Properties

Label 770.2.n.k
Level $770$
Weight $2$
Character orbit 770.n
Analytic conductor $6.148$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} + 18 x^{14} - 35 x^{13} + 89 x^{12} - 185 x^{11} + 837 x^{10} - 1660 x^{9} + 4196 x^{8} - 8420 x^{7} + 13485 x^{6} - 14630 x^{5} + 11615 x^{4} - 5200 x^{3} + 1425 x^{2} - 225 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} + 18 x^{14} - 35 x^{13} + 89 x^{12} - 185 x^{11} + 837 x^{10} - 1660 x^{9} + 4196 x^{8} - 8420 x^{7} + 13485 x^{6} - 14630 x^{5} + 11615 x^{4} - 5200 x^{3} + 1425 x^{2} - 225 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-3868393848839155704334 \nu^{15} + 137210741044597176058885 \nu^{14} - 602103152673718859142112 \nu^{13} + 1972131637204674006547975 \nu^{12} - 3456353004447917982417171 \nu^{11} + 9264041127738713907699695 \nu^{10} - 20162458808285702580751493 \nu^{9} + 94832461764688182593348230 \nu^{8} - 165086072229122240808050044 \nu^{7} + 434028121549908589683950250 \nu^{6} - 814686654062841911113066980 \nu^{5} + 1180461445072660754810361485 \nu^{4} - 1039966707279315032267324205 \nu^{3} + 624790894433310487691680075 \nu^{2} - 64970217849737633683769850 \nu - 35702600348776390967381600\)\()/ \)\(12\!\cdots\!75\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-5593447097405138343072 \nu^{15} + 33413875214437478841712 \nu^{14} - 122803399474782181558124 \nu^{13} + 270075779802333393005069 \nu^{12} - 605573839652198906139872 \nu^{11} + 1373081853780669243058672 \nu^{10} - 5295603606374634161609296 \nu^{9} + 13052663537837279424750616 \nu^{8} - 28555048582469673927375558 \nu^{7} + 62965383999451623695864288 \nu^{6} - 102777279530408178953672768 \nu^{5} + 119563094071606898301496048 \nu^{4} - 90148389712199612072837480 \nu^{3} + 41472919292194018561104250 \nu^{2} + 587321865887187362667420 \nu - 1300886355284735106090160\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{4}\)\(=\)\((\)\(9449048829314078168686 \nu^{15} - 20824039322932560170564 \nu^{14} + 41945681272404617756110 \nu^{13} + 126570490541569900696722 \nu^{12} - 21755869585093148192720 \nu^{11} + 492036846999996579746356 \nu^{10} + 3322774688200577942472545 \nu^{9} + 5785148132160743970144198 \nu^{8} - 1189168683701810071307350 \nu^{7} + 25846793353858990220439359 \nu^{6} - 81509212675922620444766648 \nu^{5} + 188295632927774547247151854 \nu^{4} - 236791388136440066980433245 \nu^{3} + 213337690670636233811688385 \nu^{2} - 83897745981371398329353255 \nu + 14625155202332126352399925\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-10090481044223853356143 \nu^{15} + 44874381353724035451156 \nu^{14} - 154073175122706287347490 \nu^{13} + 255899848895511617226063 \nu^{12} - 716019011968679529898445 \nu^{11} + 1405553866238857336774079 \nu^{10} - 7484222278249305383676960 \nu^{9} + 12240450167716810002938352 \nu^{8} - 33575245531017460833038990 \nu^{7} + 63180126347297704641070071 \nu^{6} - 92622010763878874839414531 \nu^{5} + 79739566063556449494732520 \nu^{4} - 50189133344965781444866300 \nu^{3} + 4945225676361699557165935 \nu^{2} + 256059924303019353762435 \nu - 663042186904224281201250\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-13261684353251055087598 \nu^{15} + 64889102133239984613928 \nu^{14} - 231782578275687599926012 \nu^{13} + 438151885057380705420049 \nu^{12} - 1128838501354974434094396 \nu^{11} + 2316810121766542984352397 \nu^{10} - 10831897101475339328055643 \nu^{9} + 20783869298417344986652511 \nu^{8} - 53316417725144758615329969 \nu^{7} + 105138078630046156607586703 \nu^{6} - 166700936568578517848130978 \nu^{5} + 172807259071335624468433610 \nu^{4} - 130417973960621540265911260 \nu^{3} + 48313127628625352558122130 \nu^{2} - 10830232679611655659975605 \nu + 1580257970487673865659275\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{7}\)\(=\)\((\)\(17905525670063121317389 \nu^{15} - 87642131765474328682256 \nu^{14} + 311550622005240336275953 \nu^{13} - 586294685969750729039648 \nu^{12} + 1506563372562935314402489 \nu^{11} - 3108475602158866952668444 \nu^{10} + 14548578023579894936481507 \nu^{9} - 27939374904376441549745417 \nu^{8} + 71023103816861860601170911 \nu^{7} - 140968559555443972851309926 \nu^{6} + 221591984295887668345192821 \nu^{5} - 227366323459053143745109930 \nu^{4} + 168220112919900022533647710 \nu^{3} - 61816337284676561616833085 \nu^{2} + 10612986266434190149123575 \nu - 1057051961018578043034825\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-26521687476168971248050 \nu^{15} + 142698918425068709596393 \nu^{14} - 522264755924765517916056 \nu^{13} + 1082332236788620281029240 \nu^{12} - 2616330034274550058302513 \nu^{11} + 5622531195059939210787695 \nu^{10} - 23604206283792286271391929 \nu^{9} + 51510223488689797655439960 \nu^{8} - 123525450817721813359756152 \nu^{7} + 256887854080360198741619990 \nu^{6} - 420825081963436281921024321 \nu^{5} + 480634298540230924198386031 \nu^{4} - 387788966099259050540833270 \nu^{3} + 188101908221044431934726300 \nu^{2} - 42738630329902483585637185 \nu + 5711319757834999177048815\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-42282078440743121721393 \nu^{15} + 193504866533652487289576 \nu^{14} - 673435280167901862302818 \nu^{13} + 1168322123420768923972802 \nu^{12} - 3176810295256387104164329 \nu^{11} + 6315621138974542204055216 \nu^{10} - 32281624052743125928137497 \nu^{9} + 55639672188053687121030873 \nu^{8} - 149476226232981697193219611 \nu^{7} + 284991996654195224292958149 \nu^{6} - 429205268217977023561674679 \nu^{5} + 396994823292184202438786769 \nu^{4} - 263740017630178215048869765 \nu^{3} + 51646694971964210417595890 \nu^{2} + 1564375506617613163848060 \nu - 1099518617266987761810150\)\()/ \)\(25\!\cdots\!55\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-258508759859043994521048 \nu^{15} + 1375056532814049111426385 \nu^{14} - 4972298973597692615601704 \nu^{13} + 10092984801695730818724985 \nu^{12} - 24370475562669276394399157 \nu^{11} + 52457494068312656655872945 \nu^{10} - 224495969769928143185094456 \nu^{9} + 483506076446815202534202875 \nu^{8} - 1149476165245356692342999223 \nu^{7} + 2391967661924174426968048545 \nu^{6} - 3846952453418666619181947175 \nu^{5} + 4231116010725615743098074335 \nu^{4} - 3222840733830906860603149010 \nu^{3} + 1361168335766713760685430925 \nu^{2} - 184025352587332971302400250 \nu + 8167130954661439380646450\)\()/ \)\(12\!\cdots\!75\)\( \)
\(\beta_{11}\)\(=\)\((\)\(261477095553369930256148 \nu^{15} - 1448309041734235701731575 \nu^{14} + 5312631215470010341673484 \nu^{13} - 11244779978022683839985745 \nu^{12} + 26658314415515213502390147 \nu^{11} - 58264769534611548297923925 \nu^{10} + 237552165159034732675242026 \nu^{9} - 537506071405681555025078965 \nu^{8} + 1256076823201396171521099833 \nu^{7} - 2667388566988038403416618365 \nu^{6} + 4366068837795968972301048905 \nu^{5} - 5072760815042918752027641950 \nu^{4} + 4105966374933514312805463460 \nu^{3} - 2066318520259861141160153675 \nu^{2} + 451073416178137337638597300 \nu - 54317873807241564678133200\)\()/ \)\(12\!\cdots\!75\)\( \)
\(\beta_{12}\)\(=\)\((\)\(339319163811650503003147 \nu^{15} - 1789666990494032732675620 \nu^{14} + 6512144003033947312229011 \nu^{13} - 13268700495750653991616575 \nu^{12} + 32454855171457963576183788 \nu^{11} - 69253370235610725995539735 \nu^{10} + 296369247274473964094482079 \nu^{9} - 631543835878004673560237065 \nu^{8} + 1530524918819572133376262857 \nu^{7} - 3162926717888110796649522550 \nu^{6} + 5130156247635845254631153680 \nu^{5} - 5781076872613274528860035810 \nu^{4} + 4631147780046540747060272440 \nu^{3} - 2164481658196091509688736950 \nu^{2} + 499582951364523616454552300 \nu - 74014301868065898688912325\)\()/ \)\(12\!\cdots\!75\)\( \)
\(\beta_{13}\)\(=\)\((\)\(379712132679213521462829 \nu^{15} - 1511078710666252098942985 \nu^{14} + 5015927072432154180328817 \nu^{13} - 6886384045483271064498390 \nu^{12} + 22260848963637191461671461 \nu^{11} - 39537772542636948889671540 \nu^{10} + 255985744298348567032403213 \nu^{9} - 326126887842120927398800340 \nu^{8} + 1045486712447646731457693879 \nu^{7} - 1750553630392296788793970605 \nu^{6} + 2321629221746407606229789050 \nu^{5} - 1243966782050815784470720475 \nu^{4} + 170581613417150864117653480 \nu^{3} + 1059053736980840011329395750 \nu^{2} - 374714290490823597089542725 \nu + 63766144349293326797100200\)\()/ \)\(12\!\cdots\!75\)\( \)
\(\beta_{14}\)\(=\)\((\)\(622543487914894074277751 \nu^{15} - 2890941502730216565410250 \nu^{14} + 10183348120485062064056433 \nu^{13} - 18197468782684215513838215 \nu^{12} + 49044615846688864042860564 \nu^{11} - 97904774519897579160920425 \nu^{10} + 486693006861233340448611737 \nu^{9} - 861108559827170140559790705 \nu^{8} + 2310888978777637630967857746 \nu^{7} - 4428940357440311085004190030 \nu^{6} + 6840943914004316496459157935 \nu^{5} - 6718936955434808158047384700 \nu^{4} + 4902364239042166144907342395 \nu^{3} - 1533154985435366680353721125 \nu^{2} + 344676285378747792485837375 \nu + 4918052391010093771393050\)\()/ \)\(12\!\cdots\!75\)\( \)
\(\beta_{15}\)\(=\)\((\)\(642709052819311513869406 \nu^{15} - 3115264020941943692346605 \nu^{14} + 11015802882769999493955923 \nu^{13} - 20455703524738598414450330 \nu^{12} + 52831753807897002041371509 \nu^{11} - 108633001560949968567688870 \nu^{10} + 515442348047122362760513047 \nu^{9} - 976303161566398236664359890 \nu^{8} + 2488264902210289367344574551 \nu^{7} - 4926980449625453019114604285 \nu^{6} + 7635759957537077074611333910 \nu^{5} - 7704126741573894863550125000 \nu^{4} + 5475007794353185188729008945 \nu^{3} - 1731565888274190175835146750 \nu^{2} + 119857869023157010376641325 \nu + 285014342555983582207800\)\()/ \)\(12\!\cdots\!75\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{13} - \beta_{12} - \beta_{9} - \beta_{7} - \beta_{6} + 4 \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{14} - 3 \beta_{13} - 2 \beta_{12} - \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 8 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{15} - 8 \beta_{13} + \beta_{11} + \beta_{10} - 28 \beta_{9} - 10 \beta_{8} - 2 \beta_{6} - 12 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} - 12 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-15 \beta_{15} - 13 \beta_{14} + 26 \beta_{13} + 23 \beta_{12} + 14 \beta_{11} + 23 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} + 65 \beta_{7} - 3 \beta_{6} - 25 \beta_{3} - 25 \beta_{2} - 3 \beta_{1} - 24\)
\(\nu^{6}\)\(=\)\(-85 \beta_{15} - 79 \beta_{14} + 252 \beta_{13} + 105 \beta_{12} + 105 \beta_{11} + 88 \beta_{10} + 170 \beta_{9} + 38 \beta_{8} + 125 \beta_{7} + 125 \beta_{5} - 88 \beta_{4} - 264 \beta_{3} - 170 \beta_{2} + 82 \beta_{1} - 264\)
\(\nu^{7}\)\(=\)\(-148 \beta_{15} - 105 \beta_{14} + 520 \beta_{13} + 165 \beta_{12} + 230 \beta_{11} + 444 \beta_{9} + 247 \beta_{8} + 58 \beta_{7} + 58 \beta_{6} + 606 \beta_{5} - 230 \beta_{4} - 453 \beta_{3} - 247 \beta_{2} - 444\)
\(\nu^{8}\)\(=\)\(-230 \beta_{15} + 318 \beta_{14} + 511 \beta_{13} + 223 \beta_{12} - 778 \beta_{10} + 1576 \beta_{9} + 692 \beta_{8} - 622 \beta_{7} + 620 \beta_{6} + 620 \beta_{5} - 223 \beta_{4} - 1576 \beta_{3} + 383 \beta_{2} - 622 \beta_{1} - 692\)
\(\nu^{9}\)\(=\)\(620 \beta_{15} + 1242 \beta_{14} - 1864 \beta_{13} - 2243 \beta_{11} - 2243 \beta_{10} + 293 \beta_{9} - 70 \beta_{8} + 5579 \beta_{6} + 831 \beta_{5} + 399 \beta_{4} + 70 \beta_{3} + 2484 \beta_{2} + 831 \beta_{1} + 2484\)
\(\nu^{10}\)\(=\)\(8653 \beta_{15} + 3074 \beta_{14} - 10497 \beta_{13} - 2675 \beta_{12} - 9666 \beta_{11} - 2675 \beta_{10} + 219 \beta_{9} + 219 \beta_{8} + 4419 \beta_{7} + 12112 \beta_{6} + 8254 \beta_{3} + 8254 \beta_{2} + 12112 \beta_{1} + 17581\)
\(\nu^{11}\)\(=\)\(29471 \beta_{15} + 5247 \beta_{14} - 32974 \beta_{13} - 20034 \beta_{12} - 20034 \beta_{11} + 1744 \beta_{10} - 15615 \beta_{9} - 9358 \beta_{8} - 10634 \beta_{7} - 10634 \beta_{5} - 1744 \beta_{4} + 42326 \beta_{3} + 15615 \beta_{2} + 41946 \beta_{1} + 42326\)
\(\nu^{12}\)\(=\)\(72614 \beta_{15} + 20034 \beta_{14} - 135694 \beta_{13} - 94392 \beta_{12} - 30668 \beta_{11} - 135398 \beta_{9} - 52446 \beta_{8} - 117334 \beta_{7} - 117334 \beta_{6} - 88688 \beta_{5} + 30668 \beta_{4} + 219705 \beta_{3} + 52446 \beta_{2} + 135398\)
\(\nu^{13}\)\(=\)\(30668 \beta_{15} + 59314 \beta_{14} - 359728 \beta_{13} - 211726 \beta_{12} + 2022 \beta_{10} - 490608 \beta_{9} - 208168 \beta_{8} - 376159 \beta_{7} - 504179 \beta_{6} - 504179 \beta_{5} + 211726 \beta_{4} + 490608 \beta_{3} + 123038 \beta_{2} - 376159 \beta_{1} + 208168\)
\(\nu^{14}\)\(=\)\(-504179 \beta_{15} - 128020 \beta_{14} - 248139 \beta_{13} + 341768 \beta_{11} + 341768 \beta_{10} - 1353614 \beta_{9} - 570135 \beta_{8} - 981106 \beta_{6} - 1135661 \beta_{5} + 587885 \beta_{4} + 570135 \beta_{3} - 256040 \beta_{2} - 1135661 \beta_{1} - 256040\)
\(\nu^{15}\)\(=\)\(-2458535 \beta_{15} - 1477429 \beta_{14} + 3388188 \beta_{13} + 2065314 \beta_{12} + 2252527 \beta_{11} + 2065314 \beta_{10} + 126554 \beta_{9} + 126554 \beta_{8} + 3415920 \beta_{7} - 1481154 \beta_{6} - 3046420 \beta_{3} - 3046420 \beta_{2} - 1481154 \beta_{1} - 4084652\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
2.57883 1.87363i
0.988623 0.718277i
0.295920 0.214999i
−2.05435 + 1.49258i
2.57883 + 1.87363i
0.988623 + 0.718277i
0.295920 + 0.214999i
−2.05435 1.49258i
0.897614 2.76257i
0.347506 1.06951i
0.0652271 0.200748i
−0.619365 + 1.90621i
0.897614 + 2.76257i
0.347506 + 1.06951i
0.0652271 + 0.200748i
−0.619365 1.90621i
0.309017 + 0.951057i −2.57883 1.87363i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.985024 3.03159i 0.809017 0.587785i −0.809017 0.587785i 2.21282 + 6.81035i −1.00000
71.2 0.309017 + 0.951057i −0.988623 0.718277i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.377620 1.16220i 0.809017 0.587785i −0.809017 0.587785i −0.465597 1.43296i −1.00000
71.3 0.309017 + 0.951057i −0.295920 0.214999i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.113032 0.347875i 0.809017 0.587785i −0.809017 0.587785i −0.885707 2.72592i −1.00000
71.4 0.309017 + 0.951057i 2.05435 + 1.49258i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.784693 + 2.41504i 0.809017 0.587785i −0.809017 0.587785i 1.06554 + 3.27939i −1.00000
141.1 0.309017 0.951057i −2.57883 + 1.87363i −0.809017 0.587785i −0.309017 0.951057i 0.985024 + 3.03159i 0.809017 + 0.587785i −0.809017 + 0.587785i 2.21282 6.81035i −1.00000
141.2 0.309017 0.951057i −0.988623 + 0.718277i −0.809017 0.587785i −0.309017 0.951057i 0.377620 + 1.16220i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.465597 + 1.43296i −1.00000
141.3 0.309017 0.951057i −0.295920 + 0.214999i −0.809017 0.587785i −0.309017 0.951057i 0.113032 + 0.347875i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.885707 + 2.72592i −1.00000
141.4 0.309017 0.951057i 2.05435 1.49258i −0.809017 0.587785i −0.309017 0.951057i −0.784693 2.41504i 0.809017 + 0.587785i −0.809017 + 0.587785i 1.06554 3.27939i −1.00000
421.1 −0.809017 + 0.587785i −0.897614 2.76257i 0.309017 0.951057i 0.809017 + 0.587785i 2.34998 + 1.70736i −0.309017 + 0.951057i 0.309017 + 0.951057i −4.39904 + 3.19609i −1.00000
421.2 −0.809017 + 0.587785i −0.347506 1.06951i 0.309017 0.951057i 0.809017 + 0.587785i 0.909784 + 0.660997i −0.309017 + 0.951057i 0.309017 + 0.951057i 1.40395 1.02003i −1.00000
421.3 −0.809017 + 0.587785i −0.0652271 0.200748i 0.309017 0.951057i 0.809017 + 0.587785i 0.170767 + 0.124069i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.39101 1.73717i −1.00000
421.4 −0.809017 + 0.587785i 0.619365 + 1.90621i 0.309017 0.951057i 0.809017 + 0.587785i −1.62152 1.17810i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.822965 + 0.597919i −1.00000
631.1 −0.809017 0.587785i −0.897614 + 2.76257i 0.309017 + 0.951057i 0.809017 0.587785i 2.34998 1.70736i −0.309017 0.951057i 0.309017 0.951057i −4.39904 3.19609i −1.00000
631.2 −0.809017 0.587785i −0.347506 + 1.06951i 0.309017 + 0.951057i 0.809017 0.587785i 0.909784 0.660997i −0.309017 0.951057i 0.309017 0.951057i 1.40395 + 1.02003i −1.00000
631.3 −0.809017 0.587785i −0.0652271 + 0.200748i 0.309017 + 0.951057i 0.809017 0.587785i 0.170767 0.124069i −0.309017 0.951057i 0.309017 0.951057i 2.39101 + 1.73717i −1.00000
631.4 −0.809017 0.587785i 0.619365 1.90621i 0.309017 + 0.951057i 0.809017 0.587785i −1.62152 + 1.17810i −0.309017 0.951057i 0.309017 0.951057i −0.822965 0.597919i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 631.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.k 16
11.c even 5 1 inner 770.2.n.k 16
11.c even 5 1 8470.2.a.dh 8
11.d odd 10 1 8470.2.a.dg 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.k 16 1.a even 1 1 trivial
770.2.n.k 16 11.c even 5 1 inner
8470.2.a.dg 8 11.d odd 10 1
8470.2.a.dh 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$3$ \( 25 + 225 T + 1425 T^{2} + 5200 T^{3} + 11615 T^{4} + 14630 T^{5} + 13485 T^{6} + 8420 T^{7} + 4196 T^{8} + 1660 T^{9} + 837 T^{10} + 185 T^{11} + 89 T^{12} + 35 T^{13} + 18 T^{14} + 5 T^{15} + T^{16} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$11$ \( 214358881 + 38974342 T + 70862440 T^{2} + 2254714 T^{3} + 9531291 T^{4} - 1448128 T^{5} + 747538 T^{6} - 322894 T^{7} + 53113 T^{8} - 29354 T^{9} + 6178 T^{10} - 1088 T^{11} + 651 T^{12} + 14 T^{13} + 40 T^{14} + 2 T^{15} + T^{16} \)
$13$ \( 162919696 - 731428256 T + 1345218288 T^{2} - 539126072 T^{3} + 435114896 T^{4} - 169579496 T^{5} + 68691960 T^{6} - 18462576 T^{7} + 4694084 T^{8} - 863552 T^{9} + 163100 T^{10} - 16558 T^{11} + 2011 T^{12} - 169 T^{13} + 52 T^{14} - 8 T^{15} + T^{16} \)
$17$ \( 11881 + 129383 T + 1184515 T^{2} + 5247784 T^{3} + 13035643 T^{4} + 15192444 T^{5} + 14907949 T^{6} + 2538826 T^{7} + 40170 T^{8} + 61522 T^{9} + 95633 T^{10} + 22149 T^{11} + 5387 T^{12} + 869 T^{13} + 132 T^{14} + 13 T^{15} + T^{16} \)
$19$ \( 5026810000 + 12956975000 T + 11581873000 T^{2} - 6312779000 T^{3} + 4423209025 T^{4} - 1369409850 T^{5} + 408758720 T^{6} - 83828935 T^{7} + 16315251 T^{8} - 2448760 T^{9} + 351467 T^{10} - 37600 T^{11} + 6259 T^{12} - 1035 T^{13} + 163 T^{14} - 15 T^{15} + T^{16} \)
$23$ \( ( 1280 - 5760 T + 8000 T^{2} - 3200 T^{3} - 576 T^{4} + 480 T^{5} - 36 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$29$ \( 276756496 - 508795424 T + 756290960 T^{2} - 657404208 T^{3} + 452415088 T^{4} - 203743312 T^{5} + 67204296 T^{6} - 9617332 T^{7} + 257680 T^{8} + 591374 T^{9} + 206132 T^{10} - 9418 T^{11} - 423 T^{12} + 233 T^{13} + 118 T^{14} + 14 T^{15} + T^{16} \)
$31$ \( 18317998336 - 44888732416 T + 48924917760 T^{2} - 26444750176 T^{3} + 7187970960 T^{4} - 1002377920 T^{5} + 470021392 T^{6} - 41667464 T^{7} + 10300872 T^{8} - 651792 T^{9} + 457688 T^{10} + 49712 T^{11} + 14688 T^{12} + 1080 T^{13} + 174 T^{14} + 6 T^{15} + T^{16} \)
$37$ \( 10758400 + 18236800 T + 48726400 T^{2} + 55974400 T^{3} + 78842640 T^{4} + 68781920 T^{5} + 55539040 T^{6} + 28483920 T^{7} + 8400136 T^{8} - 325696 T^{9} - 180176 T^{10} + 16328 T^{11} + 12144 T^{12} - 2954 T^{13} + 396 T^{14} - 28 T^{15} + T^{16} \)
$41$ \( 453519616 - 35185251200 T + 1037071399936 T^{2} + 705674146892 T^{3} + 276899209437 T^{4} + 63617368749 T^{5} + 13366308457 T^{6} + 1406189025 T^{7} + 254274167 T^{8} + 10900828 T^{9} + 3196273 T^{10} + 22092 T^{11} + 17061 T^{12} + 661 T^{13} + 108 T^{14} - 2 T^{15} + T^{16} \)
$43$ \( ( -591484 + 612198 T - 169261 T^{2} - 14663 T^{3} + 10984 T^{4} - 393 T^{5} - 188 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$47$ \( 377408720896 - 731305574400 T + 633516730368 T^{2} - 276814087680 T^{3} + 64273739008 T^{4} - 8658109440 T^{5} + 2532488896 T^{6} - 261088000 T^{7} + 38555280 T^{8} - 1943400 T^{9} + 1313516 T^{10} + 102750 T^{11} + 28023 T^{12} + 1695 T^{13} + 278 T^{14} + 10 T^{15} + T^{16} \)
$53$ \( 104649299070976 - 35697502798848 T + 50811733744640 T^{2} + 5001289845376 T^{3} + 787700746768 T^{4} + 33187752096 T^{5} + 22516888624 T^{6} + 70768024 T^{7} + 356938760 T^{8} + 12007528 T^{9} + 4991968 T^{10} + 167016 T^{11} + 23072 T^{12} - 644 T^{13} + 122 T^{14} + 2 T^{15} + T^{16} \)
$59$ \( 120164479345936 + 13151847874032 T + 6988115532960 T^{2} + 719421774410 T^{3} + 245445964445 T^{4} + 10678805186 T^{5} + 5820430422 T^{6} - 121922435 T^{7} + 140466515 T^{8} - 1400370 T^{9} + 1682187 T^{10} + 29654 T^{11} + 18665 T^{12} - 735 T^{13} + 105 T^{14} - 7 T^{15} + T^{16} \)
$61$ \( 3393761536 + 12943085056 T + 20132388736 T^{2} + 6525262432 T^{3} + 2948200272 T^{4} + 48221696 T^{5} + 77949312 T^{6} + 43349992 T^{7} + 16870176 T^{8} + 1579112 T^{9} + 691672 T^{10} + 85856 T^{11} + 11672 T^{12} + 792 T^{13} + 216 T^{14} - 4 T^{15} + T^{16} \)
$67$ \( ( 162964 - 333930 T - 13175 T^{2} + 80879 T^{3} - 19566 T^{4} + 435 T^{5} + 306 T^{6} - 33 T^{7} + T^{8} )^{2} \)
$71$ \( 1936 - 25872 T + 590496 T^{2} - 1707240 T^{3} + 1246568 T^{4} + 4942816 T^{5} + 27267832 T^{6} + 42004336 T^{7} + 33949932 T^{8} + 897588 T^{9} + 345108 T^{10} - 38100 T^{11} + 9245 T^{12} + 1517 T^{13} + 200 T^{14} - 2 T^{15} + T^{16} \)
$73$ \( 54253993164361 - 55880353934892 T + 52131173215320 T^{2} - 22911427477966 T^{3} + 6520285373943 T^{4} - 1195828445356 T^{5} + 164369323444 T^{6} - 13757574654 T^{7} + 1458508150 T^{8} - 125700678 T^{9} + 8388028 T^{10} - 440716 T^{11} + 65592 T^{12} - 4216 T^{13} + 222 T^{14} - 12 T^{15} + T^{16} \)
$79$ \( 228610650255376 + 157860879763120 T + 63529544211136 T^{2} + 16334074195112 T^{3} + 3461519660232 T^{4} + 527328035424 T^{5} + 85727969472 T^{6} + 9919650920 T^{7} + 1046367812 T^{8} + 74986848 T^{9} + 10730308 T^{10} + 119792 T^{11} + 41101 T^{12} + 1421 T^{13} + 68 T^{14} - 2 T^{15} + T^{16} \)
$83$ \( 185015247324961 + 13342381830241 T + 30769094799436 T^{2} + 11299376446091 T^{3} + 2197595939229 T^{4} + 158763992781 T^{5} + 38666862399 T^{6} + 3727523122 T^{7} + 353055994 T^{8} + 18116602 T^{9} + 6620729 T^{10} + 8480 T^{11} + 39457 T^{12} + 2208 T^{13} + 165 T^{14} + 5 T^{15} + T^{16} \)
$89$ \( ( 108020 - 22690 T - 62945 T^{2} + 14135 T^{3} + 6756 T^{4} - 539 T^{5} - 208 T^{6} - T^{7} + T^{8} )^{2} \)
$97$ \( 938494000081 + 1464221102960 T + 2774692495006 T^{2} + 1586676062722 T^{3} + 582113757432 T^{4} + 121718009504 T^{5} + 19503987122 T^{6} + 268405000 T^{7} + 110313662 T^{8} - 28331942 T^{9} + 9220088 T^{10} + 65382 T^{11} - 469 T^{12} - 894 T^{13} + 338 T^{14} - 22 T^{15} + T^{16} \)
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