Properties

Label 78.2.k.a
Level $78$
Weight $2$
Character orbit 78.k
Analytic conductor $0.623$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.9349208943630483456.9
Defining polynomial: \(x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + 972 x^{2} - 220 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + 972 x^{2} - 220 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1548 \nu^{15} - 13188 \nu^{14} + 76652 \nu^{13} - 318997 \nu^{12} + 1019601 \nu^{11} - 2665197 \nu^{10} + 5595909 \nu^{9} - 9912750 \nu^{8} + 14273792 \nu^{7} - 17359125 \nu^{6} + 16940634 \nu^{5} - 13645786 \nu^{4} + 8382900 \nu^{3} - 3952842 \nu^{2} + 1197759 \nu - 204955\)\()/17095\)
\(\beta_{2}\)\(=\)\((\)\(-3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + 4666596 \nu^{10} - 9554736 \nu^{9} + 15945783 \nu^{8} - 21928484 \nu^{7} + 24527176 \nu^{6} - 22138664 \nu^{5} + 15739188 \nu^{4} - 8598668 \nu^{3} + 3418428 \nu^{2} - 929924 \nu + 142555\)\()/17095\)
\(\beta_{3}\)\(=\)\((\)\(62 \nu^{14} - 434 \nu^{13} + 2541 \nu^{12} - 9604 \nu^{11} + 30137 \nu^{10} - 72992 \nu^{9} + 147747 \nu^{8} - 240948 \nu^{7} + 326020 \nu^{6} - 354283 \nu^{5} + 310030 \nu^{4} - 208129 \nu^{3} + 103708 \nu^{2} - 33855 \nu + 5945\)\()/65\)
\(\beta_{4}\)\(=\)\((\)\(-1548 \nu^{15} + 26338 \nu^{14} - 168702 \nu^{13} + 849994 \nu^{12} - 3008933 \nu^{11} + 8792571 \nu^{10} - 20191094 \nu^{9} + 38713617 \nu^{8} - 60027902 \nu^{7} + 77090107 \nu^{6} - 79360265 \nu^{5} + 65574610 \nu^{4} - 41344164 \nu^{3} + 19292843 \nu^{2} - 5858382 \nu + 979490\)\()/17095\)
\(\beta_{5}\)\(=\)\((\)\(15207 \nu^{15} - 93144 \nu^{14} + 525827 \nu^{13} - 1795712 \nu^{12} + 5254311 \nu^{11} - 11330101 \nu^{10} + 20458668 \nu^{9} - 28148715 \nu^{8} + 30952836 \nu^{7} - 24081648 \nu^{6} + 11893278 \nu^{5} - 253940 \nu^{4} - 4590360 \nu^{3} + 3846345 \nu^{2} - 1513502 \nu + 281130\)\()/17095\)
\(\beta_{6}\)\(=\)\((\)\(15977 \nu^{15} - 98393 \nu^{14} + 556086 \nu^{13} - 1902441 \nu^{12} + 5567337 \nu^{11} - 11970872 \nu^{10} + 21486123 \nu^{9} - 29032660 \nu^{8} + 30783293 \nu^{7} - 21374907 \nu^{6} + 6423019 \nu^{5} + 7076841 \nu^{4} - 11188252 \nu^{3} + 8068367 \nu^{2} - 3136623 \nu + 610830\)\()/17095\)
\(\beta_{7}\)\(=\)\((\)\(15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} - 32236863 \nu^{10} + 70027507 \nu^{9} - 125576015 \nu^{8} + 185376534 \nu^{7} - 225133368 \nu^{6} + 221440773 \nu^{5} - 173497381 \nu^{4} + 104335813 \nu^{3} - 45658062 \nu^{2} + 12999216 \nu - 1883725\)\()/17095\)
\(\beta_{8}\)\(=\)\((\)\(15207 \nu^{15} - 151267 \nu^{14} + 932688 \nu^{13} - 4151403 \nu^{12} + 14099264 \nu^{11} - 38692358 \nu^{10} + 85888071 \nu^{9} - 157951050 \nu^{8} + 238257063 \nu^{7} - 296346875 \nu^{6} + 298087511 \nu^{5} - 239467694 \nu^{4} + 147674805 \nu^{3} - 66643178 \nu^{2} + 19588566 \nu - 2984015\)\()/17095\)
\(\beta_{9}\)\(=\)\((\)\(26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} - 28666706 \nu^{10} + 56590650 \nu^{9} - 89528958 \nu^{8} + 116711850 \nu^{7} - 121073088 \nu^{6} + 99711814 \nu^{5} - 61611091 \nu^{4} + 27082796 \nu^{3} - 6949514 \nu^{2} + 510382 \nu + 218885\)\()/17095\)
\(\beta_{10}\)\(=\)\((\)\(49663 \nu^{15} - 358139 \nu^{14} + 2089429 \nu^{13} - 8006091 \nu^{12} + 25109764 \nu^{11} - 61357045 \nu^{10} + 124293850 \nu^{9} - 204357372 \nu^{8} + 277503791 \nu^{7} - 305524016 \nu^{6} + 271172221 \nu^{5} - 188678893 \nu^{4} + 99647214 \nu^{3} - 37587574 \nu^{2} + 9111403 \nu - 1090565\)\()/17095\)
\(\beta_{11}\)\(=\)\((\)\(46358 \nu^{15} - 350841 \nu^{14} + 2068305 \nu^{13} - 8157610 \nu^{12} + 26035188 \nu^{11} - 65341929 \nu^{10} + 135478704 \nu^{9} - 229565910 \nu^{8} + 321502876 \nu^{7} - 367712403 \nu^{6} + 340522371 \nu^{5} - 249596834 \nu^{4} + 140137971 \nu^{3} - 57050598 \nu^{2} + 15149982 \nu - 2064105\)\()/17095\)
\(\beta_{12}\)\(=\)\((\)\(-46358 \nu^{15} + 373985 \nu^{14} - 2230313 \nu^{13} + 9095994 \nu^{12} - 29559388 \nu^{11} + 76252747 \nu^{10} - 161588818 \nu^{9} + 281449761 \nu^{8} - 404524612 \nu^{7} + 477125400 \nu^{6} - 456031708 \nu^{5} + 346814258 \nu^{4} - 202568648 \nu^{3} + 86298565 \nu^{2} - 24026495 \nu + 3439595\)\()/17095\)
\(\beta_{13}\)\(=\)\((\)\(-54184 \nu^{15} + 399016 \nu^{14} - 2335837 \nu^{13} + 9056462 \nu^{12} - 28575749 \nu^{11} + 70523459 \nu^{10} - 143951776 \nu^{9} + 239041170 \nu^{8} - 327539991 \nu^{7} + 364427829 \nu^{6} - 326718848 \nu^{5} + 229677473 \nu^{4} - 122374471 \nu^{3} + 46393946 \nu^{2} - 11232929 \nu + 1312670\)\()/17095\)
\(\beta_{14}\)\(=\)\((\)\(10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} - 14702424 \nu^{10} + 30390552 \nu^{9} - 51252471 \nu^{8} + 71445138 \nu^{7} - 81160006 \nu^{6} + 74558856 \nu^{5} - 53991081 \nu^{4} + 29816774 \nu^{3} - 11804598 \nu^{2} + 3003630 \nu - 370592\)\()/3419\)
\(\beta_{15}\)\(=\)\((\)\(-54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + 77432206 \nu^{10} - 160446084 \nu^{9} + 271670265 \nu^{8} - 379486173 \nu^{7} + 432289456 \nu^{6} - 397583146 \nu^{5} + 288232582 \nu^{4} - 159062971 \nu^{3} + 62844859 \nu^{2} - 15915907 \nu + 1951760\)\()/17095\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{15} + 7 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} - \beta_{8} + 3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} - 5 \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\(-10 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + 8 \beta_{12} - 4 \beta_{11} - 8 \beta_{10} + 6 \beta_{9} + 6 \beta_{6} + 6 \beta_{5} - 16 \beta_{4} - 10 \beta_{3} - 3 \beta_{2} - 14 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\((\)\(-21 \beta_{15} - 46 \beta_{14} + 44 \beta_{13} - 19 \beta_{12} + 44 \beta_{11} + 19 \beta_{10} + 35 \beta_{9} + 3 \beta_{8} - 35 \beta_{7} - 5 \beta_{6} - 26 \beta_{5} + 19 \beta_{4} - \beta_{3} + 25 \beta_{2} + 3 \beta_{1} + 36\)\()/2\)
\(\nu^{6}\)\(=\)\(60 \beta_{15} - 43 \beta_{14} - 5 \beta_{13} - 70 \beta_{12} + 55 \beta_{11} + 49 \beta_{10} - 30 \beta_{9} - 10 \beta_{8} - 15 \beta_{7} - 60 \beta_{6} - 40 \beta_{5} + 122 \beta_{4} + 69 \beta_{3} + 45 \beta_{2} + 93 \beta_{1} - 50\)
\(\nu^{7}\)\(=\)\((\)\(321 \beta_{15} + 267 \beta_{14} - 365 \beta_{13} + 17 \beta_{12} - 213 \beta_{11} - 28 \beta_{10} - 336 \beta_{9} - 60 \beta_{8} + 294 \beta_{7} - 126 \beta_{6} + 136 \beta_{5} + 84 \beta_{4} + 179 \beta_{3} - 126 \beta_{2} + 162 \beta_{1} - 307\)\()/2\)
\(\nu^{8}\)\(=\)\(-256 \beta_{15} + 446 \beta_{14} - 108 \beta_{13} + 544 \beta_{12} - 528 \beta_{11} - 304 \beta_{10} + 74 \beta_{9} + 84 \beta_{8} + 280 \beta_{7} + 420 \beta_{6} + 316 \beta_{5} - 868 \beta_{4} - 392 \beta_{3} - 469 \beta_{2} - 572 \beta_{1} + 267\)
\(\nu^{9}\)\(=\)\((\)\(-3026 \beta_{15} - 1225 \beta_{14} + 2701 \beta_{13} + 906 \beta_{12} + 525 \beta_{11} - 297 \beta_{10} + 2835 \beta_{9} + 795 \beta_{8} - 1899 \beta_{7} + 2007 \beta_{6} - 606 \beta_{5} - 2265 \beta_{4} - 2202 \beta_{3} + 9 \beta_{2} - 2277 \beta_{1} + 2725\)\()/2\)
\(\nu^{10}\)\(=\)\(274 \beta_{15} - 4001 \beta_{14} + 2070 \beta_{13} - 3737 \beta_{12} + 4287 \beta_{11} + 1922 \beta_{10} + 846 \beta_{9} - 300 \beta_{8} - 3285 \beta_{7} - 2250 \beta_{6} - 2609 \beta_{5} + 5530 \beta_{4} + 1694 \beta_{3} + 3870 \beta_{2} + 3067 \beta_{1} - 932\)
\(\nu^{11}\)\(=\)\((\)\(23673 \beta_{15} + 1926 \beta_{14} - 17346 \beta_{13} - 14155 \beta_{12} + 4640 \beta_{11} + 5251 \beta_{10} - 20779 \beta_{9} - 7509 \beta_{8} + 8569 \beta_{7} - 20801 \beta_{6} + 506 \beta_{5} + 28021 \beta_{4} + 20095 \beta_{3} + 8261 \beta_{2} + 22843 \beta_{1} - 22758\)\()/2\)
\(\nu^{12}\)\(=\)\(10280 \beta_{15} + 31947 \beta_{14} - 24436 \beta_{13} + 21968 \beta_{12} - 30552 \beta_{11} - 11620 \beta_{10} - 17265 \beta_{9} - 1650 \beta_{8} + 30690 \beta_{7} + 6798 \beta_{6} + 20444 \beta_{5} - 28982 \beta_{4} - 2254 \beta_{3} - 26565 \beta_{2} - 11868 \beta_{1} - 3138\)
\(\nu^{13}\)\(=\)\((\)\(-160655 \beta_{15} + 46695 \beta_{14} + 88659 \beta_{13} + 152123 \beta_{12} - 96769 \beta_{11} - 59234 \beta_{10} + 128544 \beta_{9} + 57480 \beta_{8} - 4992 \beta_{7} + 177762 \beta_{6} + 32656 \beta_{5} - 273318 \beta_{4} - 157219 \beta_{3} - 120978 \beta_{2} - 196060 \beta_{1} + 171575\)\()/2\)
\(\nu^{14}\)\(=\)\(-160979 \beta_{15} - 225841 \beta_{14} + 234065 \beta_{13} - 95851 \beta_{12} + 187408 \beta_{11} + 60319 \beta_{10} + 200859 \beta_{9} + 43732 \beta_{8} - 244699 \beta_{7} + 38129 \beta_{6} - 144445 \beta_{5} + 90048 \beta_{4} - 62947 \beta_{3} + 146874 \beta_{2} - 5512 \beta_{1} + 106657\)
\(\nu^{15}\)\(=\)\((\)\(914760 \beta_{15} - 806121 \beta_{14} - 233687 \beta_{13} - 1368388 \beta_{12} + 1121513 \beta_{11} + 561279 \beta_{10} - 600297 \beta_{9} - 364797 \beta_{8} - 458997 \beta_{7} - 1313487 \beta_{6} - 528672 \beta_{5} + 2298683 \beta_{4} + 1078190 \beta_{3} + 1251033 \beta_{2} + 1491905 \beta_{1} - 1132093\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(-1\) \(-\beta_{9} + \beta_{14}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 1.74530i
0.500000 + 1.33108i
0.500000 0.331082i
0.500000 + 2.74530i
0.500000 0.410882i
0.500000 2.00333i
0.500000 0.589118i
0.500000 + 1.00333i
0.500000 + 0.410882i
0.500000 + 2.00333i
0.500000 + 0.589118i
0.500000 1.00333i
0.500000 + 1.74530i
0.500000 1.33108i
0.500000 + 0.331082i
0.500000 2.74530i
−0.965926 0.258819i −1.73022 + 0.0795432i 0.866025 + 0.500000i 2.76293 2.76293i 1.69185 + 0.370982i −0.657464 2.45369i −0.707107 0.707107i 2.98735 0.275255i −3.38389 + 1.95369i
11.2 −0.965926 0.258819i 1.73022 0.0795432i 0.866025 + 0.500000i −0.313444 + 0.313444i −1.69185 0.370982i −0.0745867 0.278362i −0.707107 0.707107i 2.98735 0.275255i 0.383889 0.221638i
11.3 0.965926 + 0.258819i −0.933998 + 1.45865i 0.866025 + 0.500000i 0.313444 0.313444i −1.27970 + 1.16721i −0.0745867 0.278362i 0.707107 + 0.707107i −1.25529 2.72474i 0.383889 0.221638i
11.4 0.965926 + 0.258819i 0.933998 1.45865i 0.866025 + 0.500000i −2.76293 + 2.76293i 1.27970 1.16721i −0.657464 2.45369i 0.707107 + 0.707107i −1.25529 2.72474i −3.38389 + 1.95369i
41.1 −0.258819 + 0.965926i −0.0795432 1.73022i −0.866025 0.500000i 2.02097 + 2.02097i 1.69185 + 0.370982i 3.46723 0.929042i 0.707107 0.707107i −2.98735 + 0.275255i −2.47517 + 1.42904i
41.2 −0.258819 + 0.965926i 0.0795432 + 1.73022i −0.866025 0.500000i 0.428520 + 0.428520i −1.69185 0.370982i −0.735180 + 0.196991i 0.707107 0.707107i −2.98735 + 0.275255i −0.524827 + 0.303009i
41.3 0.258819 0.965926i −1.45865 0.933998i −0.866025 0.500000i −2.02097 2.02097i −1.27970 + 1.16721i 3.46723 0.929042i −0.707107 + 0.707107i 1.25529 + 2.72474i −2.47517 + 1.42904i
41.4 0.258819 0.965926i 1.45865 + 0.933998i −0.866025 0.500000i −0.428520 0.428520i 1.27970 1.16721i −0.735180 + 0.196991i −0.707107 + 0.707107i 1.25529 + 2.72474i −0.524827 + 0.303009i
59.1 −0.258819 0.965926i −0.0795432 + 1.73022i −0.866025 + 0.500000i 2.02097 2.02097i 1.69185 0.370982i 3.46723 + 0.929042i 0.707107 + 0.707107i −2.98735 0.275255i −2.47517 1.42904i
59.2 −0.258819 0.965926i 0.0795432 1.73022i −0.866025 + 0.500000i 0.428520 0.428520i −1.69185 + 0.370982i −0.735180 0.196991i 0.707107 + 0.707107i −2.98735 0.275255i −0.524827 0.303009i
59.3 0.258819 + 0.965926i −1.45865 + 0.933998i −0.866025 + 0.500000i −2.02097 + 2.02097i −1.27970 1.16721i 3.46723 + 0.929042i −0.707107 0.707107i 1.25529 2.72474i −2.47517 1.42904i
59.4 0.258819 + 0.965926i 1.45865 0.933998i −0.866025 + 0.500000i −0.428520 + 0.428520i 1.27970 + 1.16721i −0.735180 0.196991i −0.707107 0.707107i 1.25529 2.72474i −0.524827 0.303009i
71.1 −0.965926 + 0.258819i −1.73022 0.0795432i 0.866025 0.500000i 2.76293 + 2.76293i 1.69185 0.370982i −0.657464 + 2.45369i −0.707107 + 0.707107i 2.98735 + 0.275255i −3.38389 1.95369i
71.2 −0.965926 + 0.258819i 1.73022 + 0.0795432i 0.866025 0.500000i −0.313444 0.313444i −1.69185 + 0.370982i −0.0745867 + 0.278362i −0.707107 + 0.707107i 2.98735 + 0.275255i 0.383889 + 0.221638i
71.3 0.965926 0.258819i −0.933998 1.45865i 0.866025 0.500000i 0.313444 + 0.313444i −1.27970 1.16721i −0.0745867 + 0.278362i 0.707107 0.707107i −1.25529 + 2.72474i 0.383889 + 0.221638i
71.4 0.965926 0.258819i 0.933998 + 1.45865i 0.866025 0.500000i −2.76293 2.76293i 1.27970 + 1.16721i −0.657464 + 2.45369i 0.707107 0.707107i −1.25529 + 2.72474i −3.38389 1.95369i
\(n\): 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
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.k.a 16
3.b odd 2 1 inner 78.2.k.a 16
4.b odd 2 1 624.2.cn.d 16
12.b even 2 1 624.2.cn.d 16
13.c even 3 1 1014.2.g.c 16
13.e even 6 1 1014.2.g.d 16
13.f odd 12 1 inner 78.2.k.a 16
13.f odd 12 1 1014.2.g.c 16
13.f odd 12 1 1014.2.g.d 16
39.h odd 6 1 1014.2.g.d 16
39.i odd 6 1 1014.2.g.c 16
39.k even 12 1 inner 78.2.k.a 16
39.k even 12 1 1014.2.g.c 16
39.k even 12 1 1014.2.g.d 16
52.l even 12 1 624.2.cn.d 16
156.v odd 12 1 624.2.cn.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.k.a 16 1.a even 1 1 trivial
78.2.k.a 16 3.b odd 2 1 inner
78.2.k.a 16 13.f odd 12 1 inner
78.2.k.a 16 39.k even 12 1 inner
624.2.cn.d 16 4.b odd 2 1
624.2.cn.d 16 12.b even 2 1
624.2.cn.d 16 52.l even 12 1
624.2.cn.d 16 156.v odd 12 1
1014.2.g.c 16 13.c even 3 1
1014.2.g.c 16 13.f odd 12 1
1014.2.g.c 16 39.i odd 6 1
1014.2.g.c 16 39.k even 12 1
1014.2.g.d 16 13.e even 6 1
1014.2.g.d 16 13.f odd 12 1
1014.2.g.d 16 39.h odd 6 1
1014.2.g.d 16 39.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(78, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( 6561 - 486 T^{4} - 45 T^{8} - 6 T^{12} + T^{16} \)
$5$ \( 81 + 2700 T^{4} + 15606 T^{8} + 300 T^{12} + T^{16} \)
$7$ \( ( 4 + 16 T + 68 T^{2} + 112 T^{3} + 46 T^{4} - 16 T^{5} + 2 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$11$ \( ( 36 + 180 T^{2} + 294 T^{4} - 30 T^{6} + T^{8} )^{2} \)
$13$ \( ( 28561 + 26364 T + 14872 T^{2} + 6084 T^{3} + 1875 T^{4} + 468 T^{5} + 88 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$17$ \( 151807041 + 149921928 T^{2} + 127730574 T^{4} + 18302976 T^{6} + 1834083 T^{8} + 94464 T^{10} + 3534 T^{12} + 72 T^{14} + T^{16} \)
$19$ \( ( 5476 - 8288 T + 5012 T^{2} - 1952 T^{3} + 502 T^{4} + 32 T^{5} + 2 T^{6} + 8 T^{7} + T^{8} )^{2} \)
$23$ \( 18974736 + 14113440 T^{2} + 7152192 T^{4} + 1965600 T^{6} + 391068 T^{8} + 39600 T^{10} + 2832 T^{12} + 60 T^{14} + T^{16} \)
$29$ \( 33871089681 - 11746968948 T^{2} + 3140925714 T^{4} - 275021136 T^{6} + 17095563 T^{8} - 541584 T^{10} + 12354 T^{12} - 132 T^{14} + T^{16} \)
$31$ \( ( 7744 + 2816 T + 512 T^{2} - 448 T^{3} + 1120 T^{4} + 256 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$37$ \( ( 375769 + 137312 T + 7688 T^{2} + 1460 T^{3} - 53 T^{4} - 188 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$41$ \( 65697655057281 - 2756876552352 T^{2} - 7103522178 T^{4} + 1916281152 T^{6} + 12752451 T^{8} - 540864 T^{10} - 2562 T^{12} + 96 T^{14} + T^{16} \)
$43$ \( ( 2916 - 972 T^{2} + 270 T^{4} - 18 T^{6} + T^{8} )^{2} \)
$47$ \( 41006250000 + 12276360000 T^{4} + 22120776 T^{8} + 8784 T^{12} + T^{16} \)
$53$ \( ( 522729 + 112140 T^{2} + 7374 T^{4} + 156 T^{6} + T^{8} )^{2} \)
$59$ \( 43489065701376 - 5815508742144 T^{2} + 217439797248 T^{4} + 5587439616 T^{6} - 15833664 T^{8} - 1064448 T^{10} + 3072 T^{12} + 168 T^{14} + T^{16} \)
$61$ \( ( 47961 + 178704 T + 652716 T^{2} + 54216 T^{3} + 13611 T^{4} + 912 T^{5} + 204 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$67$ \( ( 676 + 12064 T + 54308 T^{2} - 34496 T^{3} + 9382 T^{4} - 1696 T^{5} + 218 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$71$ \( 5143987297296 + 12200201106912 T^{2} + 9551502132480 T^{4} - 222311246976 T^{6} + 1727252316 T^{8} - 495936 T^{10} - 41280 T^{12} + 12 T^{14} + T^{16} \)
$73$ \( ( 16834609 - 6220148 T + 1149128 T^{2} - 78500 T^{3} + 8782 T^{4} - 2188 T^{5} + 392 T^{6} - 28 T^{7} + T^{8} )^{2} \)
$79$ \( ( 312 - 432 T + 108 T^{2} + 24 T^{3} + T^{4} )^{4} \)
$83$ \( 36804120336 + 9534767040 T^{4} + 55694088 T^{8} + 21360 T^{12} + T^{16} \)
$89$ \( 5986057479987456 - 186125299905024 T^{2} + 530849393280 T^{4} + 43475159808 T^{6} + 37529136 T^{8} - 4771008 T^{10} + 5160 T^{12} + 264 T^{14} + T^{16} \)
$97$ \( ( 43264 + 106496 T + 27008 T^{2} - 61312 T^{3} + 14320 T^{4} - 608 T^{5} + 296 T^{6} - 8 T^{7} + T^{8} )^{2} \)
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