Properties

Label 385.2.i.c
Level $385$
Weight $2$
Character orbit 385.i
Analytic conductor $3.074$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 3 x^{15} + 17 x^{14} - 28 x^{13} + 127 x^{12} - 178 x^{11} + 612 x^{10} - 527 x^{9} + 1556 x^{8} - 1065 x^{7} + 2812 x^{6} - 1224 x^{5} + 2359 x^{4} - 1154 x^{3} + 1351 x^{2} - 333 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -\beta_{4} + \beta_{7} ) q^{3} + ( -\beta_{8} + \beta_{11} ) q^{4} + ( -1 + \beta_{8} ) q^{5} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} + \beta_{15} ) q^{6} + ( -\beta_{9} - \beta_{15} ) q^{7} + ( 1 - \beta_{3} - \beta_{4} + \beta_{6} ) q^{8} + ( -3 - \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -\beta_{4} + \beta_{7} ) q^{3} + ( -\beta_{8} + \beta_{11} ) q^{4} + ( -1 + \beta_{8} ) q^{5} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} + \beta_{15} ) q^{6} + ( -\beta_{9} - \beta_{15} ) q^{7} + ( 1 - \beta_{3} - \beta_{4} + \beta_{6} ) q^{8} + ( -3 - \beta_{2} + \beta_{3} + \beta_{6} + 2 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{9} + ( \beta_{1} + \beta_{5} ) q^{10} + \beta_{8} q^{11} + ( \beta_{2} - \beta_{3} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{12} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} + \beta_{15} ) q^{14} + \beta_{4} q^{15} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{16} + ( 1 - 2 \beta_{1} - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{17} + ( 1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{5} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{15} ) q^{18} + ( \beta_{1} + \beta_{2} + \beta_{9} - \beta_{13} + \beta_{14} ) q^{19} + ( 1 - \beta_{3} ) q^{20} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{11} + \beta_{14} + \beta_{15} ) q^{21} + \beta_{5} q^{22} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{23} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} + \beta_{15} ) q^{24} -\beta_{8} q^{25} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + 2 \beta_{8} - \beta_{11} + \beta_{15} ) q^{26} + ( -1 + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{27} + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} ) q^{28} + ( 4 - \beta_{2} + \beta_{5} + \beta_{9} + \beta_{12} + \beta_{15} ) q^{29} + ( 2 - 2 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{30} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} + \beta_{13} ) q^{31} + ( -2 \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{32} + \beta_{7} q^{33} + ( -3 + \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{9} - 2 \beta_{13} - 2 \beta_{15} ) q^{34} + \beta_{15} q^{35} + ( 6 + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{12} - \beta_{13} - 3 \beta_{15} ) q^{36} + ( 2 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{14} - 3 \beta_{15} ) q^{37} + ( -2 + 2 \beta_{1} + 4 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{38} + ( -3 \beta_{4} + 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{39} + ( -\beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{40} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{12} - \beta_{15} ) q^{41} + ( 5 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{15} ) q^{42} + ( -1 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{43} + ( 1 - \beta_{3} - \beta_{8} + \beta_{11} ) q^{44} + ( 1 + \beta_{5} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{45} + ( -3 + 2 \beta_{1} - \beta_{5} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{46} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} + 3 \beta_{14} ) q^{47} + ( -3 - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{9} + 3 \beta_{13} + 3 \beta_{15} ) q^{48} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{49} -\beta_{5} q^{50} + ( -4 + 3 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} + 4 \beta_{13} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} - 5 \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{52} + ( -1 + 2 \beta_{1} + \beta_{5} - 3 \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{53} + ( -2 \beta_{2} - 6 \beta_{3} - 7 \beta_{7} - 2 \beta_{9} + 6 \beta_{11} + 2 \beta_{13} - 4 \beta_{14} ) q^{54} - q^{55} + ( -1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{56} + ( 3 - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{12} + \beta_{13} - \beta_{15} ) q^{57} + ( 2 - 5 \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{58} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{59} + ( -1 - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{60} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} + 3 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{61} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{9} - \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{62} + ( -2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} - 4 \beta_{14} + 4 \beta_{15} ) q^{63} + ( 1 - \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + \beta_{9} + 2 \beta_{12} - \beta_{13} + \beta_{15} ) q^{64} + ( -2 + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{14} + \beta_{15} ) q^{65} + ( 1 - 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{66} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{67} + ( -8 + 7 \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{6} + 3 \beta_{8} + \beta_{9} + 5 \beta_{10} - \beta_{11} + 4 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{68} + ( -2 - 3 \beta_{2} - \beta_{4} + 2 \beta_{6} + 3 \beta_{9} - 2 \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{69} + ( 3 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} ) q^{70} + ( -\beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} - 2 \beta_{12} - 2 \beta_{15} ) q^{71} + ( -4 - \beta_{1} + 5 \beta_{3} + \beta_{6} + 8 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - 5 \beta_{11} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{72} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} - \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{73} + ( -\beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{74} -\beta_{7} q^{75} + ( -1 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{12} + \beta_{13} + 3 \beta_{15} ) q^{76} -\beta_{9} q^{77} + ( -5 - 2 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{9} + 3 \beta_{12} + 3 \beta_{15} ) q^{78} + ( 4 - 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{79} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{80} + ( 4 + 2 \beta_{1} - \beta_{4} + 6 \beta_{5} + \beta_{7} - 5 \beta_{8} + 5 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} + \beta_{13} ) q^{81} + ( -2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{6} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} - 3 \beta_{13} - \beta_{14} + \beta_{15} ) q^{82} + ( 5 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{9} - \beta_{12} + \beta_{13} ) q^{83} + ( 6 - \beta_{2} - 4 \beta_{3} - 4 \beta_{4} - \beta_{5} + 3 \beta_{7} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{84} + ( 1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{12} - \beta_{13} ) q^{85} + ( -6 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{6} + 5 \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{86} + ( -3 \beta_{1} - \beta_{2} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} - \beta_{14} ) q^{87} + ( 1 - \beta_{1} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{88} + ( 1 + \beta_{2} + 2 \beta_{3} + 4 \beta_{7} - \beta_{8} + 4 \beta_{9} - 2 \beta_{11} - 4 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{89} + ( 3 + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{9} - 2 \beta_{13} - 2 \beta_{15} ) q^{90} + ( -\beta_{2} + 3 \beta_{3} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{91} + ( 2 - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 3 \beta_{9} + 3 \beta_{13} + 3 \beta_{15} ) q^{92} + ( 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{7} - 2 \beta_{9} + 4 \beta_{11} + 2 \beta_{13} - 3 \beta_{15} ) q^{93} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} + \beta_{5} - 6 \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{10} + 3 \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{94} + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{9} + \beta_{12} - \beta_{14} ) q^{95} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{13} ) q^{96} + ( \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{9} - \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{97} + ( 4 - 2 \beta_{1} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} ) q^{98} + ( -2 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{12} + \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 3q^{2} - q^{3} - 9q^{4} - 8q^{5} - 6q^{6} - q^{7} + 18q^{8} - 19q^{9} + O(q^{10}) \) \( 16q - 3q^{2} - q^{3} - 9q^{4} - 8q^{5} - 6q^{6} - q^{7} + 18q^{8} - 19q^{9} - 3q^{10} + 8q^{11} - 9q^{12} + 28q^{13} - 9q^{14} + 2q^{15} - 7q^{16} - 5q^{17} - 27q^{18} - q^{19} + 18q^{20} - 18q^{21} - 6q^{22} + 2q^{23} + 24q^{24} - 8q^{25} - 21q^{26} - 10q^{27} + 32q^{28} + 52q^{29} + 3q^{30} - 2q^{31} - 16q^{32} + q^{33} - 52q^{34} + 5q^{35} + 108q^{36} + q^{37} + 31q^{38} - 19q^{39} - 9q^{40} - 6q^{41} + 44q^{42} + 8q^{43} + 9q^{44} - 19q^{45} - 10q^{46} - q^{47} - 42q^{48} + 17q^{49} + 6q^{50} - 3q^{51} - 37q^{52} - 26q^{53} + 5q^{54} - 16q^{55} + 40q^{57} + q^{58} + 19q^{59} - 9q^{60} - 52q^{62} - 21q^{63} + 2q^{64} - 14q^{65} - 3q^{66} + 13q^{67} - 15q^{68} - 28q^{69} + 15q^{70} - 18q^{71} - 32q^{72} - 11q^{73} - 24q^{74} - q^{75} - 36q^{76} + 4q^{77} - 66q^{78} + 8q^{79} - 7q^{80} - 52q^{81} - 41q^{82} + 64q^{83} + 138q^{84} + 10q^{85} - 28q^{86} + 16q^{87} + 9q^{88} - 5q^{89} + 54q^{90} + 13q^{91} + 60q^{92} + 14q^{93} + 5q^{94} - q^{95} - q^{96} + 18q^{97} + 22q^{98} - 38q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} + 17 x^{14} - 28 x^{13} + 127 x^{12} - 178 x^{11} + 612 x^{10} - 527 x^{9} + 1556 x^{8} - 1065 x^{7} + 2812 x^{6} - 1224 x^{5} + 2359 x^{4} - 1154 x^{3} + 1351 x^{2} - 333 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(367626137496199 \nu^{15} + 22631474046638535 \nu^{14} - 52199742669658264 \nu^{13} + 351586930335553709 \nu^{12} - 374766751443962033 \nu^{11} + 2492714627655588149 \nu^{10} - 2150271080660780151 \nu^{9} + 11280851303594602768 \nu^{8} - 2881465622615418835 \nu^{7} + 25504794964966580253 \nu^{6} - 2601436697636747099 \nu^{5} + 41925436761303602253 \nu^{4} + 5747231822986549162 \nu^{3} + 24943611495551616223 \nu^{2} - 8905349555394001295 \nu + 12431796417869096070\)\()/ 3778452172261009422 \)
\(\beta_{3}\)\(=\)\((\)\(178336902592334 \nu^{15} + 477682850946139 \nu^{14} - 247188530701317 \nu^{13} + 12137664109911064 \nu^{12} - 6654964242818651 \nu^{11} + 89434448502765279 \nu^{10} - 73701315221610789 \nu^{9} + 477218131591865297 \nu^{8} - 228329506151272276 \nu^{7} + 1091510443687130179 \nu^{6} - 461339799876304613 \nu^{5} + 2333509244282819009 \nu^{4} - 542991074262630363 \nu^{3} + 1894143857464715546 \nu^{2} - 480829959382998285 \nu + 3816392733354725289\)\()/ 1259484057420336474 \)
\(\beta_{4}\)\(=\)\((\)\(100841646682033 \nu^{15} + 1131758365870985 \nu^{14} - 1472065735696314 \nu^{13} + 22141054955921807 \nu^{12} - 19130182470442783 \nu^{11} + 182670409275509469 \nu^{10} - 138291979626666300 \nu^{9} + 943306810710881266 \nu^{8} - 473777395960976375 \nu^{7} + 2676998136922459646 \nu^{6} - 919666637244686167 \nu^{5} + 4514900555121909385 \nu^{4} - 1188338065075779609 \nu^{3} + 3652397015378340307 \nu^{2} - 927000991167085971 \nu + 432978540466621161\)\()/ 629742028710168237 \)
\(\beta_{5}\)\(=\)\((\)\(-468401988811307 \nu^{15} + 1583542869026255 \nu^{14} - 7485150958846080 \nu^{13} + 12868067156015279 \nu^{12} - 47349388469124925 \nu^{11} + 76720589765593995 \nu^{10} - 197227568649754605 \nu^{9} + 173146532881948000 \nu^{8} - 251615362998528395 \nu^{7} + 270518611932769679 \nu^{6} - 225635948850265105 \nu^{5} + 111984234428735155 \nu^{4} + 1228548952676945796 \nu^{3} - 2455179174382085 \nu^{2} + 1848713160303315 \nu - 324852097108833054\)\()/ 1259484057420336474 \)
\(\beta_{6}\)\(=\)\((\)\(-1493587759288828 \nu^{15} + 9075371058793129 \nu^{14} - 33131923837478265 \nu^{13} + 107892042645815794 \nu^{12} - 234312883060203917 \nu^{11} + 761657626116160197 \nu^{10} - 1139195549073961809 \nu^{9} + 3056417884541419829 \nu^{8} - 2182345750067338606 \nu^{7} + 7527581165263128187 \nu^{6} - 3203216869766737367 \nu^{5} + 11811247292241578399 \nu^{4} + 735044548873257129 \nu^{3} + 9189117171523867820 \nu^{2} - 2327437089075956967 \nu + 2123457368432298921\)\()/ 1259484057420336474 \)
\(\beta_{7}\)\(=\)\((\)\(4916606183538715 \nu^{15} - 5619347867963844 \nu^{14} + 58155579655316312 \nu^{13} + 10015878590659691 \nu^{12} + 393309546725737132 \nu^{11} + 192751529357723849 \nu^{10} + 1567098474141707730 \nu^{9} + 2266565220568136929 \nu^{8} + 3554694907018038101 \nu^{7} + 5806298951984735370 \nu^{6} + 6138576773994425824 \nu^{5} + 10591551401704201689 \nu^{4} + 3662300355263175559 \nu^{3} + 2667406009375474513 \nu^{2} + 1388331098153583619 \nu + 274596160012850241\)\()/ 1889226086130504711 \)
\(\beta_{8}\)\(=\)\((\)\(-12031559152179002 \nu^{15} + 37499883422970927 \nu^{14} - 209287134194121799 \nu^{13} + 359339109137550296 \nu^{12} - 1566612213794779091 \nu^{11} + 2283665694495237131 \nu^{10} - 7593475970430331209 \nu^{9} + 6932314379147597869 \nu^{8} - 19240545639436371112 \nu^{7} + 13568456586066222315 \nu^{6} - 34644300171725662661 \nu^{5} + 15403536248817893763 \nu^{4} - 28718400743276471183 \nu^{3} + 10198772403583730920 \nu^{2} - 16247270877070685447 \nu + 4000963058194697721\)\()/ 3778452172261009422 \)
\(\beta_{9}\)\(=\)\((\)\(12586088327605585 \nu^{15} - 46256907117800487 \nu^{14} + 214058340904189184 \nu^{13} - 431697898210792195 \nu^{12} + 1459840549145108329 \nu^{11} - 2802057150153154225 \nu^{10} + 6571784011242209979 \nu^{9} - 8799044191925373416 \nu^{8} + 12564860647294240043 \nu^{7} - 20282204016554622525 \nu^{6} + 18825608428016736631 \nu^{5} - 30844548827464738329 \nu^{4} + 607387038607220716 \nu^{3} - 30895122813375897551 \nu^{2} + 5226961829392474843 \nu - 11995056882973354536\)\()/ 3778452172261009422 \)
\(\beta_{10}\)\(=\)\((\)\(-2943331604034800 \nu^{15} + 7413559209819278 \nu^{14} - 48122615099937959 \nu^{13} + 62938404824280609 \nu^{12} - 371073573632202290 \nu^{11} + 383821912677862137 \nu^{10} - 1808417466864290509 \nu^{9} + 906658210953875900 \nu^{8} - 4968862020924416811 \nu^{7} + 1481088621940220677 \nu^{6} - 9051530443435233842 \nu^{5} + 228486884733022177 \nu^{4} - 8554222741919966347 \nu^{3} + 391107611254047069 \nu^{2} - 4015961089211707931 \nu - 337894983353964609\)\()/ 629742028710168237 \)
\(\beta_{11}\)\(=\)\((\)\(-12031559152179002 \nu^{15} + 37499883422970927 \nu^{14} - 209287134194121799 \nu^{13} + 359339109137550296 \nu^{12} - 1566612213794779091 \nu^{11} + 2283665694495237131 \nu^{10} - 7593475970430331209 \nu^{9} + 6932314379147597869 \nu^{8} - 19240545639436371112 \nu^{7} + 13568456586066222315 \nu^{6} - 34644300171725662661 \nu^{5} + 15403536248817893763 \nu^{4} - 28718400743276471183 \nu^{3} + 11458256461004067394 \nu^{2} - 16247270877070685447 \nu + 4000963058194697721\)\()/ 1259484057420336474 \)
\(\beta_{12}\)\(=\)\((\)\(8551933388804287 \nu^{15} - 27999095694132641 \nu^{14} + 151700972031696786 \nu^{13} - 279944876026973057 \nu^{12} + 1145009852598203872 \nu^{11} - 1835251024637387546 \nu^{10} + 5590015036814905375 \nu^{9} - 6051584515173431208 \nu^{8} + 14244454649845864969 \nu^{7} - 13289691934529389864 \nu^{6} + 25250113854129857509 \nu^{5} - 18053440211641491034 \nu^{4} + 19909889364336318408 \nu^{3} - 16463172007165909627 \nu^{2} + 10068015786875302426 \nu - 4821061556606882457\)\()/ 629742028710168237 \)
\(\beta_{13}\)\(=\)\((\)\(17844671201706099 \nu^{15} - 48892044946671065 \nu^{14} + 290294010690448524 \nu^{13} - 425209591010347949 \nu^{12} + 2155241155931266597 \nu^{11} - 2650966280119342183 \nu^{10} + 10218177118037599349 \nu^{9} - 6997193887325336312 \nu^{8} + 25807128071786540205 \nu^{7} - 13610196551544816171 \nu^{6} + 45481728974185123603 \nu^{5} - 12179013888022652191 \nu^{4} + 35317599283340874048 \nu^{3} - 13635820408674512305 \nu^{2} + 15240850254840575129 \nu - 101073978047563680\)\()/ 1259484057420336474 \)
\(\beta_{14}\)\(=\)\((\)\(-19072967415584315 \nu^{15} + 46885279352668005 \nu^{14} - 289035568562647636 \nu^{13} + 348311085105864995 \nu^{12} - 2077500223607694245 \nu^{11} + 2026159022110893629 \nu^{10} - 9502594232703513045 \nu^{9} + 3499366891940833900 \nu^{8} - 23086766823411052261 \nu^{7} + 4797232759339315155 \nu^{6} - 41626034586830237375 \nu^{5} - 2067173539089385767 \nu^{4} - 32450804788505853170 \nu^{3} + 5924263032401317105 \nu^{2} - 17743579508658246707 \nu - 176109790165449390\)\()/ 1259484057420336474 \)
\(\beta_{15}\)\(=\)\((\)\(-19677115046354819 \nu^{15} + 56343566467123111 \nu^{14} - 323088510209792454 \nu^{13} + 498638151733388755 \nu^{12} - 2368838723336878595 \nu^{11} + 3092018617867607581 \nu^{10} - 11133287221234727999 \nu^{9} + 8325054805656124938 \nu^{8} - 27175161991574356349 \nu^{7} + 15594961112178206327 \nu^{6} - 47018297165226522821 \nu^{5} + 15482509079549212217 \nu^{4} - 34429850265942046872 \nu^{3} + 15912026900947961645 \nu^{2} - 15813346891594526267 \nu + 4020108277953603066\)\()/ 1259484057420336474 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} - 3 \beta_{8}\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{14} - \beta_{13} - 7 \beta_{11} + \beta_{9} + 14 \beta_{8} + \beta_{7} + 7 \beta_{3} + \beta_{2} - \beta_{1} - 14\)
\(\nu^{5}\)\(=\)\(-\beta_{15} + \beta_{14} - 7 \beta_{13} - \beta_{12} - 8 \beta_{11} - 8 \beta_{10} + \beta_{9} + 10 \beta_{8} + 8 \beta_{7} - 20 \beta_{5} - 8 \beta_{4} - 28 \beta_{1} + 8\)
\(\nu^{6}\)\(=\)\(\beta_{15} + 9 \beta_{13} - 8 \beta_{12} + \beta_{9} + 2 \beta_{6} - 10 \beta_{5} - 10 \beta_{4} - 44 \beta_{3} - \beta_{2} + 77\)
\(\nu^{7}\)\(=\)\(10 \beta_{15} - 10 \beta_{14} + 54 \beta_{13} + 57 \beta_{11} + 53 \beta_{10} - \beta_{9} - 77 \beta_{8} - 53 \beta_{7} + 53 \beta_{6} - 57 \beta_{3} - 11 \beta_{2} + 164 \beta_{1} + 24\)
\(\nu^{8}\)\(=\)\(\beta_{15} - 53 \beta_{14} + 15 \beta_{13} + 53 \beta_{12} + 272 \beta_{11} + 27 \beta_{10} - 64 \beta_{9} - 452 \beta_{8} - 76 \beta_{7} + 81 \beta_{5} + 76 \beta_{4} - 52 \beta_{2} + 108 \beta_{1} - 27\)
\(\nu^{9}\)\(=\)\(-15 \beta_{15} - 91 \beta_{13} + 76 \beta_{12} - 75 \beta_{9} - 335 \beta_{6} + 648 \beta_{5} + 334 \beta_{4} + 393 \beta_{3} + 75 \beta_{2} - 551\)
\(\nu^{10}\)\(=\)\(-106 \beta_{15} + 334 \beta_{14} - 576 \beta_{13} - 1679 \beta_{11} - 256 \beta_{10} + 320 \beta_{9} + 2730 \beta_{8} + 527 \beta_{7} - 256 \beta_{6} + 1679 \beta_{3} + 426 \beta_{2} - 866 \beta_{1} - 2474\)
\(\nu^{11}\)\(=\)\(-363 \beta_{15} + 527 \beta_{14} - 1578 \beta_{13} - 527 \beta_{12} - 2660 \beta_{11} - 2096 \beta_{10} + 682 \beta_{9} + 3822 \beta_{8} + 2070 \beta_{7} - 3882 \beta_{5} - 2070 \beta_{4} + 164 \beta_{2} - 5978 \beta_{1} + 2096\)
\(\nu^{12}\)\(=\)\(708 \beta_{15} + 2778 \beta_{13} - 2070 \beta_{12} + 837 \beta_{9} + 2109 \beta_{6} - 4412 \beta_{5} - 3515 \beta_{4} - 10389 \beta_{3} - 837 \beta_{2} + 16730\)
\(\nu^{13}\)\(=\)\(3486 \beta_{15} - 3515 \beta_{14} + 14539 \beta_{13} + 17776 \beta_{11} + 13138 \beta_{10} - 1401 \beta_{9} - 26036 \beta_{8} - 12761 \beta_{7} + 13138 \beta_{6} - 17776 \beta_{3} - 4887 \beta_{2} + 36742 \beta_{1} + 12898\)
\(\nu^{14}\)\(=\)\(995 \beta_{15} - 12761 \beta_{14} + 9902 \beta_{13} + 12761 \beta_{12} + 64506 \beta_{11} + 16161 \beta_{10} - 18025 \beta_{9} - 103434 \beta_{8} - 22999 \beta_{7} + 31051 \beta_{5} + 22999 \beta_{4} - 11766 \beta_{2} + 47212 \beta_{1} - 16161\)
\(\nu^{15}\)\(=\)\(-11187 \beta_{15} - 34186 \beta_{13} + 22999 \beta_{12} - 23289 \beta_{9} - 82821 \beta_{6} + 144941 \beta_{5} + 78598 \beta_{4} + 117687 \beta_{3} + 23289 \beta_{2} - 175151\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(276\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
1.27511 + 2.20855i
1.25936 + 2.18128i
0.827576 + 1.43340i
0.420010 + 0.727479i
0.139605 + 0.241804i
−0.531161 0.919997i
−0.735245 1.27348i
−1.15525 2.00096i
1.27511 2.20855i
1.25936 2.18128i
0.827576 1.43340i
0.420010 0.727479i
0.139605 0.241804i
−0.531161 + 0.919997i
−0.735245 + 1.27348i
−1.15525 + 2.00096i
−1.27511 2.20855i −0.937301 + 1.62345i −2.25180 + 3.90023i −0.500000 0.866025i 4.78064 −2.44383 + 1.01376i 6.38473 −0.257068 0.445255i −1.27511 + 2.20855i
221.2 −1.25936 2.18128i 1.64549 2.85007i −2.17198 + 3.76198i −0.500000 0.866025i −8.28906 −0.160806 2.64086i 5.90377 −3.91527 6.78146i −1.25936 + 2.18128i
221.3 −0.827576 1.43340i −1.71605 + 2.97229i −0.369765 + 0.640452i −0.500000 0.866025i 5.68065 2.63215 + 0.267914i −2.08627 −4.38966 7.60311i −0.827576 + 1.43340i
221.4 −0.420010 0.727479i 0.864835 1.49794i 0.647183 1.12095i −0.500000 0.866025i −1.45296 −2.55075 0.702629i −2.76733 0.00412044 + 0.00713681i −0.420010 + 0.727479i
221.5 −0.139605 0.241804i −0.137980 + 0.238988i 0.961021 1.66454i −0.500000 0.866025i 0.0770509 1.16250 + 2.37668i −1.09508 1.46192 + 2.53213i −0.139605 + 0.241804i
221.6 0.531161 + 0.919997i 1.35728 2.35087i 0.435737 0.754718i −0.500000 0.866025i 2.88373 0.964079 + 2.46385i 3.05043 −2.18440 3.78348i 0.531161 0.919997i
221.7 0.735245 + 1.27348i −0.359468 + 0.622616i −0.0811705 + 0.140591i −0.500000 0.866025i −1.05719 2.24301 1.40318i 2.70226 1.24157 + 2.15046i 0.735245 1.27348i
221.8 1.15525 + 2.00096i −1.21680 + 2.10756i −1.66923 + 2.89118i −0.500000 0.866025i −5.62286 −2.34636 + 1.22254i −3.09251 −1.46121 2.53090i 1.15525 2.00096i
331.1 −1.27511 + 2.20855i −0.937301 1.62345i −2.25180 3.90023i −0.500000 + 0.866025i 4.78064 −2.44383 1.01376i 6.38473 −0.257068 + 0.445255i −1.27511 2.20855i
331.2 −1.25936 + 2.18128i 1.64549 + 2.85007i −2.17198 3.76198i −0.500000 + 0.866025i −8.28906 −0.160806 + 2.64086i 5.90377 −3.91527 + 6.78146i −1.25936 2.18128i
331.3 −0.827576 + 1.43340i −1.71605 2.97229i −0.369765 0.640452i −0.500000 + 0.866025i 5.68065 2.63215 0.267914i −2.08627 −4.38966 + 7.60311i −0.827576 1.43340i
331.4 −0.420010 + 0.727479i 0.864835 + 1.49794i 0.647183 + 1.12095i −0.500000 + 0.866025i −1.45296 −2.55075 + 0.702629i −2.76733 0.00412044 0.00713681i −0.420010 0.727479i
331.5 −0.139605 + 0.241804i −0.137980 0.238988i 0.961021 + 1.66454i −0.500000 + 0.866025i 0.0770509 1.16250 2.37668i −1.09508 1.46192 2.53213i −0.139605 0.241804i
331.6 0.531161 0.919997i 1.35728 + 2.35087i 0.435737 + 0.754718i −0.500000 + 0.866025i 2.88373 0.964079 2.46385i 3.05043 −2.18440 + 3.78348i 0.531161 + 0.919997i
331.7 0.735245 1.27348i −0.359468 0.622616i −0.0811705 0.140591i −0.500000 + 0.866025i −1.05719 2.24301 + 1.40318i 2.70226 1.24157 2.15046i 0.735245 + 1.27348i
331.8 1.15525 2.00096i −1.21680 2.10756i −1.66923 2.89118i −0.500000 + 0.866025i −5.62286 −2.34636 1.22254i −3.09251 −1.46121 + 2.53090i 1.15525 + 2.00096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.i.c 16
7.c even 3 1 inner 385.2.i.c 16
7.c even 3 1 2695.2.a.t 8
7.d odd 6 1 2695.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.i.c 16 1.a even 1 1 trivial
385.2.i.c 16 7.c even 3 1 inner
2695.2.a.s 8 7.d odd 6 1
2695.2.a.t 8 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 81 + 333 T + 1351 T^{2} + 1154 T^{3} + 2359 T^{4} + 1224 T^{5} + 2812 T^{6} + 1065 T^{7} + 1556 T^{8} + 527 T^{9} + 612 T^{10} + 178 T^{11} + 127 T^{12} + 28 T^{13} + 17 T^{14} + 3 T^{15} + T^{16} \)
$3$ \( 2304 + 11520 T + 47424 T^{2} + 63936 T^{3} + 83776 T^{4} + 34096 T^{5} + 42820 T^{6} + 13632 T^{7} + 14613 T^{8} + 2551 T^{9} + 2590 T^{10} + 319 T^{11} + 333 T^{12} + 21 T^{13} + 22 T^{14} + T^{15} + T^{16} \)
$5$ \( ( 1 + T + T^{2} )^{8} \)
$7$ \( 5764801 + 823543 T - 941192 T^{2} + 285719 T^{3} + 64827 T^{4} - 108045 T^{5} + 14504 T^{6} + 7203 T^{7} - 4808 T^{8} + 1029 T^{9} + 296 T^{10} - 315 T^{11} + 27 T^{12} + 17 T^{13} - 8 T^{14} + T^{15} + T^{16} \)
$11$ \( ( 1 - T + T^{2} )^{8} \)
$13$ \( ( -5119 + 34 T + 3956 T^{2} - 362 T^{3} - 802 T^{4} + 150 T^{5} + 44 T^{6} - 14 T^{7} + T^{8} )^{2} \)
$17$ \( 110502144 - 166005504 T + 347905728 T^{2} + 55244736 T^{3} + 143391072 T^{4} - 1885824 T^{5} + 26248228 T^{6} + 1737292 T^{7} + 2265809 T^{8} + 133525 T^{9} + 125452 T^{10} + 9247 T^{11} + 4287 T^{12} + 237 T^{13} + 90 T^{14} + 5 T^{15} + T^{16} \)
$19$ \( 256 + 256 T + 6720 T^{2} + 1216 T^{3} + 153600 T^{4} + 69728 T^{5} + 395828 T^{6} - 218912 T^{7} + 679065 T^{8} - 45048 T^{9} + 48610 T^{10} - 762 T^{11} + 2543 T^{12} - 18 T^{13} + 59 T^{14} + T^{15} + T^{16} \)
$23$ \( 20736 - 228096 T + 1844352 T^{2} - 6000768 T^{3} + 13971952 T^{4} - 18218576 T^{5} + 17132040 T^{6} - 6426616 T^{7} + 2400513 T^{8} - 476739 T^{9} + 146871 T^{10} - 22994 T^{11} + 5777 T^{12} - 406 T^{13} + 82 T^{14} - 2 T^{15} + T^{16} \)
$29$ \( ( 1296 + 50832 T - 64020 T^{2} + 27888 T^{3} - 4025 T^{4} - 500 T^{5} + 229 T^{6} - 26 T^{7} + T^{8} )^{2} \)
$31$ \( 2332600209 - 1788582801 T + 2308984453 T^{2} + 121933676 T^{3} + 495089554 T^{4} + 36266711 T^{5} + 63114715 T^{6} + 7049714 T^{7} + 4895979 T^{8} + 476373 T^{9} + 245385 T^{10} + 23612 T^{11} + 7480 T^{12} + 362 T^{13} + 100 T^{14} + 2 T^{15} + T^{16} \)
$37$ \( 125451972864 - 29865469440 T + 170161441216 T^{2} + 35702487296 T^{3} + 205428481824 T^{4} + 1265670096 T^{5} + 8766699156 T^{6} + 162981460 T^{7} + 259845521 T^{8} + 3775278 T^{9} + 3931294 T^{10} + 27646 T^{11} + 43187 T^{12} + 126 T^{13} + 251 T^{14} - T^{15} + T^{16} \)
$41$ \( ( -82944 + 145152 T - 65408 T^{2} - 5544 T^{3} + 6475 T^{4} - 227 T^{5} - 155 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$43$ \( ( 37189 - 8798 T - 46943 T^{2} - 8940 T^{3} + 4406 T^{4} + 486 T^{5} - 137 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$47$ \( 1220194181376 - 493422285312 T + 656857760832 T^{2} + 135676798848 T^{3} + 162031057920 T^{4} + 6053573520 T^{5} + 7064222916 T^{6} + 235757040 T^{7} + 210864517 T^{8} + 4391699 T^{9} + 3504670 T^{10} + 64807 T^{11} + 41865 T^{12} + 395 T^{13} + 244 T^{14} + T^{15} + T^{16} \)
$53$ \( 427993344 + 1689464832 T + 4639598848 T^{2} + 5947076864 T^{3} + 5329490768 T^{4} + 3164591328 T^{5} + 1406222864 T^{6} + 439519880 T^{7} + 108082113 T^{8} + 20084002 T^{9} + 3234609 T^{10} + 431498 T^{11} + 55532 T^{12} + 5626 T^{13} + 489 T^{14} + 26 T^{15} + T^{16} \)
$59$ \( 1460861412921 - 1616333894673 T + 1264266280300 T^{2} - 530720541499 T^{3} + 177549134436 T^{4} - 43091432324 T^{5} + 9401598632 T^{6} - 1682786062 T^{7} + 289687881 T^{8} - 40968638 T^{9} + 5442144 T^{10} - 561688 T^{11} + 56612 T^{12} - 4365 T^{13} + 368 T^{14} - 19 T^{15} + T^{16} \)
$61$ \( 416160000 - 653779200 T + 2503381504 T^{2} + 2812766464 T^{3} + 4681440016 T^{4} + 1407871280 T^{5} + 742211136 T^{6} - 61589984 T^{7} + 57498937 T^{8} - 2549757 T^{9} + 1387653 T^{10} - 52236 T^{11} + 24163 T^{12} - 446 T^{13} + 180 T^{14} + T^{16} \)
$67$ \( 1912137984 + 7207074048 T + 20671755328 T^{2} + 21191620416 T^{3} + 16059330816 T^{4} + 6958349808 T^{5} + 2373811252 T^{6} + 386012940 T^{7} + 70586281 T^{8} - 636908 T^{9} + 3012994 T^{10} - 167862 T^{11} + 37781 T^{12} - 1774 T^{13} + 307 T^{14} - 13 T^{15} + T^{16} \)
$71$ \( ( -1485459 - 2069865 T - 458987 T^{2} + 91227 T^{3} + 22680 T^{4} - 1723 T^{5} - 267 T^{6} + 9 T^{7} + T^{8} )^{2} \)
$73$ \( 457139606641 - 124373810192 T + 166820521068 T^{2} + 863910854 T^{3} + 34441019018 T^{4} - 1114097872 T^{5} + 2804059481 T^{6} - 130976559 T^{7} + 170592639 T^{8} - 9934436 T^{9} + 3716441 T^{10} - 7215 T^{11} + 39686 T^{12} + 508 T^{13} + 317 T^{14} + 11 T^{15} + T^{16} \)
$79$ \( 747685337344 - 2287037502464 T + 6425001926464 T^{2} - 2112405113984 T^{3} + 1024090782080 T^{4} - 29949750272 T^{5} + 59584459076 T^{6} - 4166713788 T^{7} + 1292514829 T^{8} - 55614513 T^{9} + 14852483 T^{10} - 613890 T^{11} + 109031 T^{12} - 2486 T^{13} + 410 T^{14} - 8 T^{15} + T^{16} \)
$83$ \( ( 307773 - 353729 T + 71978 T^{2} + 41996 T^{3} - 18070 T^{4} + 1377 T^{5} + 230 T^{6} - 32 T^{7} + T^{8} )^{2} \)
$89$ \( 10715415980481 - 8864671361019 T + 7067645927200 T^{2} - 2021556769333 T^{3} + 912711464202 T^{4} - 244862085964 T^{5} + 81615595150 T^{6} - 14613689384 T^{7} + 2495038031 T^{8} - 210394422 T^{9} + 21603866 T^{10} - 889418 T^{11} + 119926 T^{12} - 2659 T^{13} + 438 T^{14} + 5 T^{15} + T^{16} \)
$97$ \( ( 554768 + 2219232 T + 1179328 T^{2} - 688216 T^{3} + 61467 T^{4} + 4756 T^{5} - 526 T^{6} - 9 T^{7} + T^{8} )^{2} \)
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