Properties

Label 72.2.l.b
Level 72
Weight 2
Character orbit 72.l
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.l (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: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 16 x^{12} - 12 x^{11} - 8 x^{10} + 36 x^{9} - 68 x^{8} + 72 x^{7} - 32 x^{6} - 96 x^{5} + 256 x^{4} - 384 x^{3} + 448 x^{2} - 384 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + 4 \nu^{12} - 38 \nu^{11} + 60 \nu^{10} - 104 \nu^{9} + 68 \nu^{8} + 148 \nu^{7} - 344 \nu^{6} + 440 \nu^{5} - 240 \nu^{4} - 32 \nu^{3} + 608 \nu^{2} - 1152 \nu + 1280 \)\()/896\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + \nu^{14} + 25 \nu^{13} - 38 \nu^{12} + 46 \nu^{11} - 24 \nu^{10} + 8 \nu^{9} + 68 \nu^{8} - 244 \nu^{7} + 272 \nu^{6} - 8 \nu^{5} - 128 \nu^{4} + 416 \nu^{3} - 1184 \nu^{2} + 1088 \nu - 512 \)\()/896\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 6 \nu^{14} - 11 \nu^{13} + 24 \nu^{12} - 39 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} - 124 \nu^{8} + 160 \nu^{7} - 160 \nu^{6} - 20 \nu^{5} + 240 \nu^{4} - 528 \nu^{3} + 848 \nu^{2} - 640 \nu + 512 \)\()/448\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{15} - 11 \nu^{14} + 47 \nu^{13} - 58 \nu^{12} + 40 \nu^{11} + 40 \nu^{10} - 144 \nu^{9} + 316 \nu^{8} - 340 \nu^{7} - 80 \nu^{6} + 592 \nu^{5} - 1168 \nu^{4} + 1248 \nu^{3} - 1088 \nu^{2} + 128 \nu + 1152 \)\()/896\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{15} + 23 \nu^{14} - 13 \nu^{13} - 6 \nu^{12} + 50 \nu^{11} - 132 \nu^{10} + 240 \nu^{9} - 116 \nu^{8} - 236 \nu^{7} + 656 \nu^{6} - 856 \nu^{5} + 752 \nu^{4} - 288 \nu^{3} - 1696 \nu^{2} + 2176 \nu - 1920 \)\()/896\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{15} + 4 \nu^{14} - 5 \nu^{13} + 2 \nu^{12} + 23 \nu^{11} - 40 \nu^{10} + 32 \nu^{9} + 20 \nu^{8} - 24 \nu^{7} + 80 \nu^{6} - 60 \nu^{5} + 104 \nu^{4} + 208 \nu^{3} - 144 \nu^{2} - 352 \nu + 192 \)\()/448\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{15} + 3 \nu^{14} - 9 \nu^{13} + 26 \nu^{12} - 30 \nu^{11} + 12 \nu^{10} + 52 \nu^{9} - 132 \nu^{8} + 164 \nu^{7} - 80 \nu^{6} - 136 \nu^{5} + 400 \nu^{4} - 656 \nu^{3} + 704 \nu^{2} - 320 \nu - 192 \)\()/448\)
\(\beta_{9}\)\(=\)\((\)\( -4 \nu^{15} + 17 \nu^{14} - 37 \nu^{13} + 19 \nu^{12} + 12 \nu^{11} - 86 \nu^{10} + 192 \nu^{9} - 216 \nu^{8} + 52 \nu^{7} + 340 \nu^{6} - 584 \nu^{5} + 792 \nu^{4} - 544 \nu^{3} - 192 \nu^{2} + 1248 \nu - 1088 \)\()/448\)
\(\beta_{10}\)\(=\)\((\)\( 5 \nu^{15} - 9 \nu^{14} + 27 \nu^{13} - 36 \nu^{12} + 34 \nu^{11} - 8 \nu^{10} - 72 \nu^{9} + 172 \nu^{8} - 156 \nu^{7} + 72 \nu^{6} + 184 \nu^{5} - 640 \nu^{4} + 736 \nu^{3} - 768 \nu^{2} + 512 \nu - 320 \)\()/448\)
\(\beta_{11}\)\(=\)\((\)\( 5 \nu^{15} - 2 \nu^{14} - 8 \nu^{13} + 27 \nu^{12} - 36 \nu^{11} + 48 \nu^{10} - 16 \nu^{9} - 108 \nu^{8} + 208 \nu^{7} - 236 \nu^{6} + 72 \nu^{5} + 144 \nu^{4} - 496 \nu^{3} + 800 \nu^{2} - 384 \nu + 128 \)\()/448\)
\(\beta_{12}\)\(=\)\((\)\( -6 \nu^{15} + 15 \nu^{14} - 24 \nu^{13} + 18 \nu^{12} - 3 \nu^{11} - 66 \nu^{10} + 120 \nu^{9} - 128 \nu^{8} - 20 \nu^{7} + 160 \nu^{6} - 260 \nu^{5} + 432 \nu^{4} - 480 \nu^{3} + 48 \nu^{2} + 192 \nu + 384 \)\()/448\)
\(\beta_{13}\)\(=\)\((\)\( 5 \nu^{15} - 16 \nu^{14} + 48 \nu^{13} - 71 \nu^{12} + 76 \nu^{11} - 22 \nu^{10} - 100 \nu^{9} + 284 \nu^{8} - 408 \nu^{7} + 212 \nu^{6} + 184 \nu^{5} - 920 \nu^{4} + 1520 \nu^{3} - 1888 \nu^{2} + 1632 \nu - 768 \)\()/448\)
\(\beta_{14}\)\(=\)\((\)\( 3 \nu^{14} - 7 \nu^{13} + 15 \nu^{12} - 22 \nu^{11} + 16 \nu^{10} + 4 \nu^{9} - 56 \nu^{8} + 92 \nu^{7} - 116 \nu^{6} + 32 \nu^{5} + 144 \nu^{4} - 352 \nu^{3} + 512 \nu^{2} - 448 \nu + 384 \)\()/64\)
\(\beta_{15}\)\(=\)\((\)\( 3 \nu^{15} - 18 \nu^{14} + 47 \nu^{13} - 100 \nu^{12} + 117 \nu^{11} - 100 \nu^{10} - 60 \nu^{9} + 316 \nu^{8} - 592 \nu^{7} + 648 \nu^{6} - 52 \nu^{5} - 888 \nu^{4} + 2144 \nu^{3} - 2768 \nu^{2} + 3040 \nu - 1536 \)\()/448\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{11} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1\)
\(\nu^{5}\)\(=\)\(\beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} + \beta_{7} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} - 5\)
\(\nu^{6}\)\(=\)\(2 \beta_{15} - \beta_{14} - 6 \beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{7}\)\(=\)\(-\beta_{15} - \beta_{14} - 2 \beta_{13} - 3 \beta_{12} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_{1} + 3\)
\(\nu^{8}\)\(=\)\(\beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} + 2 \beta_{10} + 2 \beta_{8} + \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{9}\)\(=\)\(-2 \beta_{15} - 3 \beta_{14} - 4 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 3 \beta_{6} + 14 \beta_{5} - 4 \beta_{3} + 5 \beta_{2} + \beta_{1} - 4\)
\(\nu^{10}\)\(=\)\(-\beta_{15} - 3 \beta_{14} - 6 \beta_{13} - 7 \beta_{12} - 7 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + 12 \beta_{5} - \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 7\)
\(\nu^{11}\)\(=\)\(-17 \beta_{15} - 4 \beta_{14} + 18 \beta_{13} + 7 \beta_{12} - 8 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} - 26 \beta_{6} - 22 \beta_{5} - 7 \beta_{4} + 22 \beta_{3} - 16 \beta_{2} + 9 \beta_{1} + 21\)
\(\nu^{12}\)\(=\)\(8 \beta_{15} - 11 \beta_{14} - 24 \beta_{13} + 18 \beta_{12} + 40 \beta_{11} - 7 \beta_{10} - 34 \beta_{9} + 18 \beta_{7} + 21 \beta_{6} + 10 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 11 \beta_{2} + 35 \beta_{1} - 22\)
\(\nu^{13}\)\(=\)\(-5 \beta_{15} - 13 \beta_{14} - 6 \beta_{13} + \beta_{12} - \beta_{10} - 18 \beta_{9} + 18 \beta_{8} + 13 \beta_{7} + 5 \beta_{6} + 31 \beta_{4} + 26 \beta_{3} - \beta_{2} + 38 \beta_{1} - 1\)
\(\nu^{14}\)\(=\)\(17 \beta_{15} + 18 \beta_{14} - 34 \beta_{13} - 43 \beta_{12} + 16 \beta_{11} + 18 \beta_{10} - 42 \beta_{8} + 25 \beta_{7} + 48 \beta_{6} + 34 \beta_{5} + 27 \beta_{4} + 2 \beta_{3} - 26 \beta_{2} + 17 \beta_{1} - 25\)
\(\nu^{15}\)\(=\)\(50 \beta_{15} + 45 \beta_{14} - 20 \beta_{13} - 20 \beta_{12} + 48 \beta_{11} + 25 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} + 20 \beta_{7} + 49 \beta_{6} + 38 \beta_{5} - 60 \beta_{4} - 56 \beta_{3} - 11 \beta_{2} - 79 \beta_{1} + 8\)

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.608741 1.27649i
−0.186766 1.40183i
−0.533474 1.30973i
1.40985 + 0.111062i
1.12063 + 0.862658i
−1.37702 0.322193i
0.867527 + 1.11687i
−0.409484 + 1.35363i
0.608741 + 1.27649i
−0.186766 + 1.40183i
−0.533474 + 1.30973i
1.40985 0.111062i
1.12063 0.862658i
−1.37702 + 0.322193i
0.867527 1.11687i
−0.409484 1.35363i
−1.40985 + 0.111062i −1.71646 0.231865i 1.97533 0.313160i 1.74322 + 3.01934i 2.44570 + 0.136260i 1.80802 + 1.04386i −2.75013 + 0.660890i 2.89248 + 0.795973i −2.79300 4.06320i
11.2 −1.12063 + 0.862658i 0.418594 1.68071i 0.511643 1.93345i −1.60936 2.78750i 0.980785 + 2.24456i 1.82223 + 1.05206i 1.09454 + 2.60806i −2.64956 1.40707i 4.20817 + 1.73544i
11.3 −0.867527 + 1.11687i 0.925606 + 1.46399i −0.494795 1.93783i 0.895377 + 1.55084i −2.43807 0.236266i −2.08793 1.20546i 2.59355 + 1.12850i −1.28651 + 2.71015i −2.50885 0.345375i
11.4 −0.608741 1.27649i −1.71646 0.231865i −1.25887 + 1.55411i −1.74322 3.01934i 0.748906 + 2.33220i −1.80802 1.04386i 2.75013 + 0.660890i 2.89248 + 0.795973i −2.79300 + 4.06320i
11.5 0.186766 1.40183i 0.418594 1.68071i −1.93024 0.523628i 1.60936 + 2.78750i −2.27788 0.900696i −1.82223 1.05206i −1.09454 + 2.60806i −2.64956 1.40707i 4.20817 1.73544i
11.6 0.409484 + 1.35363i −1.12774 + 1.31461i −1.66465 + 1.10858i −0.565188 0.978934i −2.24129 0.988231i 3.71499 + 2.14485i −2.18226 1.79937i −0.456412 2.96508i 1.09368 1.16591i
11.7 0.533474 1.30973i 0.925606 + 1.46399i −1.43081 1.39742i −0.895377 1.55084i 2.41122 0.431300i 2.08793 + 1.20546i −2.59355 + 1.12850i −1.28651 + 2.71015i −2.50885 + 0.345375i
11.8 1.37702 0.322193i −1.12774 + 1.31461i 1.79238 0.887333i 0.565188 + 0.978934i −1.12936 + 2.17360i −3.71499 2.14485i 2.18226 1.79937i −0.456412 2.96508i 1.09368 + 1.16591i
59.1 −1.40985 0.111062i −1.71646 + 0.231865i 1.97533 + 0.313160i 1.74322 3.01934i 2.44570 0.136260i 1.80802 1.04386i −2.75013 0.660890i 2.89248 0.795973i −2.79300 + 4.06320i
59.2 −1.12063 0.862658i 0.418594 + 1.68071i 0.511643 + 1.93345i −1.60936 + 2.78750i 0.980785 2.24456i 1.82223 1.05206i 1.09454 2.60806i −2.64956 + 1.40707i 4.20817 1.73544i
59.3 −0.867527 1.11687i 0.925606 1.46399i −0.494795 + 1.93783i 0.895377 1.55084i −2.43807 + 0.236266i −2.08793 + 1.20546i 2.59355 1.12850i −1.28651 2.71015i −2.50885 + 0.345375i
59.4 −0.608741 + 1.27649i −1.71646 + 0.231865i −1.25887 1.55411i −1.74322 + 3.01934i 0.748906 2.33220i −1.80802 + 1.04386i 2.75013 0.660890i 2.89248 0.795973i −2.79300 4.06320i
59.5 0.186766 + 1.40183i 0.418594 + 1.68071i −1.93024 + 0.523628i 1.60936 2.78750i −2.27788 + 0.900696i −1.82223 + 1.05206i −1.09454 2.60806i −2.64956 + 1.40707i 4.20817 + 1.73544i
59.6 0.409484 1.35363i −1.12774 1.31461i −1.66465 1.10858i −0.565188 + 0.978934i −2.24129 + 0.988231i 3.71499 2.14485i −2.18226 + 1.79937i −0.456412 + 2.96508i 1.09368 + 1.16591i
59.7 0.533474 + 1.30973i 0.925606 1.46399i −1.43081 + 1.39742i −0.895377 + 1.55084i 2.41122 + 0.431300i 2.08793 1.20546i −2.59355 1.12850i −1.28651 2.71015i −2.50885 0.345375i
59.8 1.37702 + 0.322193i −1.12774 1.31461i 1.79238 + 0.887333i 0.565188 0.978934i −1.12936 2.17360i −3.71499 + 2.14485i 2.18226 + 1.79937i −0.456412 + 2.96508i 1.09368 1.16591i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.8
Significant digits:
Format:

Inner twists

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

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \)
$3$ \( ( 1 + 3 T + 6 T^{2} + 15 T^{3} + 30 T^{4} + 45 T^{5} + 54 T^{6} + 81 T^{7} + 81 T^{8} )^{2} \)
$5$ \( 1 - 13 T^{2} + 48 T^{4} + 103 T^{6} - 1099 T^{8} + 3648 T^{10} - 5222 T^{12} - 177298 T^{14} + 1667616 T^{16} - 4432450 T^{18} - 3263750 T^{20} + 57000000 T^{22} - 429296875 T^{24} + 1005859375 T^{26} + 11718750000 T^{28} - 79345703125 T^{30} + 152587890625 T^{32} \)
$7$ \( 1 + 23 T^{2} + 204 T^{4} + 1399 T^{6} + 12029 T^{8} + 45000 T^{10} - 343382 T^{12} - 4039462 T^{14} - 24407256 T^{16} - 197933638 T^{18} - 824460182 T^{20} + 5294205000 T^{22} + 69344791229 T^{24} + 395182873351 T^{26} + 2823622589004 T^{28} + 15599130675527 T^{30} + 33232930569601 T^{32} \)
$11$ \( ( 1 - 9 T + 41 T^{2} - 108 T^{3} + 276 T^{4} - 1188 T^{5} + 4961 T^{6} - 11979 T^{7} + 14641 T^{8} )^{2}( 1 + 3 T + 38 T^{2} + 87 T^{3} + 606 T^{4} + 957 T^{5} + 4598 T^{6} + 3993 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( 1 + 47 T^{2} + 1116 T^{4} + 17239 T^{6} + 168353 T^{8} + 463056 T^{10} - 26916578 T^{12} - 784598686 T^{14} - 12569281176 T^{16} - 132597177934 T^{18} - 768764384258 T^{20} + 2235082868304 T^{22} + 137330714072513 T^{24} + 2376542540984911 T^{26} + 26000662996688796 T^{28} + 185056690127866583 T^{30} + 665416609183179841 T^{32} \)
$17$ \( ( 1 - 101 T^{2} + 4882 T^{4} - 146891 T^{6} + 2998666 T^{8} - 42451499 T^{10} + 407749522 T^{12} - 2437894469 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 + T + 64 T^{2} + 49 T^{3} + 1726 T^{4} + 931 T^{5} + 23104 T^{6} + 6859 T^{7} + 130321 T^{8} )^{4} \)
$23$ \( 1 - 85 T^{2} + 3432 T^{4} - 80441 T^{6} + 1017209 T^{8} + 1764696 T^{10} - 386280290 T^{12} + 9323832998 T^{14} - 183562178736 T^{16} + 4932307655942 T^{18} - 108097062633890 T^{20} + 261238341174744 T^{22} + 79658639026700729 T^{24} - 3332389988537139209 T^{26} + 75210991050693741672 T^{28} - \)\(98\!\cdots\!65\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 97 T^{2} + 3780 T^{4} - 80561 T^{6} + 1632089 T^{8} - 54786000 T^{10} + 1038848902 T^{12} + 18974156858 T^{14} - 1329857348520 T^{16} + 15957265917578 T^{18} + 734758090255462 T^{20} - 32587990464306000 T^{22} + 816446667883105529 T^{24} - 33892595421897492761 T^{26} + \)\(13\!\cdots\!80\)\( T^{28} - \)\(28\!\cdots\!57\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 + 131 T^{2} + 7152 T^{4} + 312151 T^{6} + 16524593 T^{8} + 706250232 T^{10} + 22761450214 T^{12} + 818587141454 T^{14} + 29243993208000 T^{16} + 786662242937294 T^{18} + 21020677263083494 T^{20} + 626799680607103992 T^{22} + 14093677267060286513 T^{24} + \)\(25\!\cdots\!51\)\( T^{26} + \)\(56\!\cdots\!72\)\( T^{28} + \)\(99\!\cdots\!51\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 - 140 T^{2} + 8596 T^{4} - 317540 T^{6} + 10470934 T^{8} - 434712260 T^{10} + 16110287956 T^{12} - 359201697260 T^{14} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 + 18 T + 292 T^{2} + 3312 T^{3} + 34963 T^{4} + 303732 T^{5} + 2503888 T^{6} + 17908938 T^{7} + 122450608 T^{8} + 734266458 T^{9} + 4209035728 T^{10} + 20933513172 T^{11} + 98797081843 T^{12} + 383715737712 T^{13} + 1387030438372 T^{14} + 3505576929858 T^{15} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 - 4 T - 96 T^{2} + 868 T^{3} + 4061 T^{4} - 55182 T^{5} + 78652 T^{6} + 1341518 T^{7} - 8451684 T^{8} + 57685274 T^{9} + 145427548 T^{10} - 4387355274 T^{11} + 13883750861 T^{12} + 127603328524 T^{13} - 606850852704 T^{14} - 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 265 T^{2} + 38172 T^{4} - 3663305 T^{6} + 256619789 T^{8} - 13483362840 T^{10} + 543517455226 T^{12} - 17957421673750 T^{14} + 672442507889160 T^{16} - 39667944477313750 T^{18} + 2652191799434662906 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(61\!\cdots\!29\)\( T^{24} - \)\(19\!\cdots\!45\)\( T^{26} + \)\(44\!\cdots\!52\)\( T^{28} - \)\(68\!\cdots\!85\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 + 196 T^{2} + 21556 T^{4} + 1665484 T^{6} + 98315734 T^{8} + 4678344556 T^{10} + 170087208436 T^{12} + 4344214781284 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( ( 1 - 6 T + 178 T^{2} - 996 T^{3} + 15871 T^{4} - 89238 T^{5} + 1194346 T^{6} - 6437052 T^{7} + 79012984 T^{8} - 379786068 T^{9} + 4157518426 T^{10} - 18327611202 T^{11} + 192314636431 T^{12} - 712064601804 T^{13} + 7508134988098 T^{14} - 14931908908914 T^{15} + 146830437604321 T^{16} )^{2} \)
$61$ \( 1 + 299 T^{2} + 43416 T^{4} + 4465735 T^{6} + 393470093 T^{8} + 31461221184 T^{10} + 2332722797890 T^{12} + 164022774523334 T^{14} + 10603540284680400 T^{16} + 610328744001325814 T^{18} + 32298508956660075490 T^{20} + \)\(16\!\cdots\!24\)\( T^{22} + \)\(75\!\cdots\!33\)\( T^{24} + \)\(31\!\cdots\!35\)\( T^{26} + \)\(11\!\cdots\!36\)\( T^{28} + \)\(29\!\cdots\!59\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 + 8 T - 138 T^{2} - 1052 T^{3} + 11279 T^{4} + 57198 T^{5} - 930218 T^{6} - 1298482 T^{7} + 71382744 T^{8} - 86998294 T^{9} - 4175748602 T^{10} + 17203042074 T^{11} + 227284493759 T^{12} - 1420331612564 T^{13} - 12483256739322 T^{14} + 48485692842584 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 + 400 T^{2} + 73324 T^{4} + 8374048 T^{6} + 685441990 T^{8} + 42213575968 T^{10} + 1863286097644 T^{12} + 51240113568400 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 + T + 214 T^{2} - 5 T^{3} + 20758 T^{4} - 365 T^{5} + 1140406 T^{6} + 389017 T^{7} + 28398241 T^{8} )^{4} \)
$79$ \( 1 + 383 T^{2} + 74724 T^{4} + 9593503 T^{6} + 916548293 T^{8} + 73505234952 T^{10} + 5745220732498 T^{12} + 475438917658202 T^{14} + 38877973133064792 T^{16} + 2967214285104838682 T^{18} + \)\(22\!\cdots\!38\)\( T^{20} + \)\(17\!\cdots\!92\)\( T^{22} + \)\(13\!\cdots\!73\)\( T^{24} + \)\(90\!\cdots\!03\)\( T^{26} + \)\(44\!\cdots\!84\)\( T^{28} + \)\(14\!\cdots\!23\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( ( 1 - 27 T + 544 T^{2} - 8127 T^{3} + 107593 T^{4} - 1304076 T^{5} + 14403022 T^{6} - 148987620 T^{7} + 1398163588 T^{8} - 12365972460 T^{9} + 99222418558 T^{10} - 745653703812 T^{11} + 5106183131353 T^{12} - 32012583305661 T^{13} + 177855563112736 T^{14} - 732673376719929 T^{15} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 - 440 T^{2} + 100348 T^{4} - 14946776 T^{6} + 1566592486 T^{8} - 118393412696 T^{10} + 6296058399868 T^{12} - 218671768022840 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 4 T - 198 T^{2} - 320 T^{3} + 21017 T^{4} + 106308 T^{5} - 1078406 T^{6} - 6931984 T^{7} + 64836612 T^{8} - 672402448 T^{9} - 10146722054 T^{10} + 97024441284 T^{11} + 1860619898777 T^{12} - 2747948882240 T^{13} - 164928456975942 T^{14} - 323193137912452 T^{15} + 7837433594376961 T^{16} )^{2} \)
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