Properties

Label 144.3.m.c
Level $144$
Weight $3$
Character orbit 144.m
Analytic conductor $3.924$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( 1 - \beta_{1} ) q^{4} + ( -\beta_{1} + \beta_{3} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{5} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{7} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( 1 - \beta_{1} ) q^{4} + ( -\beta_{1} + \beta_{3} + \beta_{9} + \beta_{12} - \beta_{13} ) q^{5} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{7} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{8} + ( -3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{10} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{11} + ( -2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{13} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{14} + ( 3 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{8} - \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{16} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 4 \beta_{8} - \beta_{10} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{19} + ( -6 - 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 6 \beta_{6} - \beta_{7} + 4 \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{20} + ( 2 \beta_{2} - 4 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{11} - 2 \beta_{13} - \beta_{15} ) q^{22} + ( 8 + 4 \beta_{1} - 4 \beta_{2} - \beta_{4} - 3 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} - 4 \beta_{15} ) q^{23} + ( 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 5 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 4 \beta_{12} + 2 \beta_{13} + 2 \beta_{15} ) q^{25} + ( 6 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} - \beta_{12} - 4 \beta_{13} + 2 \beta_{14} ) q^{26} + ( -7 + \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} - 5 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} - 2 \beta_{12} + 4 \beta_{13} ) q^{28} + ( -6 - 3 \beta_{2} + 4 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 4 \beta_{9} + \beta_{11} + 4 \beta_{12} - 3 \beta_{14} + 6 \beta_{15} ) q^{29} + ( 1 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} + 6 \beta_{12} + 4 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{31} + ( -8 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} - \beta_{7} - 4 \beta_{8} + 6 \beta_{9} + \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{32} + ( 8 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 10 \beta_{9} - 2 \beta_{12} + 6 \beta_{13} + \beta_{15} ) q^{34} + ( -5 + 8 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{4} + 5 \beta_{5} - 7 \beta_{6} + \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{35} + ( -10 - 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} + 8 \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{37} + ( -12 + 8 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} + 6 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} ) q^{38} + ( 4 + 6 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 8 \beta_{9} - 6 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} + 4 \beta_{15} ) q^{40} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} - 3 \beta_{12} + 3 \beta_{13} + 5 \beta_{14} + 2 \beta_{15} ) q^{41} + ( 13 + 8 \beta_{2} + \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} - 2 \beta_{11} - 12 \beta_{12} - 5 \beta_{13} + 5 \beta_{14} ) q^{43} + ( -4 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} - 16 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{11} - 8 \beta_{12} + 4 \beta_{13} - 2 \beta_{14} + 6 \beta_{15} ) q^{44} + ( 6 + 8 \beta_{4} - 2 \beta_{6} + 4 \beta_{8} - 8 \beta_{13} - 8 \beta_{14} - 2 \beta_{15} ) q^{46} + ( 4 \beta_{1} + 4 \beta_{2} - \beta_{4} - 24 \beta_{6} + 4 \beta_{8} - 5 \beta_{9} + 4 \beta_{11} - 4 \beta_{12} ) q^{47} + ( 7 - 2 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - 8 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + 4 \beta_{14} + 4 \beta_{15} ) q^{49} + ( 12 + 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} + 16 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 5 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{50} + ( -5 - 5 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 2 \beta_{10} - \beta_{11} + 12 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{52} + ( 6 - 11 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 14 \beta_{6} + 2 \beta_{8} + 9 \beta_{9} + 7 \beta_{12} + \beta_{13} - 8 \beta_{14} + 2 \beta_{15} ) q^{53} + ( -16 + 4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{8} - 6 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} + 8 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{55} + ( 14 - \beta_{1} + 2 \beta_{2} - 6 \beta_{4} - 16 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 8 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{56} + ( 13 - 3 \beta_{1} + 3 \beta_{2} + 9 \beta_{4} - 13 \beta_{6} + 5 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} - 3 \beta_{10} + 5 \beta_{11} - 3 \beta_{12} + 5 \beta_{13} + 5 \beta_{14} + 3 \beta_{15} ) q^{58} + ( 12 + 4 \beta_{2} - 4 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - 4 \beta_{9} - 4 \beta_{10} - 8 \beta_{11} - 12 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} - 4 \beta_{15} ) q^{59} + ( -6 - 16 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 5 \beta_{7} + \beta_{8} + 12 \beta_{9} - 5 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 5 \beta_{13} - 11 \beta_{14} - 3 \beta_{15} ) q^{61} + ( 12 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} - 20 \beta_{6} - 3 \beta_{7} + 7 \beta_{9} + 5 \beta_{10} + 5 \beta_{11} + 13 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} ) q^{62} + ( -28 - 4 \beta_{1} + 10 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 6 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} + 2 \beta_{12} - 10 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{64} + ( 2 - 11 \beta_{1} - \beta_{2} + \beta_{3} + 10 \beta_{4} - 7 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} + 10 \beta_{10} + 11 \beta_{11} + 11 \beta_{12} - \beta_{13} + 7 \beta_{14} + 6 \beta_{15} ) q^{65} + ( 18 + 8 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 22 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 8 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{67} + ( 30 - 16 \beta_{2} + 4 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 8 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} - 6 \beta_{14} - 2 \beta_{15} ) q^{68} + ( -26 + 2 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} - 8 \beta_{6} + 8 \beta_{7} + 3 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} + 8 \beta_{11} + 10 \beta_{12} - 6 \beta_{14} + 3 \beta_{15} ) q^{70} + ( -32 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} ) q^{71} + ( 8 - 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 4 \beta_{8} - 10 \beta_{9} - 2 \beta_{11} - 26 \beta_{12} - 4 \beta_{13} + 10 \beta_{14} - 2 \beta_{15} ) q^{73} + ( -20 + 10 \beta_{1} - \beta_{2} - 4 \beta_{3} - 20 \beta_{6} + 2 \beta_{8} - 18 \beta_{9} - 6 \beta_{10} + 4 \beta_{11} - 3 \beta_{12} + 8 \beta_{13} + 6 \beta_{14} - 6 \beta_{15} ) q^{74} + ( 4 + 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} - 22 \beta_{6} + 2 \beta_{8} - 6 \beta_{11} + 12 \beta_{13} - 4 \beta_{14} - 6 \beta_{15} ) q^{76} + ( -10 + 2 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} + 18 \beta_{6} + 2 \beta_{7} + 12 \beta_{9} + 2 \beta_{10} + 10 \beta_{11} - 8 \beta_{12} - 6 \beta_{13} - 4 \beta_{15} ) q^{77} + ( -1 + 3 \beta_{1} + 12 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} - \beta_{7} - 11 \beta_{9} + 3 \beta_{11} - 2 \beta_{12} + 12 \beta_{13} + 3 \beta_{14} ) q^{79} + ( -36 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 36 \beta_{6} + 6 \beta_{7} + 8 \beta_{8} + 10 \beta_{9} - 4 \beta_{11} + 2 \beta_{14} + 8 \beta_{15} ) q^{80} + ( 2 + 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} + 20 \beta_{6} + 8 \beta_{7} + 11 \beta_{8} - 20 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 18 \beta_{12} + 14 \beta_{14} - 3 \beta_{15} ) q^{82} + ( 11 + 4 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{5} + 9 \beta_{6} - 3 \beta_{7} + 3 \beta_{10} + 2 \beta_{12} - 11 \beta_{13} + 7 \beta_{14} - 4 \beta_{15} ) q^{83} + ( 6 - 10 \beta_{1} + 24 \beta_{2} + 8 \beta_{3} - 20 \beta_{4} + 2 \beta_{5} + 14 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 14 \beta_{9} + 2 \beta_{10} + 18 \beta_{12} - 4 \beta_{13} - 14 \beta_{14} + 2 \beta_{15} ) q^{85} + ( -36 + 6 \beta_{1} - 2 \beta_{2} + 14 \beta_{4} - 4 \beta_{5} + 32 \beta_{6} + 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 8 \beta_{13} + 6 \beta_{14} - 2 \beta_{15} ) q^{86} + ( 30 + 12 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} - 4 \beta_{5} + 26 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + 12 \beta_{9} - 2 \beta_{10} + 8 \beta_{12} - 6 \beta_{13} + 6 \beta_{14} - 10 \beta_{15} ) q^{88} + ( -10 \beta_{1} - 2 \beta_{2} + 10 \beta_{3} - 14 \beta_{4} + 2 \beta_{5} + 10 \beta_{6} - 8 \beta_{7} - 2 \beta_{9} - 10 \beta_{11} - 6 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{89} + ( -19 + 16 \beta_{2} - 3 \beta_{3} + 13 \beta_{5} + 41 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} - 6 \beta_{11} - 32 \beta_{12} - 13 \beta_{13} - 7 \beta_{14} - 4 \beta_{15} ) q^{91} + ( -34 - 8 \beta_{1} - 16 \beta_{2} - 38 \beta_{6} - 6 \beta_{7} - 2 \beta_{10} - 8 \beta_{11} + 16 \beta_{12} + 2 \beta_{13} - 6 \beta_{14} - 16 \beta_{15} ) q^{92} + ( 10 - 2 \beta_{6} - 6 \beta_{8} + 8 \beta_{9} + 32 \beta_{12} + 8 \beta_{13} - 8 \beta_{14} + 4 \beta_{15} ) q^{94} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 13 \beta_{4} + 8 \beta_{5} + 40 \beta_{6} + 2 \beta_{7} - 8 \beta_{8} - 5 \beta_{9} - 2 \beta_{11} + 16 \beta_{12} + 8 \beta_{13} - 6 \beta_{14} - 4 \beta_{15} ) q^{95} + ( -10 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 8 \beta_{6} + 4 \beta_{8} + 4 \beta_{9} + 8 \beta_{10} + 10 \beta_{11} + 18 \beta_{12} + 2 \beta_{13} + 14 \beta_{14} ) q^{97} + ( 40 - 6 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 40 \beta_{6} + 6 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 14 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 10 \beta_{14} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 12q^{4} + 12q^{8} + O(q^{10}) \) \( 16q + 12q^{4} + 12q^{8} - 56q^{10} - 32q^{11} + 44q^{14} + 32q^{16} - 32q^{19} - 80q^{20} + 32q^{22} + 128q^{23} + 100q^{26} - 120q^{28} - 32q^{29} - 160q^{32} + 96q^{34} - 96q^{35} - 96q^{37} - 168q^{38} + 48q^{40} + 160q^{43} - 88q^{44} + 136q^{46} + 112q^{49} + 236q^{50} - 48q^{52} + 160q^{53} - 256q^{55} + 224q^{56} + 144q^{58} + 128q^{59} - 32q^{61} + 276q^{62} - 408q^{64} + 32q^{65} + 320q^{67} + 448q^{68} - 384q^{70} - 512q^{71} - 348q^{74} + 72q^{76} - 224q^{77} - 552q^{80} - 40q^{82} + 160q^{83} + 160q^{85} - 528q^{86} + 480q^{88} - 480q^{91} - 496q^{92} + 312q^{94} + 440q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} - 6 \nu^{12} - 4 \nu^{11} + 10 \nu^{10} + 56 \nu^{9} + 88 \nu^{8} - 128 \nu^{7} - 496 \nu^{6} - 512 \nu^{5} + 1408 \nu^{4} + 3584 \nu^{3} + 2560 \nu^{2} - 4096 \nu - 20480 \)\()/4096\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 6 \nu^{13} - 4 \nu^{12} + 10 \nu^{11} + 56 \nu^{10} + 88 \nu^{9} - 128 \nu^{8} - 496 \nu^{7} - 512 \nu^{6} + 1408 \nu^{5} + 3584 \nu^{4} + 2560 \nu^{3} - 4096 \nu^{2} - 24576 \nu \)\()/16384\)
\(\beta_{3}\)\(=\)\((\)\( -19 \nu^{15} - 86 \nu^{14} - 134 \nu^{13} + 48 \nu^{12} + 618 \nu^{11} + 796 \nu^{10} - 2696 \nu^{9} - 12176 \nu^{8} - 19568 \nu^{7} - 5728 \nu^{6} + 41984 \nu^{5} + 70400 \nu^{4} - 50688 \nu^{3} - 437248 \nu^{2} - 684032 \nu - 393216 \)\()/40960\)
\(\beta_{4}\)\(=\)\((\)\(81 \nu^{15} + 268 \nu^{14} + 218 \nu^{13} - 588 \nu^{12} - 2310 \nu^{11} - 1616 \nu^{10} + 9208 \nu^{9} + 30752 \nu^{8} + 38416 \nu^{7} - 22336 \nu^{6} - 142976 \nu^{5} - 146432 \nu^{4} + 195072 \nu^{3} + 976896 \nu^{2} + 966656 \nu - 180224\)\()/122880\)
\(\beta_{5}\)\(=\)\((\)\(-172 \nu^{15} - 739 \nu^{14} - 628 \nu^{13} + 1602 \nu^{12} + 6524 \nu^{11} + 5874 \nu^{10} - 25888 \nu^{9} - 95144 \nu^{8} - 127072 \nu^{7} + 56272 \nu^{6} + 427584 \nu^{5} + 497536 \nu^{4} - 442368 \nu^{3} - 3216896 \nu^{2} - 4229120 \nu - 696320\)\()/184320\)
\(\beta_{6}\)\(=\)\((\)\(347 \nu^{15} + 626 \nu^{14} - 1234 \nu^{13} - 4536 \nu^{12} - 5530 \nu^{11} + 11868 \nu^{10} + 59096 \nu^{9} + 66544 \nu^{8} - 88528 \nu^{7} - 450272 \nu^{6} - 454272 \nu^{5} + 499456 \nu^{4} + 2271744 \nu^{3} + 3177472 \nu^{2} - 3719168 \nu - 10231808\)\()/368640\)
\(\beta_{7}\)\(=\)\((\)\(-697 \nu^{15} - 208 \nu^{14} + 5990 \nu^{13} + 11268 \nu^{12} - 1498 \nu^{11} - 56664 \nu^{10} - 128632 \nu^{9} + 41728 \nu^{8} + 695024 \nu^{7} + 1419520 \nu^{6} + 299904 \nu^{5} - 3471872 \nu^{4} - 6363648 \nu^{3} - 1679360 \nu^{2} + 25280512 \nu + 41648128\)\()/737280\)
\(\beta_{8}\)\(=\)\((\)\(188 \nu^{15} + 323 \nu^{14} - 484 \nu^{13} - 1890 \nu^{12} - 2188 \nu^{11} + 5550 \nu^{10} + 27248 \nu^{9} + 31144 \nu^{8} - 38368 \nu^{7} - 197456 \nu^{6} - 166848 \nu^{5} + 241792 \nu^{4} + 1115136 \nu^{3} + 1464832 \nu^{2} - 1869824 \nu - 4751360\)\()/184320\)
\(\beta_{9}\)\(=\)\((\)\(131 \nu^{15} + 88 \nu^{14} - 1122 \nu^{13} - 2268 \nu^{12} - 610 \nu^{11} + 9944 \nu^{10} + 27688 \nu^{9} + 1472 \nu^{8} - 117584 \nu^{7} - 278656 \nu^{6} - 125056 \nu^{5} + 588288 \nu^{4} + 1316352 \nu^{3} + 741376 \nu^{2} - 4317184 \nu - 7716864\)\()/122880\)
\(\beta_{10}\)\(=\)\((\)\(-1093 \nu^{15} - 2080 \nu^{14} + 4766 \nu^{13} + 15444 \nu^{12} + 14414 \nu^{11} - 49752 \nu^{10} - 214456 \nu^{9} - 212864 \nu^{8} + 348848 \nu^{7} + 1624576 \nu^{6} + 1527936 \nu^{5} - 2430464 \nu^{4} - 8612352 \nu^{3} - 12075008 \nu^{2} + 13115392 \nu + 38699008\)\()/737280\)
\(\beta_{11}\)\(=\)\((\)\(-187 \nu^{15} - 624 \nu^{14} - 446 \nu^{13} + 1356 \nu^{12} + 5042 \nu^{11} + 2872 \nu^{10} - 24072 \nu^{9} - 75648 \nu^{8} - 87984 \nu^{7} + 62976 \nu^{6} + 320896 \nu^{5} + 310784 \nu^{4} - 579072 \nu^{3} - 2496512 \nu^{2} - 2506752 \nu + 106496\)\()/122880\)
\(\beta_{12}\)\(=\)\((\)\(-1249 \nu^{15} - 2776 \nu^{14} + 2486 \nu^{13} + 14868 \nu^{12} + 23798 \nu^{11} - 25704 \nu^{10} - 204856 \nu^{9} - 312896 \nu^{8} + 87152 \nu^{7} + 1347712 \nu^{6} + 1843584 \nu^{5} - 842240 \nu^{4} - 7193088 \nu^{3} - 13058048 \nu^{2} + 5275648 \nu + 29753344\)\()/737280\)
\(\beta_{13}\)\(=\)\((\)\(275 \nu^{15} + 464 \nu^{14} - 1090 \nu^{13} - 3564 \nu^{12} - 3874 \nu^{11} + 10248 \nu^{10} + 46568 \nu^{9} + 44800 \nu^{8} - 87376 \nu^{7} - 365312 \nu^{6} - 329856 \nu^{5} + 497152 \nu^{4} + 1838592 \nu^{3} + 2265088 \nu^{2} - 3424256 \nu - 8830976\)\()/147456\)
\(\beta_{14}\)\(=\)\((\)\(1405 \nu^{15} + 76 \nu^{14} - 14126 \nu^{13} - 26172 \nu^{12} - 1358 \nu^{11} + 123216 \nu^{10} + 294424 \nu^{9} - 66400 \nu^{8} - 1471280 \nu^{7} - 3192640 \nu^{6} - 1079424 \nu^{5} + 6895616 \nu^{4} + 14565888 \nu^{3} + 5470208 \nu^{2} - 51503104 \nu - 84557824\)\()/737280\)
\(\beta_{15}\)\(=\)\((\)\(1037 \nu^{15} + 926 \nu^{14} - 6454 \nu^{13} - 15336 \nu^{12} - 8710 \nu^{11} + 60708 \nu^{10} + 189176 \nu^{9} + 89104 \nu^{8} - 606448 \nu^{7} - 1739552 \nu^{6} - 1020672 \nu^{5} + 3079936 \nu^{4} + 8446464 \nu^{3} + 7078912 \nu^{2} - 22089728 \nu - 41566208\)\()/368640\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{13} - \beta_{9} - 2 \beta_{6} - \beta_{4} - 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{12} + 2 \beta_{11} + \beta_{8} + \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + 2 \beta_{9} - \beta_{8} - 12 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 6 \beta_{2} + 2 \beta_{1} + 2\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{14} + 6 \beta_{11} + 2 \beta_{10} - \beta_{9} + 8 \beta_{8} + 2 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 14 \beta_{2} + 6 \beta_{1} - 24\)\()/2\)
\(\nu^{6}\)\(=\)\(-3 \beta_{15} + 6 \beta_{14} + 2 \beta_{13} + 5 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} + 7 \beta_{6} + \beta_{5} + 12 \beta_{4} - 9 \beta_{3} + 19 \beta_{2} + 6 \beta_{1} - 23\)
\(\nu^{7}\)\(=\)\(6 \beta_{15} - 3 \beta_{14} - 17 \beta_{13} - 13 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} + 8 \beta_{8} + 8 \beta_{7} - 23 \beta_{6} + 4 \beta_{4} + 16 \beta_{3} + 19 \beta_{2} - 45\)
\(\nu^{8}\)\(=\)\(11 \beta_{15} - 2 \beta_{14} - 18 \beta_{13} + 22 \beta_{12} - 38 \beta_{11} + 16 \beta_{10} + 16 \beta_{9} - 27 \beta_{8} - 16 \beta_{7} + 18 \beta_{6} - 4 \beta_{5} - 26 \beta_{4} - 2 \beta_{3} + 8 \beta_{2} + 6 \beta_{1} - 92\)
\(\nu^{9}\)\(=\)\(-28 \beta_{15} - 4 \beta_{14} - 26 \beta_{13} - 134 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 7 \beta_{9} - 36 \beta_{8} + 6 \beta_{7} + 28 \beta_{6} + 48 \beta_{5} - 71 \beta_{4} - 4 \beta_{3} + 100 \beta_{2} - 22 \beta_{1} + 10\)
\(\nu^{10}\)\(=\)\(-12 \beta_{15} - 36 \beta_{14} + 4 \beta_{13} + 118 \beta_{12} - 100 \beta_{11} - 52 \beta_{10} - 84 \beta_{9} - 118 \beta_{8} - 44 \beta_{7} + 442 \beta_{6} + 110 \beta_{5} - 88 \beta_{4} + 22 \beta_{3} - 186 \beta_{2} - 24 \beta_{1} - 374\)
\(\nu^{11}\)\(=\)\(-80 \beta_{15} + 190 \beta_{14} - 306 \beta_{13} - 234 \beta_{12} - 184 \beta_{11} - 88 \beta_{10} - 76 \beta_{9} + 28 \beta_{8} - 132 \beta_{7} - 402 \beta_{6} + 8 \beta_{5} - 38 \beta_{4} + 56 \beta_{3} - 142 \beta_{2} - 148 \beta_{1} + 686\)
\(\nu^{12}\)\(=\)\(122 \beta_{15} - 428 \beta_{14} + 348 \beta_{13} + 304 \beta_{12} - 740 \beta_{11} + 48 \beta_{10} + 588 \beta_{9} - 350 \beta_{8} - 64 \beta_{7} - 40 \beta_{6} + 244 \beta_{5} - 904 \beta_{4} + 224 \beta_{3} - 1340 \beta_{2} - 708 \beta_{1} - 20\)
\(\nu^{13}\)\(=\)\(16 \beta_{15} - 556 \beta_{14} - 272 \beta_{13} - 1536 \beta_{12} + 772 \beta_{11} - 20 \beta_{10} - 526 \beta_{9} - 96 \beta_{8} - 468 \beta_{7} + 972 \beta_{6} + 648 \beta_{5} - 70 \beta_{4} - 192 \beta_{3} - 1180 \beta_{2} + 260 \beta_{1} + 3824\)
\(\nu^{14}\)\(=\)\(-1700 \beta_{15} + 648 \beta_{14} + 1528 \beta_{13} + 3020 \beta_{12} - 1424 \beta_{11} - 840 \beta_{10} - 776 \beta_{9} + 512 \beta_{8} - 456 \beta_{7} + 4164 \beta_{6} + 220 \beta_{5} - 1440 \beta_{4} + 612 \beta_{3} - 3372 \beta_{2} - 280 \beta_{1} + 6588\)
\(\nu^{15}\)\(=\)\(1992 \beta_{15} + 460 \beta_{14} + 2596 \beta_{13} - 780 \beta_{12} + 1768 \beta_{11} + 392 \beta_{10} - 1476 \beta_{9} + 1632 \beta_{8} - 832 \beta_{7} - 14980 \beta_{6} - 1248 \beta_{5} + 5344 \beta_{4} - 672 \beta_{3} - 2412 \beta_{2} - 3040 \beta_{1} + 4276\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.96679 + 0.362960i
−1.87459 0.697079i
−1.25564 + 1.55672i
−0.455024 + 1.94755i
0.125358 1.99607i
1.78012 + 0.911682i
1.80398 0.863518i
1.84258 + 0.777752i
−1.96679 0.362960i
−1.87459 + 0.697079i
−1.25564 1.55672i
−0.455024 1.94755i
0.125358 + 1.99607i
1.78012 0.911682i
1.80398 + 0.863518i
1.84258 0.777752i
−1.96679 0.362960i 0 3.73652 + 1.42773i −1.69930 + 1.69930i 0 −5.74280 −6.83074 4.16426i 0 3.95895 2.72539i
19.2 −1.87459 + 0.697079i 0 3.02816 2.61347i 5.24354 5.24354i 0 −5.32796 −3.85476 + 7.01005i 0 −6.17431 + 13.4846i
19.3 −1.25564 1.55672i 0 −0.846753 + 3.90935i −0.909023 + 0.909023i 0 −0.654713 7.14897 3.59057i 0 2.55650 + 0.273691i
19.4 −0.455024 1.94755i 0 −3.58591 + 1.77236i 3.40572 3.40572i 0 12.1303 5.08344 + 6.17727i 0 −8.18251 5.08314i
19.5 0.125358 + 1.99607i 0 −3.96857 + 0.500444i −3.32679 + 3.32679i 0 −4.04088 −1.49641 7.85880i 0 −7.05755 6.22347i
19.6 1.78012 0.911682i 0 2.33767 3.24581i −1.00772 + 1.00772i 0 10.0236 1.20220 7.90915i 0 −0.875146 + 2.71259i
19.7 1.80398 + 0.863518i 0 2.50867 + 3.11554i −6.49473 + 6.49473i 0 3.94273 1.83527 + 7.78664i 0 −17.3247 + 6.10803i
19.8 1.84258 0.777752i 0 2.79020 2.86614i 4.78830 4.78830i 0 −10.3302 2.91202 7.45118i 0 5.09872 12.5469i
91.1 −1.96679 + 0.362960i 0 3.73652 1.42773i −1.69930 1.69930i 0 −5.74280 −6.83074 + 4.16426i 0 3.95895 + 2.72539i
91.2 −1.87459 0.697079i 0 3.02816 + 2.61347i 5.24354 + 5.24354i 0 −5.32796 −3.85476 7.01005i 0 −6.17431 13.4846i
91.3 −1.25564 + 1.55672i 0 −0.846753 3.90935i −0.909023 0.909023i 0 −0.654713 7.14897 + 3.59057i 0 2.55650 0.273691i
91.4 −0.455024 + 1.94755i 0 −3.58591 1.77236i 3.40572 + 3.40572i 0 12.1303 5.08344 6.17727i 0 −8.18251 + 5.08314i
91.5 0.125358 1.99607i 0 −3.96857 0.500444i −3.32679 3.32679i 0 −4.04088 −1.49641 + 7.85880i 0 −7.05755 + 6.22347i
91.6 1.78012 + 0.911682i 0 2.33767 + 3.24581i −1.00772 1.00772i 0 10.0236 1.20220 + 7.90915i 0 −0.875146 2.71259i
91.7 1.80398 0.863518i 0 2.50867 3.11554i −6.49473 6.49473i 0 3.94273 1.83527 7.78664i 0 −17.3247 6.10803i
91.8 1.84258 + 0.777752i 0 2.79020 + 2.86614i 4.78830 + 4.78830i 0 −10.3302 2.91202 + 7.45118i 0 5.09872 + 12.5469i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.m.c 16
3.b odd 2 1 48.3.l.a 16
4.b odd 2 1 576.3.m.c 16
8.b even 2 1 1152.3.m.f 16
8.d odd 2 1 1152.3.m.c 16
12.b even 2 1 192.3.l.a 16
16.e even 4 1 576.3.m.c 16
16.e even 4 1 1152.3.m.c 16
16.f odd 4 1 inner 144.3.m.c 16
16.f odd 4 1 1152.3.m.f 16
24.f even 2 1 384.3.l.b 16
24.h odd 2 1 384.3.l.a 16
48.i odd 4 1 192.3.l.a 16
48.i odd 4 1 384.3.l.b 16
48.k even 4 1 48.3.l.a 16
48.k even 4 1 384.3.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a 16 3.b odd 2 1
48.3.l.a 16 48.k even 4 1
144.3.m.c 16 1.a even 1 1 trivial
144.3.m.c 16 16.f odd 4 1 inner
192.3.l.a 16 12.b even 2 1
192.3.l.a 16 48.i odd 4 1
384.3.l.a 16 24.h odd 2 1
384.3.l.a 16 48.k even 4 1
384.3.l.b 16 24.f even 2 1
384.3.l.b 16 48.i odd 4 1
576.3.m.c 16 4.b odd 2 1
576.3.m.c 16 16.e even 4 1
1152.3.m.c 16 8.d odd 2 1
1152.3.m.c 16 16.e even 4 1
1152.3.m.f 16 8.b even 2 1
1152.3.m.f 16 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 65536 - 24576 T^{2} - 4096 T^{3} + 2560 T^{4} + 3584 T^{5} + 1408 T^{6} - 512 T^{7} - 496 T^{8} - 128 T^{9} + 88 T^{10} + 56 T^{11} + 10 T^{12} - 4 T^{13} - 6 T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 2117472256 + 5171462144 T + 6315081728 T^{2} + 4258177024 T^{3} + 1663893504 T^{4} + 283883520 T^{5} + 12517376 T^{6} + 4026368 T^{7} + 6221952 T^{8} + 518400 T^{9} + 512 T^{10} - 8064 T^{11} + 6656 T^{12} - 32 T^{13} + T^{16} \)
$7$ \( ( -400880 - 720640 T - 136576 T^{2} + 53248 T^{3} + 13704 T^{4} - 448 T^{5} - 224 T^{6} + T^{8} )^{2} \)
$11$ \( 25620118503424 - 29481536323584 T + 16962470019072 T^{2} + 22391525736448 T^{3} + 10536909537280 T^{4} + 2206640635904 T^{5} + 269438418944 T^{6} + 21404090368 T^{7} + 3292530688 T^{8} + 584835072 T^{9} + 68780032 T^{10} + 4242432 T^{11} + 151552 T^{12} + 4608 T^{13} + 512 T^{14} + 32 T^{15} + T^{16} \)
$13$ \( 27957043852960000 - 3777806164172800 T + 255245502513152 T^{2} + 75155778920448 T^{3} + 18426489989376 T^{4} - 1347500722176 T^{5} + 114873139200 T^{6} + 37570686976 T^{7} + 5634806368 T^{8} + 8847872 T^{9} + 5120000 T^{10} + 1717760 T^{11} + 271248 T^{12} + 3200 T^{13} + T^{16} \)
$17$ \( ( 816881920 - 9390080 T - 53986304 T^{2} - 1044480 T^{3} + 508448 T^{4} + 2944 T^{5} - 1344 T^{6} + T^{8} )^{2} \)
$19$ \( 10598900522979229696 + 1221045122401566720 T + 70335181828915200 T^{2} - 89104253114777600 T^{3} + 23939487244337152 T^{4} - 1508224010715136 T^{5} + 41927575470080 T^{6} + 1219751600128 T^{7} + 149480318464 T^{8} - 9271351296 T^{9} + 294903808 T^{10} + 15014400 T^{11} + 559552 T^{12} - 14208 T^{13} + 512 T^{14} + 32 T^{15} + T^{16} \)
$23$ \( ( -35037900800 - 425492480 T + 777799680 T^{2} - 42348544 T^{3} - 1089152 T^{4} + 109056 T^{5} - 736 T^{6} - 64 T^{7} + T^{8} )^{2} \)
$29$ \( 56446323002698240000 + 10320873959271219200 T + 943555165480583168 T^{2} - 1809064899469205504 T^{3} + 1336868519252525056 T^{4} - 34131461984319488 T^{5} + 401836469485568 T^{6} + 22033099003904 T^{7} + 4665816875136 T^{8} - 92505709824 T^{9} + 1050931712 T^{10} + 64121984 T^{11} + 3957248 T^{12} - 45280 T^{13} + 512 T^{14} + 32 T^{15} + T^{16} \)
$31$ \( \)\(41\!\cdots\!00\)\( + \)\(88\!\cdots\!60\)\( T^{2} + 7691970571770562816 T^{4} + 26353243415873536 T^{6} + 46734989650528 T^{8} + 46823575040 T^{10} + 26696080 T^{12} + 8064 T^{14} + T^{16} \)
$37$ \( \)\(38\!\cdots\!04\)\( - \)\(73\!\cdots\!12\)\( T + 70869372871016480768 T^{2} + 2884347249537718272 T^{3} + 703119961976168704 T^{4} - 97908324877949952 T^{5} + 6893443090702336 T^{6} + 6367638655488 T^{7} - 968313092000 T^{8} - 28675020544 T^{9} + 14234605568 T^{10} + 224961664 T^{11} + 1620496 T^{12} + 14528 T^{13} + 4608 T^{14} + 96 T^{15} + T^{16} \)
$41$ \( \)\(94\!\cdots\!00\)\( + \)\(70\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!56\)\( T^{4} + 365640212557922304 T^{6} + 356212041131520 T^{8} + 207527260160 T^{10} + 70874688 T^{12} + 13056 T^{14} + T^{16} \)
$43$ \( \)\(92\!\cdots\!00\)\( - \)\(28\!\cdots\!80\)\( T + \)\(44\!\cdots\!68\)\( T^{2} - \)\(32\!\cdots\!48\)\( T^{3} + \)\(14\!\cdots\!24\)\( T^{4} - 5239225685693136896 T^{5} + 245950489383796736 T^{6} - 12945845326176256 T^{7} + 537795837003264 T^{8} - 15508493912064 T^{9} + 306484011008 T^{10} - 4149187072 T^{11} + 46122944 T^{12} - 682624 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16} \)
$47$ \( \)\(11\!\cdots\!00\)\( + \)\(37\!\cdots\!36\)\( T^{2} + 4369756732867477504 T^{4} + 22735051785764864 T^{6} + 57150256209920 T^{8} + 69921902592 T^{10} + 41678592 T^{12} + 11200 T^{14} + T^{16} \)
$53$ \( \)\(96\!\cdots\!00\)\( - \)\(20\!\cdots\!60\)\( T + \)\(21\!\cdots\!52\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + 91449263140998676480 T^{4} - 5591746172921030656 T^{5} + 1685959386162102272 T^{6} - 18190042770372608 T^{7} + 100041121243264 T^{8} - 1029094838016 T^{9} + 299678376448 T^{10} - 3156153984 T^{11} + 16960000 T^{12} - 153504 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16} \)
$59$ \( \)\(23\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T + \)\(69\!\cdots\!32\)\( T^{2} - \)\(16\!\cdots\!76\)\( T^{3} + 97922953659373584384 T^{4} - 714303909442617344 T^{5} + 215142153171501056 T^{6} - 32123905497890816 T^{7} + 2068957458595840 T^{8} - 72789256044544 T^{9} + 1586277384192 T^{10} - 20765343744 T^{11} + 158699520 T^{12} - 675840 T^{13} + 8192 T^{14} - 128 T^{15} + T^{16} \)
$61$ \( \)\(12\!\cdots\!00\)\( - \)\(90\!\cdots\!60\)\( T + \)\(33\!\cdots\!72\)\( T^{2} + \)\(22\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!44\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{5} + 735866880468475904 T^{6} + 49175921370646016 T^{7} + 3370685859892320 T^{8} - 12153910400256 T^{9} + 42705459200 T^{10} + 2795215232 T^{11} + 115399952 T^{12} - 157120 T^{13} + 512 T^{14} + 32 T^{15} + T^{16} \)
$67$ \( \)\(21\!\cdots\!96\)\( + \)\(11\!\cdots\!04\)\( T + \)\(30\!\cdots\!48\)\( T^{2} - \)\(28\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!96\)\( T^{4} + 621625262253015040 T^{5} + 579595725034225664 T^{6} - 50262831949414400 T^{7} + 2061397582544896 T^{8} - 36863866568704 T^{9} + 475717435392 T^{10} - 9110568960 T^{11} + 255286272 T^{12} - 4611072 T^{13} + 51200 T^{14} - 320 T^{15} + T^{16} \)
$71$ \( ( 290924400640 + 109021003776 T + 16299499520 T^{2} + 1284915200 T^{3} + 59283584 T^{4} + 1659392 T^{5} + 27776 T^{6} + 256 T^{7} + T^{8} )^{2} \)
$73$ \( \)\(98\!\cdots\!16\)\( + \)\(13\!\cdots\!52\)\( T^{2} + \)\(45\!\cdots\!44\)\( T^{4} + 53883999480140791808 T^{6} + 25520342188187648 T^{8} + 5899646435328 T^{10} + 709382400 T^{12} + 42496 T^{14} + T^{16} \)
$79$ \( \)\(18\!\cdots\!00\)\( + \)\(25\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{4} + 1719443636116761600 T^{6} + 4201068991559776 T^{8} + 2270484756224 T^{10} + 458363920 T^{12} + 36928 T^{14} + T^{16} \)
$83$ \( \)\(14\!\cdots\!76\)\( - \)\(42\!\cdots\!84\)\( T + \)\(63\!\cdots\!28\)\( T^{2} + \)\(13\!\cdots\!24\)\( T^{3} + \)\(12\!\cdots\!56\)\( T^{4} + \)\(39\!\cdots\!64\)\( T^{5} + 6381120662228434944 T^{6} - 5903701162754048 T^{7} + 105517061441536 T^{8} + 17298459828224 T^{9} + 474865041408 T^{10} - 3104705536 T^{11} + 3373056 T^{12} + 206336 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16} \)
$89$ \( \)\(52\!\cdots\!00\)\( + \)\(68\!\cdots\!60\)\( T^{2} + \)\(23\!\cdots\!56\)\( T^{4} + 30661655762820161536 T^{6} + 18282328190707200 T^{8} + 5304960677376 T^{10} + 743027648 T^{12} + 45728 T^{14} + T^{16} \)
$97$ \( ( 409778579046400 + 2337541980160 T - 1019025981440 T^{2} - 1982349312 T^{3} + 383621120 T^{4} + 116224 T^{5} - 37056 T^{6} + T^{8} )^{2} \)
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