Properties

Label 77.2.l.b
Level 77
Weight 2
Character orbit 77.l
Analytic conductor 0.615
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{9} - \beta_{11} ) q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} + ( -1 - \beta_{4} + \beta_{5} + \beta_{11} - \beta_{14} ) q^{4} + ( \beta_{2} - \beta_{13} ) q^{5} + ( \beta_{2} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{6} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{7} + ( 2 - 3 \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{14} ) q^{8} + ( -2 + \beta_{4} + 4 \beta_{5} - \beta_{8} - \beta_{9} - 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{9} - \beta_{11} ) q^{2} + ( -\beta_{1} - \beta_{2} ) q^{3} + ( -1 - \beta_{4} + \beta_{5} + \beta_{11} - \beta_{14} ) q^{4} + ( \beta_{2} - \beta_{13} ) q^{5} + ( \beta_{2} - \beta_{6} + \beta_{10} - \beta_{15} ) q^{6} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{7} + ( 2 - 3 \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{14} ) q^{8} + ( -2 + \beta_{4} + 4 \beta_{5} - \beta_{8} - \beta_{9} - 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{9} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{13} - \beta_{15} ) q^{10} + ( -\beta_{4} + \beta_{5} + 3 \beta_{8} - \beta_{14} ) q^{11} + ( 3 \beta_{1} + \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{15} ) q^{12} + ( -\beta_{1} + \beta_{3} - \beta_{10} ) q^{13} + ( 1 - \beta_{1} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{14} + ( 3 + 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} ) q^{15} + ( -2 + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{16} + ( 3 \beta_{1} + \beta_{2} + \beta_{6} + 2 \beta_{7} + \beta_{10} - \beta_{13} ) q^{17} + ( -1 + \beta_{4} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{14} ) q^{18} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{10} + 2 \beta_{13} ) q^{19} + ( \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{10} + 3 \beta_{15} ) q^{20} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + 4 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{21} + ( 4 - \beta_{4} - 2 \beta_{5} - 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{14} ) q^{22} + ( -2 - 3 \beta_{4} - 3 \beta_{8} - 3 \beta_{11} + 3 \beta_{12} ) q^{23} + ( 2 \beta_{2} + 2 \beta_{3} + \beta_{7} - 2 \beta_{10} - 2 \beta_{13} + \beta_{15} ) q^{24} + ( -5 + \beta_{4} + 2 \beta_{5} - 4 \beta_{8} - 5 \beta_{9} - 4 \beta_{11} + \beta_{12} - \beta_{14} ) q^{25} + ( \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{15} ) q^{26} + ( 2 \beta_{1} - \beta_{3} + 2 \beta_{6} + 2 \beta_{10} + \beta_{13} ) q^{27} + ( -3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{28} + ( \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + \beta_{12} - 2 \beta_{14} ) q^{29} + ( -5 + 6 \beta_{5} + 5 \beta_{8} - 4 \beta_{9} - \beta_{11} - 8 \beta_{14} ) q^{30} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{31} + ( 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} ) q^{32} + ( -3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - 3 \beta_{7} + \beta_{13} ) q^{33} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{15} ) q^{34} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} + 3 \beta_{9} + 5 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{35} + ( 2 - 3 \beta_{4} - 3 \beta_{5} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - 3 \beta_{12} + 3 \beta_{14} ) q^{36} + ( 4 \beta_{5} + 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - 4 \beta_{12} + \beta_{14} ) q^{37} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{6} + 2 \beta_{10} - \beta_{13} ) q^{38} + ( 2 - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} + 3 \beta_{14} ) q^{39} + ( -3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + \beta_{7} + \beta_{10} - \beta_{13} - 3 \beta_{15} ) q^{40} + ( -\beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{6} - 2 \beta_{13} - 2 \beta_{15} ) q^{41} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{7} - 4 \beta_{8} - 3 \beta_{9} - \beta_{10} - 3 \beta_{11} + 5 \beta_{12} + 3 \beta_{14} + \beta_{15} ) q^{42} + ( -1 + 2 \beta_{5} ) q^{43} + ( -4 - \beta_{4} - \beta_{5} - 5 \beta_{8} - 8 \beta_{9} - 8 \beta_{11} + 5 \beta_{12} - 4 \beta_{14} ) q^{44} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{6} - 4 \beta_{7} - 3 \beta_{10} + \beta_{15} ) q^{45} + ( -1 + 3 \beta_{4} - 4 \beta_{9} - \beta_{11} - 3 \beta_{14} ) q^{46} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{7} + 2 \beta_{10} - 3 \beta_{13} + 2 \beta_{15} ) q^{47} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{7} + 3 \beta_{10} - \beta_{15} ) q^{48} + ( 2 - 2 \beta_{1} + 5 \beta_{4} + 2 \beta_{8} - 2 \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - 4 \beta_{14} - 2 \beta_{15} ) q^{49} + ( -1 + 3 \beta_{4} + 3 \beta_{5} + 7 \beta_{8} + 5 \beta_{9} + 7 \beta_{11} - 4 \beta_{12} - 3 \beta_{14} ) q^{50} + ( -1 + \beta_{4} + \beta_{5} - 2 \beta_{8} + 2 \beta_{9} + 4 \beta_{11} + 6 \beta_{14} ) q^{51} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{6} + 2 \beta_{7} + \beta_{10} + 2 \beta_{15} ) q^{52} + ( 2 - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{11} - \beta_{12} + \beta_{14} ) q^{53} + ( \beta_{6} + 2 \beta_{7} - 2 \beta_{10} ) q^{54} + ( 7 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} + 3 \beta_{7} + \beta_{10} - \beta_{13} + 3 \beta_{15} ) q^{55} + ( -2 + 4 \beta_{1} - 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} + 2 \beta_{12} - 4 \beta_{14} + 4 \beta_{15} ) q^{56} + ( -9 - 7 \beta_{4} - 4 \beta_{8} - 2 \beta_{9} - 5 \beta_{11} + 9 \beta_{12} - 2 \beta_{14} ) q^{57} + ( 1 - 2 \beta_{4} - \beta_{5} - \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{58} + ( -4 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{13} ) q^{59} + ( 8 - 5 \beta_{4} + \beta_{5} - \beta_{8} + 10 \beta_{9} + \beta_{11} + 4 \beta_{12} + 3 \beta_{14} ) q^{60} + ( -6 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - \beta_{6} - 6 \beta_{7} - 5 \beta_{10} + 3 \beta_{13} - \beta_{15} ) q^{61} + ( -\beta_{1} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{10} + \beta_{13} - \beta_{15} ) q^{62} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + 4 \beta_{11} - 5 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{63} + ( 1 + \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} - 6 \beta_{12} - 3 \beta_{14} ) q^{64} + ( 2 + 3 \beta_{4} - 4 \beta_{5} - \beta_{8} - 3 \beta_{11} - \beta_{12} ) q^{65} + ( 3 \beta_{1} + 2 \beta_{3} + 3 \beta_{6} + 5 \beta_{7} - 2 \beta_{13} + 2 \beta_{15} ) q^{66} + ( -1 + 6 \beta_{4} + 4 \beta_{5} + 3 \beta_{8} - 4 \beta_{9} + 2 \beta_{11} + \beta_{12} - 8 \beta_{14} ) q^{67} + ( 4 \beta_{1} - \beta_{3} + \beta_{10} + 3 \beta_{15} ) q^{68} + ( -4 \beta_{1} - \beta_{2} - 3 \beta_{7} - 3 \beta_{10} + 3 \beta_{13} - 3 \beta_{15} ) q^{69} + ( -3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} + \beta_{10} - 10 \beta_{11} + 7 \beta_{12} - 2 \beta_{13} + 5 \beta_{14} + \beta_{15} ) q^{70} + ( -7 + \beta_{4} + 5 \beta_{5} + \beta_{8} - 3 \beta_{9} - \beta_{11} + 3 \beta_{14} ) q^{71} + ( 7 + 3 \beta_{4} - 4 \beta_{5} + 7 \beta_{8} + 4 \beta_{9} + 7 \beta_{11} - 4 \beta_{12} + 4 \beta_{14} ) q^{72} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{6} + \beta_{7} + \beta_{10} + \beta_{13} - 4 \beta_{15} ) q^{73} + ( 7 - \beta_{4} - 6 \beta_{5} - 8 \beta_{8} + 4 \beta_{9} - 2 \beta_{11} + \beta_{12} + 9 \beta_{14} ) q^{74} + ( 6 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} + 2 \beta_{6} + 5 \beta_{7} + 5 \beta_{10} - 4 \beta_{15} ) q^{75} + ( -\beta_{2} + \beta_{3} - 2 \beta_{13} + \beta_{15} ) q^{76} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{14} - 4 \beta_{15} ) q^{77} + ( 5 \beta_{4} + 2 \beta_{5} + 6 \beta_{8} - 2 \beta_{9} + 3 \beta_{11} - 4 \beta_{12} - 4 \beta_{14} ) q^{78} + ( 5 + 2 \beta_{4} + 4 \beta_{8} + 3 \beta_{9} + \beta_{11} + 2 \beta_{12} - 4 \beta_{14} ) q^{79} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 3 \beta_{7} + 3 \beta_{13} - 2 \beta_{15} ) q^{80} + ( 2 + 2 \beta_{4} + 4 \beta_{8} + 4 \beta_{9} + 6 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{81} + ( 6 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{10} + 5 \beta_{15} ) q^{82} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{6} + 4 \beta_{7} + \beta_{10} - 2 \beta_{13} + 4 \beta_{15} ) q^{83} + ( -8 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 8 \beta_{4} + 9 \beta_{5} - \beta_{6} - 2 \beta_{7} + 5 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} - 6 \beta_{11} + \beta_{12} - 11 \beta_{14} ) q^{84} + ( -3 - 11 \beta_{4} + 3 \beta_{5} - 8 \beta_{8} - 5 \beta_{9} - 11 \beta_{11} + 11 \beta_{12} + \beta_{14} ) q^{85} + ( 1 - 2 \beta_{5} - 2 \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{14} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{6} - 2 \beta_{7} ) q^{87} + ( -7 + 6 \beta_{5} + 6 \beta_{8} - 2 \beta_{9} + 7 \beta_{11} + \beta_{12} - 3 \beta_{14} ) q^{88} + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{7} - 2 \beta_{10} - \beta_{15} ) q^{89} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{7} - \beta_{10} + 2 \beta_{13} - \beta_{15} ) q^{90} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{8} + 5 \beta_{9} - \beta_{12} + \beta_{14} ) q^{91} + ( 2 + 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{8} + 3 \beta_{9} + 4 \beta_{11} - 6 \beta_{12} - \beta_{14} ) q^{92} + ( -6 \beta_{4} - 9 \beta_{5} - 5 \beta_{8} - 4 \beta_{9} + 5 \beta_{11} + \beta_{12} + \beta_{14} ) q^{93} + ( -3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + \beta_{13} - \beta_{15} ) q^{94} + ( 2 - 2 \beta_{4} - 10 \beta_{5} - 5 \beta_{8} + 5 \beta_{9} - 8 \beta_{12} + 5 \beta_{14} ) q^{95} + ( -6 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 2 \beta_{6} - 5 \beta_{7} - \beta_{10} + 3 \beta_{13} - \beta_{15} ) q^{96} + ( -8 \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{6} - 6 \beta_{7} - \beta_{10} - \beta_{15} ) q^{97} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 7 \beta_{5} + 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 5 \beta_{11} ) q^{98} + ( -2 + 4 \beta_{4} + 9 \beta_{5} - \beta_{8} + 2 \beta_{9} - 10 \beta_{11} - 5 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 10q^{2} - 10q^{4} - 10q^{7} + 10q^{8} + 8q^{9} + O(q^{10}) \) \( 16q - 10q^{2} - 10q^{4} - 10q^{7} + 10q^{8} + 8q^{9} + 2q^{11} + 8q^{14} - 14q^{16} - 20q^{18} + 42q^{22} - 8q^{23} - 30q^{25} - 10q^{28} + 10q^{29} - 40q^{30} + 40q^{35} + 20q^{36} + 4q^{37} + 30q^{39} + 50q^{42} - 10q^{44} - 10q^{46} + 8q^{49} - 60q^{50} - 10q^{51} - 4q^{56} - 90q^{57} - 2q^{58} + 120q^{60} - 20q^{63} - 38q^{64} - 4q^{67} - 56q^{71} + 30q^{72} + 90q^{74} + 2q^{77} - 20q^{78} + 50q^{79} - 16q^{81} - 70q^{84} + 80q^{85} + 6q^{86} - 86q^{88} - 30q^{91} + 20q^{92} - 40q^{93} - 60q^{95} + 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 20 x^{14} + 260 x^{12} + 2030 x^{10} + 11605 x^{8} + 42100 x^{6} + 106925 x^{4} + 113575 x^{2} + 87025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-473414 \nu^{15} - 121962046 \nu^{13} - 1914724533 \nu^{11} - 19540502651 \nu^{9} - 110976178125 \nu^{7} - 427139590595 \nu^{5} - 804725306550 \nu^{3} - 700136714820 \nu\)\()/ 141563707035 \)
\(\beta_{3}\)\(=\)\((\)\(-1553143 \nu^{15} - 83115610 \nu^{13} - 738702950 \nu^{11} - 5169909042 \nu^{9} - 7773695495 \nu^{7} + 12526757215 \nu^{5} + 284343731710 \nu^{3} + 389963137595 \nu\)\()/ 141563707035 \)
\(\beta_{4}\)\(=\)\((\)\(2026557 \nu^{14} + 205077656 \nu^{12} + 2653427483 \nu^{10} + 24710411693 \nu^{8} + 118749873620 \nu^{6} + 414612833380 \nu^{4} + 520381574840 \nu^{2} + 310173577225\)\()/ 141563707035 \)
\(\beta_{5}\)\(=\)\((\)\(-2307412 \nu^{14} - 42954924 \nu^{12} - 546435360 \nu^{10} - 4022687807 \nu^{8} - 22443730260 \nu^{6} - 74163373020 \nu^{4} - 200532703805 \nu^{2} - 70674046800\)\()/ 141563707035 \)
\(\beta_{6}\)\(=\)\((\)\( 2307412 \nu^{15} + 42954924 \nu^{13} + 546435360 \nu^{11} + 4022687807 \nu^{9} + 22443730260 \nu^{7} + 74163373020 \nu^{5} + 200532703805 \nu^{3} + 70674046800 \nu \)\()/ 141563707035 \)
\(\beta_{7}\)\(=\)\((\)\(-8126617 \nu^{15} + 107242387 \nu^{13} + 1727473773 \nu^{11} + 21511248812 \nu^{9} + 103285230840 \nu^{7} + 401725517900 \nu^{5} + 425884080360 \nu^{3} + 534354646485 \nu\)\()/ 141563707035 \)
\(\beta_{8}\)\(=\)\((\)\(9624825 \nu^{14} + 326813127 \nu^{12} + 4112826637 \nu^{10} + 33537013348 \nu^{8} + 159160366925 \nu^{6} + 503717613335 \nu^{4} + 695040238750 \nu^{2} + 564950170875\)\()/ 141563707035 \)
\(\beta_{9}\)\(=\)\((\)\( -85328 \nu^{14} - 3129227 \nu^{12} - 37034577 \nu^{10} - 295472690 \nu^{8} - 1274305820 \nu^{6} - 3749548520 \nu^{4} - 3135087840 \nu^{2} - 2416417865 \)\()/ 1080638985 \)
\(\beta_{10}\)\(=\)\((\)\( -85328 \nu^{15} - 3129227 \nu^{13} - 37034577 \nu^{11} - 295472690 \nu^{9} - 1274305820 \nu^{7} - 3749548520 \nu^{5} - 3135087840 \nu^{3} - 2416417865 \nu \)\()/ 1080638985 \)
\(\beta_{11}\)\(=\)\((\)\(19576647 \nu^{14} + 220271693 \nu^{12} + 1820750374 \nu^{10} + 4608222307 \nu^{8} + 1502139270 \nu^{6} - 102149491525 \nu^{4} - 183298295855 \nu^{2} - 312330974765\)\()/ 141563707035 \)
\(\beta_{12}\)\(=\)\((\)\(-23957031 \nu^{14} - 304074935 \nu^{12} - 2842215611 \nu^{10} - 12038362652 \nu^{8} - 33691071920 \nu^{6} + 17998043615 \nu^{4} + 171049346710 \nu^{2} + 147206723390\)\()/ 141563707035 \)
\(\beta_{13}\)\(=\)\((\)\( -198348 \nu^{15} - 3886661 \nu^{13} - 41951474 \nu^{11} - 280524995 \nu^{9} - 1163671340 \nu^{7} - 3027593815 \nu^{5} - 2666658230 \nu^{3} - 1244021785 \nu \)\()/ 1080638985 \)
\(\beta_{14}\)\(=\)\((\)\(-26536505 \nu^{14} - 264615202 \nu^{12} - 2085345531 \nu^{10} - 3473599188 \nu^{8} + 6011890130 \nu^{6} + 135449500340 \nu^{4} + 138627365440 \nu^{2} + 29496038150\)\()/ 141563707035 \)
\(\beta_{15}\)\(=\)\((\)\(-38187887 \nu^{15} - 796505985 \nu^{13} - 8851599651 \nu^{11} - 61721024229 \nu^{9} - 271898350415 \nu^{7} - 782880946375 \nu^{5} - 1076794448150 \nu^{3} - 987191416985 \nu\)\()/ 141563707035 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{14} - 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{8} - 5 \beta_{5} - \beta_{4} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{15} - 2 \beta_{13} + \beta_{10} + 2 \beta_{7} + 7 \beta_{6} + \beta_{3} + 2 \beta_{2} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-7 \beta_{14} + 8 \beta_{12} + \beta_{11} + 7 \beta_{9} + 16 \beta_{8} + 37 \beta_{5} - 2 \beta_{4} - 30\)
\(\nu^{5}\)\(=\)\(-8 \beta_{15} + 9 \beta_{13} - 2 \beta_{10} - \beta_{7} - 38 \beta_{6} + 9 \beta_{3} + \beta_{2} - 47 \beta_{1}\)
\(\nu^{6}\)\(=\)\(84 \beta_{14} - 29 \beta_{12} + 98 \beta_{11} + 42 \beta_{9} - 13 \beta_{8} - 42 \beta_{5} + 56 \beta_{4} + 231\)
\(\nu^{7}\)\(=\)\(-14 \beta_{15} + 69 \beta_{13} - 29 \beta_{10} - 98 \beta_{7} - 56 \beta_{6} - 138 \beta_{3} - 111 \beta_{2} + 146 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-245 \beta_{14} - 260 \beta_{12} - 750 \beta_{11} - 490 \beta_{9} - 630 \beta_{8} - 1210 \beta_{5} - 125 \beta_{4} - 245\)
\(\nu^{9}\)\(=\)\(505 \beta_{15} - 1010 \beta_{13} + 385 \beta_{10} + 750 \beta_{7} + 1960 \beta_{6} + 505 \beta_{3} + 630 \beta_{2} + 1135 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-1420 \beta_{14} + 2645 \beta_{12} + 965 \beta_{11} + 1420 \beta_{9} + 5030 \beta_{8} + 9235 \beta_{5} - 2190 \beta_{4} - 7815\)
\(\nu^{11}\)\(=\)\(-2385 \beta_{15} + 3610 \beta_{13} - 2190 \beta_{10} - 965 \beta_{7} - 10200 \beta_{6} + 3610 \beta_{3} + 965 \beta_{2} - 13810 \beta_{1}\)
\(\nu^{12}\)\(=\)\(16410 \beta_{14} - 995 \beta_{12} + 26405 \beta_{11} + 8205 \beta_{9} - 7210 \beta_{8} - 8205 \beta_{5} + 18200 \beta_{4} + 59020\)
\(\nu^{13}\)\(=\)\(-9995 \beta_{15} + 25410 \beta_{13} - 995 \beta_{10} - 26405 \beta_{7} - 18200 \beta_{6} - 50820 \beta_{3} - 33615 \beta_{2} + 39825 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-47235 \beta_{14} - 129635 \beta_{12} - 224105 \beta_{11} - 94470 \beta_{9} - 172640 \beta_{8} - 332295 \beta_{5} + 4230 \beta_{4} - 47235\)
\(\nu^{15}\)\(=\)\(176870 \beta_{15} - 353740 \beta_{13} + 125405 \beta_{10} + 224105 \beta_{7} + 556400 \beta_{6} + 176870 \beta_{3} + 172640 \beta_{2} + 349510 \beta_{1}\)

Character values

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

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(1 + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
1.29877 2.24954i
−1.29877 + 2.24954i
−1.17141 2.02895i
1.17141 + 2.02895i
1.29877 + 2.24954i
−1.29877 2.24954i
−1.17141 + 2.02895i
1.17141 2.02895i
1.27939 2.21596i
−1.27939 + 2.21596i
0.551501 + 0.955228i
−0.551501 0.955228i
1.27939 + 2.21596i
−1.27939 2.21596i
0.551501 0.955228i
−0.551501 + 0.955228i
−1.16913 + 0.379874i −0.767604 + 1.05652i −0.395472 + 0.287327i −2.26926 0.737329i 0.496087 1.52680i −2.39697 + 1.12006i 1.79833 2.47520i 0.400040 + 1.23120i 2.93316
6.2 −1.16913 + 0.379874i 0.767604 1.05652i −0.395472 + 0.287327i 2.26926 + 0.737329i −0.496087 + 1.52680i 1.80595 1.93354i 1.79833 2.47520i 0.400040 + 1.23120i −2.93316
6.3 0.478148 0.155360i −1.69284 + 2.32999i −1.41355 + 1.02700i 0.572621 + 0.186056i −0.447440 + 1.37708i 2.61795 0.382556i −1.10735 + 1.52414i −1.63611 5.03542i 0.302703
6.4 0.478148 0.155360i 1.69284 2.32999i −1.41355 + 1.02700i −0.572621 0.186056i 0.447440 1.37708i −1.17282 + 2.37160i −1.10735 + 1.52414i −1.63611 5.03542i −0.302703
13.1 −1.16913 0.379874i −0.767604 1.05652i −0.395472 0.287327i −2.26926 + 0.737329i 0.496087 + 1.52680i −2.39697 1.12006i 1.79833 + 2.47520i 0.400040 1.23120i 2.93316
13.2 −1.16913 0.379874i 0.767604 + 1.05652i −0.395472 0.287327i 2.26926 0.737329i −0.496087 1.52680i 1.80595 + 1.93354i 1.79833 + 2.47520i 0.400040 1.23120i −2.93316
13.3 0.478148 + 0.155360i −1.69284 2.32999i −1.41355 1.02700i 0.572621 0.186056i −0.447440 1.37708i 2.61795 + 0.382556i −1.10735 1.52414i −1.63611 + 5.03542i 0.302703
13.4 0.478148 + 0.155360i 1.69284 + 2.32999i −1.41355 1.02700i −0.572621 + 0.186056i 0.447440 + 1.37708i −1.17282 2.37160i −1.10735 1.52414i −1.63611 + 5.03542i −0.302703
41.1 −1.41355 1.94558i −1.63732 + 0.531999i −1.16913 + 3.59821i −1.97962 + 2.72471i 3.34948 + 2.43354i 0.150818 2.64145i 4.07890 1.32531i −0.0292442 + 0.0212472i 8.09942
41.2 −1.41355 1.94558i 1.63732 0.531999i −1.16913 + 3.59821i 1.97962 2.72471i −3.34948 2.43354i −1.43059 + 2.22563i 4.07890 1.32531i −0.0292442 + 0.0212472i −8.09942
41.3 −0.395472 0.544320i −2.52275 + 0.819690i 0.478148 1.47159i 2.08654 2.87188i 1.44385 + 1.04902i −1.94632 1.79216i −2.26988 + 0.737529i 3.26531 2.37239i −2.38839
41.4 −0.395472 0.544320i 2.52275 0.819690i 0.478148 1.47159i −2.08654 + 2.87188i −1.44385 1.04902i −2.62801 + 0.305873i −2.26988 + 0.737529i 3.26531 2.37239i 2.38839
62.1 −1.41355 + 1.94558i −1.63732 0.531999i −1.16913 3.59821i −1.97962 2.72471i 3.34948 2.43354i 0.150818 + 2.64145i 4.07890 + 1.32531i −0.0292442 0.0212472i 8.09942
62.2 −1.41355 + 1.94558i 1.63732 + 0.531999i −1.16913 3.59821i 1.97962 + 2.72471i −3.34948 + 2.43354i −1.43059 2.22563i 4.07890 + 1.32531i −0.0292442 0.0212472i −8.09942
62.3 −0.395472 + 0.544320i −2.52275 0.819690i 0.478148 + 1.47159i 2.08654 + 2.87188i 1.44385 1.04902i −1.94632 + 1.79216i −2.26988 0.737529i 3.26531 + 2.37239i −2.38839
62.4 −0.395472 + 0.544320i 2.52275 + 0.819690i 0.478148 + 1.47159i −2.08654 2.87188i −1.44385 + 1.04902i −2.62801 0.305873i −2.26988 0.737529i 3.26531 + 2.37239i 2.38839
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.l.b 16
3.b odd 2 1 693.2.bu.d 16
7.b odd 2 1 inner 77.2.l.b 16
7.c even 3 1 539.2.s.b 16
7.c even 3 1 539.2.s.c 16
7.d odd 6 1 539.2.s.b 16
7.d odd 6 1 539.2.s.c 16
11.b odd 2 1 847.2.l.i 16
11.c even 5 1 847.2.b.f 16
11.c even 5 1 847.2.l.e 16
11.c even 5 1 847.2.l.i 16
11.c even 5 1 847.2.l.j 16
11.d odd 10 1 inner 77.2.l.b 16
11.d odd 10 1 847.2.b.f 16
11.d odd 10 1 847.2.l.e 16
11.d odd 10 1 847.2.l.j 16
21.c even 2 1 693.2.bu.d 16
33.f even 10 1 693.2.bu.d 16
77.b even 2 1 847.2.l.i 16
77.j odd 10 1 847.2.b.f 16
77.j odd 10 1 847.2.l.e 16
77.j odd 10 1 847.2.l.i 16
77.j odd 10 1 847.2.l.j 16
77.l even 10 1 inner 77.2.l.b 16
77.l even 10 1 847.2.b.f 16
77.l even 10 1 847.2.l.e 16
77.l even 10 1 847.2.l.j 16
77.n even 30 1 539.2.s.b 16
77.n even 30 1 539.2.s.c 16
77.o odd 30 1 539.2.s.b 16
77.o odd 30 1 539.2.s.c 16
231.r odd 10 1 693.2.bu.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.l.b 16 1.a even 1 1 trivial
77.2.l.b 16 7.b odd 2 1 inner
77.2.l.b 16 11.d odd 10 1 inner
77.2.l.b 16 77.l even 10 1 inner
539.2.s.b 16 7.c even 3 1
539.2.s.b 16 7.d odd 6 1
539.2.s.b 16 77.n even 30 1
539.2.s.b 16 77.o odd 30 1
539.2.s.c 16 7.c even 3 1
539.2.s.c 16 7.d odd 6 1
539.2.s.c 16 77.n even 30 1
539.2.s.c 16 77.o odd 30 1
693.2.bu.d 16 3.b odd 2 1
693.2.bu.d 16 21.c even 2 1
693.2.bu.d 16 33.f even 10 1
693.2.bu.d 16 231.r odd 10 1
847.2.b.f 16 11.c even 5 1
847.2.b.f 16 11.d odd 10 1
847.2.b.f 16 77.j odd 10 1
847.2.b.f 16 77.l even 10 1
847.2.l.e 16 11.c even 5 1
847.2.l.e 16 11.d odd 10 1
847.2.l.e 16 77.j odd 10 1
847.2.l.e 16 77.l even 10 1
847.2.l.i 16 11.b odd 2 1
847.2.l.i 16 11.c even 5 1
847.2.l.i 16 77.b even 2 1
847.2.l.i 16 77.j odd 10 1
847.2.l.j 16 11.c even 5 1
847.2.l.j 16 11.d odd 10 1
847.2.l.j 16 77.j odd 10 1
847.2.l.j 16 77.l even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 5 T_{2}^{7} + 13 T_{2}^{6} + 15 T_{2}^{5} + 4 T_{2}^{4} - 5 T_{2}^{3} - 3 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} )^{4}( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )^{2} \)
$3$ \( 1 + 2 T^{2} + 10 T^{6} + 65 T^{8} + 466 T^{10} + 912 T^{12} + 1220 T^{14} + 6865 T^{16} + 10980 T^{18} + 73872 T^{20} + 339714 T^{22} + 426465 T^{24} + 590490 T^{26} + 9565938 T^{30} + 43046721 T^{32} \)
$5$ \( 1 + 25 T^{2} + 235 T^{4} + 830 T^{6} - 985 T^{8} - 8750 T^{10} + 76100 T^{12} + 846725 T^{14} + 4674775 T^{16} + 21168125 T^{18} + 47562500 T^{20} - 136718750 T^{22} - 384765625 T^{24} + 8105468750 T^{26} + 57373046875 T^{28} + 152587890625 T^{30} + 152587890625 T^{32} \)
$7$ \( 1 + 10 T + 46 T^{2} + 110 T^{3} + 67 T^{4} - 430 T^{5} - 1612 T^{6} - 3140 T^{7} - 6135 T^{8} - 21980 T^{9} - 78988 T^{10} - 147490 T^{11} + 160867 T^{12} + 1848770 T^{13} + 5411854 T^{14} + 8235430 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 - T - 20 T^{2} + T^{3} + 309 T^{4} + 11 T^{5} - 2420 T^{6} - 1331 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( 1 - 37 T^{2} + 905 T^{4} - 17590 T^{6} + 252535 T^{8} - 3649636 T^{10} + 49961932 T^{12} - 672618165 T^{14} + 9550335135 T^{16} - 113672469885 T^{18} + 1426962739852 T^{20} - 17616095891524 T^{22} + 206000557627735 T^{24} - 2424930871623910 T^{26} + 21084767035845305 T^{28} - 145682926270873693 T^{30} + 665416609183179841 T^{32} \)
$17$ \( 1 - 53 T^{2} + 935 T^{4} + 10075 T^{6} - 695885 T^{8} + 10961626 T^{10} - 1176848 T^{12} - 2883558960 T^{14} + 66310473675 T^{16} - 833348539440 T^{18} - 98291521808 T^{20} + 264587003927194 T^{22} - 4854324966830285 T^{24} + 20311138547023675 T^{26} + 544751791809826535 T^{28} - 8924024807648249237 T^{30} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 131 T^{2} + 7835 T^{4} - 269555 T^{6} + 5344225 T^{8} - 36384548 T^{10} - 1284840662 T^{12} + 54300385290 T^{14} - 1231352034765 T^{16} + 19602439089690 T^{18} - 167441719912502 T^{20} - 1711743115446788 T^{22} + 90763982192788225 T^{24} - 1652659565121548555 T^{26} + 17341322390883371435 T^{28} - \)\(10\!\cdots\!51\)\( T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( ( 1 + T + 35 T^{2} + 23 T^{3} + 529 T^{4} )^{8} \)
$29$ \( ( 1 - 5 T + 66 T^{2} - 555 T^{3} + 3920 T^{4} - 28200 T^{5} + 189134 T^{6} - 1026380 T^{7} + 6343899 T^{8} - 29765020 T^{9} + 159061694 T^{10} - 687769800 T^{11} + 2772541520 T^{12} - 11383687695 T^{13} + 39258339186 T^{14} - 86249381545 T^{15} + 500246412961 T^{16} )^{2} \)
$31$ \( 1 + 39 T^{2} + 3430 T^{4} + 163065 T^{6} + 6773220 T^{8} + 296986872 T^{10} + 10144743998 T^{12} + 345739802400 T^{14} + 11501584169495 T^{16} + 332255950106400 T^{18} + 9368884121776958 T^{20} + 263576942108675832 T^{22} + 5776818632616130020 T^{24} + \)\(13\!\cdots\!65\)\( T^{26} + \)\(27\!\cdots\!30\)\( T^{28} + \)\(29\!\cdots\!19\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 - 2 T - 31 T^{2} + 88 T^{3} - 122 T^{4} - 18302 T^{5} + 81031 T^{6} + 289434 T^{7} - 1619457 T^{8} + 10709058 T^{9} + 110931439 T^{10} - 927051206 T^{11} - 228647642 T^{12} + 6102268216 T^{13} - 79537518679 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2} \)
$41$ \( 1 - 54 T^{2} + 5160 T^{4} - 368270 T^{6} + 20358825 T^{8} - 1151359622 T^{10} + 57899888128 T^{12} - 2567003976540 T^{14} + 113908617753985 T^{16} - 4315133684563740 T^{18} + 163611245776465408 T^{20} - 5469078223378356902 T^{22} + \)\(16\!\cdots\!25\)\( T^{24} - \)\(49\!\cdots\!70\)\( T^{26} + \)\(11\!\cdots\!60\)\( T^{28} - \)\(20\!\cdots\!94\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )^{8}( 1 + 13 T + 43 T^{2} )^{8} \)
$47$ \( 1 + 173 T^{2} + 13870 T^{4} + 773020 T^{6} + 37466570 T^{8} + 1409514809 T^{10} + 35630259662 T^{12} + 1022803179550 T^{14} + 50909696806195 T^{16} + 2259372223625950 T^{18} + 173864301097727822 T^{20} + 15193463635625307161 T^{22} + \)\(89\!\cdots\!70\)\( T^{24} + \)\(40\!\cdots\!80\)\( T^{26} + \)\(16\!\cdots\!70\)\( T^{28} + \)\(44\!\cdots\!37\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 - 71 T^{2} - 150 T^{3} + 6432 T^{4} + 1950 T^{5} - 488983 T^{6} + 192300 T^{7} + 26002355 T^{8} + 10191900 T^{9} - 1373553247 T^{10} + 290310150 T^{11} + 50751573792 T^{12} - 62729323950 T^{13} - 1573669640159 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( 1 + 181 T^{2} + 12520 T^{4} + 229955 T^{6} - 15999940 T^{8} - 623745812 T^{10} + 48156426998 T^{12} + 5123867326280 T^{14} + 307569246051235 T^{16} + 17836182162780680 T^{18} + 583528810404912278 T^{20} - 26309931206498861492 T^{22} - \)\(23\!\cdots\!40\)\( T^{24} + \)\(11\!\cdots\!55\)\( T^{26} + \)\(22\!\cdots\!20\)\( T^{28} + \)\(11\!\cdots\!41\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 189 T^{2} + 16680 T^{4} - 1152085 T^{6} + 74808120 T^{8} - 4782904102 T^{10} + 347814490078 T^{12} - 24590631974850 T^{14} + 1546838296855255 T^{16} - 91501741578416850 T^{18} + 4815784127116065598 T^{20} - \)\(24\!\cdots\!22\)\( T^{22} + \)\(14\!\cdots\!20\)\( T^{24} - \)\(82\!\cdots\!85\)\( T^{26} + \)\(44\!\cdots\!80\)\( T^{28} - \)\(18\!\cdots\!49\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 + T + 159 T^{2} - 238 T^{3} + 12569 T^{4} - 15946 T^{5} + 713751 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{4} \)
$71$ \( ( 1 + 28 T + 311 T^{2} + 1907 T^{3} + 10031 T^{4} - 8251 T^{5} - 1509411 T^{6} - 19087514 T^{7} - 158934393 T^{8} - 1355213494 T^{9} - 7608940851 T^{10} - 2953123661 T^{11} + 254904572111 T^{12} + 3440665372357 T^{13} + 39839188299431 T^{14} + 254663364434948 T^{15} + 645753531245761 T^{16} )^{2} \)
$73$ \( 1 + 23 T^{2} + 14710 T^{4} + 27475 T^{6} + 102964580 T^{8} - 4851528886 T^{10} + 427407600722 T^{12} - 53265322210310 T^{14} + 1562285407061275 T^{16} - 283850902058741990 T^{18} + 12137624050535130002 T^{20} - \)\(73\!\cdots\!54\)\( T^{22} + \)\(83\!\cdots\!80\)\( T^{24} + \)\(11\!\cdots\!75\)\( T^{26} + \)\(33\!\cdots\!10\)\( T^{28} + \)\(28\!\cdots\!07\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( ( 1 - 25 T + 501 T^{2} - 8100 T^{3} + 108715 T^{4} - 1313450 T^{5} + 14241744 T^{6} - 141629175 T^{7} + 1311396489 T^{8} - 11188704825 T^{9} + 88882724304 T^{10} - 647582074550 T^{11} + 4234458055915 T^{12} - 24924156831900 T^{13} + 121786815216021 T^{14} - 480097724653975 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 - 97 T^{2} + 16850 T^{4} - 1726555 T^{6} + 145607140 T^{8} - 9460226116 T^{10} + 616698856582 T^{12} - 21019950912420 T^{14} + 1372062609692295 T^{16} - 144806441835661380 T^{18} + 29267492296001518822 T^{20} - \)\(30\!\cdots\!04\)\( T^{22} + \)\(32\!\cdots\!40\)\( T^{24} - \)\(26\!\cdots\!95\)\( T^{26} + \)\(18\!\cdots\!50\)\( T^{28} - \)\(71\!\cdots\!13\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 - 547 T^{2} + 136828 T^{4} - 20978449 T^{6} + 2212310245 T^{8} - 166170294529 T^{10} + 8584895351548 T^{12} - 271848766155667 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 + 378 T^{2} + 43195 T^{4} - 1829100 T^{6} - 951404880 T^{8} - 97555724586 T^{10} - 1788063513133 T^{12} + 794596480873200 T^{14} + 120216740714426075 T^{16} + 7476358288535938800 T^{18} - \)\(15\!\cdots\!73\)\( T^{20} - \)\(81\!\cdots\!94\)\( T^{22} - \)\(74\!\cdots\!80\)\( T^{24} - \)\(13\!\cdots\!00\)\( T^{26} + \)\(29\!\cdots\!95\)\( T^{28} + \)\(24\!\cdots\!82\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \)
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