Properties

Label 150.2.h.b
Level 150
Weight 2
Character orbit 150.h
Analytic conductor 1.198
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.h (of order \(10\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(4\) \(x^{15}\mathstrut -\mathstrut \) \(24\) \(x^{14}\mathstrut +\mathstrut \) \(94\) \(x^{13}\mathstrut +\mathstrut \) \(262\) \(x^{12}\mathstrut -\mathstrut \) \(936\) \(x^{11}\mathstrut -\mathstrut \) \(1584\) \(x^{10}\mathstrut +\mathstrut \) \(4642\) \(x^{9}\mathstrut +\mathstrut \) \(6259\) \(x^{8}\mathstrut -\mathstrut \) \(11958\) \(x^{7}\mathstrut -\mathstrut \) \(15752\) \(x^{6}\mathstrut +\mathstrut \) \(14670\) \(x^{5}\mathstrut +\mathstrut \) \(18271\) \(x^{4}\mathstrut -\mathstrut \) \(10440\) \(x^{3}\mathstrut +\mathstrut \) \(1135\) \(x^{2}\mathstrut +\mathstrut \) \(21080\) \(x\mathstrut +\mathstrut \) \(11105\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(216062652991087486517\) \(\nu^{15}\mathstrut -\mathstrut \) \(222689963418511649230\) \(\nu^{14}\mathstrut +\mathstrut \) \(11692850809057402771308\) \(\nu^{13}\mathstrut -\mathstrut \) \(4830492818371367785816\) \(\nu^{12}\mathstrut -\mathstrut \) \(200158344597704877408533\) \(\nu^{11}\mathstrut +\mathstrut \) \(142199510139739938929725\) \(\nu^{10}\mathstrut +\mathstrut \) \(1696861859405690275607183\) \(\nu^{9}\mathstrut -\mathstrut \) \(1203647771351280118822772\) \(\nu^{8}\mathstrut -\mathstrut \) \(7616568797732981879968331\) \(\nu^{7}\mathstrut +\mathstrut \) \(3146071202213920505926562\) \(\nu^{6}\mathstrut +\mathstrut \) \(19090952137993148860203137\) \(\nu^{5}\mathstrut -\mathstrut \) \(132407938274590801116617\) \(\nu^{4}\mathstrut -\mathstrut \) \(24907817546557276804454900\) \(\nu^{3}\mathstrut +\mathstrut \) \(3697181661562006774932655\) \(\nu^{2}\mathstrut -\mathstrut \) \(2518379054720136093673150\) \(\nu\mathstrut -\mathstrut \) \(36766787356071608007667585\)\()/\)\(30\!\cdots\!75\)
\(\beta_{2}\)\(=\)\((\)\(-\)\(265732875567395073657\) \(\nu^{15}\mathstrut +\mathstrut \) \(1990174463411413335455\) \(\nu^{14}\mathstrut +\mathstrut \) \(15780868218717044920838\) \(\nu^{13}\mathstrut -\mathstrut \) \(82635447635890840435206\) \(\nu^{12}\mathstrut -\mathstrut \) \(310570846147264249920748\) \(\nu^{11}\mathstrut +\mathstrut \) \(1241691800858697588974155\) \(\nu^{10}\mathstrut +\mathstrut \) \(3010621887147134394227693\) \(\nu^{9}\mathstrut -\mathstrut \) \(9299193782108641569228897\) \(\nu^{8}\mathstrut -\mathstrut \) \(15187750120575825142764691\) \(\nu^{7}\mathstrut +\mathstrut \) \(37039168692149977669870742\) \(\nu^{6}\mathstrut +\mathstrut \) \(41504619394117749813534187\) \(\nu^{5}\mathstrut -\mathstrut \) \(78953875227996269190814922\) \(\nu^{4}\mathstrut -\mathstrut \) \(49901182956943689204688600\) \(\nu^{3}\mathstrut +\mathstrut \) \(104941588005445229750025955\) \(\nu^{2}\mathstrut -\mathstrut \) \(24344868121453920674255425\) \(\nu\mathstrut -\mathstrut \) \(91429957070907564671351610\)\()/\)\(30\!\cdots\!75\)
\(\beta_{3}\)\(=\)\((\)\(702375526475979478783\) \(\nu^{15}\mathstrut -\mathstrut \) \(3304044074082752777785\) \(\nu^{14}\mathstrut -\mathstrut \) \(25795447997562939235427\) \(\nu^{13}\mathstrut +\mathstrut \) \(114953003304995515650009\) \(\nu^{12}\mathstrut +\mathstrut \) \(368633707984163431786287\) \(\nu^{11}\mathstrut -\mathstrut \) \(1482042781994674721659060\) \(\nu^{10}\mathstrut -\mathstrut \) \(2812846670822251548242917\) \(\nu^{9}\mathstrut +\mathstrut \) \(9176301900507627049359998\) \(\nu^{8}\mathstrut +\mathstrut \) \(12537511618422589653976589\) \(\nu^{7}\mathstrut -\mathstrut \) \(26170693115741185316682238\) \(\nu^{6}\mathstrut -\mathstrut \) \(40039221089263675036570378\) \(\nu^{5}\mathstrut +\mathstrut \) \(32685303511358818643459718\) \(\nu^{4}\mathstrut +\mathstrut \) \(72705390025681627618678875\) \(\nu^{3}\mathstrut -\mathstrut \) \(35152535417255483308930345\) \(\nu^{2}\mathstrut -\mathstrut \) \(30211511665347684930563925\) \(\nu\mathstrut +\mathstrut \) \(49504201727591742141925840\)\()/\)\(30\!\cdots\!75\)
\(\beta_{4}\)\(=\)\((\)\(2296292574754742295778\) \(\nu^{15}\mathstrut -\mathstrut \) \(18777167771030970954085\) \(\nu^{14}\mathstrut -\mathstrut \) \(5742569515547694072792\) \(\nu^{13}\mathstrut +\mathstrut \) \(376031136320790011131049\) \(\nu^{12}\mathstrut -\mathstrut \) \(424999022122650057110578\) \(\nu^{11}\mathstrut -\mathstrut \) \(3188418937056165399714235\) \(\nu^{10}\mathstrut +\mathstrut \) \(5262312369001650000411893\) \(\nu^{9}\mathstrut +\mathstrut \) \(12651496691912944253088318\) \(\nu^{8}\mathstrut -\mathstrut \) \(21294724748266832493330656\) \(\nu^{7}\mathstrut -\mathstrut \) \(31019540923068154312589818\) \(\nu^{6}\mathstrut +\mathstrut \) \(37963741339531390922508492\) \(\nu^{5}\mathstrut +\mathstrut \) \(52805059704158952082659513\) \(\nu^{4}\mathstrut -\mathstrut \) \(32920117107035471964771150\) \(\nu^{3}\mathstrut -\mathstrut \) \(38527298141204740926854120\) \(\nu^{2}\mathstrut +\mathstrut \) \(44218902932902401592517525\) \(\nu\mathstrut -\mathstrut \) \(10609792203969280509060435\)\()/\)\(30\!\cdots\!75\)
\(\beta_{5}\)\(=\)\((\)\(102309325228650587446\) \(\nu^{15}\mathstrut -\mathstrut \) \(425968439454909023504\) \(\nu^{14}\mathstrut -\mathstrut \) \(2318583371847014159396\) \(\nu^{13}\mathstrut +\mathstrut \) \(9842957759035774370007\) \(\nu^{12}\mathstrut +\mathstrut \) \(23588922138538677715666\) \(\nu^{11}\mathstrut -\mathstrut \) \(96439403899597794479889\) \(\nu^{10}\mathstrut -\mathstrut \) \(132289916480158900957958\) \(\nu^{9}\mathstrut +\mathstrut \) \(469952553315433857842339\) \(\nu^{8}\mathstrut +\mathstrut \) \(521913912002145220106872\) \(\nu^{7}\mathstrut -\mathstrut \) \(1201274203759528179191999\) \(\nu^{6}\mathstrut -\mathstrut \) \(1439503235730439393390400\) \(\nu^{5}\mathstrut +\mathstrut \) \(1561882257402276944726861\) \(\nu^{4}\mathstrut +\mathstrut \) \(1751624753872932949676820\) \(\nu^{3}\mathstrut -\mathstrut \) \(1056349544697974528792295\) \(\nu^{2}\mathstrut +\mathstrut \) \(648738016230841178303300\) \(\nu\mathstrut +\mathstrut \) \(2136427940493114503422100\)\()/\)\(12\!\cdots\!55\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(3180944021456305418003\) \(\nu^{15}\mathstrut +\mathstrut \) \(12108798860040862926030\) \(\nu^{14}\mathstrut +\mathstrut \) \(76658914050049775735647\) \(\nu^{13}\mathstrut -\mathstrut \) \(279795475163391518337309\) \(\nu^{12}\mathstrut -\mathstrut \) \(843282840396799929838402\) \(\nu^{11}\mathstrut +\mathstrut \) \(2772410277587660664967595\) \(\nu^{10}\mathstrut +\mathstrut \) \(5120683136367422331041022\) \(\nu^{9}\mathstrut -\mathstrut \) \(13952896772257730298480513\) \(\nu^{8}\mathstrut -\mathstrut \) \(20136486077065159637670929\) \(\nu^{7}\mathstrut +\mathstrut \) \(38669929503840247602021988\) \(\nu^{6}\mathstrut +\mathstrut \) \(48847780672259798172146818\) \(\nu^{5}\mathstrut -\mathstrut \) \(57387686031561159003451668\) \(\nu^{4}\mathstrut -\mathstrut \) \(56558593741097731772595025\) \(\nu^{3}\mathstrut +\mathstrut \) \(52324310540791207677909920\) \(\nu^{2}\mathstrut -\mathstrut \) \(9360398580013835322450250\) \(\nu\mathstrut -\mathstrut \) \(54250932322302201472419340\)\()/\)\(30\!\cdots\!75\)
\(\beta_{7}\)\(=\)\((\)\(3754084957894051064894\) \(\nu^{15}\mathstrut -\mathstrut \) \(15083470458469574064070\) \(\nu^{14}\mathstrut -\mathstrut \) \(89761163865409232604666\) \(\nu^{13}\mathstrut +\mathstrut \) \(347746317243990280362907\) \(\nu^{12}\mathstrut +\mathstrut \) \(998563896155868111731741\) \(\nu^{11}\mathstrut -\mathstrut \) \(3429976210567685569153495\) \(\nu^{10}\mathstrut -\mathstrut \) \(6202656224864865816953881\) \(\nu^{9}\mathstrut +\mathstrut \) \(17064490132080048666621894\) \(\nu^{8}\mathstrut +\mathstrut \) \(24890198037032388014893387\) \(\nu^{7}\mathstrut -\mathstrut \) \(46215997398926518335783349\) \(\nu^{6}\mathstrut -\mathstrut \) \(60890899573543787233277629\) \(\nu^{5}\mathstrut +\mathstrut \) \(60090136619662887556863749\) \(\nu^{4}\mathstrut +\mathstrut \) \(77529726343056299821308475\) \(\nu^{3}\mathstrut -\mathstrut \) \(40267892959128470588816910\) \(\nu^{2}\mathstrut -\mathstrut \) \(36825642304869580862835650\) \(\nu\mathstrut +\mathstrut \) \(68909263042569036125855870\)\()/\)\(30\!\cdots\!75\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(3973936680890235791447\) \(\nu^{15}\mathstrut +\mathstrut \) \(19337803181486775124430\) \(\nu^{14}\mathstrut +\mathstrut \) \(73342780711804225674553\) \(\nu^{13}\mathstrut -\mathstrut \) \(416649754235775402919971\) \(\nu^{12}\mathstrut -\mathstrut \) \(558606750221222446608433\) \(\nu^{11}\mathstrut +\mathstrut \) \(3759928827840084662619945\) \(\nu^{10}\mathstrut +\mathstrut \) \(1783092343379774194213858\) \(\nu^{9}\mathstrut -\mathstrut \) \(16064794943644897425418542\) \(\nu^{8}\mathstrut -\mathstrut \) \(4220420776903807591816446\) \(\nu^{7}\mathstrut +\mathstrut \) \(35426011517516138530176972\) \(\nu^{6}\mathstrut +\mathstrut \) \(10251508388168254787967127\) \(\nu^{5}\mathstrut -\mathstrut \) \(38968246371604410909723242\) \(\nu^{4}\mathstrut -\mathstrut \) \(1287569082167687474870500\) \(\nu^{3}\mathstrut +\mathstrut \) \(19285624254963832677634030\) \(\nu^{2}\mathstrut -\mathstrut \) \(37529598314145175899342950\) \(\nu\mathstrut -\mathstrut \) \(3543808081464618883649710\)\()/\)\(30\!\cdots\!75\)
\(\beta_{9}\)\(=\)\((\)\(161168803239414780910\) \(\nu^{15}\mathstrut -\mathstrut \) \(661637860075799467724\) \(\nu^{14}\mathstrut -\mathstrut \) \(3827536126025598552754\) \(\nu^{13}\mathstrut +\mathstrut \) \(15994665739743758299833\) \(\nu^{12}\mathstrut +\mathstrut \) \(40151303735551147100534\) \(\nu^{11}\mathstrut -\mathstrut \) \(164810455922834007902603\) \(\nu^{10}\mathstrut -\mathstrut \) \(224574784032425709435836\) \(\nu^{9}\mathstrut +\mathstrut \) \(865180285576144364779891\) \(\nu^{8}\mathstrut +\mathstrut \) \(794296728242089068127598\) \(\nu^{7}\mathstrut -\mathstrut \) \(2456515842661058092401641\) \(\nu^{6}\mathstrut -\mathstrut \) \(1829541259881354437224758\) \(\nu^{5}\mathstrut +\mathstrut \) \(3808469691277882506316428\) \(\nu^{4}\mathstrut +\mathstrut \) \(1780293400073412257719180\) \(\nu^{3}\mathstrut -\mathstrut \) \(3317734863222409081306840\) \(\nu^{2}\mathstrut +\mathstrut \) \(1107814885746452854956660\) \(\nu\mathstrut +\mathstrut \) \(2686892356943722466767320\)\()/\)\(12\!\cdots\!55\)
\(\beta_{10}\)\(=\)\((\)\(164194301487618633410\) \(\nu^{15}\mathstrut -\mathstrut \) \(719766109199837227876\) \(\nu^{14}\mathstrut -\mathstrut \) \(3474010950605366952410\) \(\nu^{13}\mathstrut +\mathstrut \) \(16401138606042229504145\) \(\nu^{12}\mathstrut +\mathstrut \) \(31059323892781834514230\) \(\nu^{11}\mathstrut -\mathstrut \) \(157188061964900911061375\) \(\nu^{10}\mathstrut -\mathstrut \) \(130154545341889076617028\) \(\nu^{9}\mathstrut +\mathstrut \) \(733989392081838127266340\) \(\nu^{8}\mathstrut +\mathstrut \) \(317412747752376679716350\) \(\nu^{7}\mathstrut -\mathstrut \) \(1736893910036273955685895\) \(\nu^{6}\mathstrut -\mathstrut \) \(552527714012012839873230\) \(\nu^{5}\mathstrut +\mathstrut \) \(1984064979606023800874943\) \(\nu^{4}\mathstrut +\mathstrut \) \(47852376229771545163160\) \(\nu^{3}\mathstrut -\mathstrut \) \(1332489074800974413015230\) \(\nu^{2}\mathstrut +\mathstrut \) \(2233536035317308402213160\) \(\nu\mathstrut +\mathstrut \) \(1025511991580005661458970\)\()/\)\(12\!\cdots\!55\)
\(\beta_{11}\)\(=\)\((\)\(838230261608330797746\) \(\nu^{15}\mathstrut -\mathstrut \) \(4786382082628259634787\) \(\nu^{14}\mathstrut -\mathstrut \) \(13869542625980282893647\) \(\nu^{13}\mathstrut +\mathstrut \) \(106085643722406510377459\) \(\nu^{12}\mathstrut +\mathstrut \) \(90720060109888652320392\) \(\nu^{11}\mathstrut -\mathstrut \) \(1002540084989697030577461\) \(\nu^{10}\mathstrut -\mathstrut \) \(234297600527699664967938\) \(\nu^{9}\mathstrut +\mathstrut \) \(4699509630594445307398133\) \(\nu^{8}\mathstrut +\mathstrut \) \(1109399877527339581698789\) \(\nu^{7}\mathstrut -\mathstrut \) \(12618721201525299962308298\) \(\nu^{6}\mathstrut -\mathstrut \) \(6217938413303324609066387\) \(\nu^{5}\mathstrut +\mathstrut \) \(20000504884661427693942799\) \(\nu^{4}\mathstrut +\mathstrut \) \(10638060647839174881601105\) \(\nu^{3}\mathstrut -\mathstrut \) \(18066278410794456581746620\) \(\nu^{2}\mathstrut +\mathstrut \) \(3148595090397016687536785\) \(\nu\mathstrut +\mathstrut \) \(17988954213833427432880420\)\()/\)\(60\!\cdots\!75\)
\(\beta_{12}\)\(=\)\((\)\(908669311439016539378\) \(\nu^{15}\mathstrut -\mathstrut \) \(5174938240432228690153\) \(\nu^{14}\mathstrut -\mathstrut \) \(15283113254819993188609\) \(\nu^{13}\mathstrut +\mathstrut \) \(120512535486778277787298\) \(\nu^{12}\mathstrut +\mathstrut \) \(82750556881744744497304\) \(\nu^{11}\mathstrut -\mathstrut \) \(1186706887477185726218359\) \(\nu^{10}\mathstrut +\mathstrut \) \(90637788457715278790657\) \(\nu^{9}\mathstrut +\mathstrut \) \(5770412096942933836755541\) \(\nu^{8}\mathstrut -\mathstrut \) \(1561116277429088140696022\) \(\nu^{7}\mathstrut -\mathstrut \) \(14822422913804415887485051\) \(\nu^{6}\mathstrut +\mathstrut \) \(1820874604516396099156933\) \(\nu^{5}\mathstrut +\mathstrut \) \(20462537822506322798652590\) \(\nu^{4}\mathstrut +\mathstrut \) \(1296008565761348543038545\) \(\nu^{3}\mathstrut -\mathstrut \) \(11744901364273461227406290\) \(\nu^{2}\mathstrut +\mathstrut \) \(12987872084176881105529190\) \(\nu\mathstrut +\mathstrut \) \(4308719570906786186270775\)\()/\)\(60\!\cdots\!75\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(6533138943972814245933\) \(\nu^{15}\mathstrut +\mathstrut \) \(31619240120523939425865\) \(\nu^{14}\mathstrut +\mathstrut \) \(125897227911835136338677\) \(\nu^{13}\mathstrut -\mathstrut \) \(708040835635361844344624\) \(\nu^{12}\mathstrut -\mathstrut \) \(1007374201550559641412517\) \(\nu^{11}\mathstrut +\mathstrut \) \(6724682741769758012533880\) \(\nu^{10}\mathstrut +\mathstrut \) \(3470026494541648226365932\) \(\nu^{9}\mathstrut -\mathstrut \) \(31333706626178340309002363\) \(\nu^{8}\mathstrut -\mathstrut \) \(7246181075197585096864139\) \(\nu^{7}\mathstrut +\mathstrut \) \(77592234545423436266661168\) \(\nu^{6}\mathstrut +\mathstrut \) \(11636953568620512039028188\) \(\nu^{5}\mathstrut -\mathstrut \) \(98305813076082701994022623\) \(\nu^{4}\mathstrut +\mathstrut \) \(10820382804042389176367050\) \(\nu^{3}\mathstrut +\mathstrut \) \(67691544638516502492240320\) \(\nu^{2}\mathstrut -\mathstrut \) \(84912508264192029168955175\) \(\nu\mathstrut -\mathstrut \) \(61955069294991105282252615\)\()/\)\(30\!\cdots\!75\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(9361640845971301396397\) \(\nu^{15}\mathstrut +\mathstrut \) \(52366780732480415686465\) \(\nu^{14}\mathstrut +\mathstrut \) \(159762414534900794176883\) \(\nu^{13}\mathstrut -\mathstrut \) \(1190582299946926644582166\) \(\nu^{12}\mathstrut -\mathstrut \) \(1024978252277545106715408\) \(\nu^{11}\mathstrut +\mathstrut \) \(11575346968422089033871410\) \(\nu^{10}\mathstrut +\mathstrut \) \(1598877636542195838733293\) \(\nu^{9}\mathstrut -\mathstrut \) \(55828465749972912641212972\) \(\nu^{8}\mathstrut +\mathstrut \) \(613642773665136143529419\) \(\nu^{7}\mathstrut +\mathstrut \) \(145008784517773349507323362\) \(\nu^{6}\mathstrut +\mathstrut \) \(9914427702507185104901877\) \(\nu^{5}\mathstrut -\mathstrut \) \(183985152384127705242957177\) \(\nu^{4}\mathstrut -\mathstrut \) \(15715753986137640451935575\) \(\nu^{3}\mathstrut +\mathstrut \) \(81809532792011322933042280\) \(\nu^{2}\mathstrut -\mathstrut \) \(134474994846987328679889500\) \(\nu\mathstrut -\mathstrut \) \(86034467893737742078614385\)\()/\)\(30\!\cdots\!75\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(1965198321477020977800\) \(\nu^{15}\mathstrut +\mathstrut \) \(9926479283735303812942\) \(\nu^{14}\mathstrut +\mathstrut \) \(40235447863223328373118\) \(\nu^{13}\mathstrut -\mathstrut \) \(237493265324739844946336\) \(\nu^{12}\mathstrut -\mathstrut \) \(348172097985448217050033\) \(\nu^{11}\mathstrut +\mathstrut \) \(2410564961185097493334441\) \(\nu^{10}\mathstrut +\mathstrut \) \(1429316307775823908404854\) \(\nu^{9}\mathstrut -\mathstrut \) \(12215850222197537282508032\) \(\nu^{8}\mathstrut -\mathstrut \) \(3751634380400785138202611\) \(\nu^{7}\mathstrut +\mathstrut \) \(32442715594977735558622037\) \(\nu^{6}\mathstrut +\mathstrut \) \(8620102019274457312300996\) \(\nu^{5}\mathstrut -\mathstrut \) \(42222846117366993415189833\) \(\nu^{4}\mathstrut -\mathstrut \) \(4541857374012864958803405\) \(\nu^{3}\mathstrut +\mathstrut \) \(21432392262237651257289005\) \(\nu^{2}\mathstrut -\mathstrut \) \(32435795750138591078265085\) \(\nu\mathstrut -\mathstrut \) \(25977835089457219631295215\)\()/\)\(60\!\cdots\!75\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(3\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut -\mathstrut \) \(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(17\) \(\beta_{15}\mathstrut -\mathstrut \) \(6\) \(\beta_{14}\mathstrut -\mathstrut \) \(26\) \(\beta_{13}\mathstrut +\mathstrut \) \(29\) \(\beta_{12}\mathstrut -\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(53\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(34\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(19\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(12\) \(\beta_{15}\mathstrut +\mathstrut \) \(31\) \(\beta_{14}\mathstrut -\mathstrut \) \(89\) \(\beta_{13}\mathstrut +\mathstrut \) \(21\) \(\beta_{12}\mathstrut -\mathstrut \) \(77\) \(\beta_{11}\mathstrut -\mathstrut \) \(107\) \(\beta_{10}\mathstrut -\mathstrut \) \(78\) \(\beta_{9}\mathstrut -\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut -\mathstrut \) \(77\) \(\beta_{6}\mathstrut +\mathstrut \) \(111\) \(\beta_{5}\mathstrut +\mathstrut \) \(22\) \(\beta_{4}\mathstrut +\mathstrut \) \(53\) \(\beta_{3}\mathstrut -\mathstrut \) \(21\) \(\beta_{2}\mathstrut +\mathstrut \) \(153\) \(\beta_{1}\mathstrut +\mathstrut \) \(66\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(102\) \(\beta_{15}\mathstrut -\mathstrut \) \(31\) \(\beta_{14}\mathstrut -\mathstrut \) \(356\) \(\beta_{13}\mathstrut +\mathstrut \) \(179\) \(\beta_{12}\mathstrut -\mathstrut \) \(158\) \(\beta_{11}\mathstrut -\mathstrut \) \(593\) \(\beta_{10}\mathstrut -\mathstrut \) \(97\) \(\beta_{9}\mathstrut -\mathstrut \) \(244\) \(\beta_{8}\mathstrut -\mathstrut \) \(74\) \(\beta_{7}\mathstrut -\mathstrut \) \(108\) \(\beta_{6}\mathstrut +\mathstrut \) \(159\) \(\beta_{5}\mathstrut -\mathstrut \) \(177\) \(\beta_{4}\mathstrut +\mathstrut \) \(32\) \(\beta_{3}\mathstrut -\mathstrut \) \(99\) \(\beta_{2}\mathstrut +\mathstrut \) \(112\) \(\beta_{1}\mathstrut +\mathstrut \) \(69\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(67\) \(\beta_{15}\mathstrut +\mathstrut \) \(396\) \(\beta_{14}\mathstrut -\mathstrut \) \(1679\) \(\beta_{13}\mathstrut +\mathstrut \) \(411\) \(\beta_{12}\mathstrut -\mathstrut \) \(757\) \(\beta_{11}\mathstrut -\mathstrut \) \(1647\) \(\beta_{10}\mathstrut -\mathstrut \) \(1303\) \(\beta_{9}\mathstrut +\mathstrut \) \(284\) \(\beta_{8}\mathstrut -\mathstrut \) \(86\) \(\beta_{7}\mathstrut -\mathstrut \) \(877\) \(\beta_{6}\mathstrut +\mathstrut \) \(1561\) \(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(508\) \(\beta_{3}\mathstrut -\mathstrut \) \(276\) \(\beta_{2}\mathstrut +\mathstrut \) \(1658\) \(\beta_{1}\mathstrut -\mathstrut \) \(104\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(322\) \(\beta_{15}\mathstrut +\mathstrut \) \(209\) \(\beta_{14}\mathstrut -\mathstrut \) \(4661\) \(\beta_{13}\mathstrut +\mathstrut \) \(1249\) \(\beta_{12}\mathstrut -\mathstrut \) \(1198\) \(\beta_{11}\mathstrut -\mathstrut \) \(6368\) \(\beta_{10}\mathstrut -\mathstrut \) \(2737\) \(\beta_{9}\mathstrut -\mathstrut \) \(1044\) \(\beta_{8}\mathstrut -\mathstrut \) \(1629\) \(\beta_{7}\mathstrut -\mathstrut \) \(3763\) \(\beta_{6}\mathstrut +\mathstrut \) \(1859\) \(\beta_{5}\mathstrut -\mathstrut \) \(1652\) \(\beta_{4}\mathstrut +\mathstrut \) \(1272\) \(\beta_{3}\mathstrut -\mathstrut \) \(399\) \(\beta_{2}\mathstrut +\mathstrut \) \(3612\) \(\beta_{1}\mathstrut +\mathstrut \) \(659\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(722\) \(\beta_{15}\mathstrut +\mathstrut \) \(4821\) \(\beta_{14}\mathstrut -\mathstrut \) \(22869\) \(\beta_{13}\mathstrut +\mathstrut \) \(6081\) \(\beta_{12}\mathstrut -\mathstrut \) \(6637\) \(\beta_{11}\mathstrut -\mathstrut \) \(22292\) \(\beta_{10}\mathstrut -\mathstrut \) \(17653\) \(\beta_{9}\mathstrut +\mathstrut \) \(5054\) \(\beta_{8}\mathstrut -\mathstrut \) \(1361\) \(\beta_{7}\mathstrut -\mathstrut \) \(11897\) \(\beta_{6}\mathstrut +\mathstrut \) \(17361\) \(\beta_{5}\mathstrut -\mathstrut \) \(2488\) \(\beta_{4}\mathstrut +\mathstrut \) \(4463\) \(\beta_{3}\mathstrut -\mathstrut \) \(2791\) \(\beta_{2}\mathstrut +\mathstrut \) \(17423\) \(\beta_{1}\mathstrut -\mathstrut \) \(6464\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(4648\) \(\beta_{15}\mathstrut +\mathstrut \) \(10089\) \(\beta_{14}\mathstrut -\mathstrut \) \(61441\) \(\beta_{13}\mathstrut +\mathstrut \) \(10719\) \(\beta_{12}\mathstrut -\mathstrut \) \(10283\) \(\beta_{11}\mathstrut -\mathstrut \) \(69933\) \(\beta_{10}\mathstrut -\mathstrut \) \(49987\) \(\beta_{9}\mathstrut +\mathstrut \) \(6701\) \(\beta_{8}\mathstrut -\mathstrut \) \(21274\) \(\beta_{7}\mathstrut -\mathstrut \) \(64498\) \(\beta_{6}\mathstrut +\mathstrut \) \(23899\) \(\beta_{5}\mathstrut -\mathstrut \) \(13622\) \(\beta_{4}\mathstrut +\mathstrut \) \(20527\) \(\beta_{3}\mathstrut +\mathstrut \) \(561\) \(\beta_{2}\mathstrut +\mathstrut \) \(58037\) \(\beta_{1}\mathstrut +\mathstrut \) \(1354\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(17042\) \(\beta_{15}\mathstrut +\mathstrut \) \(61161\) \(\beta_{14}\mathstrut -\mathstrut \) \(277069\) \(\beta_{13}\mathstrut +\mathstrut \) \(72921\) \(\beta_{12}\mathstrut -\mathstrut \) \(50587\) \(\beta_{11}\mathstrut -\mathstrut \) \(273867\) \(\beta_{10}\mathstrut -\mathstrut \) \(221948\) \(\beta_{9}\mathstrut +\mathstrut \) \(71349\) \(\beta_{8}\mathstrut -\mathstrut \) \(28166\) \(\beta_{7}\mathstrut -\mathstrut \) \(175417\) \(\beta_{6}\mathstrut +\mathstrut \) \(166606\) \(\beta_{5}\mathstrut -\mathstrut \) \(42628\) \(\beta_{4}\mathstrut +\mathstrut \) \(45218\) \(\beta_{3}\mathstrut -\mathstrut \) \(18106\) \(\beta_{2}\mathstrut +\mathstrut \) \(182258\) \(\beta_{1}\mathstrut -\mathstrut \) \(101659\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(134263\) \(\beta_{15}\mathstrut +\mathstrut \) \(202349\) \(\beta_{14}\mathstrut -\mathstrut \) \(797386\) \(\beta_{13}\mathstrut +\mathstrut \) \(111324\) \(\beta_{12}\mathstrut -\mathstrut \) \(86558\) \(\beta_{11}\mathstrut -\mathstrut \) \(792838\) \(\beta_{10}\mathstrut -\mathstrut \) \(749067\) \(\beta_{9}\mathstrut +\mathstrut \) \(261221\) \(\beta_{8}\mathstrut -\mathstrut \) \(241464\) \(\beta_{7}\mathstrut -\mathstrut \) \(907323\) \(\beta_{6}\mathstrut +\mathstrut \) \(302154\) \(\beta_{5}\mathstrut -\mathstrut \) \(100422\) \(\beta_{4}\mathstrut +\mathstrut \) \(265777\) \(\beta_{3}\mathstrut +\mathstrut \) \(51356\) \(\beta_{2}\mathstrut +\mathstrut \) \(747912\) \(\beta_{1}\mathstrut -\mathstrut \) \(106706\)\()/5\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(378617\) \(\beta_{15}\mathstrut +\mathstrut \) \(805976\) \(\beta_{14}\mathstrut -\mathstrut \) \(3193009\) \(\beta_{13}\mathstrut +\mathstrut \) \(749601\) \(\beta_{12}\mathstrut -\mathstrut \) \(291027\) \(\beta_{11}\mathstrut -\mathstrut \) \(3144842\) \(\beta_{10}\mathstrut -\mathstrut \) \(2726573\) \(\beta_{9}\mathstrut +\mathstrut \) \(1034699\) \(\beta_{8}\mathstrut -\mathstrut \) \(507941\) \(\beta_{7}\mathstrut -\mathstrut \) \(2581267\) \(\beta_{6}\mathstrut +\mathstrut \) \(1434546\) \(\beta_{5}\mathstrut -\mathstrut \) \(498313\) \(\beta_{4}\mathstrut +\mathstrut \) \(553503\) \(\beta_{3}\mathstrut +\mathstrut \) \(29859\) \(\beta_{2}\mathstrut +\mathstrut \) \(1963313\) \(\beta_{1}\mathstrut -\mathstrut \) \(1309799\)\()/5\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(2303643\) \(\beta_{15}\mathstrut +\mathstrut \) \(3165794\) \(\beta_{14}\mathstrut -\mathstrut \) \(9996001\) \(\beta_{13}\mathstrut +\mathstrut \) \(1254869\) \(\beta_{12}\mathstrut -\mathstrut \) \(527138\) \(\beta_{11}\mathstrut -\mathstrut \) \(9072163\) \(\beta_{10}\mathstrut -\mathstrut \) \(10035822\) \(\beta_{9}\mathstrut +\mathstrut \) \(4755751\) \(\beta_{8}\mathstrut -\mathstrut \) \(2651744\) \(\beta_{7}\mathstrut -\mathstrut \) \(11712898\) \(\beta_{6}\mathstrut +\mathstrut \) \(3497229\) \(\beta_{5}\mathstrut -\mathstrut \) \(655547\) \(\beta_{4}\mathstrut +\mathstrut \) \(3100882\) \(\beta_{3}\mathstrut +\mathstrut \) \(1093781\) \(\beta_{2}\mathstrut +\mathstrut \) \(8564457\) \(\beta_{1}\mathstrut -\mathstrut \) \(2885261\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(-\)\(6955377\) \(\beta_{15}\mathstrut +\mathstrut \) \(10771406\) \(\beta_{14}\mathstrut -\mathstrut \) \(35942739\) \(\beta_{13}\mathstrut +\mathstrut \) \(6786036\) \(\beta_{12}\mathstrut -\mathstrut \) \(208367\) \(\beta_{11}\mathstrut -\mathstrut \) \(34294982\) \(\beta_{10}\mathstrut -\mathstrut \) \(33244988\) \(\beta_{9}\mathstrut +\mathstrut \) \(15468999\) \(\beta_{8}\mathstrut -\mathstrut \) \(7800891\) \(\beta_{7}\mathstrut -\mathstrut \) \(36336197\) \(\beta_{6}\mathstrut +\mathstrut \) \(11168716\) \(\beta_{5}\mathstrut -\mathstrut \) \(4560408\) \(\beta_{4}\mathstrut +\mathstrut \) \(7312018\) \(\beta_{3}\mathstrut +\mathstrut \) \(3446899\) \(\beta_{2}\mathstrut +\mathstrut \) \(21728638\) \(\beta_{1}\mathstrut -\mathstrut \) \(16212534\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(33676153\) \(\beta_{15}\mathstrut +\mathstrut \) \(43741979\) \(\beta_{14}\mathstrut -\mathstrut \) \(119837181\) \(\beta_{13}\mathstrut +\mathstrut \) \(13712654\) \(\beta_{12}\mathstrut +\mathstrut \) \(1164252\) \(\beta_{11}\mathstrut -\mathstrut \) \(101919263\) \(\beta_{10}\mathstrut -\mathstrut \) \(125083137\) \(\beta_{9}\mathstrut +\mathstrut \) \(70977541\) \(\beta_{8}\mathstrut -\mathstrut \) \(29532179\) \(\beta_{7}\mathstrut -\mathstrut \) \(145033883\) \(\beta_{6}\mathstrut +\mathstrut \) \(35437464\) \(\beta_{5}\mathstrut -\mathstrut \) \(3225587\) \(\beta_{4}\mathstrut +\mathstrut \) \(34156632\) \(\beta_{3}\mathstrut +\mathstrut \) \(18966791\) \(\beta_{2}\mathstrut +\mathstrut \) \(90702602\) \(\beta_{1}\mathstrut -\mathstrut \) \(51277636\)\()/5\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.80334 0.309017i
3.42137 0.309017i
2.32349 + 0.309017i
−0.705457 + 0.309017i
−1.80334 + 0.309017i
3.42137 + 0.309017i
2.32349 0.309017i
−0.705457 0.309017i
2.17199 0.809017i
−2.79002 0.809017i
0.543374 + 0.809017i
−1.16141 + 0.809017i
2.17199 + 0.809017i
−2.79002 + 0.809017i
0.543374 0.809017i
−1.16141 0.809017i
−0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i −1.73558 + 1.40988i −0.809017 0.587785i 2.61995i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.21496 1.87720i
19.2 −0.951057 + 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i 2.23558 0.0466062i −0.809017 0.587785i 3.52206i −0.587785 + 0.809017i −0.309017 + 0.951057i −2.11176 + 0.735158i
19.3 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i −1.47959 1.67655i −0.809017 0.587785i 3.23143i 0.587785 0.809017i −0.309017 + 0.951057i −1.92526 1.13727i
19.4 0.951057 0.309017i −0.587785 0.809017i 0.809017 0.587785i 1.97959 + 1.03982i −0.809017 0.587785i 0.329315i 0.587785 0.809017i −0.309017 + 0.951057i 2.20402 + 0.377200i
79.1 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i −1.73558 1.40988i −0.809017 + 0.587785i 2.61995i −0.587785 0.809017i −0.309017 0.951057i 1.21496 + 1.87720i
79.2 −0.951057 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i 2.23558 + 0.0466062i −0.809017 + 0.587785i 3.52206i −0.587785 0.809017i −0.309017 0.951057i −2.11176 0.735158i
79.3 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i −1.47959 + 1.67655i −0.809017 + 0.587785i 3.23143i 0.587785 + 0.809017i −0.309017 0.951057i −1.92526 + 1.13727i
79.4 0.951057 + 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 1.97959 1.03982i −0.809017 + 0.587785i 0.329315i 0.587785 + 0.809017i −0.309017 0.951057i 2.20402 0.377200i
109.1 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i −1.53938 1.62182i 0.309017 0.951057i 4.63137i 0.951057 + 0.309017i 0.809017 0.587785i 2.21691 0.292102i
109.2 −0.587785 + 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i 2.03938 0.917020i 0.309017 0.951057i 4.80694i 0.951057 + 0.309017i 0.809017 0.587785i −0.456833 + 2.18890i
109.3 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i −1.36682 1.76969i 0.309017 0.951057i 0.533559i −0.951057 0.309017i 0.809017 0.587785i −2.23511 + 0.0655797i
109.4 0.587785 0.809017i 0.951057 0.309017i −0.309017 0.951057i 1.86682 + 1.23085i 0.309017 0.951057i 2.70913i −0.951057 0.309017i 0.809017 0.587785i 2.09307 0.786811i
139.1 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i −1.53938 + 1.62182i 0.309017 + 0.951057i 4.63137i 0.951057 0.309017i 0.809017 + 0.587785i 2.21691 + 0.292102i
139.2 −0.587785 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i 2.03938 + 0.917020i 0.309017 + 0.951057i 4.80694i 0.951057 0.309017i 0.809017 + 0.587785i −0.456833 2.18890i
139.3 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i −1.36682 + 1.76969i 0.309017 + 0.951057i 0.533559i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.23511 0.0655797i
139.4 0.587785 + 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 1.86682 1.23085i 0.309017 + 0.951057i 2.70913i −0.951057 + 0.309017i 0.809017 + 0.587785i 2.09307 + 0.786811i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
25.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(150, [\chi])\).