Properties

Label 77.2.f.b
Level 77
Weight 2
Character orbit 77.f
Analytic conductor 0.615
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} - 200 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} - 3234633146529285 \nu^{12} + 4494612950061958 \nu^{11} - 20835947263746904 \nu^{10} + 25956864481307657 \nu^{9} - 65098048812157320 \nu^{8} + 30869277708222543 \nu^{7} - 160665000409949210 \nu^{6} + 100524107363786610 \nu^{5} - 560336279122558890 \nu^{4} + 546963632352958070 \nu^{3} - 434474896057227745 \nu^{2} + 120133073008741205 \nu + 88528161377575350\)\()/ 454580475630153760 \)
\(\beta_{2}\)\(=\)\((\)\(-686277383720070 \nu^{15} - 1360333162376645 \nu^{14} + 444742620384542 \nu^{13} - 24553779304461089 \nu^{12} + 45808084826141647 \nu^{11} - 179218800902324756 \nu^{10} + 292393929070812298 \nu^{9} - 592228758909440833 \nu^{8} + 274072637813433960 \nu^{7} - 1226946693965489055 \nu^{6} + 2313198962618101340 \nu^{5} - 5164313069616781590 \nu^{4} + 5505326380247646890 \nu^{3} - 4265354995237286060 \nu^{2} + 1164454349991851775 \nu - 1569079812936964435\)\()/ 454580475630153760 \)
\(\beta_{3}\)\(=\)\((\)\(-874936745197272 \nu^{15} - 514706296938285 \nu^{14} - 5527222230184928 \nu^{13} - 8597109141630243 \nu^{12} - 9687442744709873 \nu^{11} - 69609203132183412 \nu^{10} + 41469078152400808 \nu^{9} - 248365491806816213 \nu^{8} - 292096370375405502 \nu^{7} - 583922916320059255 \nu^{6} + 395306407846606840 \nu^{5} - 2070346335621533450 \nu^{4} + 1021707328353270230 \nu^{3} - 1173049633586345730 \nu^{2} + 368306587037923705 \nu - 928639412535219835\)\()/ 454580475630153760 \)
\(\beta_{4}\)\(=\)\((\)\(-1565624412275039 \nu^{15} + 4832412997019746 \nu^{14} - 21672367495495401 \nu^{13} + 49577176484430299 \nu^{12} - 129795369286809034 \nu^{11} + 214474805906670840 \nu^{10} - 347621794857624183 \nu^{9} + 312652728515985000 \nu^{8} - 887685879699924177 \nu^{7} + 1770758763681867190 \nu^{6} - 3159142418183242750 \nu^{5} + 2957115228664641670 \nu^{4} - 1979053855141622330 \nu^{3} + 366777001484809055 \nu^{2} - 739274040704445675 \nu + 95214593589332550\)\()/ 454580475630153760 \)
\(\beta_{5}\)\(=\)\((\)\(1676337845170790 \nu^{15} - 5194753922189191 \nu^{14} + 23996414540642882 \nu^{13} - 55793399026072099 \nu^{12} + 149577475159066613 \nu^{11} - 254981761232364844 \nu^{10} + 433123005758956374 \nu^{9} - 436889234085257675 \nu^{8} + 1101433166283370416 \nu^{7} - 2030901870569891045 \nu^{6} + 3769863137668677300 \nu^{5} - 4085868252644216690 \nu^{4} + 3585858111334770990 \nu^{3} - 1476295694818421740 \nu^{2} + 1479871527393288525 \nu - 189874590198959025\)\()/ 454580475630153760 \)
\(\beta_{6}\)\(=\)\((\)\(-1676337845170790 \nu^{15} + 5194753922189191 \nu^{14} - 23996414540642882 \nu^{13} + 55793399026072099 \nu^{12} - 149577475159066613 \nu^{11} + 254981761232364844 \nu^{10} - 433123005758956374 \nu^{9} + 436889234085257675 \nu^{8} - 1101433166283370416 \nu^{7} + 2030901870569891045 \nu^{6} - 3769863137668677300 \nu^{5} + 4085868252644216690 \nu^{4} - 3585858111334770990 \nu^{3} + 1476295694818421740 \nu^{2} - 1025291051763134765 \nu + 189874590198959025\)\()/ 454580475630153760 \)
\(\beta_{7}\)\(=\)\((\)\(-3541126455103014 \nu^{15} + 10589657246428067 \nu^{14} - 49791707208255586 \nu^{13} + 113470349300804375 \nu^{12} - 307771508285388489 \nu^{11} + 517957948939998988 \nu^{10} - 888411928763985334 \nu^{9} + 893532845981546087 \nu^{8} - 2331418980078086280 \nu^{7} + 4191692862454263993 \nu^{6} - 7720969982054884100 \nu^{5} + 8245114954100718810 \nu^{4} - 7111420221063134790 \nu^{3} + 3822505603323246020 \nu^{2} - 2913263414629337545 \nu + 828358364029344005\)\()/ 454580475630153760 \)
\(\beta_{8}\)\(=\)\((\)\(-3808583743573302 \nu^{15} + 9860126818444867 \nu^{14} - 48487759413006482 \nu^{13} + 100202312298850263 \nu^{12} - 277961025462873673 \nu^{11} + 422449273531319756 \nu^{10} - 718628211268788150 \nu^{9} + 585481222317834807 \nu^{8} - 2124840867370928280 \nu^{7} + 3587400018998705673 \nu^{6} - 6360567528847132580 \nu^{5} + 5600600192035351850 \nu^{4} - 4088764696945967030 \nu^{3} + 1543886107663682020 \nu^{2} - 2299231619016502345 \nu + 22442708010214725\)\()/ 454580475630153760 \)
\(\beta_{9}\)\(=\)\((\)\(4147870816217860 \nu^{15} - 9098179878471885 \nu^{14} + 48642002600512068 \nu^{13} - 88958591156351927 \nu^{12} + 262233539631762351 \nu^{11} - 349871456206210068 \nu^{10} + 623455379026927044 \nu^{9} - 381247122554708893 \nu^{8} + 2150910817199786310 \nu^{7} - 3112048315103081495 \nu^{6} + 5535211606668989440 \nu^{5} - 3712550967753264370 \nu^{4} + 2767110524781279550 \nu^{3} - 1335895153189756830 \nu^{2} + 2975718288695695725 \nu - 29045982804284875\)\()/ 454580475630153760 \)
\(\beta_{10}\)\(=\)\((\)\(7327526319488073 \nu^{15} - 21838143406687493 \nu^{14} + 102438964384476967 \nu^{13} - 232311097915978782 \nu^{12} + 627185674080861763 \nu^{11} - 1047203823403574964 \nu^{10} + 1775572940212630785 \nu^{9} - 1735699601854553351 \nu^{8} + 4630478387715251365 \nu^{7} - 8426965416523262079 \nu^{6} + 15544790585628961210 \nu^{5} - 16343113922700919960 \nu^{4} + 12987506946195918920 \nu^{3} - 5718121221686276935 \nu^{2} + 4454224626365266160 \nu - 845040850217512955\)\()/ 454580475630153760 \)
\(\beta_{11}\)\(=\)\((\)\(-5481281688546819 \nu^{15} + 16493950936678582 \nu^{14} - 76461534295292341 \nu^{13} + 174584775915853027 \nu^{12} - 466393479528723526 \nu^{11} + 782325573976058616 \nu^{10} - 1307780222870847083 \nu^{9} + 1276995934654789444 \nu^{8} - 3395227103875767205 \nu^{7} + 6340299488823316290 \nu^{6} - 11480502622702818310 \nu^{5} + 12158198218393584390 \nu^{4} - 9161804501671007210 \nu^{3} + 3888216659798809835 \nu^{2} - 2965788298433688075 \nu + 864066394282347650\)\()/ 227290237815076880 \)
\(\beta_{12}\)\(=\)\((\)\(-11668286039317543 \nu^{15} + 32209452556198329 \nu^{14} - 155422622027191961 \nu^{13} + 335017774059801892 \nu^{12} - 918061503999498391 \nu^{11} + 1456783386223745044 \nu^{10} - 2469198487594261567 \nu^{9} + 2179375478325730279 \nu^{8} - 6793193349504836855 \nu^{7} + 11856118418058363915 \nu^{6} - 21771993760224740470 \nu^{5} + 21003963276182506660 \nu^{4} - 15299497235328621900 \nu^{3} + 5270338292583504165 \nu^{2} - 5226852994856502950 \nu + 59418569180970775\)\()/ 454580475630153760 \)
\(\beta_{13}\)\(=\)\((\)\(-16296167766296901 \nu^{15} + 46657645630438849 \nu^{14} - 220693813391828059 \nu^{13} + 490255203026234886 \nu^{12} - 1324081568271118439 \nu^{11} + 2171662124721910708 \nu^{10} - 3646287371621077981 \nu^{9} + 3462410165728884427 \nu^{8} - 9848220683260194529 \nu^{7} + 17797250626139187779 \nu^{6} - 31784433905471092610 \nu^{5} + 32902654497697219800 \nu^{4} - 24876887445936951080 \nu^{3} + 11522918100224582795 \nu^{2} - 8819030598088243120 \nu + 1667167131638041055\)\()/ 454580475630153760 \)
\(\beta_{14}\)\(=\)\((\)\(16576781827677814 \nu^{15} - 45453688208119129 \nu^{14} + 219284175557671186 \nu^{13} - 470575312000685073 \nu^{12} + 1289205279920317419 \nu^{11} - 2038073676823909444 \nu^{10} + 3445236405584454006 \nu^{9} - 3032211154151816061 \nu^{8} + 9585093068819660548 \nu^{7} - 16816185325336642299 \nu^{6} + 30431971572347844100 \nu^{5} - 29120858031490839590 \nu^{4} + 20842243045981534410 \nu^{3} - 8417551544734639360 \nu^{2} + 7973236770827366535 \nu - 1025213227739747135\)\()/ 454580475630153760 \)
\(\beta_{15}\)\(=\)\((\)\(-31742657223153564 \nu^{15} + 90930749048576029 \nu^{14} - 431347940936808556 \nu^{13} + 955931937354509231 \nu^{12} - 2590293343054165503 \nu^{11} + 4236554416589668948 \nu^{10} - 7143632052250437628 \nu^{9} + 6742540241983255677 \nu^{8} - 19234203982822940782 \nu^{7} + 34610408690995495111 \nu^{6} - 62595008084751532400 \nu^{5} + 64110835668533354210 \nu^{4} - 48322167102593217550 \nu^{3} + 21980961042064417270 \nu^{2} - 18197077383887239845 \nu + 3492178872565443195\)\()/ 454580475630153760 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{6} + \beta_{5}\)
\(\nu^{2}\)\(=\)\(\beta_{11} + 2 \beta_{10} - 2 \beta_{8} + 3 \beta_{7} + \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5 \beta_{6} - \beta_{5} - \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(-7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta_{1} - 3\)
\(\nu^{6}\)\(=\)\(7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{1}\)
\(\nu^{7}\)\(=\)\(42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28\)
\(\nu^{8}\)\(=\)\(-9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta_{1} - 98\)
\(\nu^{9}\)\(=\)\(-346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta_{1} - 94\)
\(\nu^{10}\)\(=\)\(-56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta_{1} + 1406\)
\(\nu^{11}\)\(=\)\(2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} - 3461 \beta_{1}\)
\(\nu^{12}\)\(=\)\(284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} - 8669\)
\(\nu^{13}\)\(=\)\(-8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta_{1} + 5394\)
\(\nu^{14}\)\(=\)\(-3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} - 50750 \beta_{2} - 13793 \beta_{1} + 31232\)
\(\nu^{15}\)\(=\)\(-54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} - 96554 \beta_{1} - 67213\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7} - \beta_{8} + \beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
1.60551 + 1.16647i
0.901622 + 0.655067i
0.183009 + 0.132964i
−1.38112 1.00344i
1.60551 1.16647i
0.901622 0.655067i
0.183009 0.132964i
−1.38112 + 1.00344i
0.751051 2.31150i
0.435488 1.34029i
−0.206962 + 0.636964i
−0.788594 + 2.42704i
0.751051 + 2.31150i
0.435488 + 1.34029i
−0.206962 0.636964i
−0.788594 2.42704i
−1.60551 1.16647i 0.861043 2.65002i 0.598967 + 1.84343i 0.0217822 0.0158257i −4.47357 + 3.25024i 0.309017 + 0.951057i −0.0378378 + 0.116453i −3.85415 2.80020i −0.0534317
15.2 −0.901622 0.655067i −0.883423 + 2.71890i −0.234224 0.720867i −2.79603 + 2.03143i 2.57757 1.87272i 0.309017 + 0.951057i −0.949813 + 2.92322i −4.18492 3.04052i 3.85168
15.3 −0.183009 0.132964i −0.0677147 + 0.208405i −0.602221 1.85345i 2.01892 1.46683i 0.0401026 0.0291363i 0.309017 + 0.951057i −0.276036 + 0.849550i 2.38820 + 1.73513i −0.564516
15.4 1.38112 + 1.00344i 0.708129 2.17940i 0.282562 + 0.869638i −3.28976 + 2.39015i 3.16491 2.29944i 0.309017 + 0.951057i 0.572703 1.76260i −1.82128 1.32323i −6.94194
36.1 −1.60551 + 1.16647i 0.861043 + 2.65002i 0.598967 1.84343i 0.0217822 + 0.0158257i −4.47357 3.25024i 0.309017 0.951057i −0.0378378 0.116453i −3.85415 + 2.80020i −0.0534317
36.2 −0.901622 + 0.655067i −0.883423 2.71890i −0.234224 + 0.720867i −2.79603 2.03143i 2.57757 + 1.87272i 0.309017 0.951057i −0.949813 2.92322i −4.18492 + 3.04052i 3.85168
36.3 −0.183009 + 0.132964i −0.0677147 0.208405i −0.602221 + 1.85345i 2.01892 + 1.46683i 0.0401026 + 0.0291363i 0.309017 0.951057i −0.276036 0.849550i 2.38820 1.73513i −0.564516
36.4 1.38112 1.00344i 0.708129 + 2.17940i 0.282562 0.869638i −3.28976 2.39015i 3.16491 + 2.29944i 0.309017 0.951057i 0.572703 + 1.76260i −1.82128 + 1.32323i −6.94194
64.1 −0.751051 + 2.31150i −1.16030 + 0.843005i −3.16091 2.29654i 0.388938 + 1.19703i −1.07716 3.31516i −0.809017 0.587785i 3.74989 2.72445i −0.291419 + 0.896896i −3.05904
64.2 −0.435488 + 1.34029i 1.75021 1.27160i 0.0112975 + 0.00820814i −0.565930 1.74175i 0.942126 + 2.89957i −0.809017 0.587785i −2.29616 + 1.66826i 0.519216 1.59798i 2.58091
64.3 0.206962 0.636964i −2.54013 + 1.84551i 1.25514 + 0.911915i 0.662464 + 2.03885i 0.649815 + 1.99992i −0.809017 0.587785i 1.92429 1.39808i 2.11929 6.52251i 1.43578
64.4 0.788594 2.42704i 0.332181 0.241344i −3.65062 2.65233i 1.05961 + 3.26115i −0.323795 0.996539i −0.809017 0.587785i −5.18703 + 3.76860i −0.874954 + 2.69283i 8.75055
71.1 −0.751051 2.31150i −1.16030 0.843005i −3.16091 + 2.29654i 0.388938 1.19703i −1.07716 + 3.31516i −0.809017 + 0.587785i 3.74989 + 2.72445i −0.291419 0.896896i −3.05904
71.2 −0.435488 1.34029i 1.75021 + 1.27160i 0.0112975 0.00820814i −0.565930 + 1.74175i 0.942126 2.89957i −0.809017 + 0.587785i −2.29616 1.66826i 0.519216 + 1.59798i 2.58091
71.3 0.206962 + 0.636964i −2.54013 1.84551i 1.25514 0.911915i 0.662464 2.03885i 0.649815 1.99992i −0.809017 + 0.587785i 1.92429 + 1.39808i 2.11929 + 6.52251i 1.43578
71.4 0.788594 + 2.42704i 0.332181 + 0.241344i −3.65062 + 2.65233i 1.05961 3.26115i −0.323795 + 0.996539i −0.809017 + 0.587785i −5.18703 3.76860i −0.874954 2.69283i 8.75055
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.f.b 16
3.b odd 2 1 693.2.m.i 16
7.b odd 2 1 539.2.f.e 16
7.c even 3 2 539.2.q.g 32
7.d odd 6 2 539.2.q.f 32
11.b odd 2 1 847.2.f.x 16
11.c even 5 1 inner 77.2.f.b 16
11.c even 5 1 847.2.a.p 8
11.c even 5 2 847.2.f.w 16
11.d odd 10 1 847.2.a.o 8
11.d odd 10 2 847.2.f.v 16
11.d odd 10 1 847.2.f.x 16
33.f even 10 1 7623.2.a.cw 8
33.h odd 10 1 693.2.m.i 16
33.h odd 10 1 7623.2.a.ct 8
77.j odd 10 1 539.2.f.e 16
77.j odd 10 1 5929.2.a.bt 8
77.l even 10 1 5929.2.a.bs 8
77.m even 15 2 539.2.q.g 32
77.p odd 30 2 539.2.q.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 1.a even 1 1 trivial
77.2.f.b 16 11.c even 5 1 inner
539.2.f.e 16 7.b odd 2 1
539.2.f.e 16 77.j odd 10 1
539.2.q.f 32 7.d odd 6 2
539.2.q.f 32 77.p odd 30 2
539.2.q.g 32 7.c even 3 2
539.2.q.g 32 77.m even 15 2
693.2.m.i 16 3.b odd 2 1
693.2.m.i 16 33.h odd 10 1
847.2.a.o 8 11.d odd 10 1
847.2.a.p 8 11.c even 5 1
847.2.f.v 16 11.d odd 10 2
847.2.f.w 16 11.c even 5 2
847.2.f.x 16 11.b odd 2 1
847.2.f.x 16 11.d odd 10 1
5929.2.a.bs 8 77.l even 10 1
5929.2.a.bt 8 77.j odd 10 1
7623.2.a.ct 8 33.h odd 10 1
7623.2.a.cw 8 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 6 T^{2} + 12 T^{3} + 18 T^{4} + 17 T^{5} + 15 T^{6} - T^{7} - 58 T^{8} - 119 T^{9} - 185 T^{10} - 292 T^{11} - 242 T^{12} + 33 T^{13} + 312 T^{14} + 878 T^{15} + 1785 T^{16} + 1756 T^{17} + 1248 T^{18} + 264 T^{19} - 3872 T^{20} - 9344 T^{21} - 11840 T^{22} - 15232 T^{23} - 14848 T^{24} - 512 T^{25} + 15360 T^{26} + 34816 T^{27} + 73728 T^{28} + 98304 T^{29} + 98304 T^{30} + 98304 T^{31} + 65536 T^{32} \)
$3$ \( 1 + 2 T + 2 T^{2} - 4 T^{3} - 8 T^{4} + 4 T^{5} + 42 T^{6} + 104 T^{7} + 161 T^{8} + 8 T^{9} - 278 T^{10} - 122 T^{11} + 1632 T^{12} + 3996 T^{13} + 4420 T^{14} - 908 T^{15} - 3527 T^{16} - 2724 T^{17} + 39780 T^{18} + 107892 T^{19} + 132192 T^{20} - 29646 T^{21} - 202662 T^{22} + 17496 T^{23} + 1056321 T^{24} + 2047032 T^{25} + 2480058 T^{26} + 708588 T^{27} - 4251528 T^{28} - 6377292 T^{29} + 9565938 T^{30} + 28697814 T^{31} + 43046721 T^{32} \)
$5$ \( 1 + 5 T - T^{2} - 66 T^{3} - 156 T^{4} + 267 T^{5} + 1789 T^{6} + 1241 T^{7} - 10249 T^{8} - 25094 T^{9} + 25181 T^{10} + 188691 T^{11} + 137316 T^{12} - 851484 T^{13} - 1968914 T^{14} + 1709452 T^{15} + 12417791 T^{16} + 8547260 T^{17} - 49222850 T^{18} - 106435500 T^{19} + 85822500 T^{20} + 589659375 T^{21} + 393453125 T^{22} - 1960468750 T^{23} - 4003515625 T^{24} + 2423828125 T^{25} + 17470703125 T^{26} + 13037109375 T^{27} - 38085937500 T^{28} - 80566406250 T^{29} - 6103515625 T^{30} + 152587890625 T^{31} + 152587890625 T^{32} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$11$ \( 1 + 3 T + 19 T^{2} - 76 T^{3} - 286 T^{4} - 2245 T^{5} + 1052 T^{6} + 7668 T^{7} + 114073 T^{8} + 84348 T^{9} + 127292 T^{10} - 2988095 T^{11} - 4187326 T^{12} - 12239876 T^{13} + 33659659 T^{14} + 58461513 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 7 T - 13 T^{2} - 114 T^{3} + 302 T^{4} + 619 T^{5} - 3203 T^{6} + 11659 T^{7} - 57039 T^{8} - 415732 T^{9} + 983197 T^{10} + 802283 T^{11} - 7060468 T^{12} + 40515806 T^{13} - 160505050 T^{14} - 179149458 T^{15} + 5116354083 T^{16} - 2328942954 T^{17} - 27125353450 T^{18} + 89013225782 T^{19} - 201654026548 T^{20} + 297882061919 T^{21} + 4745704128373 T^{22} - 26086566469444 T^{23} - 46528464595119 T^{24} + 123637858189807 T^{25} - 441560749392347 T^{26} + 1109347283908903 T^{27} + 7036021706989262 T^{28} - 34527762151516842 T^{29} - 51185893014090757 T^{30} + 358301251098635299 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 + 5 T + 13 T^{2} + 163 T^{3} + 1572 T^{4} + 7139 T^{5} + 24321 T^{6} + 178695 T^{7} + 1262499 T^{8} + 5104293 T^{9} + 17336797 T^{10} + 107294017 T^{11} + 637972982 T^{12} + 2309382396 T^{13} + 7576691718 T^{14} + 42485576390 T^{15} + 218631742615 T^{16} + 722254798630 T^{17} + 2189663906502 T^{18} + 11345995711548 T^{19} + 53284141429622 T^{20} + 152342161095569 T^{21} + 418468133826493 T^{22} + 2094488816223189 T^{23} + 8806886793505059 T^{24} + 21191060590631415 T^{25} + 49030987652820129 T^{26} + 244667067740191987 T^{27} + 915882156925184292 T^{28} + 1614446219363667731 T^{29} + 2188911745272212077 T^{30} + 14312115257549078965 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 19 T + 113 T^{2} + 15 T^{3} - 2596 T^{4} + 2631 T^{5} + 106245 T^{6} - 756527 T^{7} + 1484429 T^{8} + 6206619 T^{9} - 26235105 T^{10} - 133461535 T^{11} + 1444663142 T^{12} - 5677361136 T^{13} + 11109732198 T^{14} - 13150474478 T^{15} + 38443302841 T^{16} - 249859015082 T^{17} + 4010613323478 T^{18} - 38941020031824 T^{19} + 188269945328582 T^{20} - 330463973351965 T^{21} - 1234253627852505 T^{22} + 5547921318840441 T^{23} + 25210893501388589 T^{24} - 244121955937653533 T^{25} + 651395134560067245 T^{26} + 306485871161214189 T^{27} - 5745765529895753956 T^{28} + 630794751933855885 T^{29} + 90287755493465905673 T^{30} - \)\(28\!\cdots\!81\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( ( 1 - 16 T + 224 T^{2} - 2234 T^{3} + 19567 T^{4} - 141712 T^{5} + 926670 T^{6} - 5230592 T^{7} + 26816927 T^{8} - 120303616 T^{9} + 490208430 T^{10} - 1724209904 T^{11} + 5475648847 T^{12} - 14378790262 T^{13} + 33160039136 T^{14} - 54477207152 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( 1 - 3 T - 75 T^{3} + 2441 T^{4} - 473 T^{5} - 37550 T^{6} - 114715 T^{7} + 3177286 T^{8} + 2623287 T^{9} - 24972736 T^{10} - 263064707 T^{11} + 3471423716 T^{12} + 694532172 T^{13} - 11302302466 T^{14} - 87801320252 T^{15} + 2485262854281 T^{16} - 2546238287308 T^{17} - 9505236373906 T^{18} + 16938945142908 T^{19} + 2455272037276196 T^{20} - 5395759401918343 T^{21} - 14854365761976256 T^{22} + 45251376273007683 T^{23} + 1589425924451203846 T^{24} - 1664187250621812335 T^{25} - 15797556610422547550 T^{26} - 5770841119178857117 T^{27} + \)\(86\!\cdots\!81\)\( T^{28} - \)\(76\!\cdots\!75\)\( T^{29} - \)\(25\!\cdots\!47\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 + 7 T + 13 T^{3} + 77 T^{4} - 8801 T^{5} - 22254 T^{6} + 56119 T^{7} - 947868 T^{8} + 3202093 T^{9} + 57615842 T^{10} - 102279009 T^{11} - 159564504 T^{12} + 5764151836 T^{13} - 36979480986 T^{14} - 181717554252 T^{15} + 841105835367 T^{16} - 5633244181812 T^{17} - 35537281227546 T^{18} + 171719847346276 T^{19} - 147361170298584 T^{20} - 2928161192791359 T^{21} + 51134271858914402 T^{22} + 88097949056534323 T^{23} - 808428121877125788 T^{24} + 1483765156034695849 T^{25} - 18240007898470745454 T^{26} - \)\(22\!\cdots\!31\)\( T^{27} + 60650034351718331597 T^{28} + \)\(31\!\cdots\!83\)\( T^{29} + \)\(16\!\cdots\!57\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 - 4 T - 64 T^{2} + 392 T^{3} + 2308 T^{4} - 10820 T^{5} - 47691 T^{6} - 31178 T^{7} + 2748163 T^{8} + 12045860 T^{9} - 124307808 T^{10} - 75116270 T^{11} + 4767089611 T^{12} - 456957494 T^{13} - 122520873497 T^{14} + 217199596514 T^{15} + 4709853143282 T^{16} + 8036385071018 T^{17} - 167731075817393 T^{18} - 23146267943582 T^{19} + 8934293432441371 T^{20} - 5208859396880390 T^{21} - 318939825830501472 T^{22} + 1143536101481319380 T^{23} + 9652866073525897123 T^{24} - 4051947123330910706 T^{25} - \)\(22\!\cdots\!59\)\( T^{26} - \)\(19\!\cdots\!60\)\( T^{27} + \)\(15\!\cdots\!48\)\( T^{28} + \)\(95\!\cdots\!24\)\( T^{29} - \)\(57\!\cdots\!96\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 + 10 T + 32 T^{2} + 192 T^{3} + 5438 T^{4} + 48920 T^{5} + 140400 T^{6} + 769008 T^{7} + 14682039 T^{8} + 96831496 T^{9} + 182717296 T^{10} + 1985070190 T^{11} + 32206741548 T^{12} + 131499578440 T^{13} + 27334729480 T^{14} + 4391234846920 T^{15} + 62202943867805 T^{16} + 180040628723720 T^{17} + 45949680255880 T^{18} + 9063082445663240 T^{19} + 91008554203418028 T^{20} + 229982690931748190 T^{21} + 867926202633652336 T^{22} + 18858347692290955976 T^{23} + \)\(11\!\cdots\!19\)\( T^{24} + \)\(25\!\cdots\!88\)\( T^{25} + \)\(18\!\cdots\!00\)\( T^{26} + \)\(26\!\cdots\!20\)\( T^{27} + \)\(12\!\cdots\!78\)\( T^{28} + \)\(17\!\cdots\!32\)\( T^{29} + \)\(12\!\cdots\!52\)\( T^{30} + \)\(15\!\cdots\!10\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 + 4 T + 239 T^{2} + 936 T^{3} + 27462 T^{4} + 98508 T^{5} + 2007760 T^{6} + 6271668 T^{7} + 102278657 T^{8} + 269681724 T^{9} + 3712348240 T^{10} + 7832075556 T^{11} + 93887113062 T^{12} + 137599902648 T^{13} + 1510805768711 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 + 23 T + 186 T^{2} + 322 T^{3} - 4127 T^{4} - 30138 T^{5} - 70920 T^{6} + 1094344 T^{7} + 18527582 T^{8} + 119631636 T^{9} + 262096090 T^{10} - 2200503287 T^{11} - 30693701772 T^{12} - 88025997822 T^{13} + 1410058928022 T^{14} + 15969608141898 T^{15} + 104939392199145 T^{16} + 750571582669206 T^{17} + 3114820172000598 T^{18} - 9139123171873506 T^{19} - 149775473356494732 T^{20} - 504674441760538009 T^{21} + 2825190190998963610 T^{22} + 60608152736413767468 T^{23} + \)\(44\!\cdots\!02\)\( T^{24} + \)\(12\!\cdots\!48\)\( T^{25} - \)\(37\!\cdots\!80\)\( T^{26} - \)\(74\!\cdots\!14\)\( T^{27} - \)\(47\!\cdots\!07\)\( T^{28} + \)\(17\!\cdots\!94\)\( T^{29} + \)\(47\!\cdots\!34\)\( T^{30} + \)\(27\!\cdots\!89\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 - 4 T - 22 T^{2} - 440 T^{3} + 660 T^{4} + 49868 T^{5} + 140113 T^{6} - 2285596 T^{7} - 15873165 T^{8} - 76638860 T^{9} + 1323931162 T^{10} + 10376469922 T^{11} - 35381234389 T^{12} - 446229246850 T^{13} - 3132652820265 T^{14} + 9035271835264 T^{15} + 292112805840914 T^{16} + 478869407268992 T^{17} - 8799621772124385 T^{18} - 66433271583287450 T^{19} - 279174957702951109 T^{20} + 4339392954630461546 T^{21} + 29344088384504601898 T^{22} - 90028522586408265820 T^{23} - \)\(98\!\cdots\!65\)\( T^{24} - \)\(75\!\cdots\!68\)\( T^{25} + \)\(24\!\cdots\!37\)\( T^{26} + \)\(46\!\cdots\!96\)\( T^{27} + \)\(32\!\cdots\!60\)\( T^{28} - \)\(11\!\cdots\!20\)\( T^{29} - \)\(30\!\cdots\!18\)\( T^{30} - \)\(29\!\cdots\!28\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 17 T - 50 T^{2} + 1985 T^{3} + 451 T^{4} - 76307 T^{5} - 426220 T^{6} - 1429235 T^{7} + 60743716 T^{8} + 309308083 T^{9} - 1984484046 T^{10} - 33011892753 T^{11} - 101353409674 T^{12} + 2740413120218 T^{13} + 8464635562844 T^{14} - 86256705224328 T^{15} - 317724345351369 T^{16} - 5089145608235352 T^{17} + 29465396394259964 T^{18} + 562823306217252622 T^{19} - 1228135853600750314 T^{20} - 23601004285101705147 T^{21} - 83706596062330791486 T^{22} + \)\(76\!\cdots\!77\)\( T^{23} + \)\(89\!\cdots\!36\)\( T^{24} - \)\(12\!\cdots\!65\)\( T^{25} - \)\(21\!\cdots\!20\)\( T^{26} - \)\(23\!\cdots\!13\)\( T^{27} + \)\(80\!\cdots\!31\)\( T^{28} + \)\(20\!\cdots\!15\)\( T^{29} - \)\(30\!\cdots\!50\)\( T^{30} - \)\(62\!\cdots\!83\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 7 T - 20 T^{2} + 403 T^{3} + 10067 T^{4} + 48739 T^{5} - 116314 T^{6} + 1172749 T^{7} + 26082242 T^{8} + 24039263 T^{9} - 379016678 T^{10} - 5077729829 T^{11} - 113601259714 T^{12} - 664603591174 T^{13} + 819501759744 T^{14} - 46004665392052 T^{15} - 871290620425703 T^{16} - 2806284588915172 T^{17} + 3049366048007424 T^{18} - 150852387728265694 T^{19} - 1572904979399749474 T^{20} - 4288631831050762529 T^{21} - 19527081139622592758 T^{22} + 75549221576474692523 T^{23} + \)\(50\!\cdots\!02\)\( T^{24} + \)\(13\!\cdots\!09\)\( T^{25} - \)\(82\!\cdots\!14\)\( T^{26} + \)\(21\!\cdots\!79\)\( T^{27} + \)\(26\!\cdots\!07\)\( T^{28} + \)\(65\!\cdots\!43\)\( T^{29} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(42\!\cdots\!07\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 + 19 T + 520 T^{2} + 7751 T^{3} + 119005 T^{4} + 1415171 T^{5} + 15702408 T^{6} + 150444400 T^{7} + 1308159665 T^{8} + 10079774800 T^{9} + 70488109512 T^{10} + 425631075473 T^{11} + 2398084154605 T^{12} + 10464819704357 T^{13} + 47038358727880 T^{14} + 115153520501137 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( 1 + 14 T + 140 T^{2} + 763 T^{3} + 10437 T^{4} + 25566 T^{5} + 32310 T^{6} - 3862236 T^{7} + 10586088 T^{8} - 330657408 T^{9} - 2081383949 T^{10} - 39603959100 T^{11} + 15217861524 T^{12} - 1721196695007 T^{13} + 1593568125167 T^{14} - 22773634065006 T^{15} + 1780879231377332 T^{16} - 1616928018615426 T^{17} + 8033176918966847 T^{18} - 616035230306650377 T^{19} + 386711442550061844 T^{20} - 71454625424023544100 T^{21} - \)\(26\!\cdots\!29\)\( T^{22} - \)\(30\!\cdots\!28\)\( T^{23} + \)\(68\!\cdots\!68\)\( T^{24} - \)\(17\!\cdots\!16\)\( T^{25} + \)\(10\!\cdots\!10\)\( T^{26} + \)\(59\!\cdots\!86\)\( T^{27} + \)\(17\!\cdots\!17\)\( T^{28} + \)\(88\!\cdots\!93\)\( T^{29} + \)\(11\!\cdots\!40\)\( T^{30} + \)\(82\!\cdots\!14\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 + 35 T + 622 T^{2} + 8881 T^{3} + 123217 T^{4} + 1506367 T^{5} + 15880754 T^{6} + 159131205 T^{7} + 1535260024 T^{8} + 13342640169 T^{9} + 108277262748 T^{10} + 880406179959 T^{11} + 6960680213462 T^{12} + 52467783616198 T^{13} + 405906809006112 T^{14} + 3284917386947140 T^{15} + 27500270697009885 T^{16} + 239798969247141220 T^{17} + 2163077385193570848 T^{18} + 20410859779022497366 T^{19} + \)\(19\!\cdots\!42\)\( T^{20} + \)\(18\!\cdots\!87\)\( T^{21} + \)\(16\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!93\)\( T^{23} + \)\(12\!\cdots\!44\)\( T^{24} + \)\(93\!\cdots\!65\)\( T^{25} + \)\(68\!\cdots\!46\)\( T^{26} + \)\(47\!\cdots\!59\)\( T^{27} + \)\(28\!\cdots\!57\)\( T^{28} + \)\(14\!\cdots\!73\)\( T^{29} + \)\(75\!\cdots\!98\)\( T^{30} + \)\(31\!\cdots\!95\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 15 T + 94 T^{2} - 2495 T^{3} + 38855 T^{4} - 414785 T^{5} + 5676295 T^{6} - 62322940 T^{7} + 690670255 T^{8} - 8423294280 T^{9} + 83337602737 T^{10} - 856819337855 T^{11} + 8989269459608 T^{12} - 82967566755485 T^{13} + 808287092858990 T^{14} - 7591629044633355 T^{15} + 65763672844350810 T^{16} - 599738694526035045 T^{17} + 5044519746532956590 T^{18} - 40906246145557568915 T^{19} + \)\(35\!\cdots\!48\)\( T^{20} - \)\(26\!\cdots\!45\)\( T^{21} + \)\(20\!\cdots\!77\)\( T^{22} - \)\(16\!\cdots\!20\)\( T^{23} + \)\(10\!\cdots\!55\)\( T^{24} - \)\(74\!\cdots\!60\)\( T^{25} + \)\(53\!\cdots\!95\)\( T^{26} - \)\(31\!\cdots\!15\)\( T^{27} + \)\(22\!\cdots\!55\)\( T^{28} - \)\(11\!\cdots\!05\)\( T^{29} + \)\(34\!\cdots\!14\)\( T^{30} - \)\(43\!\cdots\!85\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 5 T - 318 T^{2} + 3741 T^{3} + 48517 T^{4} - 994133 T^{5} - 3255786 T^{6} + 170309105 T^{7} - 263285806 T^{8} - 21750909411 T^{9} + 119250152558 T^{10} + 2097202277489 T^{11} - 20999477370338 T^{12} - 141328121959992 T^{13} + 2545394345691772 T^{14} + 4513257769614610 T^{15} - 237031046695965905 T^{16} + 374600394878012630 T^{17} + 17535221647470617308 T^{18} - 80809582871137945704 T^{19} - \)\(99\!\cdots\!98\)\( T^{20} + \)\(82\!\cdots\!27\)\( T^{21} + \)\(38\!\cdots\!02\)\( T^{22} - \)\(59\!\cdots\!97\)\( T^{23} - \)\(59\!\cdots\!46\)\( T^{24} + \)\(31\!\cdots\!15\)\( T^{25} - \)\(50\!\cdots\!14\)\( T^{26} - \)\(12\!\cdots\!11\)\( T^{27} + \)\(51\!\cdots\!37\)\( T^{28} + \)\(33\!\cdots\!83\)\( T^{29} - \)\(23\!\cdots\!22\)\( T^{30} - \)\(30\!\cdots\!35\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 - 37 T + 1032 T^{2} - 20901 T^{3} + 357823 T^{4} - 5152639 T^{5} + 65606528 T^{6} - 731972999 T^{7} + 7339853952 T^{8} - 65145596911 T^{9} + 519669308288 T^{10} - 3632450763191 T^{11} + 22450616901343 T^{12} - 116712426543549 T^{13} + 512884692271752 T^{14} - 1636559391134573 T^{15} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 - 20 T + 123 T^{2} + 28 T^{3} + 4997 T^{4} - 239296 T^{5} + 2683386 T^{6} - 13918670 T^{7} + 12108314 T^{8} + 81926108 T^{9} + 9118270907 T^{10} - 123570735168 T^{11} + 816960697702 T^{12} + 3552108922056 T^{13} - 86512314985382 T^{14} - 43450963654390 T^{15} + 5424229096525225 T^{16} - 4214743474475830 T^{17} - 813994371697459238 T^{18} + 3241913906219615688 T^{19} + 72324943172816412262 T^{20} - \)\(10\!\cdots\!76\)\( T^{21} + \)\(75\!\cdots\!03\)\( T^{22} + \)\(66\!\cdots\!04\)\( T^{23} + \)\(94\!\cdots\!54\)\( T^{24} - \)\(10\!\cdots\!90\)\( T^{25} + \)\(19\!\cdots\!14\)\( T^{26} - \)\(17\!\cdots\!88\)\( T^{27} + \)\(34\!\cdots\!77\)\( T^{28} + \)\(18\!\cdots\!56\)\( T^{29} + \)\(80\!\cdots\!87\)\( T^{30} - \)\(12\!\cdots\!60\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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