Properties

Label 126.2.l.a
Level $126$
Weight $2$
Character orbit 126.l
Analytic conductor $1.006$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + 16767 x^{2} - 17496 x + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + 2785 x^{8} - 2640 x^{7} - 2601 x^{6} + 10260 x^{5} - 10611 x^{4} - 1944 x^{3} + 16767 x^{2} - 17496 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-50 \nu^{15} - 1352 \nu^{14} + 6827 \nu^{13} - 7676 \nu^{12} - 27422 \nu^{11} + 107246 \nu^{10} - 107467 \nu^{9} - 206194 \nu^{8} + 757363 \nu^{7} - 724572 \nu^{6} - 756198 \nu^{5} + 2730942 \nu^{4} - 2372247 \nu^{3} - 1318518 \nu^{2} + 4444713 \nu - 2825604\)\()/142155\)
\(\beta_{2}\)\(=\)\((\)\(1292 \nu^{15} - 10486 \nu^{14} + 25660 \nu^{13} + 10145 \nu^{12} - 192280 \nu^{11} + 408694 \nu^{10} - 51650 \nu^{9} - 1459361 \nu^{8} + 2979638 \nu^{7} - 1138791 \nu^{6} - 5534208 \nu^{5} + 10987326 \nu^{4} - 5516586 \nu^{3} - 9628389 \nu^{2} + 17565255 \nu - 9270693\)\()/142155\)
\(\beta_{3}\)\(=\)\((\)\(2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + 878875 \nu^{10} - 166676 \nu^{9} - 3023495 \nu^{8} + 6386423 \nu^{7} - 2756469 \nu^{6} - 11333493 \nu^{5} + 23486949 \nu^{4} - 12667509 \nu^{3} - 19623951 \nu^{2} + 37934244 \nu - 20783061\)\()/142155\)
\(\beta_{4}\)\(=\)\((\)\(-2782 \nu^{15} + 15918 \nu^{14} - 26947 \nu^{13} - 42629 \nu^{12} + 270897 \nu^{11} - 425335 \nu^{10} - 220488 \nu^{9} + 2001225 \nu^{8} - 3082271 \nu^{7} - 70562 \nu^{6} + 7458951 \nu^{5} - 11186748 \nu^{4} + 2636253 \nu^{3} + 12673422 \nu^{2} - 17180343 \nu + 7405182\)\()/47385\)
\(\beta_{5}\)\(=\)\((\)\(-3648 \nu^{15} + 23120 \nu^{14} - 45376 \nu^{13} - 47012 \nu^{12} + 401996 \nu^{11} - 719704 \nu^{10} - 150629 \nu^{9} + 2988436 \nu^{8} - 5221301 \nu^{7} + 915140 \nu^{6} + 11160666 \nu^{5} - 19038915 \nu^{4} + 7000290 \nu^{3} + 19081575 \nu^{2} - 29765799 \nu + 14281839\)\()/47385\)
\(\beta_{6}\)\(=\)\((\)\(-16948 \nu^{15} + 107120 \nu^{14} - 210206 \nu^{13} - 216202 \nu^{12} + 1856546 \nu^{11} - 3329594 \nu^{10} - 672599 \nu^{9} + 13772146 \nu^{8} - 24135496 \nu^{7} + 4376130 \nu^{6} + 51333696 \nu^{5} - 87961680 \nu^{4} + 32792850 \nu^{3} + 87596640 \nu^{2} - 137624994 \nu + 66620394\)\()/142155\)
\(\beta_{7}\)\(=\)\((\)\(-4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} - 771188 \nu^{10} - 200900 \nu^{9} + 3269857 \nu^{8} - 5585140 \nu^{7} + 796053 \nu^{6} + 12187116 \nu^{5} - 20322468 \nu^{4} + 7028856 \nu^{3} + 20790351 \nu^{2} - 31653180 \nu + 14935023\)\()/28431\)
\(\beta_{8}\)\(=\)\((\)\(24706 \nu^{15} - 155855 \nu^{14} + 305147 \nu^{13} + 316579 \nu^{12} - 2700647 \nu^{11} + 4831208 \nu^{10} + 1007453 \nu^{9} - 20043787 \nu^{8} + 35017972 \nu^{7} - 6181845 \nu^{6} - 74736702 \nu^{5} + 127624545 \nu^{4} - 47130660 \nu^{3} - 127644255 \nu^{2} + 199674558 \nu - 96324228\)\()/142155\)
\(\beta_{9}\)\(=\)\((\)\(2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + 93322 \nu^{9} - 1605386 \nu^{8} + 2760692 \nu^{7} - 423483 \nu^{6} - 5980032 \nu^{5} + 10047753 \nu^{4} - 3553308 \nu^{3} - 10193607 \nu^{2} + 15657462 \nu - 7453296\)\()/10935\)
\(\beta_{10}\)\(=\)\((\)\(-26777 \nu^{15} + 172141 \nu^{14} - 345205 \nu^{13} - 329150 \nu^{12} + 2992915 \nu^{11} - 5473354 \nu^{10} - 873115 \nu^{9} + 22225271 \nu^{8} - 39691043 \nu^{7} + 8309796 \nu^{6} + 82849923 \nu^{5} - 144806481 \nu^{4} + 56755161 \nu^{3} + 141385419 \nu^{2} - 227291265 \nu + 112005018\)\()/142155\)
\(\beta_{11}\)\(=\)\((\)\(32006 \nu^{15} - 201190 \nu^{14} + 390892 \nu^{13} + 415394 \nu^{12} - 3479542 \nu^{11} + 6180118 \nu^{10} + 1385833 \nu^{9} - 25793657 \nu^{8} + 44738537 \nu^{7} - 7406040 \nu^{6} - 96117777 \nu^{5} + 162796500 \nu^{4} - 58760640 \nu^{3} - 164045655 \nu^{2} + 253970478 \nu - 121376313\)\()/142155\)
\(\beta_{12}\)\(=\)\((\)\(-10990 \nu^{15} + 68842 \nu^{14} - 133062 \nu^{13} - 143724 \nu^{12} + 1190062 \nu^{11} - 2105781 \nu^{10} - 489283 \nu^{9} + 8822519 \nu^{8} - 15253998 \nu^{7} + 2459777 \nu^{6} + 32873688 \nu^{5} - 55540512 \nu^{4} + 19941417 \nu^{3} + 56061558 \nu^{2} - 86668623 \nu + 41433444\)\()/47385\)
\(\beta_{13}\)\(=\)\((\)\(-35897 \nu^{15} + 219337 \nu^{14} - 409846 \nu^{13} - 492632 \nu^{12} + 3774151 \nu^{11} - 6478657 \nu^{10} - 1950109 \nu^{9} + 27930713 \nu^{8} - 46902542 \nu^{7} + 5479452 \nu^{6} + 103995882 \nu^{5} - 170423757 \nu^{4} + 56059857 \nu^{3} + 177093783 \nu^{2} - 264354354 \nu + 122931270\)\()/142155\)
\(\beta_{14}\)\(=\)\((\)\(-14807 \nu^{15} + 92828 \nu^{14} - 179757 \nu^{13} - 193044 \nu^{12} + 1605257 \nu^{11} - 2844540 \nu^{10} - 652043 \nu^{9} + 11900965 \nu^{8} - 20604021 \nu^{7} + 3361183 \nu^{6} + 44347701 \nu^{5} - 75011778 \nu^{4} + 26997138 \nu^{3} + 75667932 \nu^{2} - 117064278 \nu + 55960227\)\()/47385\)
\(\beta_{15}\)\(=\)\((\)\(-68699 \nu^{15} + 433198 \nu^{14} - 844756 \nu^{13} - 889547 \nu^{12} + 7506256 \nu^{11} - 13372516 \nu^{10} - 2912329 \nu^{9} + 55693424 \nu^{8} - 96903605 \nu^{7} + 16506963 \nu^{6} + 207637920 \nu^{5} - 352968678 \nu^{4} + 128769183 \nu^{3} + 354359367 \nu^{2} - 551274174 \nu + 264515463\)\()/142155\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} + 4 \beta_{14} + \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 3\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{15} - \beta_{14} + 5 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{3} - 6 \beta_{2} - \beta_{1} - 4\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{15} + 2 \beta_{14} + 11 \beta_{13} - 4 \beta_{12} + 7 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} - 4 \beta_{8} - \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + 7 \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 15 \beta_{1} + 7\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{15} - 6 \beta_{14} + 9 \beta_{13} + 12 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + \beta_{8} + 23 \beta_{7} + 17 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} - 16 \beta_{2} - 31 \beta_{1} + 10\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{15} + 26 \beta_{14} + 2 \beta_{13} - 31 \beta_{12} + 28 \beta_{11} + 13 \beta_{10} - 14 \beta_{9} + 4 \beta_{8} + 39 \beta_{7} - 20 \beta_{6} - 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 10 \beta_{2} - 20 \beta_{1} + 15\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-15 \beta_{15} + 16 \beta_{14} - 20 \beta_{13} + 16 \beta_{12} + 29 \beta_{11} - 13 \beta_{10} - 60 \beta_{9} + 6 \beta_{8} + 26 \beta_{7} - 5 \beta_{6} - 10 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 30 \beta_{2} - 14 \beta_{1} - 11\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(43 \beta_{15} + 28 \beta_{14} - 8 \beta_{13} + 10 \beta_{12} + 44 \beta_{11} - 16 \beta_{10} + 25 \beta_{9} - 92 \beta_{8} - 65 \beta_{7} - 114 \beta_{6} - 52 \beta_{5} - 7 \beta_{4} + 31 \beta_{3} + 26 \beta_{2} - 27 \beta_{1} - 1\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(50 \beta_{15} - 12 \beta_{14} - 18 \beta_{13} + 102 \beta_{12} + 102 \beta_{11} - 144 \beta_{10} + 80 \beta_{9} - 106 \beta_{8} - 104 \beta_{7} + 229 \beta_{6} - 84 \beta_{5} + 34 \beta_{4} + 44 \beta_{3} - 8 \beta_{2} - 194 \beta_{1} + 134\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(140 \beta_{15} + 139 \beta_{14} + 31 \beta_{13} - 224 \beta_{12} + 95 \beta_{11} - 46 \beta_{10} + 320 \beta_{9} - 220 \beta_{8} - 117 \beta_{7} + 251 \beta_{6} - 337 \beta_{5} + 10 \beta_{4} + 224 \beta_{3} - 80 \beta_{2} - 109 \beta_{1}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(60 \beta_{15} + 503 \beta_{14} + 23 \beta_{13} - 313 \beta_{12} + 133 \beta_{11} - 218 \beta_{10} + 114 \beta_{9} + 171 \beta_{8} - 482 \beta_{7} + 779 \beta_{6} - 524 \beta_{5} + 270 \beta_{4} - 12 \beta_{3} - 399 \beta_{2} - 166 \beta_{1} - 193\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(-94 \beta_{15} + 722 \beta_{14} + 191 \beta_{13} - 343 \beta_{12} - 182 \beta_{11} + 238 \beta_{10} + 569 \beta_{9} - 454 \beta_{8} - 1069 \beta_{7} - 192 \beta_{6} - 674 \beta_{5} + 382 \beta_{4} + 398 \beta_{3} - 719 \beta_{2} - 84 \beta_{1} - 1112\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(-425 \beta_{15} + 1353 \beta_{14} + 255 \beta_{13} + 1086 \beta_{12} + 1068 \beta_{11} - 588 \beta_{10} + 691 \beta_{9} - 122 \beta_{8} - 1675 \beta_{7} + 662 \beta_{6} + 498 \beta_{5} + 707 \beta_{4} - 206 \beta_{3} - 454 \beta_{2} - 1759 \beta_{1} + 727\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(-953 \beta_{15} + 446 \beta_{14} + 35 \beta_{13} + 2231 \beta_{12} + 1294 \beta_{11} + 82 \beta_{10} + 1990 \beta_{9} - 1151 \beta_{8} + 1755 \beta_{7} + 91 \beta_{6} + 688 \beta_{5} - 970 \beta_{4} + 1378 \beta_{3} - 484 \beta_{2} - 1505 \beta_{1} + 1896\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(1563 \beta_{15} + 1963 \beta_{14} - 725 \beta_{13} + 2308 \beta_{12} + 3086 \beta_{11} - 1306 \beta_{10} + 714 \beta_{9} + 4854 \beta_{8} + 2906 \beta_{7} - 1550 \beta_{6} + 32 \beta_{5} - 1380 \beta_{4} - 2391 \beta_{3} + 1551 \beta_{2} + 1375 \beta_{1} + 6694\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-\beta_{7}\) \(-\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.73109 0.0577511i
0.320287 + 1.70218i
−1.68301 + 0.409224i
0.765614 1.55365i
1.71298 + 0.256290i
1.27866 1.16834i
−1.70672 0.295146i
1.58110 + 0.707199i
1.71298 0.256290i
1.27866 + 1.16834i
−1.70672 + 0.295146i
1.58110 0.707199i
1.73109 + 0.0577511i
0.320287 1.70218i
−1.68301 0.409224i
0.765614 + 1.55365i
1.00000i −0.890915 + 1.48535i −1.00000 1.14095 + 1.97618i 1.48535 + 0.890915i 1.42337 + 2.23025i 1.00000i −1.41254 2.64665i 1.97618 1.14095i
5.2 1.00000i −0.290993 1.70743i −1.00000 −0.0338034 0.0585493i −1.70743 + 0.290993i 1.19767 2.35915i 1.00000i −2.83065 + 0.993700i −0.0585493 + 0.0338034i
5.3 1.00000i 1.38631 + 1.03834i −1.00000 0.714925 + 1.23829i 1.03834 1.38631i 0.327442 2.62541i 1.00000i 0.843698 + 2.87892i 1.23829 0.714925i
5.4 1.00000i 1.52765 0.816261i −1.00000 −1.82207 3.15592i −0.816261 1.52765i −1.58246 + 2.12034i 1.00000i 1.66744 2.49392i −3.15592 + 1.82207i
5.5 1.00000i −1.56012 0.752355i −1.00000 −1.80966 3.13442i 0.752355 1.56012i −2.41308 1.08492i 1.00000i 1.86792 + 2.34752i 3.13442 1.80966i
5.6 1.00000i −1.08509 1.35003i −1.00000 1.77612 + 3.07634i 1.35003 1.08509i 2.63804 0.201867i 1.00000i −0.645160 + 2.92981i −3.07634 + 1.77612i
5.7 1.00000i −0.734581 + 1.56856i −1.00000 0.483662 + 0.837727i −1.56856 0.734581i −2.16249 + 1.52435i 1.00000i −1.92078 2.30447i −0.837727 + 0.483662i
5.8 1.00000i 1.64774 + 0.533822i −1.00000 −0.450129 0.779646i −0.533822 + 1.64774i 1.57151 + 2.12847i 1.00000i 2.43007 + 1.75919i 0.779646 0.450129i
101.1 1.00000i −1.56012 + 0.752355i −1.00000 −1.80966 + 3.13442i 0.752355 + 1.56012i −2.41308 + 1.08492i 1.00000i 1.86792 2.34752i 3.13442 + 1.80966i
101.2 1.00000i −1.08509 + 1.35003i −1.00000 1.77612 3.07634i 1.35003 + 1.08509i 2.63804 + 0.201867i 1.00000i −0.645160 2.92981i −3.07634 1.77612i
101.3 1.00000i −0.734581 1.56856i −1.00000 0.483662 0.837727i −1.56856 + 0.734581i −2.16249 1.52435i 1.00000i −1.92078 + 2.30447i −0.837727 0.483662i
101.4 1.00000i 1.64774 0.533822i −1.00000 −0.450129 + 0.779646i −0.533822 1.64774i 1.57151 2.12847i 1.00000i 2.43007 1.75919i 0.779646 + 0.450129i
101.5 1.00000i −0.890915 1.48535i −1.00000 1.14095 1.97618i 1.48535 0.890915i 1.42337 2.23025i 1.00000i −1.41254 + 2.64665i 1.97618 + 1.14095i
101.6 1.00000i −0.290993 + 1.70743i −1.00000 −0.0338034 + 0.0585493i −1.70743 0.290993i 1.19767 + 2.35915i 1.00000i −2.83065 0.993700i −0.0585493 0.0338034i
101.7 1.00000i 1.38631 1.03834i −1.00000 0.714925 1.23829i 1.03834 + 1.38631i 0.327442 + 2.62541i 1.00000i 0.843698 2.87892i 1.23829 + 0.714925i
101.8 1.00000i 1.52765 + 0.816261i −1.00000 −1.82207 + 3.15592i −0.816261 + 1.52765i −1.58246 2.12034i 1.00000i 1.66744 + 2.49392i −3.15592 1.82207i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.l.a 16
3.b odd 2 1 378.2.l.a 16
4.b odd 2 1 1008.2.ca.c 16
7.b odd 2 1 882.2.l.b 16
7.c even 3 1 882.2.m.b 16
7.c even 3 1 882.2.t.a 16
7.d odd 6 1 126.2.t.a yes 16
7.d odd 6 1 882.2.m.a 16
9.c even 3 1 378.2.t.a 16
9.c even 3 1 1134.2.k.a 16
9.d odd 6 1 126.2.t.a yes 16
9.d odd 6 1 1134.2.k.b 16
12.b even 2 1 3024.2.ca.c 16
21.c even 2 1 2646.2.l.a 16
21.g even 6 1 378.2.t.a 16
21.g even 6 1 2646.2.m.a 16
21.h odd 6 1 2646.2.m.b 16
21.h odd 6 1 2646.2.t.b 16
28.f even 6 1 1008.2.df.c 16
36.f odd 6 1 3024.2.df.c 16
36.h even 6 1 1008.2.df.c 16
63.g even 3 1 2646.2.m.a 16
63.h even 3 1 2646.2.l.a 16
63.i even 6 1 inner 126.2.l.a 16
63.j odd 6 1 882.2.l.b 16
63.k odd 6 1 1134.2.k.b 16
63.k odd 6 1 2646.2.m.b 16
63.l odd 6 1 2646.2.t.b 16
63.n odd 6 1 882.2.m.a 16
63.o even 6 1 882.2.t.a 16
63.s even 6 1 882.2.m.b 16
63.s even 6 1 1134.2.k.a 16
63.t odd 6 1 378.2.l.a 16
84.j odd 6 1 3024.2.df.c 16
252.r odd 6 1 1008.2.ca.c 16
252.bj even 6 1 3024.2.ca.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a 16 1.a even 1 1 trivial
126.2.l.a 16 63.i even 6 1 inner
126.2.t.a yes 16 7.d odd 6 1
126.2.t.a yes 16 9.d odd 6 1
378.2.l.a 16 3.b odd 2 1
378.2.l.a 16 63.t odd 6 1
378.2.t.a 16 9.c even 3 1
378.2.t.a 16 21.g even 6 1
882.2.l.b 16 7.b odd 2 1
882.2.l.b 16 63.j odd 6 1
882.2.m.a 16 7.d odd 6 1
882.2.m.a 16 63.n odd 6 1
882.2.m.b 16 7.c even 3 1
882.2.m.b 16 63.s even 6 1
882.2.t.a 16 7.c even 3 1
882.2.t.a 16 63.o even 6 1
1008.2.ca.c 16 4.b odd 2 1
1008.2.ca.c 16 252.r odd 6 1
1008.2.df.c 16 28.f even 6 1
1008.2.df.c 16 36.h even 6 1
1134.2.k.a 16 9.c even 3 1
1134.2.k.a 16 63.s even 6 1
1134.2.k.b 16 9.d odd 6 1
1134.2.k.b 16 63.k odd 6 1
2646.2.l.a 16 21.c even 2 1
2646.2.l.a 16 63.h even 3 1
2646.2.m.a 16 21.g even 6 1
2646.2.m.a 16 63.g even 3 1
2646.2.m.b 16 21.h odd 6 1
2646.2.m.b 16 63.k odd 6 1
2646.2.t.b 16 21.h odd 6 1
2646.2.t.b 16 63.l odd 6 1
3024.2.ca.c 16 12.b even 2 1
3024.2.ca.c 16 252.bj even 6 1
3024.2.df.c 16 36.f odd 6 1
3024.2.df.c 16 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( 6561 - 2916 T^{3} + 729 T^{4} + 486 T^{5} + 729 T^{6} - 270 T^{7} - 126 T^{8} - 90 T^{9} + 81 T^{10} + 18 T^{11} + 9 T^{12} - 12 T^{13} + T^{16} \)
$5$ \( 81 + 1134 T + 16929 T^{2} - 11826 T^{3} + 32724 T^{4} - 19494 T^{5} + 42201 T^{6} - 25002 T^{7} + 22536 T^{8} - 5814 T^{9} + 3582 T^{10} - 450 T^{11} + 423 T^{12} - 24 T^{13} + 24 T^{14} + T^{16} \)
$7$ \( 5764801 - 1647086 T + 705894 T^{2} + 134456 T^{3} + 55223 T^{4} - 72030 T^{5} + 22687 T^{6} - 8036 T^{7} - 1350 T^{8} - 1148 T^{9} + 463 T^{10} - 210 T^{11} + 23 T^{12} + 8 T^{13} + 6 T^{14} - 2 T^{15} + T^{16} \)
$11$ \( 61732449 + 105645222 T + 43505991 T^{2} - 28680318 T^{3} - 16460820 T^{4} + 5642460 T^{5} + 3600531 T^{6} - 868968 T^{7} - 395118 T^{8} + 92826 T^{9} + 30132 T^{10} - 7776 T^{11} - 1017 T^{12} + 360 T^{13} + 18 T^{14} - 12 T^{15} + T^{16} \)
$13$ \( 390971529 + 543480678 T + 88818120 T^{2} - 226594584 T^{3} - 17730819 T^{4} + 46095318 T^{5} + 2433294 T^{6} - 6581682 T^{7} + 452916 T^{8} + 423144 T^{9} - 40167 T^{10} - 19584 T^{11} + 2682 T^{12} + 414 T^{13} - 57 T^{14} - 6 T^{15} + T^{16} \)
$17$ \( 56070144 + 73322496 T + 91570176 T^{2} + 66023424 T^{3} + 53021952 T^{4} + 31774464 T^{5} + 19253376 T^{6} + 8227008 T^{7} + 3161232 T^{8} + 875952 T^{9} + 241380 T^{10} + 53892 T^{11} + 11493 T^{12} + 1794 T^{13} + 231 T^{14} + 18 T^{15} + T^{16} \)
$19$ \( 9199089 + 6551280 T - 11629251 T^{2} - 9389520 T^{3} + 14697288 T^{4} + 6237810 T^{5} - 3302019 T^{6} - 1418796 T^{7} + 634680 T^{8} + 162594 T^{9} - 55782 T^{10} - 10116 T^{11} + 4167 T^{12} - 72 T^{14} + T^{16} \)
$23$ \( 187388721 - 48787596 T - 100363617 T^{2} + 27232524 T^{3} + 39946284 T^{4} - 10493226 T^{5} - 6579225 T^{6} + 1751058 T^{7} + 815022 T^{8} - 187596 T^{9} - 54108 T^{10} + 11340 T^{11} + 2763 T^{12} - 396 T^{13} - 54 T^{14} + 6 T^{15} + T^{16} \)
$29$ \( 1108809 + 4264650 T + 2880279 T^{2} - 9950850 T^{3} + 3174066 T^{4} + 3639168 T^{5} - 1095687 T^{6} - 912708 T^{7} + 299376 T^{8} + 121986 T^{9} - 19278 T^{10} - 7938 T^{11} + 1197 T^{12} + 288 T^{13} - 36 T^{14} - 6 T^{15} + T^{16} \)
$31$ \( 65610000 + 157901400 T^{2} + 125744481 T^{4} + 42206670 T^{6} + 6469713 T^{8} + 444582 T^{10} + 14238 T^{12} + 204 T^{14} + T^{16} \)
$37$ \( 32746159681 - 85008385594 T + 184692416508 T^{2} - 97465928012 T^{3} + 46677522614 T^{4} - 7527670110 T^{5} + 2287630168 T^{6} - 226939114 T^{7} + 74228751 T^{8} - 4587118 T^{9} + 1458136 T^{10} - 40410 T^{11} + 20726 T^{12} - 212 T^{13} + 180 T^{14} + 2 T^{15} + T^{16} \)
$41$ \( 81 - 1620 T + 22518 T^{2} - 157140 T^{3} + 789021 T^{4} - 2008098 T^{5} + 3629772 T^{6} - 2659284 T^{7} + 1500462 T^{8} - 435366 T^{9} + 114903 T^{10} - 10656 T^{11} + 4086 T^{12} - 210 T^{13} + 105 T^{14} + 6 T^{15} + T^{16} \)
$43$ \( 2999643361 + 3327326288 T + 7799247270 T^{2} - 4349128328 T^{3} + 5436205979 T^{4} - 541864944 T^{5} + 414160000 T^{6} - 31678138 T^{7} + 21475854 T^{8} - 1079260 T^{9} + 594199 T^{10} - 12744 T^{11} + 11744 T^{12} - 86 T^{13} + 135 T^{14} + 2 T^{15} + T^{16} \)
$47$ \( ( 766944 + 832824 T + 290241 T^{2} + 19818 T^{3} - 8910 T^{4} - 1650 T^{5} + 3 T^{6} + 18 T^{7} + T^{8} )^{2} \)
$53$ \( 36759242529 - 192322120608 T + 216276115095 T^{2} + 623279673504 T^{3} + 467177028516 T^{4} + 151911504432 T^{5} + 18102553515 T^{6} - 1559379114 T^{7} - 432814590 T^{8} + 18378090 T^{9} + 9260244 T^{10} + 312498 T^{11} - 59697 T^{12} - 3240 T^{13} + 342 T^{14} + 36 T^{15} + T^{16} \)
$59$ \( ( 465300 - 452880 T + 12501 T^{2} + 91044 T^{3} - 24165 T^{4} + 1254 T^{5} + 228 T^{6} - 30 T^{7} + T^{8} )^{2} \)
$61$ \( 547560000 + 82792432800 T^{2} + 57490686441 T^{4} + 6740334162 T^{6} + 325742625 T^{8} + 7706430 T^{10} + 91530 T^{12} + 504 T^{14} + T^{16} \)
$67$ \( ( 51028 + 97988 T - 96551 T^{2} + 13262 T^{3} + 5338 T^{4} - 970 T^{5} - 101 T^{6} + 14 T^{7} + T^{8} )^{2} \)
$71$ \( 65610000 + 1520839800 T^{2} + 5897814849 T^{4} + 1363509936 T^{6} + 118814850 T^{8} + 4631418 T^{10} + 77553 T^{12} + 486 T^{14} + T^{16} \)
$73$ \( 71115489 + 461301966 T + 804430197 T^{2} - 1251964674 T^{3} + 184027950 T^{4} + 381956796 T^{5} + 2268459 T^{6} - 52748334 T^{7} + 5074506 T^{8} + 2724732 T^{9} - 293184 T^{10} - 99126 T^{11} + 18909 T^{12} - 150 T^{14} + T^{16} \)
$79$ \( ( -985100 + 1252640 T - 126107 T^{2} - 92278 T^{3} + 9004 T^{4} + 2306 T^{5} - 149 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$83$ \( 953512641 - 3606173136 T + 10746004968 T^{2} - 10462495572 T^{3} + 8139710007 T^{4} - 2738647692 T^{5} + 898947882 T^{6} + 16891146 T^{7} + 59460768 T^{8} + 1501128 T^{9} + 1367865 T^{10} + 35334 T^{11} + 22680 T^{12} + 312 T^{13} + 177 T^{14} + T^{16} \)
$89$ \( 131145120363321 + 115458464232270 T + 66081140272737 T^{2} + 24192458322894 T^{3} + 6794512511634 T^{4} + 1393795430352 T^{5} + 232008422067 T^{6} + 29227238586 T^{7} + 3371130738 T^{8} + 313712028 T^{9} + 32872068 T^{10} + 2265570 T^{11} + 187029 T^{12} + 9180 T^{13} + 702 T^{14} + 24 T^{15} + T^{16} \)
$97$ \( 9120206721024 + 42337728663552 T + 71879395230720 T^{2} + 29552832820224 T^{3} + 3933195148800 T^{4} - 297673761024 T^{5} - 99905635008 T^{6} + 1664083008 T^{7} + 1709674848 T^{8} + 28723248 T^{9} - 15309036 T^{10} - 354204 T^{11} + 102717 T^{12} + 2358 T^{13} - 381 T^{14} - 6 T^{15} + T^{16} \)
show more
show less