Properties

Label 720.2.t.d
Level $720$
Weight $2$
Character orbit 720.t
Analytic conductor $5.749$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + 136 x^{10} + 80 x^{9} - 168 x^{8} - 288 x^{7} + 96 x^{6} + 512 x^{5} + 448 x^{4} - 512 x^{3} - 512 x^{2} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + 136 x^{10} + 80 x^{9} - 168 x^{8} - 288 x^{7} + 96 x^{6} + 512 x^{5} + 448 x^{4} - 512 x^{3} - 512 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{19} + 2 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 16 \nu^{14} - 6 \nu^{13} + 36 \nu^{12} + 42 \nu^{11} - 40 \nu^{10} - 136 \nu^{9} - 80 \nu^{8} + 168 \nu^{7} + 288 \nu^{6} - 96 \nu^{5} - 512 \nu^{4} - 448 \nu^{3} + 512 \nu^{2} + 512 \nu \)\()/512\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{18} + 2 \nu^{16} + 4 \nu^{15} - 7 \nu^{14} - 16 \nu^{13} - 6 \nu^{12} + 36 \nu^{11} + 42 \nu^{10} - 40 \nu^{9} - 136 \nu^{8} - 80 \nu^{7} + 168 \nu^{6} + 288 \nu^{5} - 96 \nu^{4} - 512 \nu^{3} - 448 \nu^{2} + 512 \nu + 512 \)\()/256\)
\(\beta_{3}\)\(=\)\((\)\( -18 \nu^{19} - 33 \nu^{18} - 8 \nu^{17} + 104 \nu^{16} + 94 \nu^{15} - 199 \nu^{14} - 616 \nu^{13} - 364 \nu^{12} + 768 \nu^{11} + 1426 \nu^{10} - 152 \nu^{9} - 3332 \nu^{8} - 3808 \nu^{7} + 1024 \nu^{6} + 5536 \nu^{5} + 2976 \nu^{4} - 7040 \nu^{3} - 8448 \nu^{2} - 2816 \nu + 5632 \)\()/512\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{19} - 36 \nu^{18} - 44 \nu^{17} + 28 \nu^{16} + 131 \nu^{15} + 12 \nu^{14} - 464 \nu^{13} - 836 \nu^{12} - 266 \nu^{11} + 1096 \nu^{10} + 1356 \nu^{9} - 1184 \nu^{8} - 4816 \nu^{7} - 4448 \nu^{6} + 992 \nu^{5} + 5440 \nu^{4} + 1024 \nu^{3} - 8448 \nu^{2} - 11264 \nu - 5632 \)\()/512\)
\(\beta_{5}\)\(=\)\((\)\( 8 \nu^{19} + 25 \nu^{18} + 30 \nu^{17} - 20 \nu^{16} - 84 \nu^{15} + 7 \nu^{14} + 330 \nu^{13} + 568 \nu^{12} + 156 \nu^{11} - 746 \nu^{10} - 876 \nu^{9} + 940 \nu^{8} + 3352 \nu^{7} + 3024 \nu^{6} - 800 \nu^{5} - 3744 \nu^{4} - 448 \nu^{3} + 6016 \nu^{2} + 7936 \nu + 4096 \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( 8 \nu^{19} + 25 \nu^{18} + 30 \nu^{17} - 20 \nu^{16} - 84 \nu^{15} + 7 \nu^{14} + 330 \nu^{13} + 568 \nu^{12} + 156 \nu^{11} - 746 \nu^{10} - 876 \nu^{9} + 940 \nu^{8} + 3352 \nu^{7} + 3024 \nu^{6} - 800 \nu^{5} - 3744 \nu^{4} - 448 \nu^{3} + 6016 \nu^{2} + 8448 \nu + 4096 \)\()/256\)
\(\beta_{7}\)\(=\)\((\)\(11 \nu^{19} - 22 \nu^{18} - 94 \nu^{17} - 132 \nu^{16} + 133 \nu^{15} + 438 \nu^{14} + 90 \nu^{13} - 1324 \nu^{12} - 2134 \nu^{11} - 92 \nu^{10} + 3688 \nu^{9} + 3592 \nu^{8} - 4216 \nu^{7} - 12800 \nu^{6} - 7840 \nu^{5} + 7616 \nu^{4} + 15808 \nu^{3} - 3584 \nu^{2} - 22528 \nu - 22528\)\()/512\)
\(\beta_{8}\)\(=\)\((\)\(31 \nu^{19} + 39 \nu^{18} - 30 \nu^{17} - 204 \nu^{16} - 67 \nu^{15} + 497 \nu^{14} + 946 \nu^{13} - 64 \nu^{12} - 2082 \nu^{11} - 2086 \nu^{10} + 2008 \nu^{9} + 6636 \nu^{8} + 3800 \nu^{7} - 7264 \nu^{6} - 11680 \nu^{5} - 736 \nu^{4} + 17728 \nu^{3} + 11776 \nu^{2} - 5120 \nu - 17920\)\()/512\)
\(\beta_{9}\)\(=\)\((\)\(-21 \nu^{19} + 10 \nu^{18} + 104 \nu^{17} + 192 \nu^{16} - 123 \nu^{15} - 594 \nu^{14} - 364 \nu^{13} + 1376 \nu^{12} + 2786 \nu^{11} + 652 \nu^{10} - 4436 \nu^{9} - 5696 \nu^{8} + 3232 \nu^{7} + 15248 \nu^{6} + 11424 \nu^{5} - 7936 \nu^{4} - 21760 \nu^{3} - 128 \nu^{2} + 24064 \nu + 28672\)\()/512\)
\(\beta_{10}\)\(=\)\((\)\(7 \nu^{19} - 35 \nu^{18} - 116 \nu^{17} - 116 \nu^{16} + 205 \nu^{15} + 483 \nu^{14} - 120 \nu^{13} - 1808 \nu^{12} - 2354 \nu^{11} + 550 \nu^{10} + 4804 \nu^{9} + 3412 \nu^{8} - 6848 \nu^{7} - 15968 \nu^{6} - 7840 \nu^{5} + 12000 \nu^{4} + 18560 \nu^{3} - 6912 \nu^{2} - 30208 \nu - 26624\)\()/512\)
\(\beta_{11}\)\(=\)\((\)\( 35 \nu^{19} + 62 \nu^{18} + 8 \nu^{17} - 200 \nu^{16} - 163 \nu^{15} + 426 \nu^{14} + 1204 \nu^{13} + 632 \nu^{12} - 1598 \nu^{11} - 2764 \nu^{10} + 588 \nu^{9} + 6816 \nu^{8} + 7392 \nu^{7} - 2480 \nu^{6} - 11168 \nu^{5} - 5440 \nu^{4} + 14208 \nu^{3} + 17536 \nu^{2} + 5632 \nu - 10240 \)\()/512\)
\(\beta_{12}\)\(=\)\((\)\( -10 \nu^{19} - 35 \nu^{18} - 42 \nu^{17} + 32 \nu^{16} + 130 \nu^{15} + 3 \nu^{14} - 486 \nu^{13} - 844 \nu^{12} - 212 \nu^{11} + 1198 \nu^{10} + 1404 \nu^{9} - 1388 \nu^{8} - 5136 \nu^{7} - 4512 \nu^{6} + 1520 \nu^{5} + 6048 \nu^{4} + 960 \nu^{3} - 9088 \nu^{2} - 12416 \nu - 5632 \)\()/256\)
\(\beta_{13}\)\(=\)\((\)\( -5 \nu^{19} + 22 \nu^{18} + 70 \nu^{17} + 80 \nu^{16} - 119 \nu^{15} - 286 \nu^{14} + 54 \nu^{13} + 1056 \nu^{12} + 1446 \nu^{11} - 220 \nu^{10} - 2736 \nu^{9} - 2080 \nu^{8} + 3888 \nu^{7} + 9408 \nu^{6} + 4752 \nu^{5} - 6368 \nu^{4} - 10624 \nu^{3} + 4480 \nu^{2} + 17024 \nu + 16128 \)\()/256\)
\(\beta_{14}\)\(=\)\((\)\(2 \nu^{19} - 65 \nu^{18} - 150 \nu^{17} - 116 \nu^{16} + 314 \nu^{15} + 497 \nu^{14} - 458 \nu^{13} - 2496 \nu^{12} - 2672 \nu^{11} + 1338 \nu^{10} + 5956 \nu^{9} + 2644 \nu^{8} - 10808 \nu^{7} - 20112 \nu^{6} - 7200 \nu^{5} + 16032 \nu^{4} + 20416 \nu^{3} - 14464 \nu^{2} - 39168 \nu - 34304\)\()/512\)
\(\beta_{15}\)\(=\)\((\)\(-28 \nu^{19} - 21 \nu^{18} + 66 \nu^{17} + 216 \nu^{16} - 4 \nu^{15} - 571 \nu^{14} - 762 \nu^{13} + 644 \nu^{12} + 2552 \nu^{11} + 1666 \nu^{10} - 3156 \nu^{9} - 6676 \nu^{8} - 992 \nu^{7} + 11296 \nu^{6} + 12560 \nu^{5} - 2912 \nu^{4} - 20480 \nu^{3} - 7424 \nu^{2} + 14208 \nu + 24064\)\()/256\)
\(\beta_{16}\)\(=\)\((\)\( -5 \nu^{19} - 31 \nu^{18} - 54 \nu^{17} - 12 \nu^{16} + 129 \nu^{15} + 119 \nu^{14} - 310 \nu^{13} - 952 \nu^{12} - 746 \nu^{11} + 790 \nu^{10} + 2008 \nu^{9} + 148 \nu^{8} - 4600 \nu^{7} - 6720 \nu^{6} - 1408 \nu^{5} + 6176 \nu^{4} + 5440 \nu^{3} - 6656 \nu^{2} - 14336 \nu - 10752 \)\()/128\)
\(\beta_{17}\)\(=\)\((\)\(-31 \nu^{19} - 7 \nu^{18} + 110 \nu^{17} + 264 \nu^{16} - 85 \nu^{15} - 753 \nu^{14} - 722 \nu^{13} + 1348 \nu^{12} + 3482 \nu^{11} + 1462 \nu^{10} - 5032 \nu^{9} - 8068 \nu^{8} + 1688 \nu^{7} + 17680 \nu^{6} + 15808 \nu^{5} - 7584 \nu^{4} - 28096 \nu^{3} - 4736 \nu^{2} + 26368 \nu + 35840\)\()/256\)
\(\beta_{18}\)\(=\)\((\)\(47 \nu^{19} + 56 \nu^{18} - 52 \nu^{17} - 320 \nu^{16} - 103 \nu^{15} + 760 \nu^{14} + 1424 \nu^{13} - 168 \nu^{12} - 3262 \nu^{11} - 3224 \nu^{10} + 3060 \nu^{9} + 10072 \nu^{8} + 5456 \nu^{7} - 11552 \nu^{6} - 18080 \nu^{5} - 1056 \nu^{4} + 26880 \nu^{3} + 16896 \nu^{2} - 9216 \nu - 28416\)\()/256\)
\(\beta_{19}\)\(=\)\((\)\(74 \nu^{19} - 11 \nu^{18} - 318 \nu^{17} - 652 \nu^{16} + 338 \nu^{15} + 1963 \nu^{14} + 1454 \nu^{13} - 4176 \nu^{12} - 9168 \nu^{11} - 2706 \nu^{10} + 14228 \nu^{9} + 19708 \nu^{8} - 8680 \nu^{7} - 49392 \nu^{6} - 39040 \nu^{5} + 24544 \nu^{4} + 73536 \nu^{3} + 4480 \nu^{2} - 76800 \nu - 96256\)\()/512\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{19} - \beta_{17} - \beta_{15} + \beta_{14} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{19} + 2 \beta_{17} - \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{1} + 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{19} + 2 \beta_{18} - \beta_{17} - \beta_{16} + \beta_{15} + \beta_{14} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - \beta_{1} - 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{19} + 2 \beta_{17} + \beta_{16} - 3 \beta_{15} + \beta_{14} - 2 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 3 \beta_{8} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 3 \beta_{1} - 3\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-\beta_{19} + 3 \beta_{17} + 2 \beta_{16} - 3 \beta_{15} - \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - 5 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} - 5\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{19} + 2 \beta_{18} - 2 \beta_{16} + 3 \beta_{15} + \beta_{14} - 2 \beta_{13} - 4 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} - \beta_{9} + 7 \beta_{8} - 4 \beta_{7} - 6 \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \beta_{1} - 5\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(3 \beta_{19} - 4 \beta_{18} - \beta_{17} + 5 \beta_{16} + \beta_{15} - 7 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 13 \beta_{11} - 5 \beta_{10} + \beta_{9} - 15 \beta_{8} + \beta_{6} + \beta_{5} - 5 \beta_{4} - 12 \beta_{3} + 2 \beta_{2} - 5 \beta_{1} - 1\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-3 \beta_{19} - 8 \beta_{18} + 2 \beta_{17} + 3 \beta_{16} + \beta_{15} - 7 \beta_{14} - 4 \beta_{13} + 14 \beta_{12} + 15 \beta_{11} - \beta_{10} + \beta_{9} + 25 \beta_{8} + 4 \beta_{7} - 5 \beta_{6} - 6 \beta_{5} - 7 \beta_{4} - 14 \beta_{3} + 10 \beta_{2} + \beta_{1} - 7\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-11 \beta_{19} - 12 \beta_{18} - 23 \beta_{17} + 6 \beta_{16} + 7 \beta_{15} - 15 \beta_{14} - 14 \beta_{13} + 6 \beta_{12} - 13 \beta_{11} - 17 \beta_{10} - 15 \beta_{9} + 13 \beta_{8} + 10 \beta_{7} + 32 \beta_{6} - 12 \beta_{5} - 13 \beta_{4} - 18 \beta_{3} - 6 \beta_{2} + 3 \beta_{1} + 13\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(5 \beta_{19} - 14 \beta_{18} - 12 \beta_{17} - 6 \beta_{16} + 13 \beta_{15} + 7 \beta_{14} + 2 \beta_{13} - 8 \beta_{12} + 37 \beta_{11} + 3 \beta_{10} + 17 \beta_{9} - 23 \beta_{8} + 20 \beta_{7} + 6 \beta_{6} - 17 \beta_{5} + 5 \beta_{4} - 24 \beta_{3} + 10 \beta_{2} + 11 \beta_{1} + 1\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(\beta_{19} - 16 \beta_{18} - 19 \beta_{17} - 13 \beta_{16} + 27 \beta_{15} - 13 \beta_{14} - 14 \beta_{13} + 26 \beta_{12} + 19 \beta_{11} - 27 \beta_{10} - \beta_{9} + 47 \beta_{8} + 20 \beta_{7} + 27 \beta_{6} - 33 \beta_{5} + 21 \beta_{4} - 4 \beta_{3} - 34 \beta_{2} + 9 \beta_{1} + 21\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-\beta_{19} - 26 \beta_{17} + 13 \beta_{16} + 3 \beta_{15} - 13 \beta_{14} + 28 \beta_{13} + 2 \beta_{12} - 27 \beta_{11} - 27 \beta_{10} + 27 \beta_{9} + 11 \beta_{8} + 52 \beta_{7} + 41 \beta_{6} - 18 \beta_{5} - 13 \beta_{4} + 30 \beta_{3} + 14 \beta_{2} - 53 \beta_{1} + 19\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-5 \beta_{19} + 12 \beta_{18} - \beta_{17} - 30 \beta_{16} + 9 \beta_{15} + 39 \beta_{14} - 2 \beta_{13} - 14 \beta_{12} + 5 \beta_{11} + 33 \beta_{10} + 39 \beta_{9} - 21 \beta_{8} + 30 \beta_{7} + 40 \beta_{6} - 4 \beta_{5} + 69 \beta_{4} + 18 \beta_{3} - 50 \beta_{2} + 53 \beta_{1} - 101\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(27 \beta_{19} + 54 \beta_{18} + 12 \beta_{17} - 74 \beta_{16} - 5 \beta_{15} + 97 \beta_{14} + 62 \beta_{13} - 64 \beta_{12} - 77 \beta_{11} + 21 \beta_{10} + 39 \beta_{9} + 31 \beta_{8} - 68 \beta_{7} - 126 \beta_{6} + 105 \beta_{5} + 155 \beta_{4} + 208 \beta_{3} - 18 \beta_{2} + 5 \beta_{1} + 79\)\()/2\)
\(\nu^{16}\)\(=\)\((\)\(167 \beta_{19} + 104 \beta_{18} + 195 \beta_{17} + 21 \beta_{16} + 45 \beta_{15} - 75 \beta_{14} + 118 \beta_{13} - 50 \beta_{12} - 3 \beta_{11} + 123 \beta_{10} + 113 \beta_{9} - 31 \beta_{8} - 12 \beta_{7} - 235 \beta_{6} + 81 \beta_{5} + 43 \beta_{4} + 68 \beta_{3} + 178 \beta_{2} - 97 \beta_{1} - 157\)\()/2\)
\(\nu^{17}\)\(=\)\((\)\(-135 \beta_{19} + 192 \beta_{18} + 2 \beta_{17} + 19 \beta_{16} - 27 \beta_{15} + 101 \beta_{14} - 124 \beta_{13} + 142 \beta_{12} - 77 \beta_{11} + 99 \beta_{10} + 189 \beta_{9} + 93 \beta_{8} - 164 \beta_{7} + 31 \beta_{6} + 170 \beta_{5} + 5 \beta_{4} + 322 \beta_{3} - 46 \beta_{2} + 141 \beta_{1} - 59\)\()/2\)
\(\nu^{18}\)\(=\)\((\)\(149 \beta_{19} - 60 \beta_{18} + 233 \beta_{17} - 58 \beta_{16} - 361 \beta_{15} + 89 \beta_{14} + 66 \beta_{13} - 210 \beta_{12} - 277 \beta_{11} + 127 \beta_{10} + 73 \beta_{9} - 27 \beta_{8} - 302 \beta_{7} - 384 \beta_{6} + 324 \beta_{5} + 123 \beta_{4} - 50 \beta_{3} + 226 \beta_{2} + 123 \beta_{1} + 133\)\()/2\)
\(\nu^{19}\)\(=\)\((\)\(149 \beta_{19} + 266 \beta_{18} + 188 \beta_{17} - 70 \beta_{16} + 373 \beta_{15} - 49 \beta_{14} + 162 \beta_{13} + 32 \beta_{12} + 125 \beta_{11} + 299 \beta_{10} - 423 \beta_{9} - 255 \beta_{8} - 300 \beta_{7} - 482 \beta_{6} + 95 \beta_{5} - 155 \beta_{4} + 48 \beta_{3} + 482 \beta_{2} + 299 \beta_{1} + 97\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(\beta_{9}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.32147 0.503713i
1.19834 + 0.750988i
1.18701 0.768775i
1.15787 + 0.811989i
−0.0861743 + 1.41159i
−0.491956 + 1.32589i
−0.720859 1.21670i
−1.04932 0.948122i
−1.13207 + 0.847599i
−1.38431 + 0.289262i
1.32147 + 0.503713i
1.19834 0.750988i
1.18701 + 0.768775i
1.15787 0.811989i
−0.0861743 1.41159i
−0.491956 1.32589i
−0.720859 + 1.21670i
−1.04932 + 0.948122i
−1.13207 0.847599i
−1.38431 0.289262i
−1.32147 0.503713i 0 1.49255 + 1.33128i 0.707107 + 0.707107i 0 2.69529i −1.30176 2.51106i 0 −0.578239 1.29060i
181.2 −1.19834 + 0.750988i 0 0.872033 1.79988i 0.707107 + 0.707107i 0 3.79862i 0.306697 + 2.81175i 0 −1.37838 0.316325i
181.3 −1.18701 0.768775i 0 0.817970 + 1.82508i −0.707107 0.707107i 0 4.92824i 0.432142 2.79522i 0 0.295735 + 1.38295i
181.4 −1.15787 + 0.811989i 0 0.681349 1.88036i −0.707107 0.707107i 0 2.18060i 0.737916 + 2.73047i 0 1.39290 + 0.244579i
181.5 0.0861743 + 1.41159i 0 −1.98515 + 0.243285i 0.707107 + 0.707107i 0 2.76462i −0.514486 2.78124i 0 −0.937207 + 1.05908i
181.6 0.491956 + 1.32589i 0 −1.51596 + 1.30456i −0.707107 0.707107i 0 3.46600i −2.47548 1.36821i 0 0.589679 1.28541i
181.7 0.720859 1.21670i 0 −0.960724 1.75414i −0.707107 0.707107i 0 0.0588949i −2.82681 0.0955746i 0 −1.37006 + 0.350613i
181.8 1.04932 0.948122i 0 0.202128 1.98976i 0.707107 + 0.707107i 0 0.740019i −1.67444 2.27953i 0 1.41240 + 0.0715547i
181.9 1.13207 + 0.847599i 0 0.563151 + 1.91908i −0.707107 0.707107i 0 4.27253i −0.989085 + 2.64985i 0 −0.201149 1.39984i
181.10 1.38431 + 0.289262i 0 1.83266 + 0.800859i 0.707107 + 0.707107i 0 2.60796i 2.30531 + 1.63876i 0 0.774320 + 1.18340i
541.1 −1.32147 + 0.503713i 0 1.49255 1.33128i 0.707107 0.707107i 0 2.69529i −1.30176 + 2.51106i 0 −0.578239 + 1.29060i
541.2 −1.19834 0.750988i 0 0.872033 + 1.79988i 0.707107 0.707107i 0 3.79862i 0.306697 2.81175i 0 −1.37838 + 0.316325i
541.3 −1.18701 + 0.768775i 0 0.817970 1.82508i −0.707107 + 0.707107i 0 4.92824i 0.432142 + 2.79522i 0 0.295735 1.38295i
541.4 −1.15787 0.811989i 0 0.681349 + 1.88036i −0.707107 + 0.707107i 0 2.18060i 0.737916 2.73047i 0 1.39290 0.244579i
541.5 0.0861743 1.41159i 0 −1.98515 0.243285i 0.707107 0.707107i 0 2.76462i −0.514486 + 2.78124i 0 −0.937207 1.05908i
541.6 0.491956 1.32589i 0 −1.51596 1.30456i −0.707107 + 0.707107i 0 3.46600i −2.47548 + 1.36821i 0 0.589679 + 1.28541i
541.7 0.720859 + 1.21670i 0 −0.960724 + 1.75414i −0.707107 + 0.707107i 0 0.0588949i −2.82681 + 0.0955746i 0 −1.37006 0.350613i
541.8 1.04932 + 0.948122i 0 0.202128 + 1.98976i 0.707107 0.707107i 0 0.740019i −1.67444 + 2.27953i 0 1.41240 0.0715547i
541.9 1.13207 0.847599i 0 0.563151 1.91908i −0.707107 + 0.707107i 0 4.27253i −0.989085 2.64985i 0 −0.201149 + 1.39984i
541.10 1.38431 0.289262i 0 1.83266 0.800859i 0.707107 0.707107i 0 2.60796i 2.30531 1.63876i 0 0.774320 1.18340i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.t.d 20
3.b odd 2 1 240.2.s.c 20
4.b odd 2 1 2880.2.t.d 20
12.b even 2 1 960.2.s.c 20
16.e even 4 1 inner 720.2.t.d 20
16.f odd 4 1 2880.2.t.d 20
24.f even 2 1 1920.2.s.f 20
24.h odd 2 1 1920.2.s.e 20
48.i odd 4 1 240.2.s.c 20
48.i odd 4 1 1920.2.s.e 20
48.k even 4 1 960.2.s.c 20
48.k even 4 1 1920.2.s.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.c 20 3.b odd 2 1
240.2.s.c 20 48.i odd 4 1
720.2.t.d 20 1.a even 1 1 trivial
720.2.t.d 20 16.e even 4 1 inner
960.2.s.c 20 12.b even 2 1
960.2.s.c 20 48.k even 4 1
1920.2.s.e 20 24.h odd 2 1
1920.2.s.e 20 48.i odd 4 1
1920.2.s.f 20 24.f even 2 1
1920.2.s.f 20 48.k even 4 1
2880.2.t.d 20 4.b odd 2 1
2880.2.t.d 20 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{20} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1024 - 512 T^{2} + 512 T^{3} + 448 T^{4} - 512 T^{5} + 96 T^{6} + 288 T^{7} - 168 T^{8} - 80 T^{9} + 136 T^{10} - 40 T^{11} - 42 T^{12} + 36 T^{13} + 6 T^{14} - 16 T^{15} + 7 T^{16} + 4 T^{17} - 2 T^{18} + T^{20} \)
$3$ \( T^{20} \)
$5$ \( ( 1 + T^{4} )^{5} \)
$7$ \( 262144 + 76283904 T^{2} + 204587008 T^{4} + 145248256 T^{6} + 49590528 T^{8} + 9697664 T^{10} + 1161104 T^{12} + 86368 T^{14} + 3880 T^{16} + 96 T^{18} + T^{20} \)
$11$ \( 1048576 - 115343360 T + 6343884800 T^{2} + 15595208704 T^{3} + 18952028160 T^{4} + 11886264320 T^{5} + 4560683008 T^{6} + 1115578368 T^{7} + 251479040 T^{8} + 84426752 T^{9} + 30799872 T^{10} + 7170048 T^{11} + 1093696 T^{12} + 173952 T^{13} + 57984 T^{14} + 13696 T^{15} + 1808 T^{16} + 112 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$13$ \( 167981940736 - 262517686272 T + 205127811072 T^{2} - 90035355648 T^{3} + 25928249344 T^{4} - 7034568704 T^{5} + 3460349952 T^{6} - 1511043072 T^{7} + 413145856 T^{8} - 60063232 T^{9} + 12435968 T^{10} - 5323392 T^{11} + 1636752 T^{12} - 175680 T^{13} + 10368 T^{14} - 4192 T^{15} + 2088 T^{16} - 144 T^{17} + T^{20} \)
$17$ \( ( -20032 - 151040 T - 145472 T^{2} + 106144 T^{3} + 25976 T^{4} - 15824 T^{5} - 676 T^{6} + 792 T^{7} - 34 T^{8} - 12 T^{9} + T^{10} )^{2} \)
$19$ \( 19716653056 + 59527397376 T + 89860866048 T^{2} + 68702330880 T^{3} + 29049219072 T^{4} + 7396435968 T^{5} + 9631844352 T^{6} + 7963704832 T^{7} + 3224447552 T^{8} - 73408640 T^{9} + 28386432 T^{10} + 20915520 T^{11} + 7892400 T^{12} - 209856 T^{13} + 29888 T^{14} + 17152 T^{15} + 5164 T^{16} - 72 T^{17} + 8 T^{18} + 4 T^{19} + T^{20} \)
$23$ \( 217558810624 + 1211058880512 T^{2} + 1734629007360 T^{4} + 505220718592 T^{6} + 64474080320 T^{8} + 4370425280 T^{10} + 169125872 T^{12} + 3828512 T^{14} + 49980 T^{16} + 348 T^{18} + T^{20} \)
$29$ \( 19723262623744 + 2510316109824 T + 159752650752 T^{2} + 1868828770304 T^{3} + 2276071636992 T^{4} + 779917459456 T^{5} + 169368092672 T^{6} + 67031334912 T^{7} + 46757634048 T^{8} + 18135728128 T^{9} + 4155932672 T^{10} + 541822976 T^{11} + 52284928 T^{12} + 7973888 T^{13} + 1863680 T^{14} + 231168 T^{15} + 15632 T^{16} + 576 T^{17} + 128 T^{18} + 16 T^{19} + T^{20} \)
$31$ \( ( -28698368 - 4962560 T + 6148800 T^{2} + 621056 T^{3} - 468112 T^{4} - 19536 T^{5} + 14932 T^{6} + 168 T^{7} - 204 T^{8} + T^{10} )^{2} \)
$37$ \( 18939904 + 171573248 T + 777125888 T^{2} + 1848475648 T^{3} + 2665852928 T^{4} + 2241855488 T^{5} + 1128316928 T^{6} + 422612992 T^{7} + 451001088 T^{8} + 394871296 T^{9} + 184033792 T^{10} + 12409216 T^{11} - 2713200 T^{12} - 581952 T^{13} + 678528 T^{14} - 81440 T^{15} + 4904 T^{16} - 80 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$41$ \( 3311118843904 + 11279143534592 T^{2} + 4549133533184 T^{4} + 806185730048 T^{6} + 78819631104 T^{8} + 4616884224 T^{10} + 166691328 T^{12} + 3697152 T^{14} + 48592 T^{16} + 344 T^{18} + T^{20} \)
$43$ \( 3288334336 + 31474057216 T + 150625845248 T^{2} + 354645180416 T^{3} + 430934327296 T^{4} + 72320286720 T^{5} + 76928778240 T^{6} + 118577430528 T^{7} + 83002048512 T^{8} + 32131072000 T^{9} + 7850893312 T^{10} + 1219307520 T^{11} + 121272576 T^{12} + 8994816 T^{13} + 1246208 T^{14} + 208640 T^{15} + 20960 T^{16} + 704 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$47$ \( ( -12544 - 145152 T + 283712 T^{2} - 35072 T^{3} - 97448 T^{4} + 11520 T^{5} + 7980 T^{6} - 160 T^{7} - 166 T^{8} + T^{10} )^{2} \)
$53$ \( 4398046511104 - 37383395344384 T + 158879430213632 T^{2} - 217797791580160 T^{3} + 164036679303168 T^{4} - 64499805585408 T^{5} + 15258273972224 T^{6} - 2110865276928 T^{7} + 243227688960 T^{8} - 46965587968 T^{9} + 11259117568 T^{10} - 1463160832 T^{11} + 113316096 T^{12} - 8978432 T^{13} + 2199552 T^{14} - 279040 T^{15} + 18720 T^{16} - 512 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$59$ \( 16482977321058304 + 15913855322423296 T + 7682191945367552 T^{2} - 225773976551424 T^{3} - 19284448182272 T^{4} + 8752961814528 T^{5} + 18984867758080 T^{6} - 1690337058816 T^{7} + 60533785600 T^{8} - 4783368192 T^{9} + 14857752576 T^{10} - 1869032960 T^{11} + 117256256 T^{12} - 208256 T^{13} + 2543744 T^{14} - 332800 T^{15} + 21776 T^{16} + 112 T^{17} + 128 T^{18} - 16 T^{19} + T^{20} \)
$61$ \( 74350019584 - 90727790592 T + 55356622848 T^{2} + 405824488448 T^{3} + 635588871424 T^{4} + 400694880256 T^{5} + 145373775872 T^{6} + 32054490112 T^{7} + 8596122240 T^{8} + 3542995712 T^{9} + 1229804288 T^{10} + 246492544 T^{11} + 31363232 T^{12} + 3663488 T^{13} + 926848 T^{14} + 186688 T^{15} + 21588 T^{16} + 840 T^{17} + 8 T^{18} + 4 T^{19} + T^{20} \)
$67$ \( 210453397504 + 2284922601472 T + 12403865550848 T^{2} + 27655294418944 T^{3} + 28263770488832 T^{4} - 13407377948672 T^{5} + 5667880435712 T^{6} + 2867528204288 T^{7} + 724321959936 T^{8} - 392953856 T^{9} + 1266810880 T^{10} + 1069056000 T^{11} + 402550784 T^{12} - 10567680 T^{13} + 430080 T^{14} + 164864 T^{15} + 42176 T^{16} - 960 T^{17} + 32 T^{18} + 8 T^{19} + T^{20} \)
$71$ \( 84783728164864 + 286334329552896 T^{2} + 119444957298688 T^{4} + 19751533281280 T^{6} + 1633300643840 T^{8} + 72970174464 T^{10} + 1802900480 T^{12} + 24765952 T^{14} + 184320 T^{16} + 688 T^{18} + T^{20} \)
$73$ \( 1224592353918976 + 9210388927217664 T^{2} + 5934864196173824 T^{4} + 620748281806848 T^{6} + 26910405677056 T^{8} + 619035629568 T^{10} + 8344750848 T^{12} + 68179456 T^{14} + 332272 T^{16} + 888 T^{18} + T^{20} \)
$79$ \( ( 46268416 - 77168640 T - 50255872 T^{2} + 3687424 T^{3} + 3393408 T^{4} - 266432 T^{5} - 70364 T^{6} + 9352 T^{7} - 104 T^{8} - 28 T^{9} + T^{10} )^{2} \)
$83$ \( 761801736963751936 - 875555940458299392 T + 503148107752538112 T^{2} - 158089032079769600 T^{3} + 31122539714969600 T^{4} - 4222033294524416 T^{5} + 700248334794752 T^{6} - 152230226034688 T^{7} + 27414010302464 T^{8} - 3259538489344 T^{9} + 322663907328 T^{10} - 40555229184 T^{11} + 6124317440 T^{12} - 684328960 T^{13} + 51499008 T^{14} - 2781696 T^{15} + 181424 T^{16} - 15936 T^{17} + 1152 T^{18} - 48 T^{19} + T^{20} \)
$89$ \( 1518596177800462336 + 469438536991899648 T^{2} + 52248775078248448 T^{4} + 2876984841338880 T^{6} + 88667808735232 T^{8} + 1624662126592 T^{10} + 18159960576 T^{12} + 123870720 T^{14} + 500432 T^{16} + 1096 T^{18} + T^{20} \)
$97$ \( ( -37943296 + 87896064 T - 52876288 T^{2} + 5846528 T^{3} + 2172480 T^{4} - 392640 T^{5} - 22480 T^{6} + 6080 T^{7} - 44 T^{8} - 28 T^{9} + T^{10} )^{2} \)
show more
show less