Properties

Label 585.2.i.e
Level $585$
Weight $2$
Character orbit 585.i
Analytic conductor $4.671$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + 770 x + 268\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + 770 x + 268\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-89 \nu^{15} - 1135 \nu^{14} + 7576 \nu^{13} - 31264 \nu^{12} + 72320 \nu^{11} - 147579 \nu^{10} + 190360 \nu^{9} - 259593 \nu^{8} + 151587 \nu^{7} - 267712 \nu^{6} - 48005 \nu^{5} - 224641 \nu^{4} - 207308 \nu^{3} + 176473 \nu^{2} - 45495 \nu + 194918\)\()/205578\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{14} + 16 \nu^{13} - 28 \nu^{12} + 68 \nu^{11} - 39 \nu^{10} + 139 \nu^{9} + 120 \nu^{8} + 351 \nu^{7} + 677 \nu^{6} + 1228 \nu^{5} + 2087 \nu^{4} + 3205 \nu^{3} + 4420 \nu^{2} + 3336 \nu + 6293 \)\()/2187\)
\(\beta_{3}\)\(=\)\((\)\(-53 \nu^{15} + 1221 \nu^{14} - 4880 \nu^{13} + 14280 \nu^{12} - 18634 \nu^{11} + 24185 \nu^{10} + 23394 \nu^{9} - 18231 \nu^{8} + 152463 \nu^{7} + 42206 \nu^{6} + 312111 \nu^{5} + 354893 \nu^{4} + 509676 \nu^{3} + 567013 \nu^{2} + 436375 \nu + 125592\)\()/68526\)
\(\beta_{4}\)\(=\)\((\)\(-265 \nu^{15} - 851 \nu^{14} - 2404 \nu^{13} + 18478 \nu^{12} - 122780 \nu^{11} + 267189 \nu^{10} - 630706 \nu^{9} + 600027 \nu^{8} - 1096911 \nu^{7} - 12878 \nu^{6} - 1628113 \nu^{5} - 1542137 \nu^{4} - 2561554 \nu^{3} - 2836237 \nu^{2} - 2196081 \nu - 1138394\)\()/205578\)
\(\beta_{5}\)\(=\)\((\)\(-547 \nu^{15} - 2825 \nu^{14} + 11696 \nu^{13} - 42716 \nu^{12} + 9478 \nu^{11} + 20721 \nu^{10} - 417232 \nu^{9} + 363147 \nu^{8} - 1222965 \nu^{7} - 271472 \nu^{6} - 2073673 \nu^{5} - 2132363 \nu^{4} - 3222844 \nu^{3} - 3007411 \nu^{2} - 2176341 \nu - 568472\)\()/205578\)
\(\beta_{6}\)\(=\)\((\)\(685 \nu^{15} - 2275 \nu^{14} + 10492 \nu^{13} - 25438 \nu^{12} + 90950 \nu^{11} - 164457 \nu^{10} + 406510 \nu^{9} - 423375 \nu^{8} + 877113 \nu^{7} - 182266 \nu^{6} + 1218265 \nu^{5} + 955763 \nu^{4} + 1546546 \nu^{3} + 1686079 \nu^{2} + 1281639 \nu + 752246\)\()/205578\)
\(\beta_{7}\)\(=\)\((\)\(-556 \nu^{15} + 910 \nu^{14} + 1274 \nu^{13} - 22988 \nu^{12} + 57808 \nu^{11} - 136806 \nu^{10} + 137726 \nu^{9} - 222348 \nu^{8} + 10989 \nu^{7} - 315041 \nu^{6} - 213625 \nu^{5} - 501563 \nu^{4} - 330217 \nu^{3} - 274471 \nu^{2} - 28941 \nu + 188851\)\()/102789\)
\(\beta_{8}\)\(=\)\((\)\(1099 \nu^{15} - 4885 \nu^{14} + 28834 \nu^{13} - 92848 \nu^{12} + 290714 \nu^{11} - 524601 \nu^{10} + 991078 \nu^{9} - 865473 \nu^{8} + 1367433 \nu^{7} + 197588 \nu^{6} + 1766167 \nu^{5} + 2184713 \nu^{4} + 3032698 \nu^{3} + 3272689 \nu^{2} + 2521281 \nu + 1087838\)\()/205578\)
\(\beta_{9}\)\(=\)\((\)\( -14 \nu^{15} + 44 \nu^{14} - 53 \nu^{13} - 331 \nu^{12} + 1253 \nu^{11} - 3357 \nu^{10} + 4663 \nu^{9} - 6243 \nu^{8} + 2448 \nu^{7} - 3007 \nu^{6} - 4508 \nu^{5} + 1643 \nu^{4} - 1982 \nu^{3} + 5971 \nu^{2} + 8634 \nu + 3587 \)\()/2187\)
\(\beta_{10}\)\(=\)\((\)\(195 \nu^{15} - 931 \nu^{14} + 2748 \nu^{13} - 3688 \nu^{12} + 3582 \nu^{11} + 2953 \nu^{10} - 3464 \nu^{9} + 5595 \nu^{8} + 21477 \nu^{7} - 29046 \nu^{6} + 43807 \nu^{5} - 79347 \nu^{4} - 86552 \nu^{3} - 151197 \nu^{2} - 216913 \nu - 138346\)\()/22842\)
\(\beta_{11}\)\(=\)\((\)\(-938 \nu^{15} + 5045 \nu^{14} - 19022 \nu^{13} + 40454 \nu^{12} - 78373 \nu^{11} + 83322 \nu^{10} - 119705 \nu^{9} + 16041 \nu^{8} - 150696 \nu^{7} - 141799 \nu^{6} - 375791 \nu^{5} - 287764 \nu^{4} - 648728 \nu^{3} - 371333 \nu^{2} - 454875 \nu - 220840\)\()/102789\)
\(\beta_{12}\)\(=\)\((\)\(405 \nu^{15} - 2212 \nu^{14} + 6369 \nu^{13} - 7450 \nu^{12} - 3732 \nu^{11} + 40360 \nu^{10} - 89195 \nu^{9} + 120006 \nu^{8} - 108444 \nu^{7} + 31020 \nu^{6} - 74402 \nu^{5} - 167943 \nu^{4} - 271589 \nu^{3} - 340359 \nu^{2} - 435703 \nu - 177730\)\()/34263\)
\(\beta_{13}\)\(=\)\((\)\(523 \nu^{15} - 1973 \nu^{14} + 5068 \nu^{13} - 650 \nu^{12} - 10177 \nu^{11} + 56326 \nu^{10} - 68467 \nu^{9} + 139773 \nu^{8} - 14142 \nu^{7} + 172772 \nu^{6} + 180167 \nu^{5} + 203561 \nu^{4} + 277010 \nu^{3} + 101233 \nu^{2} + 19619 \nu - 53165\)\()/34263\)
\(\beta_{14}\)\(=\)\((\)\(-4021 \nu^{15} + 14221 \nu^{14} - 42610 \nu^{13} + 32740 \nu^{12} - 46970 \nu^{11} - 188259 \nu^{10} + 63962 \nu^{9} - 750819 \nu^{8} - 383103 \nu^{7} - 1812884 \nu^{6} - 1944757 \nu^{5} - 3449927 \nu^{4} - 3468346 \nu^{3} - 3279607 \nu^{2} - 1943709 \nu - 512798\)\()/205578\)
\(\beta_{15}\)\(=\)\((\)\(4721 \nu^{15} - 27797 \nu^{14} + 107414 \nu^{13} - 251234 \nu^{12} + 501802 \nu^{11} - 668505 \nu^{10} + 915602 \nu^{9} - 643377 \nu^{8} + 951039 \nu^{7} + 92026 \nu^{6} + 839147 \nu^{5} + 597901 \nu^{4} + 482192 \nu^{3} + 7385 \nu^{2} - 493101 \nu - 191036\)\()/205578\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{15} + 2 \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{2} + 2 \beta_{1} - 3\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + 4 \beta_{8} - \beta_{7} + 5 \beta_{5} - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 2 \beta_{1} - 12\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{15} + \beta_{14} + 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 7 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} - 3 \beta_{7} + 5 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} + 21 \beta_{2} - 31\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{15} - 10 \beta_{14} - 12 \beta_{13} - 6 \beta_{12} - 21 \beta_{11} + 21 \beta_{10} + 12 \beta_{9} - \beta_{7} + 10 \beta_{6} - 14 \beta_{5} + 22 \beta_{4} - 28 \beta_{3} + 43 \beta_{2} + 7 \beta_{1} - 56\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-21 \beta_{15} - 23 \beta_{14} - 76 \beta_{13} + 41 \beta_{12} - 78 \beta_{11} + 8 \beta_{10} + 23 \beta_{9} - 16 \beta_{8} - \beta_{7} - 10 \beta_{6} - 72 \beta_{5} + 36 \beta_{4} - 35 \beta_{3} + 25 \beta_{2} - 11 \beta_{1} - 13\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(8 \beta_{15} - 14 \beta_{14} - 183 \beta_{13} + 156 \beta_{12} - 153 \beta_{11} - 60 \beta_{10} + 18 \beta_{9} - 36 \beta_{8} - 5 \beta_{7} - 106 \beta_{6} - 133 \beta_{5} + 14 \beta_{4} + 103 \beta_{3} - 115 \beta_{2} - 88 \beta_{1} + 299\)\()/3\)
\(\nu^{8}\)\(=\)\(36 \beta_{15} + 16 \beta_{14} - 56 \beta_{13} + 89 \beta_{12} - 45 \beta_{11} - 52 \beta_{10} + 19 \beta_{9} - 28 \beta_{8} + 7 \beta_{7} - 103 \beta_{6} - 32 \beta_{5} - 42 \beta_{4} + 163 \beta_{3} - 143 \beta_{2} - 62 \beta_{1} + 368\)
\(\nu^{9}\)\(=\)\((\)\(267 \beta_{15} + 200 \beta_{14} + 415 \beta_{13} + 136 \beta_{12} + 309 \beta_{11} - 194 \beta_{10} + 256 \beta_{9} - 179 \beta_{8} + 139 \beta_{7} - 407 \beta_{6} + 258 \beta_{5} - 534 \beta_{4} + 875 \beta_{3} - 835 \beta_{2} - 172 \beta_{1} + 2065\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(271 \beta_{15} + 495 \beta_{14} + 2219 \beta_{13} - 718 \beta_{12} + 1797 \beta_{11} - 94 \beta_{10} + 401 \beta_{9} - 25 \beta_{8} + 187 \beta_{7} + 483 \beta_{6} + 1432 \beta_{5} - 1343 \beta_{4} + 15 \beta_{3} - 973 \beta_{2} + 200 \beta_{1} + 966\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-562 \beta_{15} + 819 \beta_{14} + 5284 \beta_{13} - 3086 \beta_{12} + 4665 \beta_{11} + 250 \beta_{10} - 785 \beta_{9} + 1402 \beta_{8} - 643 \beta_{7} + 4206 \beta_{6} + 3965 \beta_{5} - 1933 \beta_{4} - 5007 \beta_{3} + 145 \beta_{2} + 1642 \beta_{1} - 7629\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(-3293 \beta_{15} + 13 \beta_{14} + 6391 \beta_{13} - 7562 \beta_{12} + 6747 \beta_{11} + 1306 \beta_{10} - 5876 \beta_{9} + 5200 \beta_{8} - 3246 \beta_{7} + 12074 \beta_{6} + 6283 \beta_{5} + 1096 \beta_{4} - 17279 \beta_{3} + 4887 \beta_{2} + 5874 \beta_{1} - 30547\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(-7271 \beta_{15} - 5467 \beta_{14} - 6057 \beta_{13} - 11703 \beta_{12} - 993 \beta_{11} + 4242 \beta_{10} - 16398 \beta_{9} + 8862 \beta_{8} - 5269 \beta_{7} + 17608 \beta_{6} - 410 \beta_{5} + 15937 \beta_{4} - 30142 \beta_{3} + 17758 \beta_{2} + 13723 \beta_{1} - 63437\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(-4191 \beta_{15} - 20177 \beta_{14} - 55156 \beta_{13} - 2431 \beta_{12} - 38424 \beta_{11} + 10259 \beta_{10} - 23467 \beta_{9} - 1204 \beta_{8} + 4397 \beta_{7} - 9829 \beta_{6} - 36678 \beta_{5} + 50877 \beta_{4} - 1727 \beta_{3} + 43735 \beta_{2} + 15172 \beta_{1} - 56845\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(28025 \beta_{15} - 38420 \beta_{14} - 157983 \beta_{13} + 55743 \beta_{12} - 129234 \beta_{11} + 17493 \beta_{10} + 15225 \beta_{9} - 55227 \beta_{8} + 45550 \beta_{7} - 138565 \beta_{6} - 124777 \beta_{5} + 88751 \beta_{4} + 172129 \beta_{3} + 75692 \beta_{2} - 31399 \beta_{1} + 136049\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
196.1
−0.724143 0.165319i
−0.628312 + 0.590424i
−0.317019 1.12493i
0.172467 1.52157i
0.252952 + 1.56266i
0.466399 + 1.64781i
1.48460 1.66288i
1.79305 + 1.53983i
−0.724143 + 0.165319i
−0.628312 0.590424i
−0.317019 + 1.12493i
0.172467 + 1.52157i
0.252952 1.56266i
0.466399 1.64781i
1.48460 + 1.66288i
1.79305 1.53983i
−1.22414 + 2.12028i −1.71693 0.228383i −1.99705 3.45900i 0.500000 + 0.866025i 2.58600 3.36079i 1.96726 3.40740i 4.88214 2.89568 + 0.784233i −2.44829
196.2 −1.12831 + 1.95429i −0.604199 1.62325i −1.54617 2.67805i 0.500000 + 0.866025i 3.85403 + 0.650751i −0.353285 + 0.611908i 2.46502 −2.26989 + 1.96153i −2.25662
196.3 −0.817019 + 1.41512i 1.40359 + 1.01486i −0.335039 0.580304i 0.500000 + 0.866025i −2.58290 + 1.15709i −1.06506 + 1.84473i −2.17314 0.940129 + 2.84889i −1.63404
196.4 −0.327533 + 0.567303i 0.997116 1.41625i 0.785445 + 1.36043i 0.500000 + 0.866025i 0.476854 + 1.02954i 0.388592 0.673061i −2.33917 −1.01152 2.82433i −0.655066
196.5 −0.247048 + 0.427900i 0.674928 + 1.59514i 0.877935 + 1.52063i 0.500000 + 0.866025i −0.849299 0.105275i 2.14269 3.71125i −1.85576 −2.08895 + 2.15321i −0.494096
196.6 −0.0336011 + 0.0581988i −0.332146 1.69991i 0.997742 + 1.72814i 0.500000 + 0.866025i 0.110093 + 0.0377882i −1.23179 + 2.13352i −0.268505 −2.77936 + 1.12923i −0.0672022
196.7 0.984603 1.70538i −1.60495 0.651266i −0.938888 1.62620i 0.500000 + 0.866025i −2.69089 + 2.09581i 1.51414 2.62256i 0.240686 2.15171 + 2.09049i 1.96921
196.8 1.29305 2.23963i 1.68259 + 0.410979i −2.34397 4.05987i 0.500000 + 0.866025i 3.09611 3.23696i 2.13745 3.70217i −6.95128 2.66219 + 1.38302i 2.58610
391.1 −1.22414 2.12028i −1.71693 + 0.228383i −1.99705 + 3.45900i 0.500000 0.866025i 2.58600 + 3.36079i 1.96726 + 3.40740i 4.88214 2.89568 0.784233i −2.44829
391.2 −1.12831 1.95429i −0.604199 + 1.62325i −1.54617 + 2.67805i 0.500000 0.866025i 3.85403 0.650751i −0.353285 0.611908i 2.46502 −2.26989 1.96153i −2.25662
391.3 −0.817019 1.41512i 1.40359 1.01486i −0.335039 + 0.580304i 0.500000 0.866025i −2.58290 1.15709i −1.06506 1.84473i −2.17314 0.940129 2.84889i −1.63404
391.4 −0.327533 0.567303i 0.997116 + 1.41625i 0.785445 1.36043i 0.500000 0.866025i 0.476854 1.02954i 0.388592 + 0.673061i −2.33917 −1.01152 + 2.82433i −0.655066
391.5 −0.247048 0.427900i 0.674928 1.59514i 0.877935 1.52063i 0.500000 0.866025i −0.849299 + 0.105275i 2.14269 + 3.71125i −1.85576 −2.08895 2.15321i −0.494096
391.6 −0.0336011 0.0581988i −0.332146 + 1.69991i 0.997742 1.72814i 0.500000 0.866025i 0.110093 0.0377882i −1.23179 2.13352i −0.268505 −2.77936 1.12923i −0.0672022
391.7 0.984603 + 1.70538i −1.60495 + 0.651266i −0.938888 + 1.62620i 0.500000 0.866025i −2.69089 2.09581i 1.51414 + 2.62256i 0.240686 2.15171 2.09049i 1.96921
391.8 1.29305 + 2.23963i 1.68259 0.410979i −2.34397 + 4.05987i 0.500000 0.866025i 3.09611 + 3.23696i 2.13745 + 3.70217i −6.95128 2.66219 1.38302i 2.58610
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.i.e 16
3.b odd 2 1 1755.2.i.f 16
9.c even 3 1 inner 585.2.i.e 16
9.c even 3 1 5265.2.a.bf 8
9.d odd 6 1 1755.2.i.f 16
9.d odd 6 1 5265.2.a.ba 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e 16 1.a even 1 1 trivial
585.2.i.e 16 9.c even 3 1 inner
1755.2.i.f 16 3.b odd 2 1
1755.2.i.f 16 9.d odd 6 1
5265.2.a.ba 8 9.d odd 6 1
5265.2.a.bf 8 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\(T_{2}^{16} + \cdots\)
\(T_{7}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 19 T + 295 T^{2} + 1114 T^{3} + 3027 T^{4} + 4627 T^{5} + 5539 T^{6} + 3873 T^{7} + 2812 T^{8} + 1368 T^{9} + 875 T^{10} + 324 T^{11} + 158 T^{12} + 38 T^{13} + 17 T^{14} + 3 T^{15} + T^{16} \)
$3$ \( 6561 - 2187 T + 1215 T^{3} - 648 T^{4} + 405 T^{5} - 81 T^{6} - 135 T^{7} + 153 T^{8} - 45 T^{9} - 9 T^{10} + 15 T^{11} - 8 T^{12} + 5 T^{13} - T^{15} + T^{16} \)
$5$ \( ( 1 - T + T^{2} )^{8} \)
$7$ \( 395641 - 18870 T + 856340 T^{2} - 251056 T^{3} + 1614395 T^{4} - 359094 T^{5} + 544052 T^{6} - 191361 T^{7} + 140146 T^{8} - 44604 T^{9} + 20644 T^{10} - 6158 T^{11} + 2127 T^{12} - 476 T^{13} + 97 T^{14} - 11 T^{15} + T^{16} \)
$11$ \( 401956 - 7607366 T + 144611269 T^{2} + 7940038 T^{3} + 39378942 T^{4} + 6526234 T^{5} + 7734292 T^{6} + 1288626 T^{7} + 792871 T^{8} + 123459 T^{9} + 56870 T^{10} + 7764 T^{11} + 2531 T^{12} + 272 T^{13} + 74 T^{14} + 6 T^{15} + T^{16} \)
$13$ \( ( 1 - T + T^{2} )^{8} \)
$17$ \( ( 892 + 1275 T - 5408 T^{2} + 748 T^{3} + 1532 T^{4} - 73 T^{5} - 78 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$19$ \( ( -1584 - 12735 T + 3552 T^{2} + 4746 T^{3} - 413 T^{4} - 541 T^{5} - 37 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$23$ \( 144360225 + 597554010 T + 1941770961 T^{2} + 1887936012 T^{3} + 1313009082 T^{4} + 587467485 T^{5} + 211331268 T^{6} + 55708785 T^{7} + 13261735 T^{8} + 2511180 T^{9} + 487609 T^{10} + 70845 T^{11} + 10650 T^{12} + 930 T^{13} + 115 T^{14} + 6 T^{15} + T^{16} \)
$29$ \( 1243225 - 7254190 T + 30973991 T^{2} - 55499768 T^{3} + 72427298 T^{4} - 49775321 T^{5} + 26908734 T^{6} - 8229927 T^{7} + 2295865 T^{8} - 222180 T^{9} + 93171 T^{10} + 199 T^{11} + 5900 T^{12} + 922 T^{13} + 191 T^{14} + 14 T^{15} + T^{16} \)
$31$ \( 2547422784 + 37540417464 T + 587937427425 T^{2} - 534856998123 T^{3} + 303695449497 T^{4} - 109219863132 T^{5} + 29604069633 T^{6} - 6073006854 T^{7} + 1010969710 T^{8} - 136136800 T^{9} + 15745783 T^{10} - 1550944 T^{11} + 138475 T^{12} - 10450 T^{13} + 685 T^{14} - 31 T^{15} + T^{16} \)
$37$ \( ( -33660 - 47061 T - 10002 T^{2} + 7773 T^{3} + 2722 T^{4} - 211 T^{5} - 120 T^{6} - T^{7} + T^{8} )^{2} \)
$41$ \( 371525625 + 3772676475 T + 41639867541 T^{2} - 35355151656 T^{3} + 21996268305 T^{4} - 7559441862 T^{5} + 2158964724 T^{6} - 417073164 T^{7} + 78999628 T^{8} - 11135433 T^{9} + 1807906 T^{10} - 196602 T^{11} + 25611 T^{12} - 1896 T^{13} + 226 T^{14} - 12 T^{15} + T^{16} \)
$43$ \( 62415940941376 + 23481805661864 T + 11927486290657 T^{2} + 1951758905744 T^{3} + 767331823674 T^{4} + 123323953388 T^{5} + 30698577001 T^{6} + 3560936037 T^{7} + 622298584 T^{8} + 57518151 T^{9} + 8484905 T^{10} + 620805 T^{11} + 71942 T^{12} + 3841 T^{13} + 392 T^{14} + 15 T^{15} + T^{16} \)
$47$ \( 424482219529 - 1279662839530 T + 3248273932210 T^{2} - 1680643045456 T^{3} + 645954819852 T^{4} - 152813912530 T^{5} + 30711240460 T^{6} - 4512065298 T^{7} + 650712493 T^{8} - 73943418 T^{9} + 8953964 T^{10} - 783498 T^{11} + 75620 T^{12} - 4832 T^{13} + 410 T^{14} - 18 T^{15} + T^{16} \)
$53$ \( ( 111438 - 1441287 T - 527853 T^{2} + 40404 T^{3} + 24010 T^{4} + 34 T^{5} - 282 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$59$ \( 3524698346724 + 3147714689742 T + 2038429931949 T^{2} + 792853210602 T^{3} + 260063674326 T^{4} + 63722633094 T^{5} + 15226016148 T^{6} + 2976805512 T^{7} + 550097847 T^{8} + 77940117 T^{9} + 9912708 T^{10} + 958500 T^{11} + 90951 T^{12} + 6732 T^{13} + 516 T^{14} + 24 T^{15} + T^{16} \)
$61$ \( 106354483234561 + 72104199877038 T + 60778402377829 T^{2} - 1176151107174 T^{3} + 3239300036677 T^{4} - 158759663706 T^{5} + 128696126584 T^{6} - 9023364243 T^{7} + 2219192077 T^{8} - 99722991 T^{9} + 20750026 T^{10} - 811188 T^{11} + 129298 T^{12} - 3108 T^{13} + 457 T^{14} - 9 T^{15} + T^{16} \)
$67$ \( 974523752673769 - 640510517953695 T + 417224699042386 T^{2} - 47825089843203 T^{3} + 15845363589646 T^{4} - 1340432509050 T^{5} + 413771541892 T^{6} - 26434479180 T^{7} + 5056878430 T^{8} - 279036495 T^{9} + 42945253 T^{10} - 2006577 T^{11} + 212176 T^{12} - 7338 T^{13} + 670 T^{14} - 18 T^{15} + T^{16} \)
$71$ \( ( -127950 + 436287 T - 45522 T^{2} - 71802 T^{3} + 8011 T^{4} + 1915 T^{5} - 211 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$73$ \( ( 1196532 + 541647 T - 546264 T^{2} - 22572 T^{3} + 24390 T^{4} + 747 T^{5} - 300 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$79$ \( 17219921365191204 + 58768849774602 T + 770768004795957 T^{2} + 16509823136514 T^{3} + 23351454457443 T^{4} + 476048343240 T^{5} + 373725005112 T^{6} + 4277853648 T^{7} + 4247252091 T^{8} + 25818183 T^{9} + 31293864 T^{10} - 33129 T^{11} + 165771 T^{12} - 396 T^{13} + 507 T^{14} - 3 T^{15} + T^{16} \)
$83$ \( 469225 + 915845 T + 3493904 T^{2} + 3751063 T^{3} + 13211249 T^{4} + 13993630 T^{5} + 24758769 T^{6} + 5983284 T^{7} + 5334472 T^{8} - 1313994 T^{9} + 1113264 T^{10} - 133184 T^{11} + 24254 T^{12} - 1028 T^{13} + 215 T^{14} - 10 T^{15} + T^{16} \)
$89$ \( ( 1032219 - 887229 T - 283113 T^{2} + 86394 T^{3} + 15982 T^{4} - 2243 T^{5} - 228 T^{6} + 13 T^{7} + T^{8} )^{2} \)
$97$ \( 122483314259524 - 27999585649626 T + 18066194226929 T^{2} - 1185041673992 T^{3} + 1243109191223 T^{4} - 88195342548 T^{5} + 45607413938 T^{6} - 3068221395 T^{7} + 1161895990 T^{8} - 91851144 T^{9} + 17734885 T^{10} - 1399039 T^{11} + 194409 T^{12} - 13762 T^{13} + 994 T^{14} - 34 T^{15} + T^{16} \)
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