \beta_{4} - 556107 \beta_{2} - 862116 \beta_{1} - 279525\)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.94768i
1.13420i
1.04753i
0.631711i
2.07247i
0.0834312i
2.07247i
0.0834312i
1.04753i
0.631711i
1.94768i
1.13420i
2.86648i
1.38019i
2.86648i
1.38019i
−0.809017 + 0.587785i −0.104528 + 0.994522i 0.309017 0.951057i 0.500000 0.866025i −0.500000 0.866025i 1.12135 0.238351i 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 2.37487 0.504794i 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 −0.195942 1.86427i −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 0.153192 + 1.45753i −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 −1.16565 + 1.29459i 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 0.978446 1.08667i 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 −1.16565 1.29459i 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 0.978446 + 1.08667i 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 −0.195942 + 1.86427i −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 0.153192 1.45753i −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 1.12135 + 0.238351i 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 2.37487 + 0.504794i 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 −0.461080 + 0.205286i −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 2.69481 1.19981i −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 −0.461080 0.205286i −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 2.69481 + 1.19981i −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.d 16
31.g even 15 1 inner 930.2.bg.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.d 16 1.a even 1 1 trivial
930.2.bg.d 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

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$3$ \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{8} \)
$7$ \( 841 + 261 T - 2261 T^{2} + 386 T^{3} + 3906 T^{4} - 7042 T^{5} + 7726 T^{6} - 6337 T^{7} + 4307 T^{8} - 2623 T^{9} + 1556 T^{10} - 848 T^{11} + 401 T^{12} - 166 T^{13} + 54 T^{14} - 11 T^{15} + T^{16} \)
$11$ \( 132043081 - 192060574 T + 133702289 T^{2} - 99657119 T^{3} - 7358299 T^{4} + 36774958 T^{5} + 12571011 T^{6} + 1117103 T^{7} + 36522 T^{8} + 55987 T^{9} + 20721 T^{10} + 3437 T^{11} + 176 T^{12} - 91 T^{13} - 16 T^{14} + 4 T^{15} + T^{16} \)
$13$ \( 525625 + 3770000 T + 6410625 T^{2} - 9260000 T^{3} + 1293375 T^{4} + 285375 T^{5} + 53750 T^{6} + 122250 T^{7} + 5450 T^{8} + 28825 T^{9} + 45200 T^{10} + 19700 T^{11} + 5485 T^{12} + 1180 T^{13} + 190 T^{14} + 20 T^{15} + T^{16} \)
$17$ \( 42784681 + 17150502 T - 56763099 T^{2} - 22591702 T^{3} + 34627191 T^{4} - 9134344 T^{5} + 6749104 T^{6} - 301506 T^{7} + 195487 T^{8} + 223014 T^{9} + 20764 T^{10} + 7766 T^{11} + 2346 T^{12} + 128 T^{13} + 36 T^{14} + 12 T^{15} + T^{16} \)
$19$ \( 7946761 + 11693212 T + 8329276 T^{2} + 10940808 T^{3} + 17166471 T^{4} + 17847691 T^{5} + 16018199 T^{6} + 9165574 T^{7} + 2778592 T^{8} + 335794 T^{9} - 34721 T^{10} - 16719 T^{11} - 1984 T^{12} - 77 T^{13} + 16 T^{14} + 7 T^{15} + T^{16} \)
$23$ \( 259499881 - 818160001 T + 1195033370 T^{2} - 961472390 T^{3} + 546223070 T^{4} - 247927403 T^{5} + 96358788 T^{6} - 31584940 T^{7} + 8702145 T^{8} - 2022260 T^{9} + 407958 T^{10} - 70663 T^{11} + 10670 T^{12} - 1330 T^{13} + 140 T^{14} - 11 T^{15} + T^{16} \)
$29$ \( 2598042841 - 3138131557 T + 2249924435 T^{2} - 1021180400 T^{3} + 351212405 T^{4} - 100486676 T^{5} + 38372922 T^{6} - 15995965 T^{7} + 6177150 T^{8} - 1905095 T^{9} + 508062 T^{10} - 109429 T^{11} + 20690 T^{12} - 2950 T^{13} + 320 T^{14} - 23 T^{15} + T^{16} \)
$31$ \( 852891037441 + 467714439887 T + 98512908591 T^{2} + 3091948308 T^{3} - 2437171919 T^{4} - 450856994 T^{5} - 25376166 T^{6} - 2217771 T^{7} - 735033 T^{8} - 71541 T^{9} - 26406 T^{10} - 15134 T^{11} - 2639 T^{12} + 108 T^{13} + 111 T^{14} + 17 T^{15} + T^{16} \)
$37$ \( 28425622801 + 55621990293 T + 160506297996 T^{2} - 119715668263 T^{3} + 74698934126 T^{4} - 21184979196 T^{5} + 5560182729 T^{6} - 907446084 T^{7} + 174079137 T^{8} - 22892254 T^{9} + 3581139 T^{10} - 353496 T^{11} + 42936 T^{12} - 3293 T^{13} + 336 T^{14} - 17 T^{15} + T^{16} \)
$41$ \( 419881081 - 1067232753 T + 1362549046 T^{2} - 349886482 T^{3} - 176699469 T^{4} + 66102761 T^{5} + 15799894 T^{6} - 9853216 T^{7} + 1976552 T^{8} - 489061 T^{9} + 125924 T^{10} - 19334 T^{11} + 3581 T^{12} - 757 T^{13} + 81 T^{14} - 8 T^{15} + T^{16} \)
$43$ \( 61763881 + 86661193 T - 190449299 T^{2} - 94266193 T^{3} + 296203166 T^{4} - 149879501 T^{5} + 118085939 T^{6} - 48479074 T^{7} + 11327542 T^{8} - 2391064 T^{9} + 683384 T^{10} - 178226 T^{11} + 34421 T^{12} - 4768 T^{13} + 481 T^{14} - 32 T^{15} + T^{16} \)
$47$ \( 531836191441 + 1934466442413 T + 2791683909530 T^{2} + 516110022480 T^{3} + 210710116730 T^{4} + 5119358229 T^{5} + 3914489547 T^{6} - 159194090 T^{7} + 115748105 T^{8} + 8355760 T^{9} + 2187687 T^{10} + 56291 T^{11} + 9260 T^{12} - 155 T^{13} + 135 T^{14} + 7 T^{15} + T^{16} \)
$53$ \( 3799220807281 + 4697757767214 T + 363609245619 T^{2} - 1203526231251 T^{3} + 392066022401 T^{4} - 78477383943 T^{5} + 18798073021 T^{6} - 5462861328 T^{7} + 1305515322 T^{8} - 239787867 T^{9} + 35501236 T^{10} - 4230042 T^{11} + 391476 T^{12} - 27669 T^{13} + 1474 T^{14} - 54 T^{15} + T^{16} \)
$59$ \( 708454521342481 + 252498282467398 T + 60958509145866 T^{2} + 12384230329437 T^{3} + 1877270844101 T^{4} + 255381666699 T^{5} + 38837309114 T^{6} + 4960296931 T^{7} + 483529137 T^{8} + 33476831 T^{9} - 1263271 T^{10} - 751416 T^{11} - 80854 T^{12} - 1188 T^{13} + 456 T^{14} + 38 T^{15} + T^{16} \)
$61$ \( ( -9157649 - 3704039 T + 161057 T^{2} + 191683 T^{3} + 7645 T^{4} - 3058 T^{5} - 193 T^{6} + 14 T^{7} + T^{8} )^{2} \)
$67$ \( 86869874800801 - 20787001417869 T + 15380448694074 T^{2} + 1297895917681 T^{3} + 1088009463551 T^{4} + 69650956173 T^{5} + 33412667301 T^{6} + 2034581388 T^{7} + 700386852 T^{8} + 29440552 T^{9} + 8381181 T^{10} + 236157 T^{11} + 71676 T^{12} + 989 T^{13} + 324 T^{14} - T^{15} + T^{16} \)
$71$ \( 3171245078401 - 1443549504581 T + 567337426839 T^{2} + 34082403599 T^{3} + 12042563751 T^{4} + 11008367477 T^{5} + 205461981 T^{6} - 370254063 T^{7} - 24949888 T^{8} - 9440202 T^{9} - 1040279 T^{10} + 369568 T^{11} + 65641 T^{12} + 1771 T^{13} - 86 T^{14} + 11 T^{15} + T^{16} \)
$73$ \( 132043081 - 296042633 T + 2198310421 T^{2} - 2913810162 T^{3} + 1385227011 T^{4} - 491888219 T^{5} + 450873119 T^{6} - 315385826 T^{7} + 132104767 T^{8} - 38154446 T^{9} + 8420329 T^{10} - 1463649 T^{11} + 195791 T^{12} - 19202 T^{13} + 1291 T^{14} - 53 T^{15} + T^{16} \)
$79$ \( 8570379205441 - 16920362919918 T + 13374016675896 T^{2} - 5844176494687 T^{3} + 1821946293141 T^{4} - 454130766544 T^{5} + 84116809834 T^{6} - 10640498301 T^{7} + 858772297 T^{8} - 18366381 T^{9} - 4203686 T^{10} + 359846 T^{11} + 44781 T^{12} - 11197 T^{13} + 1026 T^{14} - 48 T^{15} + T^{16} \)
$83$ \( 241031182982041 + 449654036082131 T + 62915406958899 T^{2} - 140102717398109 T^{3} + 49762541383476 T^{4} - 10005310545302 T^{5} + 1496985802266 T^{6} - 182403916242 T^{7} + 18070566557 T^{8} - 1498310868 T^{9} + 114819556 T^{10} - 8568148 T^{11} + 592726 T^{12} - 35266 T^{13} + 1669 T^{14} - 56 T^{15} + T^{16} \)
$89$ \( 52098210332281 - 2879635320913 T + 21569187300080 T^{2} - 3585670614560 T^{3} + 3537702353900 T^{4} - 427442606189 T^{5} + 47725721412 T^{6} - 3579348640 T^{7} + 1319955570 T^{8} + 89456050 T^{9} + 2174157 T^{10} - 226861 T^{11} + 60275 T^{12} + 10505 T^{13} + 950 T^{14} + 43 T^{15} + T^{16} \)
$97$ \( 5282049589441 + 6468534291462 T + 19716608788895 T^{2} + 5674569572730 T^{3} + 10613452192490 T^{4} - 5197323031134 T^{5} + 1099453637277 T^{6} - 132376947410 T^{7} + 10894219340 T^{8} - 719979560 T^{9} + 45213747 T^{10} - 2215306 T^{11} + 88325 T^{12} - 3965 T^{13} + 345 T^{14} - 17 T^{15} + T^{16} \)
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