Properties

Label 72.2.n.b
Level $72$
Weight $2$
Character orbit 72.n
Analytic conductor $0.575$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 8 \nu^{10} - 16 \nu^{9} - 48 \nu^{8} - 12 \nu^{7} + 16 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} + 256 \nu^{2} + 384 \nu - 128 \)\()/384\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 3 \nu^{13} - 6 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + 24 \nu^{4} + 80 \nu^{3} + 32 \nu^{2} + 32 \nu + 64 \)\()/192\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} - 24 \nu^{6} - 32 \nu^{5} - 72 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 96 \nu - 256 \)\()/192\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} - \nu^{13} - 2 \nu^{11} + 4 \nu^{10} + 8 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + 32 \nu^{4} - 32 \nu^{3} - 64 \nu + 128 \)\()/128\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} - 24 \nu^{8} - 20 \nu^{7} + 64 \nu^{6} + 96 \nu^{5} + 160 \nu^{4} + 256 \nu^{3} + 384 \nu^{2} + 448 \nu + 512 \)\()/384\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + 4 \nu^{7} + 16 \nu^{6} + 20 \nu^{5} + 64 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 64 \nu + 128 \)\()/96\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{15} + \nu^{14} + 7 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + 20 \nu^{10} + 24 \nu^{9} + 16 \nu^{8} + 20 \nu^{7} - 40 \nu^{6} - 160 \nu^{5} - 272 \nu^{4} - 160 \nu^{3} - 128 \nu^{2} - 384 \nu - 640 \)\()/384\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 2 \nu^{12} - 10 \nu^{11} - 4 \nu^{10} - 16 \nu^{9} + 36 \nu^{7} + 88 \nu^{6} + 64 \nu^{5} + 112 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} - 128 \)\()/384\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 7 \nu^{13} - 14 \nu^{12} - 22 \nu^{11} - 20 \nu^{10} - 24 \nu^{9} - 16 \nu^{8} - 20 \nu^{7} + 40 \nu^{6} + 160 \nu^{5} + 272 \nu^{4} + 160 \nu^{3} + 512 \nu^{2} + 384 \nu + 640 \)\()/384\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{15} - \nu^{14} + 4 \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} + 8 \nu^{7} + 20 \nu^{5} - 52 \nu^{4} - 56 \nu^{3} - 48 \nu^{2} + 16 \nu - 288 \)\()/96\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} - 4 \nu^{11} - \nu^{10} - 4 \nu^{9} + 2 \nu^{8} - 8 \nu^{7} - 20 \nu^{5} + 52 \nu^{4} - 40 \nu^{3} + 48 \nu^{2} - 16 \nu + 192 \)\()/96\)
\(\beta_{13}\)\(=\)\((\)\( 3 \nu^{15} - \nu^{14} + 9 \nu^{13} + 6 \nu^{12} + 26 \nu^{11} + 12 \nu^{10} + 16 \nu^{9} - 16 \nu^{8} + 44 \nu^{7} - 40 \nu^{6} - 32 \nu^{5} - 240 \nu^{4} - 64 \nu^{3} - 256 \nu^{2} - 128 \nu - 896 \)\()/384\)
\(\beta_{14}\)\(=\)\((\)\( -2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} - 8 \nu^{8} - 32 \nu^{7} + 12 \nu^{6} + 64 \nu^{5} + 160 \nu^{4} + 96 \nu^{3} + 256 \nu^{2} + 256 \nu + 704 \)\()/192\)
\(\beta_{15}\)\(=\)\((\)\( -9 \nu^{15} + \nu^{14} - 9 \nu^{13} - 8 \nu^{12} - 42 \nu^{11} - 12 \nu^{10} - 24 \nu^{9} + 40 \nu^{8} - 52 \nu^{7} + 48 \nu^{6} + 96 \nu^{5} + 384 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} + 64 \nu + 1664 \)\()/384\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{8}\)
\(\nu^{3}\)\(=\)\(-\beta_{12} - \beta_{11} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_{1}\)
\(\nu^{5}\)\(=\)\(\beta_{15} + \beta_{14} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{2}\)
\(\nu^{6}\)\(=\)\(-\beta_{15} + \beta_{13} - 2 \beta_{10} + \beta_{9} - \beta_{7} + 5 \beta_{6} + 5 \beta_{5} - \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{7}\)\(=\)\(-2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - 10 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1}\)
\(\nu^{8}\)\(=\)\(-3 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2\)
\(\nu^{9}\)\(=\)\(5 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} - \beta_{9} - 7 \beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{3} + \beta_{2} + 7 \beta_{1} + 2\)
\(\nu^{10}\)\(=\)\(6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} - 3 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} - 11 \beta_{6} + 6 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-7 \beta_{15} - \beta_{14} + 12 \beta_{13} + \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} + 13 \beta_{5} + 26 \beta_{4} + 20 \beta_{3} + \beta_{2} + 26\)
\(\nu^{12}\)\(=\)\(9 \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 19 \beta_{9} - 27 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 9 \beta_{3} - 19 \beta_{2} + 27 \beta_{1} - 10\)
\(\nu^{13}\)\(=\)\(14 \beta_{15} - 3 \beta_{14} + 45 \beta_{13} + 13 \beta_{12} - 7 \beta_{11} - 11 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} + 29 \beta_{6} + 14 \beta_{4} - 29 \beta_{3} - 6 \beta_{2} + 5 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-47 \beta_{15} + 31 \beta_{14} - 28 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} + 41 \beta_{10} - 42 \beta_{9} + 58 \beta_{8} - 13 \beta_{7} + 53 \beta_{5} - 78 \beta_{4} - 12 \beta_{3} - 31 \beta_{2} - 78\)
\(\nu^{15}\)\(=\)\(-55 \beta_{15} + 12 \beta_{14} - 65 \beta_{13} - 22 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + 19 \beta_{9} + 5 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} - 55 \beta_{3} - 19 \beta_{2} - 5 \beta_{1} + 46\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1 - \beta_{4}\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.05026 0.947078i
−0.179748 1.40274i
1.41411 0.0174668i
−1.12494 0.857038i
0.820200 + 1.15207i
0.587625 + 1.28635i
−1.34532 + 0.436011i
−0.722180 + 1.21592i
1.05026 + 0.947078i
−0.179748 + 1.40274i
1.41411 + 0.0174668i
−1.12494 + 0.857038i
0.820200 1.15207i
0.587625 1.28635i
−1.34532 0.436011i
−0.722180 1.21592i
−1.34532 + 0.436011i 1.52768 + 0.816201i 1.61979 1.17315i −0.602794 0.348023i −2.41110 0.431967i 0.795065 + 1.37709i −1.66763 + 2.28452i 1.66763 + 2.49379i 0.962695 + 0.205379i
13.2 −1.12494 0.857038i −0.986088 + 1.42395i 0.530970 + 1.92823i 1.19115 + 0.687709i 2.32967 0.756738i 1.80469 + 3.12581i 1.05526 2.62420i −1.05526 2.80828i −0.750573 1.79449i
13.3 −0.722180 + 1.21592i 0.294546 1.70682i −0.956913 1.75622i 3.17262 + 1.83171i 1.86264 + 1.59078i −0.191926 0.332426i 2.82649 + 0.104780i −2.82649 1.00547i −4.51841 + 2.53482i
13.4 −0.179748 1.40274i 0.986088 1.42395i −1.93538 + 0.504281i −1.19115 0.687709i −2.17468 1.12728i 1.80469 + 3.12581i 1.05526 + 2.62420i −1.05526 2.80828i −0.750573 + 1.79449i
13.5 0.587625 + 1.28635i 1.69028 0.378078i −1.30939 + 1.51178i −1.97542 1.14051i 1.47959 + 1.95213i −0.907824 1.57240i −2.71411 0.795980i 2.71411 1.27812i 0.306290 3.21128i
13.6 0.820200 + 1.15207i −1.69028 + 0.378078i −0.654545 + 1.88986i 1.97542 + 1.14051i −1.82194 1.63723i −0.907824 1.57240i −2.71411 + 0.795980i 2.71411 1.27812i 0.306290 + 3.21128i
13.7 1.05026 0.947078i −1.52768 0.816201i 0.206086 1.98935i 0.602794 + 0.348023i −2.37747 + 0.589613i 0.795065 + 1.37709i −1.66763 2.28452i 1.66763 + 2.49379i 0.962695 0.205379i
13.8 1.41411 0.0174668i −0.294546 + 1.70682i 1.99939 0.0493999i −3.17262 1.83171i −0.386706 + 2.41877i −0.191926 0.332426i 2.82649 0.104780i −2.82649 1.00547i −4.51841 2.53482i
61.1 −1.34532 0.436011i 1.52768 0.816201i 1.61979 + 1.17315i −0.602794 + 0.348023i −2.41110 + 0.431967i 0.795065 1.37709i −1.66763 2.28452i 1.66763 2.49379i 0.962695 0.205379i
61.2 −1.12494 + 0.857038i −0.986088 1.42395i 0.530970 1.92823i 1.19115 0.687709i 2.32967 + 0.756738i 1.80469 3.12581i 1.05526 + 2.62420i −1.05526 + 2.80828i −0.750573 + 1.79449i
61.3 −0.722180 1.21592i 0.294546 + 1.70682i −0.956913 + 1.75622i 3.17262 1.83171i 1.86264 1.59078i −0.191926 + 0.332426i 2.82649 0.104780i −2.82649 + 1.00547i −4.51841 2.53482i
61.4 −0.179748 + 1.40274i 0.986088 + 1.42395i −1.93538 0.504281i −1.19115 + 0.687709i −2.17468 + 1.12728i 1.80469 3.12581i 1.05526 2.62420i −1.05526 + 2.80828i −0.750573 1.79449i
61.5 0.587625 1.28635i 1.69028 + 0.378078i −1.30939 1.51178i −1.97542 + 1.14051i 1.47959 1.95213i −0.907824 + 1.57240i −2.71411 + 0.795980i 2.71411 + 1.27812i 0.306290 + 3.21128i
61.6 0.820200 1.15207i −1.69028 0.378078i −0.654545 1.88986i 1.97542 1.14051i −1.82194 + 1.63723i −0.907824 + 1.57240i −2.71411 0.795980i 2.71411 + 1.27812i 0.306290 3.21128i
61.7 1.05026 + 0.947078i −1.52768 + 0.816201i 0.206086 + 1.98935i 0.602794 0.348023i −2.37747 0.589613i 0.795065 1.37709i −1.66763 + 2.28452i 1.66763 2.49379i 0.962695 + 0.205379i
61.8 1.41411 + 0.0174668i −0.294546 1.70682i 1.99939 + 0.0493999i −3.17262 + 1.83171i −0.386706 2.41877i −0.191926 + 0.332426i 2.82649 + 0.104780i −2.82649 + 1.00547i −4.51841 + 2.53482i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.n.b 16
3.b odd 2 1 216.2.n.b 16
4.b odd 2 1 288.2.r.b 16
8.b even 2 1 inner 72.2.n.b 16
8.d odd 2 1 288.2.r.b 16
9.c even 3 1 inner 72.2.n.b 16
9.c even 3 1 648.2.d.j 8
9.d odd 6 1 216.2.n.b 16
9.d odd 6 1 648.2.d.k 8
12.b even 2 1 864.2.r.b 16
24.f even 2 1 864.2.r.b 16
24.h odd 2 1 216.2.n.b 16
36.f odd 6 1 288.2.r.b 16
36.f odd 6 1 2592.2.d.j 8
36.h even 6 1 864.2.r.b 16
36.h even 6 1 2592.2.d.k 8
72.j odd 6 1 216.2.n.b 16
72.j odd 6 1 648.2.d.k 8
72.l even 6 1 864.2.r.b 16
72.l even 6 1 2592.2.d.k 8
72.n even 6 1 inner 72.2.n.b 16
72.n even 6 1 648.2.d.j 8
72.p odd 6 1 288.2.r.b 16
72.p odd 6 1 2592.2.d.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b 16 1.a even 1 1 trivial
72.2.n.b 16 8.b even 2 1 inner
72.2.n.b 16 9.c even 3 1 inner
72.2.n.b 16 72.n even 6 1 inner
216.2.n.b 16 3.b odd 2 1
216.2.n.b 16 9.d odd 6 1
216.2.n.b 16 24.h odd 2 1
216.2.n.b 16 72.j odd 6 1
288.2.r.b 16 4.b odd 2 1
288.2.r.b 16 8.d odd 2 1
288.2.r.b 16 36.f odd 6 1
288.2.r.b 16 72.p odd 6 1
648.2.d.j 8 9.c even 3 1
648.2.d.j 8 72.n even 6 1
648.2.d.k 8 9.d odd 6 1
648.2.d.k 8 72.j odd 6 1
864.2.r.b 16 12.b even 2 1
864.2.r.b 16 24.f even 2 1
864.2.r.b 16 36.h even 6 1
864.2.r.b 16 72.l even 6 1
2592.2.d.j 8 36.f odd 6 1
2592.2.d.j 8 72.p odd 6 1
2592.2.d.k 8 36.h even 6 1
2592.2.d.k 8 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 128 T + 64 T^{2} + 32 T^{4} - 32 T^{5} - 16 T^{7} + 4 T^{8} - 8 T^{9} - 4 T^{11} + 2 T^{12} + T^{14} - T^{15} + T^{16} \)
$3$ \( 6561 - 729 T^{2} - 162 T^{4} + 81 T^{6} + 18 T^{8} + 9 T^{10} - 2 T^{12} - T^{14} + T^{16} \)
$5$ \( 4096 - 11712 T^{2} + 26129 T^{4} - 18357 T^{6} + 9318 T^{8} - 2049 T^{10} + 326 T^{12} - 21 T^{14} + T^{16} \)
$7$ \( ( 16 + 36 T + 101 T^{2} - 21 T^{3} + 48 T^{4} - 3 T^{5} + 14 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$11$ \( 10556001 - 6822900 T^{2} + 2895966 T^{4} - 718680 T^{6} + 129907 T^{8} - 14440 T^{10} + 1134 T^{12} - 40 T^{14} + T^{16} \)
$13$ \( 20736 - 119664 T^{2} + 587601 T^{4} - 578901 T^{6} + 467038 T^{8} - 36233 T^{10} + 2094 T^{12} - 53 T^{14} + T^{16} \)
$17$ \( ( 36 - 48 T - 2 T^{2} + 7 T^{3} + T^{4} )^{4} \)
$19$ \( ( 5184 + 7056 T^{2} + 1660 T^{4} + 83 T^{6} + T^{8} )^{2} \)
$23$ \( ( 90000 + 67500 T + 40125 T^{2} + 10875 T^{3} + 2650 T^{4} + 275 T^{5} + 60 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$29$ \( 4032758016 - 1453416048 T^{2} + 356481729 T^{4} - 46463373 T^{6} + 4385038 T^{8} - 241441 T^{10} + 9246 T^{12} - 109 T^{14} + T^{16} \)
$31$ \( ( 92416 - 62320 T + 57529 T^{2} + 7415 T^{3} + 3322 T^{4} + 155 T^{5} + 76 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$37$ \( ( 1327104 + 177264 T^{2} + 8080 T^{4} + 152 T^{6} + T^{8} )^{2} \)
$41$ \( ( 335241 + 264024 T + 165090 T^{2} + 38376 T^{3} + 7879 T^{4} + 616 T^{5} + 90 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$43$ \( 81 - 1080 T^{2} + 13446 T^{4} - 12360 T^{6} + 8827 T^{8} - 1880 T^{10} + 294 T^{12} - 20 T^{14} + T^{16} \)
$47$ \( ( 2178576 + 225828 T + 142965 T^{2} - 3537 T^{3} + 5544 T^{4} - 63 T^{5} + 90 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$53$ \( ( 451584 + 113520 T^{2} + 7744 T^{4} + 160 T^{6} + T^{8} )^{2} \)
$59$ \( 131079601 - 286866144 T^{2} + 591967766 T^{4} - 75952296 T^{6} + 7079403 T^{8} - 287928 T^{10} + 8534 T^{12} - 108 T^{14} + T^{16} \)
$61$ \( 176319369216 - 57650719680 T^{2} + 15331541409 T^{4} - 1011826485 T^{6} + 47134062 T^{8} - 1107945 T^{10} + 18846 T^{12} - 165 T^{14} + T^{16} \)
$67$ \( 9881774573841 - 1672043075100 T^{2} + 223335161334 T^{4} - 8572738680 T^{6} + 228454587 T^{8} - 3485160 T^{10} + 38646 T^{12} - 240 T^{14} + T^{16} \)
$71$ \( ( -864 + 252 T + 48 T^{2} - 18 T^{3} + T^{4} )^{4} \)
$73$ \( ( 36 - 84 T - 14 T^{2} + 11 T^{3} + T^{4} )^{4} \)
$79$ \( ( 3136 + 34440 T + 377609 T^{2} + 8445 T^{3} + 9402 T^{4} + 1065 T^{5} + 236 T^{6} + 15 T^{7} + T^{8} )^{2} \)
$83$ \( 31713911056 - 8048506380 T^{2} + 1398102029 T^{4} - 126163065 T^{6} + 8173602 T^{8} - 289605 T^{10} + 7406 T^{12} - 105 T^{14} + T^{16} \)
$89$ \( ( -504 + 156 T + 52 T^{2} - 16 T^{3} + T^{4} )^{4} \)
$97$ \( ( 1324801 + 897780 T + 454166 T^{2} + 104520 T^{3} + 19107 T^{4} + 1560 T^{5} + 134 T^{6} + T^{8} )^{2} \)
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