Properties

Label 273.2.bd.b
Level $273$
Weight $2$
Character orbit 273.bd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} - 132 x^{7} - 45 x^{6} + 864 x^{5} - 243 x^{4} - 1944 x^{3} + 7290 x^{2} - 8748 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{14} q^{2} -\beta_{5} q^{3} + ( 2 + 2 \beta_{5} + \beta_{7} + \beta_{13} ) q^{4} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{9} + \beta_{12} ) q^{5} + ( \beta_{4} - \beta_{14} ) q^{6} + ( -\beta_{3} + \beta_{15} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} ) q^{8} + ( -1 - \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{14} q^{2} -\beta_{5} q^{3} + ( 2 + 2 \beta_{5} + \beta_{7} + \beta_{13} ) q^{4} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{9} + \beta_{12} ) q^{5} + ( \beta_{4} - \beta_{14} ) q^{6} + ( -\beta_{3} + \beta_{15} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{13} ) q^{8} + ( -1 - \beta_{5} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{10} + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{11} + ( 2 + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} ) q^{12} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{13} + ( -\beta_{6} - \beta_{11} ) q^{14} + ( -1 - \beta_{2} - \beta_{5} - \beta_{9} + \beta_{15} ) q^{15} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{10} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{16} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{17} + \beta_{4} q^{18} + ( -3 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{19} + ( -\beta_{1} - \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{20} -\beta_{3} q^{21} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 5 \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 2 \beta_{13} - 3 \beta_{15} ) q^{22} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{23} + ( \beta_{1} + \beta_{5} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{24} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{25} + ( 1 - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{26} - q^{27} + ( 1 - \beta_{1} + \beta_{2} - \beta_{12} + 2 \beta_{15} ) q^{28} + ( -2 - 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{5} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + 2 \beta_{15} ) q^{29} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{30} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{31} + ( 2 - 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{9} - 2 \beta_{11} + 4 \beta_{15} ) q^{32} + ( 1 + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} + 2 \beta_{15} ) q^{33} + ( 2 + \beta_{1} + \beta_{2} - 7 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{34} + ( 1 + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{35} + ( -2 \beta_{5} + \beta_{8} + \beta_{10} ) q^{36} + ( -2 - 2 \beta_{2} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{13} - \beta_{15} ) q^{37} + ( -6 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{38} + ( 2 + \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{15} ) q^{39} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} - 6 \beta_{15} ) q^{40} + ( -5 + \beta_{1} - 3 \beta_{2} - 3 \beta_{5} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{41} -\beta_{11} q^{42} + ( 3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - 2 \beta_{12} + \beta_{13} + 2 \beta_{15} ) q^{43} + ( -3 + 8 \beta_{3} + \beta_{4} - 6 \beta_{5} + \beta_{6} + 3 \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - 2 \beta_{13} ) q^{44} + ( \beta_{1} - \beta_{3} + \beta_{5} - \beta_{12} + \beta_{15} ) q^{45} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{46} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + 4 \beta_{10} + \beta_{12} - 4 \beta_{13} ) q^{47} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{13} - \beta_{15} ) q^{48} -\beta_{5} q^{49} + ( 8 - \beta_{1} + 4 \beta_{2} + 5 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} + 4 \beta_{15} ) q^{50} + ( -3 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{51} + ( -2 - 2 \beta_{1} + 4 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{52} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + \beta_{12} + 3 \beta_{13} - 4 \beta_{15} ) q^{53} + \beta_{14} q^{54} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{7} + 4 \beta_{8} - 3 \beta_{9} - \beta_{11} + \beta_{12} + 4 \beta_{13} - \beta_{14} - \beta_{15} ) q^{55} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{13} ) q^{56} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} ) q^{57} + ( 4 - 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 5 \beta_{8} + 5 \beta_{10} - 4 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{58} + ( \beta_{2} + 6 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{14} - 6 \beta_{15} ) q^{59} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{60} + ( 3 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{61} + ( 2 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + \beta_{11} + \beta_{12} + 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{62} -\beta_{15} q^{63} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} + \beta_{7} + \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{64} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + 3 \beta_{9} - 4 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 4 \beta_{15} ) q^{65} + ( -1 + 3 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - 6 \beta_{15} ) q^{66} + ( 1 + 3 \beta_{1} + \beta_{2} + 4 \beta_{5} + 3 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{14} + 3 \beta_{15} ) q^{67} + ( 3 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 5 \beta_{11} - \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{68} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{14} ) q^{69} + ( 1 - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{12} ) q^{70} + ( -1 - \beta_{1} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} - 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{71} + ( -1 - \beta_{2} - \beta_{5} + 2 \beta_{7} + \beta_{9} + \beta_{12} - 2 \beta_{13} ) q^{72} + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} + 3 \beta_{13} ) q^{73} + ( \beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - 6 \beta_{15} ) q^{74} + ( 1 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{75} + ( -3 + \beta_{2} - \beta_{5} + 4 \beta_{6} + \beta_{9} + 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{76} + ( 1 + \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{12} + 2 \beta_{14} ) q^{77} + ( 3 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 4 \beta_{12} + \beta_{13} + \beta_{15} ) q^{78} + ( 9 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{79} + ( 10 + \beta_{1} + 2 \beta_{2} + 7 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{80} + \beta_{5} q^{81} + ( -6 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - 5 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} - 2 \beta_{12} - 4 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{82} + ( 2 - \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} + 4 \beta_{5} - 2 \beta_{9} + \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{9} + 2 \beta_{15} ) q^{84} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{9} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{85} + ( -1 - 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 2 \beta_{11} ) q^{86} + ( -1 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{87} + ( -3 - 5 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} - 12 \beta_{5} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} + 5 \beta_{12} - \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{88} + ( 5 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{5} + 4 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} + 4 \beta_{13} - \beta_{14} + 5 \beta_{15} ) q^{89} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{90} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{91} + ( -7 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{7} - 4 \beta_{8} - 3 \beta_{9} - \beta_{10} - 3 \beta_{12} - \beta_{13} ) q^{92} + ( -1 + \beta_{1} + \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{93} + ( -5 - 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} + 7 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 7 \beta_{13} - 2 \beta_{15} ) q^{94} + ( -5 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 7 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{95} + ( 1 - 4 \beta_{3} + 2 \beta_{5} + \beta_{6} - 3 \beta_{9} - \beta_{11} + 3 \beta_{12} ) q^{96} + ( 2 + \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{12} + \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{97} + ( \beta_{4} - \beta_{14} ) q^{98} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{3} + 14q^{4} - 8q^{9} + O(q^{10}) \) \( 16q + 8q^{3} + 14q^{4} - 8q^{9} - 4q^{10} + 28q^{12} - 12q^{13} - 4q^{14} - 12q^{15} - 10q^{16} - 2q^{17} + 18q^{20} - 18q^{22} - 6q^{23} - 20q^{25} + 20q^{26} - 16q^{27} - 12q^{29} + 4q^{30} - 30q^{32} + 6q^{35} + 14q^{36} - 6q^{37} - 24q^{38} - 28q^{40} - 30q^{41} - 2q^{42} + 14q^{43} - 12q^{45} - 42q^{46} + 10q^{48} + 8q^{49} + 84q^{50} - 4q^{51} + 30q^{52} + 28q^{53} + 2q^{55} - 12q^{56} + 66q^{58} - 24q^{59} + 2q^{61} - 20q^{62} - 48q^{64} - 44q^{65} - 36q^{66} + 30q^{67} + 36q^{68} + 6q^{69} - 6q^{71} + 6q^{74} - 10q^{75} - 24q^{76} + 32q^{77} + 10q^{78} + 92q^{79} + 114q^{80} - 8q^{81} - 42q^{82} + 48q^{85} + 12q^{87} + 62q^{88} + 18q^{89} + 8q^{90} - 116q^{92} - 6q^{93} - 24q^{94} - 24q^{95} - 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} - 132 x^{7} - 45 x^{6} + 864 x^{5} - 243 x^{4} - 1944 x^{3} + 7290 x^{2} - 8748 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{14} + 10 \nu^{13} - 8 \nu^{12} - 3 \nu^{11} + 32 \nu^{10} - 5 \nu^{9} - 44 \nu^{8} + 214 \nu^{7} - 132 \nu^{6} - 45 \nu^{5} + 864 \nu^{4} - 243 \nu^{3} - 1944 \nu^{2} + 7290 \nu - 8748 \)\()/2187\)
\(\beta_{3}\)\(=\)\((\)\(57 \nu^{15} - 5518 \nu^{14} + 7381 \nu^{13} - 18694 \nu^{12} - 28900 \nu^{11} - 3540 \nu^{10} - 97601 \nu^{9} - 227974 \nu^{8} - 68965 \nu^{7} - 678274 \nu^{6} - 899556 \nu^{5} - 99108 \nu^{4} - 2301399 \nu^{3} - 4692006 \nu^{2} + 5341869 \nu - 11910402\)\()/64152\)
\(\beta_{4}\)\(=\)\((\)\(-2194 \nu^{15} + 19537 \nu^{14} - 30118 \nu^{13} + 44909 \nu^{12} + 84636 \nu^{11} - 34820 \nu^{10} + 195806 \nu^{9} + 639719 \nu^{8} - 100294 \nu^{7} + 1629003 \nu^{6} + 2560212 \nu^{5} - 867564 \nu^{4} + 4904226 \nu^{3} + 15869601 \nu^{2} - 21366990 \nu + 35549685\)\()/192456\)
\(\beta_{5}\)\(=\)\((\)\(686 \nu^{15} - 4953 \nu^{14} + 7898 \nu^{13} - 10141 \nu^{12} - 20180 \nu^{11} + 11796 \nu^{10} - 40674 \nu^{9} - 152359 \nu^{8} + 47706 \nu^{7} - 369947 \nu^{6} - 611332 \nu^{5} + 281604 \nu^{4} - 1069686 \nu^{3} - 4003857 \nu^{2} + 5544450 \nu - 8790525\)\()/42768\)
\(\beta_{6}\)\(=\)\((\)\(-2551 \nu^{15} - 30666 \nu^{14} + 35541 \nu^{13} - 130226 \nu^{12} - 178900 \nu^{11} - 67220 \nu^{10} - 709257 \nu^{9} - 1424186 \nu^{8} - 727301 \nu^{7} - 4565062 \nu^{6} - 5476404 \nu^{5} - 1553364 \nu^{4} - 16204887 \nu^{3} - 25765938 \nu^{2} + 27258525 \nu - 73435086\)\()/128304\)
\(\beta_{7}\)\(=\)\((\)\( 515 \nu^{15} - 348 \nu^{14} + 1311 \nu^{13} + 3844 \nu^{12} + 2060 \nu^{11} + 9484 \nu^{10} + 27621 \nu^{9} + 20896 \nu^{8} + 68905 \nu^{7} + 127508 \nu^{6} + 67356 \nu^{5} + 229788 \nu^{4} + 591435 \nu^{3} - 194724 \nu^{2} + 827415 \nu + 769824 \)\()/11664\)
\(\beta_{8}\)\(=\)\((\)\(533 \nu^{15} - 2036 \nu^{14} + 3545 \nu^{13} - 1876 \nu^{12} - 6828 \nu^{11} + 8356 \nu^{10} - 2509 \nu^{9} - 49216 \nu^{8} + 47039 \nu^{7} - 80580 \nu^{6} - 210588 \nu^{5} + 196740 \nu^{4} - 117747 \nu^{3} - 1626156 \nu^{2} + 2440449 \nu - 2881008\)\()/11664\)
\(\beta_{9}\)\(=\)\((\)\(2209 \nu^{15} - 1038 \nu^{14} + 4653 \nu^{13} + 18122 \nu^{12} + 10156 \nu^{11} + 37244 \nu^{10} + 122175 \nu^{9} + 99098 \nu^{8} + 279395 \nu^{7} + 580462 \nu^{6} + 311100 \nu^{5} + 902988 \nu^{4} + 2670273 \nu^{3} - 543510 \nu^{2} + 2746629 \nu + 4543614\)\()/42768\)
\(\beta_{10}\)\(=\)\((\)\(25793 \nu^{15} - 43421 \nu^{14} + 101321 \nu^{13} + 100655 \nu^{12} - 37320 \nu^{11} + 444640 \nu^{10} + 869603 \nu^{9} - 115039 \nu^{8} + 2984339 \nu^{7} + 2850081 \nu^{6} - 1361736 \nu^{5} + 10317888 \nu^{4} + 17012997 \nu^{3} - 33436557 \nu^{2} + 64514313 \nu - 21167973\)\()/384912\)
\(\beta_{11}\)\(=\)\((\)\(-35555 \nu^{15} + 209918 \nu^{14} - 340439 \nu^{13} + 377302 \nu^{12} + 822876 \nu^{11} - 555316 \nu^{10} + 1446739 \nu^{9} + 6248686 \nu^{8} - 2436425 \nu^{7} + 14366802 \nu^{6} + 25605324 \nu^{5} - 12546900 \nu^{4} + 39127293 \nu^{3} + 171173574 \nu^{2} - 235676223 \nu + 358488666\)\()/384912\)
\(\beta_{12}\)\(=\)\((\)\(36175 \nu^{15} - 126178 \nu^{14} + 228019 \nu^{13} - 76154 \nu^{12} - 382332 \nu^{11} + 612740 \nu^{10} + 137617 \nu^{9} - 2689898 \nu^{8} + 3627757 \nu^{7} - 3487038 \nu^{6} - 11616444 \nu^{5} + 14749236 \nu^{4} - 1187217 \nu^{3} - 99205722 \nu^{2} + 155611611 \nu - 166373838\)\()/384912\)
\(\beta_{13}\)\(=\)\((\)\(13197 \nu^{15} - 41905 \nu^{14} + 78133 \nu^{13} - 14485 \nu^{12} - 117448 \nu^{11} + 226080 \nu^{10} + 116311 \nu^{9} - 824875 \nu^{8} + 1357079 \nu^{7} - 830875 \nu^{6} - 3648648 \nu^{5} + 5321808 \nu^{4} + 960417 \nu^{3} - 33076593 \nu^{2} + 52548021 \nu - 53118585\)\()/128304\)
\(\beta_{14}\)\(=\)\((\)\(-47866 \nu^{15} + 127777 \nu^{14} - 247654 \nu^{13} - 30859 \nu^{12} + 311940 \nu^{11} - 813620 \nu^{10} - 843826 \nu^{9} + 2022815 \nu^{8} - 5148238 \nu^{7} - 23901 \nu^{6} + 9382500 \nu^{5} - 19635588 \nu^{4} - 14476806 \nu^{3} + 98903673 \nu^{2} - 167505246 \nu + 134472069\)\()/384912\)
\(\beta_{15}\)\(=\)\((\)\(-5481 \nu^{15} + 12006 \nu^{14} - 25157 \nu^{13} - 12274 \nu^{12} + 21460 \nu^{11} - 95516 \nu^{10} - 141111 \nu^{9} + 126350 \nu^{8} - 619643 \nu^{7} - 302534 \nu^{6} + 673252 \nu^{5} - 2246412 \nu^{4} - 2622393 \nu^{3} + 9246798 \nu^{2} - 16556157 \nu + 10314378\)\()/42768\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14} + \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{15} - \beta_{13} - \beta_{12} - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(\beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_{1}\)
\(\nu^{5}\)\(=\)\(7 \beta_{15} - 2 \beta_{14} + \beta_{12} - 5 \beta_{11} - 5 \beta_{10} + 5 \beta_{9} + \beta_{8} - 6 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + 4 \beta_{2} - 2 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{15} - 5 \beta_{14} - 9 \beta_{13} + 13 \beta_{12} - 6 \beta_{11} + 12 \beta_{10} - 10 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} - 20 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} - 11 \beta_{2} - 11 \beta_{1} - 16\)
\(\nu^{7}\)\(=\)\(-15 \beta_{15} - 2 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 9 \beta_{10} - 6 \beta_{9} - 5 \beta_{8} - 8 \beta_{7} + 10 \beta_{6} + 21 \beta_{5} + 16 \beta_{4} + 4 \beta_{3} + \beta_{2} + 2 \beta_{1} + 14\)
\(\nu^{8}\)\(=\)\(-37 \beta_{15} + 22 \beta_{14} - 10 \beta_{13} + 26 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} - 18 \beta_{9} + 6 \beta_{8} + \beta_{7} + 4 \beta_{6} - 18 \beta_{5} + 39 \beta_{3} - 20 \beta_{2} - 10\)
\(\nu^{9}\)\(=\)\(22 \beta_{15} + 40 \beta_{14} + 46 \beta_{13} - 18 \beta_{12} + 14 \beta_{11} + 2 \beta_{10} + 26 \beta_{9} + 36 \beta_{8} + 44 \beta_{7} + 2 \beta_{6} + 48 \beta_{5} - 22 \beta_{4} + 15 \beta_{3} - 26 \beta_{2} + 22 \beta_{1} + 36\)
\(\nu^{10}\)\(=\)\(-11 \beta_{15} + 14 \beta_{14} - 24 \beta_{13} - 18 \beta_{12} - 44 \beta_{11} - 50 \beta_{10} + 34 \beta_{9} + 15 \beta_{8} + 14 \beta_{7} - 22 \beta_{6} - 22 \beta_{5} - 62 \beta_{4} - 62 \beta_{3} - 40 \beta_{2} + 4 \beta_{1} + 112\)
\(\nu^{11}\)\(=\)\(41 \beta_{15} - 78 \beta_{14} + 89 \beta_{13} - 90 \beta_{12} + 2 \beta_{11} + 68 \beta_{10} - 42 \beta_{9} - 92 \beta_{8} - 5 \beta_{7} + 61 \beta_{6} - 117 \beta_{5} - 18 \beta_{4} - 73 \beta_{3} + 66 \beta_{2} + 59 \beta_{1} + 153\)
\(\nu^{12}\)\(=\)\(-30 \beta_{15} + 5 \beta_{14} + 24 \beta_{13} - 78 \beta_{12} - 28 \beta_{11} - 211 \beta_{10} + 267 \beta_{9} + 101 \beta_{8} - 299 \beta_{7} - 122 \beta_{6} + 356 \beta_{5} + 19 \beta_{4} - 74 \beta_{3} + 293 \beta_{2} + 481 \beta_{1} - 232\)
\(\nu^{13}\)\(=\)\(27 \beta_{15} + 238 \beta_{14} - 349 \beta_{13} + 305 \beta_{12} - 146 \beta_{11} + 220 \beta_{10} + 121 \beta_{9} - 164 \beta_{8} + 265 \beta_{7} - 17 \beta_{6} - 271 \beta_{5} - 486 \beta_{4} - 3 \beta_{3} + 166 \beta_{2} - 384 \beta_{1} - 643\)
\(\nu^{14}\)\(=\)\(-289 \beta_{15} - 753 \beta_{14} + 145 \beta_{13} - 851 \beta_{12} + 332 \beta_{11} + 383 \beta_{10} - 608 \beta_{9} - 481 \beta_{8} - 73 \beta_{7} + 314 \beta_{6} + 209 \beta_{5} + 146 \beta_{4} + 626 \beta_{3} - 9 \beta_{2} - 653 \beta_{1} - 687\)
\(\nu^{15}\)\(=\)\(786 \beta_{15} - 652 \beta_{14} + 1048 \beta_{13} - 316 \beta_{12} + 169 \beta_{11} - 773 \beta_{10} - 385 \beta_{9} + 1261 \beta_{8} - 590 \beta_{7} - 644 \beta_{6} + 33 \beta_{5} + 1406 \beta_{4} + 493 \beta_{3} + 1169 \beta_{2} - 1460 \beta_{1} + 1595\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1 + \beta_{5}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.67549 + 0.438998i
1.35201 1.08262i
1.33452 1.10411i
−1.36010 1.07244i
0.750089 + 1.56121i
−0.306536 1.70471i
0.590887 1.62814i
1.31463 + 1.12772i
−1.67549 0.438998i
1.35201 + 1.08262i
1.33452 + 1.10411i
−1.36010 + 1.07244i
0.750089 1.56121i
−0.306536 + 1.70471i
0.590887 + 1.62814i
1.31463 1.12772i
−2.20165 1.27113i 0.500000 0.866025i 2.23152 + 3.86511i 4.07309i −2.20165 + 1.27113i 0.866025 0.500000i 6.26168i −0.500000 0.866025i −5.17741 + 8.96754i
43.2 −1.98604 1.14664i 0.500000 0.866025i 1.62956 + 2.82249i 0.692320i −1.98604 + 1.14664i −0.866025 + 0.500000i 2.88753i −0.500000 0.866025i 0.793841 1.37497i
43.3 −1.44724 0.835563i 0.500000 0.866025i 0.396329 + 0.686463i 2.68351i −1.44724 + 0.835563i 0.866025 0.500000i 2.01762i −0.500000 0.866025i 2.24224 3.88368i
43.4 −0.654865 0.378087i 0.500000 0.866025i −0.714101 1.23686i 4.01537i −0.654865 + 0.378087i −0.866025 + 0.500000i 2.59231i −0.500000 0.866025i −1.51816 + 2.62953i
43.5 0.924500 + 0.533760i 0.500000 0.866025i −0.430200 0.745128i 0.994065i 0.924500 0.533760i 0.866025 0.500000i 3.05354i −0.500000 0.866025i 0.530592 0.919013i
43.6 1.15033 + 0.664145i 0.500000 0.866025i −0.117823 0.204075i 1.55828i 1.15033 0.664145i −0.866025 + 0.500000i 2.96959i −0.500000 0.866025i 1.03492 1.79254i
43.7 1.85837 + 1.07293i 0.500000 0.866025i 1.30235 + 2.25573i 1.91954i 1.85837 1.07293i 0.866025 0.500000i 1.29759i −0.500000 0.866025i −2.05953 + 3.56721i
43.8 2.35660 + 1.36058i 0.500000 0.866025i 2.70236 + 4.68063i 1.58278i 2.35660 1.36058i −0.866025 + 0.500000i 9.26480i −0.500000 0.866025i 2.15350 3.72996i
127.1 −2.20165 + 1.27113i 0.500000 + 0.866025i 2.23152 3.86511i 4.07309i −2.20165 1.27113i 0.866025 + 0.500000i 6.26168i −0.500000 + 0.866025i −5.17741 8.96754i
127.2 −1.98604 + 1.14664i 0.500000 + 0.866025i 1.62956 2.82249i 0.692320i −1.98604 1.14664i −0.866025 0.500000i 2.88753i −0.500000 + 0.866025i 0.793841 + 1.37497i
127.3 −1.44724 + 0.835563i 0.500000 + 0.866025i 0.396329 0.686463i 2.68351i −1.44724 0.835563i 0.866025 + 0.500000i 2.01762i −0.500000 + 0.866025i 2.24224 + 3.88368i
127.4 −0.654865 + 0.378087i 0.500000 + 0.866025i −0.714101 + 1.23686i 4.01537i −0.654865 0.378087i −0.866025 0.500000i 2.59231i −0.500000 + 0.866025i −1.51816 2.62953i
127.5 0.924500 0.533760i 0.500000 + 0.866025i −0.430200 + 0.745128i 0.994065i 0.924500 + 0.533760i 0.866025 + 0.500000i 3.05354i −0.500000 + 0.866025i 0.530592 + 0.919013i
127.6 1.15033 0.664145i 0.500000 + 0.866025i −0.117823 + 0.204075i 1.55828i 1.15033 + 0.664145i −0.866025 0.500000i 2.96959i −0.500000 + 0.866025i 1.03492 + 1.79254i
127.7 1.85837 1.07293i 0.500000 + 0.866025i 1.30235 2.25573i 1.91954i 1.85837 + 1.07293i 0.866025 + 0.500000i 1.29759i −0.500000 + 0.866025i −2.05953 3.56721i
127.8 2.35660 1.36058i 0.500000 + 0.866025i 2.70236 4.68063i 1.58278i 2.35660 + 1.36058i −0.866025 0.500000i 9.26480i −0.500000 + 0.866025i 2.15350 + 3.72996i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bd.b 16
3.b odd 2 1 819.2.ct.c 16
13.e even 6 1 inner 273.2.bd.b 16
13.f odd 12 1 3549.2.a.ba 8
13.f odd 12 1 3549.2.a.bc 8
39.h odd 6 1 819.2.ct.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.b 16 1.a even 1 1 trivial
273.2.bd.b 16 13.e even 6 1 inner
819.2.ct.c 16 3.b odd 2 1
819.2.ct.c 16 39.h odd 6 1
3549.2.a.ba 8 13.f odd 12 1
3549.2.a.bc 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3721 + 1464 T - 7616 T^{2} - 3072 T^{3} + 11883 T^{4} + 768 T^{5} - 7502 T^{6} - 78 T^{7} + 3348 T^{8} - 24 T^{9} - 839 T^{10} + 6 T^{11} + 152 T^{12} - 15 T^{14} + T^{16} \)
$3$ \( ( 1 - T + T^{2} )^{8} \)
$5$ \( 20449 + 90832 T^{2} + 142366 T^{4} + 105026 T^{6} + 40755 T^{8} + 8554 T^{10} + 943 T^{12} + 50 T^{14} + T^{16} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$11$ \( 583696 + 5931696 T + 24604652 T^{2} + 45846420 T^{3} + 38322277 T^{4} + 5590014 T^{5} - 7562693 T^{6} - 1779870 T^{7} + 1354161 T^{8} + 55608 T^{9} - 78130 T^{10} - 1506 T^{11} + 3295 T^{12} - 68 T^{14} + T^{16} \)
$13$ \( 815730721 + 752982204 T + 419932383 T^{2} + 173765124 T^{3} + 56293731 T^{4} + 14803386 T^{5} + 3118557 T^{6} + 541398 T^{7} + 110993 T^{8} + 41646 T^{9} + 18453 T^{10} + 6738 T^{11} + 1971 T^{12} + 468 T^{13} + 87 T^{14} + 12 T^{15} + T^{16} \)
$17$ \( 238177489 + 161892170 T + 334590250 T^{2} - 101145012 T^{3} + 191836527 T^{4} - 27233098 T^{5} + 35681508 T^{6} - 6300552 T^{7} + 4486990 T^{8} - 433552 T^{9} + 196663 T^{10} - 7166 T^{11} + 5952 T^{12} - 90 T^{13} + 95 T^{14} + 2 T^{15} + T^{16} \)
$19$ \( 9412624 - 42817008 T + 82659420 T^{2} - 80679636 T^{3} + 34496909 T^{4} + 3169812 T^{5} - 7195555 T^{6} + 319776 T^{7} + 1206690 T^{8} - 167436 T^{9} - 84679 T^{10} + 10092 T^{11} + 4962 T^{12} - 79 T^{14} + T^{16} \)
$23$ \( 22886656 - 8993920 T + 38443248 T^{2} + 36624152 T^{3} + 44487729 T^{4} + 23350938 T^{5} + 13767342 T^{6} + 5468296 T^{7} + 2504193 T^{8} + 790278 T^{9} + 230400 T^{10} + 43284 T^{11} + 7319 T^{12} + 718 T^{13} + 93 T^{14} + 6 T^{15} + T^{16} \)
$29$ \( 81511963009 + 44493358526 T + 46472881591 T^{2} + 15480683942 T^{3} + 14027206436 T^{4} + 4681122176 T^{5} + 2038109827 T^{6} + 322530472 T^{7} + 79451914 T^{8} + 8625846 T^{9} + 1971772 T^{10} + 167952 T^{11} + 27535 T^{12} + 1712 T^{13} + 250 T^{14} + 12 T^{15} + T^{16} \)
$31$ \( 1539149824 + 1959532416 T^{2} + 893375705 T^{4} + 180839080 T^{6} + 17762874 T^{8} + 871282 T^{10} + 21081 T^{12} + 238 T^{14} + T^{16} \)
$37$ \( 223550241721 + 299817020076 T + 151888656134 T^{2} + 23945488392 T^{3} - 5919188267 T^{4} - 2184800070 T^{5} + 184089388 T^{6} + 145053996 T^{7} + 7563018 T^{8} - 3894930 T^{9} - 349861 T^{10} + 80784 T^{11} + 10306 T^{12} - 786 T^{13} - 119 T^{14} + 6 T^{15} + T^{16} \)
$41$ \( 7256313856 - 10483765248 T - 16322056192 T^{2} + 30876303360 T^{3} + 66645882112 T^{4} + 25257766656 T^{5} + 2203861888 T^{6} - 562260672 T^{7} - 85318704 T^{8} + 13399152 T^{9} + 3478244 T^{10} + 97116 T^{11} - 29147 T^{12} - 1590 T^{13} + 247 T^{14} + 30 T^{15} + T^{16} \)
$43$ \( 599528101264 - 992348113040 T + 1172048967724 T^{2} - 635201723716 T^{3} + 254780946785 T^{4} - 68763606212 T^{5} + 15187009598 T^{6} - 2512139128 T^{7} + 384369919 T^{8} - 47357182 T^{9} + 6077732 T^{10} - 572594 T^{11} + 57419 T^{12} - 3514 T^{13} + 319 T^{14} - 14 T^{15} + T^{16} \)
$47$ \( 1683953296 + 17007610936 T^{2} + 32974980985 T^{4} + 7226156720 T^{6} + 517335786 T^{8} + 11917774 T^{10} + 120949 T^{12} + 566 T^{14} + T^{16} \)
$53$ \( ( -8807 + 7238 T + 28748 T^{2} - 17386 T^{3} - 6458 T^{4} + 2434 T^{5} - 124 T^{6} - 14 T^{7} + T^{8} )^{2} \)
$59$ \( 2315546542864 + 1720284979536 T + 123773082572 T^{2} - 224544890484 T^{3} - 16048208927 T^{4} + 20155762644 T^{5} + 3391153420 T^{6} - 532299372 T^{7} - 138231129 T^{8} + 10043916 T^{9} + 5093834 T^{10} + 403314 T^{11} - 15611 T^{12} - 3000 T^{13} + 67 T^{14} + 24 T^{15} + T^{16} \)
$61$ \( 114864732889 + 47457191842 T + 53208867724 T^{2} - 7338928308 T^{3} + 9101071194 T^{4} - 681160754 T^{5} + 576233976 T^{6} - 31785078 T^{7} + 25590091 T^{8} - 826730 T^{9} + 673816 T^{10} - 17470 T^{11} + 12834 T^{12} - 204 T^{13} + 140 T^{14} - 2 T^{15} + T^{16} \)
$67$ \( 119295633664 + 381437109120 T + 235521335280 T^{2} - 546807288600 T^{3} + 298239165665 T^{4} - 72757590504 T^{5} + 6004645558 T^{6} + 850305960 T^{7} - 168190977 T^{8} - 14825850 T^{9} + 5706748 T^{10} - 330294 T^{11} - 30909 T^{12} + 3030 T^{13} + 199 T^{14} - 30 T^{15} + T^{16} \)
$71$ \( 672053776 - 5140729200 T + 13558681676 T^{2} - 3450221700 T^{3} - 6851424155 T^{4} + 1837238664 T^{5} + 3695865109 T^{6} + 1216979016 T^{7} + 104319321 T^{8} - 17574768 T^{9} - 2428390 T^{10} + 237816 T^{11} + 40585 T^{12} - 1416 T^{13} - 224 T^{14} + 6 T^{15} + T^{16} \)
$73$ \( 9572078569 + 114981929964 T^{2} + 253081686350 T^{4} + 22257380302 T^{6} + 762202839 T^{8} + 13078486 T^{10} + 119307 T^{12} + 550 T^{14} + T^{16} \)
$79$ \( ( -7916 + 328532 T - 535107 T^{2} + 170512 T^{3} - 7678 T^{4} - 4238 T^{5} + 741 T^{6} - 46 T^{7} + T^{8} )^{2} \)
$83$ \( 1430416 + 47083672 T^{2} + 314617561 T^{4} + 403086002 T^{6} + 139400385 T^{8} + 7510042 T^{10} + 107158 T^{12} + 572 T^{14} + T^{16} \)
$89$ \( 29787842814976 + 46703036593920 T + 30642677991264 T^{2} + 9775274465880 T^{3} + 1287408394433 T^{4} - 73774715394 T^{5} - 36459678523 T^{6} - 94054650 T^{7} + 907825674 T^{8} + 70583022 T^{9} - 7084771 T^{10} - 802794 T^{11} + 51858 T^{12} + 5742 T^{13} - 211 T^{14} - 18 T^{15} + T^{16} \)
$97$ \( 43264 + 1287936 T + 15364272 T^{2} + 76923216 T^{3} + 193814129 T^{4} + 237275004 T^{5} + 117317600 T^{6} - 14658288 T^{7} - 17412681 T^{8} + 2738022 T^{9} + 3824426 T^{10} + 782040 T^{11} + 52281 T^{12} - 1446 T^{13} - 229 T^{14} + 6 T^{15} + T^{16} \)
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