Properties

Label 483.2.i.g
Level $483$
Weight $2$
Character orbit 483.i
Analytic conductor $3.857$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} + 16 x^{14} - 7 x^{13} + 161 x^{12} - 50 x^{11} + 929 x^{10} - 47 x^{9} + 3741 x^{8} - 158 x^{7} + 8993 x^{6} + 265 x^{5} + 15176 x^{4} + 383 x^{3} + 12649 x^{2} + 2960 x + 6400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + 16 x^{14} - 7 x^{13} + 161 x^{12} - 50 x^{11} + 929 x^{10} - 47 x^{9} + 3741 x^{8} - 158 x^{7} + 8993 x^{6} + 265 x^{5} + 15176 x^{4} + 383 x^{3} + 12649 x^{2} + 2960 x + 6400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(489314876170653131 \nu^{15} + 2877632859699594789 \nu^{14} - 6884602150261440304 \nu^{13} + 58504771735189737963 \nu^{12} - 114737068170720275469 \nu^{11} + 498717248495520599930 \nu^{10} - 1301862983958083383101 \nu^{9} + 2549173510407032663203 \nu^{8} - 6528849731796473787609 \nu^{7} + 5350053001764707687942 \nu^{6} - 23885699882955850201917 \nu^{5} + 2194297061269817695195 \nu^{4} - 40662880506656600462384 \nu^{3} - 13115209331314937492827 \nu^{2} - 36429158393557767027461 \nu - 21126534364691541271520\)\()/ \)\(81\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(8651766457060096 \nu^{15} - 19472619575046966 \nu^{14} + 135181850968244616 \nu^{13} - 195067167613471280 \nu^{12} + 1251126373131039744 \nu^{11} - 1774318828702555233 \nu^{10} + 6613806712314092352 \nu^{9} - 6619957156550926992 \nu^{8} + 22727111648800356108 \nu^{7} - 26318672382549128250 \nu^{6} + 46218521419258600752 \nu^{5} - 19084021506243017904 \nu^{4} + 62321311775619753656 \nu^{3} - 3514392651161626455 \nu^{2} + 20320246137469222800 \nu + 326485123714847271188\)\()/ \)\(10\!\cdots\!97\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-730342106328004561 \nu^{15} + 1760264332511008401 \nu^{14} - 12023254610866269136 \nu^{13} + 20033340797220336087 \nu^{12} - 113979986624630191921 \nu^{11} + 186729210322786143890 \nu^{10} - 629893818237383249649 \nu^{9} + 849178320825222363087 \nu^{8} - 2251511627418538855021 \nu^{7} + 3438736528426367926718 \nu^{6} - 4912525342040631135153 \nu^{5} + 6963056131282902064455 \nu^{4} - 7113260702154613139976 \nu^{3} + 13823656957330199096657 \nu^{2} - 3857932148265258928889 \nu + 2406944155975240387440\)\()/ \)\(81\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-12874027827287548 \nu^{15} + 4222261370227452 \nu^{14} - 186511825661553802 \nu^{13} - 45063656177231780 \nu^{12} - 1877651312579823948 \nu^{11} - 607424981766662344 \nu^{10} - 10185653022847576859 \nu^{9} - 6008727404431577596 \nu^{8} - 41541780945331790076 \nu^{7} - 20693015252088923524 \nu^{6} - 89457459868247790914 \nu^{5} - 49630138793489800972 \nu^{4} - 176292224800672810544 \nu^{3} - 67252064433470884540 \nu^{2} - 57109459883826673600 \nu - 58427368506240364880\)\()/ \)\(10\!\cdots\!97\)\( \)
\(\beta_{6}\)\(=\)\((\)\(176629053076158417 \nu^{15} - 514016679708002577 \nu^{14} + 3371616148898613712 \nu^{13} - 7549042069958174167 \nu^{12} + 33907898744243648305 \nu^{11} - 73172805285822608978 \nu^{10} + 206610188649281339121 \nu^{9} - 425501062805963035599 \nu^{8} + 796976481895316064109 \nu^{7} - 1754341495943223718718 \nu^{6} + 1954539971208080687025 \nu^{5} - 4923920951264373761823 \nu^{4} + 3124696748514948905992 \nu^{3} - 7056673398704053749329 \nu^{2} + 3375194199086203826465 \nu - 5311320394108012294400\)\()/ \)\(81\!\cdots\!76\)\( \)
\(\beta_{7}\)\(=\)\((\)\(1990481780621992017 \nu^{15} - 892608534935747497 \nu^{14} + 32541743308144375832 \nu^{13} - 20023363027746932439 \nu^{12} + 355684555011313854737 \nu^{11} - 243755343368410296650 \nu^{10} + 2074438367360940997553 \nu^{9} - 1994832573743560473039 \nu^{8} + 8610364965781190044877 \nu^{7} - 9994588985317586395046 \nu^{6} + 17079396076094111445041 \nu^{5} - 34615848082827694480935 \nu^{4} + 25340664685060108284872 \nu^{3} - 47256397130882187712129 \nu^{2} + 18101351853785260990833 \nu - 36981248440812946694400\)\()/ \)\(81\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-68862515097890950 \nu^{15} + 280864108697835052 \nu^{14} - 1182133815949820917 \nu^{13} + 3388229179044614558 \nu^{12} - 10201178214071112342 \nu^{11} + 30437002962504678345 \nu^{10} - 52219127545580459425 \nu^{9} + 131835695108422278818 \nu^{8} - 149006442271052472906 \nu^{7} + 437289602831923128039 \nu^{6} - 278719953895353618745 \nu^{5} + 537828881712279372410 \nu^{4} - 448316108161737896000 \nu^{3} + 382827975341476926845 \nu^{2} - 35719637487542161200 \nu - 375696728354443679418\)\()/ \)\(20\!\cdots\!94\)\( \)
\(\beta_{9}\)\(=\)\((\)\(730342106328004561 \nu^{15} - 1760264332511008401 \nu^{14} + 12023254610866269136 \nu^{13} - 20033340797220336087 \nu^{12} + 113979986624630191921 \nu^{11} - 186729210322786143890 \nu^{10} + 629893818237383249649 \nu^{9} - 849178320825222363087 \nu^{8} + 2251511627418538855021 \nu^{7} - 3438736528426367926718 \nu^{6} + 4912525342040631135153 \nu^{5} - 6963056131282902064455 \nu^{4} + 7113260702154613139976 \nu^{3} - 11779262448282761204717 \nu^{2} + 3857932148265258928889 \nu - 2406944155975240387440\)\()/ \)\(20\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-1493096424743868911 \nu^{15} + 1308208354551848691 \nu^{14} - 30418784632791464096 \nu^{13} + 17005084627917901257 \nu^{12} - 328185435331309965811 \nu^{11} + 103570869247560874150 \nu^{10} - 2145766677577365257439 \nu^{9} + 373989125175825256177 \nu^{8} - 8914990132760330976511 \nu^{7} + 15053501726201448178 \nu^{6} - 22799600743619327584383 \nu^{5} + 1777368612314732230825 \nu^{4} - 32363147275523802428876 \nu^{3} - 2216141300764903538693 \nu^{2} - 25558337389001352064999 \nu + 1617875710154318213080\)\()/ \)\(40\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-4244463682271324767 \nu^{15} + 2948565287745468527 \nu^{14} - 72491084357911351632 \nu^{13} + 5073656239697606329 \nu^{12} - 732474535305719009767 \nu^{11} - 143849160888437465770 \nu^{10} - 4341323539653214643743 \nu^{9} - 2389455948755203148671 \nu^{8} - 16940508891835892525947 \nu^{7} - 12884421364697389898374 \nu^{6} - 39132037154857671577951 \nu^{5} - 35594116643551721289655 \nu^{4} - 49441103395432417238112 \nu^{3} - 54833388125285265941241 \nu^{2} - 24254147880130243329463 \nu - 33285552629352457301840\)\()/ \)\(81\!\cdots\!60\)\( \)
\(\beta_{12}\)\(=\)\((\)\(5336359747375319137 \nu^{15} + 105066438082346983 \nu^{14} + 74265424757468254752 \nu^{13} + 24695737262458158041 \nu^{12} + 766941351734913617937 \nu^{11} + 208887873099685436470 \nu^{10} + 3965049750376255211233 \nu^{9} + 1087660854111020324001 \nu^{8} + 15642539697531252087837 \nu^{7} + 224274810090053004154 \nu^{6} + 23274198018655780556641 \nu^{5} - 12165220077269899454775 \nu^{4} + 26701563288582634420472 \nu^{3} - 17793828449400485530929 \nu^{2} - 28997717432180621382647 \nu - 18433946269191271174400\)\()/ \)\(81\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(68020894021064526 \nu^{15} - 121374370496863868 \nu^{14} + 1050412635125966928 \nu^{13} - 1115046838584202823 \nu^{12} + 9921935315434061568 \nu^{11} - 9747877234313079770 \nu^{10} + 52948379957753469757 \nu^{9} - 31421811615920590752 \nu^{8} + 190062165014009870904 \nu^{7} - 113921171335612123910 \nu^{6} + 393095261321858854240 \nu^{5} - 47872510994473140960 \nu^{4} + 589891157971935879498 \nu^{3} + 68006887506055600170 \nu^{2} + 196329848887636976800 \nu + 490727858227952821250\)\()/ \)\(10\!\cdots\!97\)\( \)
\(\beta_{14}\)\(=\)\((\)\(176930487264112036 \nu^{15} + 184674370801341722 \nu^{14} + 2432658535035399818 \nu^{13} + 4034463201465252080 \nu^{12} + 26093614209699680316 \nu^{11} + 40272164728359612577 \nu^{10} + 146811629178910485156 \nu^{9} + 238689839627235297980 \nu^{8} + 649658393051330338944 \nu^{7} + 811446374720153081263 \nu^{6} + 1438769630275880390202 \nu^{5} + 1451548536747267436076 \nu^{4} + 2197365631174578900614 \nu^{3} + 1640526067702568741675 \nu^{2} + 1052686454348098798000 \nu + 761247511315299727062\)\()/ \)\(20\!\cdots\!94\)\( \)
\(\beta_{15}\)\(=\)\((\)\(7233783127086583691 \nu^{15} - 8957806889978802411 \nu^{14} + 99573166185059135696 \nu^{13} - 53595123591584353077 \nu^{12} + 893951122460543165811 \nu^{11} - 426380885780929683550 \nu^{10} + 4144766640720614645379 \nu^{9} - 323982902425894602077 \nu^{8} + 12943120061904193083591 \nu^{7} - 3346972546214658454258 \nu^{6} + 16482215978867850480003 \nu^{5} - 1095405802718313611045 \nu^{4} + 15772016567136634299856 \nu^{3} - 4740079011825765673027 \nu^{2} - 15933532865158649475461 \nu + 6244503128971517652880\)\()/ \)\(81\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 4 \beta_{4}\)
\(\nu^{3}\)\(=\)\(\beta_{15} - \beta_{14} + 2 \beta_{8} - 5 \beta_{5} + \beta_{3} - \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{9} + \beta_{6} - 22 \beta_{4} + 8 \beta_{3} - \beta_{1} - 22\)
\(\nu^{5}\)\(=\)\(-10 \beta_{15} + 8 \beta_{14} - 8 \beta_{12} - 10 \beta_{11} + 10 \beta_{10} - 10 \beta_{9} - 18 \beta_{8} + 18 \beta_{7} + 29 \beta_{5} - 20 \beta_{4} + 10 \beta_{2} - 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{15} + 12 \beta_{13} + 12 \beta_{5} - 57 \beta_{3} + 2 \beta_{2} + 138\)
\(\nu^{7}\)\(=\)\(2 \beta_{15} + 55 \beta_{12} + 81 \beta_{11} - 83 \beta_{10} + 81 \beta_{9} - 134 \beta_{7} - 4 \beta_{6} + 164 \beta_{4} - 81 \beta_{3} + 2 \beta_{2} + 183 \beta_{1} + 164\)
\(\nu^{8}\)\(=\)\(38 \beta_{15} - 2 \beta_{14} - 109 \beta_{13} + 2 \beta_{12} + 38 \beta_{11} - 36 \beta_{10} + 402 \beta_{9} + 6 \beta_{8} - 6 \beta_{7} - 109 \beta_{6} - 107 \beta_{5} + 918 \beta_{4} - 36 \beta_{2} + 107 \beta_{1}\)
\(\nu^{9}\)\(=\)\(620 \beta_{15} - 368 \beta_{14} - 72 \beta_{13} + 32 \beta_{11} + 32 \beta_{10} + 948 \beta_{8} - 1205 \beta_{5} + 624 \beta_{3} - 652 \beta_{2} - 1264\)
\(\nu^{10}\)\(=\)\(-40 \beta_{15} - 36 \beta_{12} - 432 \beta_{11} + 472 \beta_{10} - 2849 \beta_{9} + 120 \beta_{7} + 904 \beta_{6} - 6288 \beta_{4} + 2849 \beta_{3} - 40 \beta_{2} - 868 \beta_{1} - 6288\)
\(\nu^{11}\)\(=\)\(-5009 \beta_{15} + 2457 \beta_{14} + 868 \beta_{13} - 2457 \beta_{12} - 5009 \beta_{11} + 4661 \beta_{10} - 4741 \beta_{9} - 6602 \beta_{8} + 6602 \beta_{7} + 868 \beta_{6} + 8113 \beta_{5} - 9526 \beta_{4} + 4661 \beta_{2} - 8113 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-4368 \beta_{15} + 428 \beta_{14} + 7213 \beta_{13} - 516 \beta_{11} - 516 \beta_{10} - 1548 \beta_{8} + 6797 \beta_{5} - 20324 \beta_{3} + 4884 \beta_{2} + 43790\)
\(\nu^{13}\)\(=\)\(3248 \beta_{15} + 16472 \beta_{12} + 34838 \beta_{11} - 38086 \beta_{10} + 35882 \beta_{9} - 45822 \beta_{7} - 8824 \beta_{6} + 71260 \beta_{4} - 35882 \beta_{3} + 3248 \beta_{2} + 55353 \beta_{1} + 71260\)
\(\nu^{14}\)\(=\)\(45794 \beta_{15} - 4292 \beta_{14} - 56452 \beta_{13} + 4292 \beta_{12} + 45794 \beta_{11} - 40306 \beta_{10} + 145881 \beta_{9} + 16488 \beta_{8} - 16488 \beta_{7} - 56452 \beta_{6} - 52496 \beta_{5} + 308294 \beta_{4} - 40306 \beta_{2} + 52496 \beta_{1}\)
\(\nu^{15}\)\(=\)\(259981 \beta_{15} - 111063 \beta_{14} - 81808 \beta_{13} + 28110 \beta_{11} + 28110 \beta_{10} + 318566 \beta_{8} - 381235 \beta_{5} + 271317 \beta_{3} - 288091 \beta_{2} - 532272\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.37292 2.37797i
1.25839 2.17960i
0.736956 1.27644i
0.595971 1.03225i
−0.409272 + 0.708880i
−0.838903 + 1.45302i
−0.939958 + 1.62805i
−1.27611 + 2.21029i
1.37292 + 2.37797i
1.25839 + 2.17960i
0.736956 + 1.27644i
0.595971 + 1.03225i
−0.409272 0.708880i
−0.838903 1.45302i
−0.939958 1.62805i
−1.27611 2.21029i
−1.37292 2.37797i −0.500000 + 0.866025i −2.76984 + 4.79751i 0.957403 + 1.65827i 2.74585 1.61005 2.09946i 9.71944 −0.500000 0.866025i 2.62888 4.55336i
277.2 −1.25839 2.17960i −0.500000 + 0.866025i −2.16710 + 3.75353i −1.20917 2.09434i 2.51679 −0.750699 + 2.53702i 5.87470 −0.500000 0.866025i −3.04322 + 5.27101i
277.3 −0.736956 1.27644i −0.500000 + 0.866025i −0.0862077 + 0.149316i −1.81170 3.13796i 1.47391 −0.366412 2.62026i −2.69370 −0.500000 0.866025i −2.67029 + 4.62508i
277.4 −0.595971 1.03225i −0.500000 + 0.866025i 0.289637 0.501666i 1.89410 + 3.28068i 1.19194 2.62328 + 0.344130i −3.07435 −0.500000 0.866025i 2.25766 3.91038i
277.5 0.409272 + 0.708880i −0.500000 + 0.866025i 0.664993 1.15180i −1.86020 3.22197i −0.818544 −2.34029 + 1.23412i 2.72574 −0.500000 0.866025i 1.52266 2.63732i
277.6 0.838903 + 1.45302i −0.500000 + 0.866025i −0.407518 + 0.705841i −1.47412 2.55325i −1.67781 2.59341 0.523657i 1.98814 −0.500000 0.866025i 2.47329 4.28386i
277.7 0.939958 + 1.62805i −0.500000 + 0.866025i −0.767041 + 1.32855i 1.32777 + 2.29977i −1.87992 −1.98456 1.74973i 0.875886 −0.500000 0.866025i −2.49610 + 4.32337i
277.8 1.27611 + 2.21029i −0.500000 + 0.866025i −2.25692 + 3.90910i −0.324080 0.561322i −2.55222 −2.38478 + 1.14579i −6.41586 −0.500000 0.866025i 0.827123 1.43262i
415.1 −1.37292 + 2.37797i −0.500000 0.866025i −2.76984 4.79751i 0.957403 1.65827i 2.74585 1.61005 + 2.09946i 9.71944 −0.500000 + 0.866025i 2.62888 + 4.55336i
415.2 −1.25839 + 2.17960i −0.500000 0.866025i −2.16710 3.75353i −1.20917 + 2.09434i 2.51679 −0.750699 2.53702i 5.87470 −0.500000 + 0.866025i −3.04322 5.27101i
415.3 −0.736956 + 1.27644i −0.500000 0.866025i −0.0862077 0.149316i −1.81170 + 3.13796i 1.47391 −0.366412 + 2.62026i −2.69370 −0.500000 + 0.866025i −2.67029 4.62508i
415.4 −0.595971 + 1.03225i −0.500000 0.866025i 0.289637 + 0.501666i 1.89410 3.28068i 1.19194 2.62328 0.344130i −3.07435 −0.500000 + 0.866025i 2.25766 + 3.91038i
415.5 0.409272 0.708880i −0.500000 0.866025i 0.664993 + 1.15180i −1.86020 + 3.22197i −0.818544 −2.34029 1.23412i 2.72574 −0.500000 + 0.866025i 1.52266 + 2.63732i
415.6 0.838903 1.45302i −0.500000 0.866025i −0.407518 0.705841i −1.47412 + 2.55325i −1.67781 2.59341 + 0.523657i 1.98814 −0.500000 + 0.866025i 2.47329 + 4.28386i
415.7 0.939958 1.62805i −0.500000 0.866025i −0.767041 1.32855i 1.32777 2.29977i −1.87992 −1.98456 + 1.74973i 0.875886 −0.500000 + 0.866025i −2.49610 4.32337i
415.8 1.27611 2.21029i −0.500000 0.866025i −2.25692 3.90910i −0.324080 + 0.561322i −2.55222 −2.38478 1.14579i −6.41586 −0.500000 + 0.866025i 0.827123 + 1.43262i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.g 16
7.c even 3 1 inner 483.2.i.g 16
7.c even 3 1 3381.2.a.bf 8
7.d odd 6 1 3381.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.g 16 1.a even 1 1 trivial
483.2.i.g 16 7.c even 3 1 inner
3381.2.a.be 8 7.d odd 6 1
3381.2.a.bf 8 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\):

\(T_{2}^{16} + \cdots\)
\(T_{5}^{16} + \cdots\)
\(T_{11}^{16} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6400 - 2960 T + 12649 T^{2} - 383 T^{3} + 15176 T^{4} - 265 T^{5} + 8993 T^{6} + 158 T^{7} + 3741 T^{8} + 47 T^{9} + 929 T^{10} + 50 T^{11} + 161 T^{12} + 7 T^{13} + 16 T^{14} + T^{15} + T^{16} \)
$3$ \( ( 1 + T + T^{2} )^{8} \)
$5$ \( 1440000 + 2415600 T + 4401369 T^{2} + 1653417 T^{3} + 2060010 T^{4} + 750009 T^{5} + 641667 T^{6} + 185760 T^{7} + 112233 T^{8} + 28341 T^{9} + 13747 T^{10} + 2790 T^{11} + 995 T^{12} + 149 T^{13} + 46 T^{14} + 5 T^{15} + T^{16} \)
$7$ \( 5764801 + 1647086 T - 705894 T^{2} - 705894 T^{3} - 115248 T^{4} + 34986 T^{5} + 35672 T^{6} + 322 T^{7} - 2565 T^{8} + 46 T^{9} + 728 T^{10} + 102 T^{11} - 48 T^{12} - 42 T^{13} - 6 T^{14} + 2 T^{15} + T^{16} \)
$11$ \( 19219456 + 13397504 T + 21193472 T^{2} + 8080128 T^{3} + 11675264 T^{4} + 4304512 T^{5} + 3549248 T^{6} + 786176 T^{7} + 523680 T^{8} + 116720 T^{9} + 51088 T^{10} + 8968 T^{11} + 2880 T^{12} + 472 T^{13} + 104 T^{14} + 10 T^{15} + T^{16} \)
$13$ \( ( 2127 + 2370 T - 4602 T^{2} + 930 T^{3} + 660 T^{4} - 158 T^{5} - 38 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$17$ \( 4513421124 + 3967298646 T + 3112985887 T^{2} + 1499429067 T^{3} + 727577126 T^{4} + 276560671 T^{5} + 105350229 T^{6} + 32421344 T^{7} + 9624099 T^{8} + 2399799 T^{9} + 576231 T^{10} + 115298 T^{11} + 20499 T^{12} + 2773 T^{13} + 302 T^{14} + 21 T^{15} + T^{16} \)
$19$ \( 9822395664 + 10206538272 T + 9446933520 T^{2} + 3857011872 T^{3} + 1761026776 T^{4} + 401704040 T^{5} + 177968888 T^{6} + 31320028 T^{7} + 10474560 T^{8} + 1178280 T^{9} + 367680 T^{10} + 33120 T^{11} + 8724 T^{12} + 478 T^{13} + 119 T^{14} + 5 T^{15} + T^{16} \)
$23$ \( ( 1 - T + T^{2} )^{8} \)
$29$ \( ( 34688 + 26816 T - 22272 T^{2} - 4960 T^{3} + 2736 T^{4} + 232 T^{5} - 108 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$31$ \( 118722816 + 196825344 T + 362569984 T^{2} + 118054400 T^{3} + 162820160 T^{4} + 47971456 T^{5} + 52990528 T^{6} + 6975248 T^{7} + 5078880 T^{8} - 548400 T^{9} + 330456 T^{10} - 32632 T^{11} + 9592 T^{12} - 1058 T^{13} + 191 T^{14} - 13 T^{15} + T^{16} \)
$37$ \( 391090176 + 851791872 T + 1599533056 T^{2} + 1108031488 T^{3} + 790641152 T^{4} - 62722048 T^{5} + 176325376 T^{6} - 2727104 T^{7} + 10982400 T^{8} + 278592 T^{9} + 519984 T^{10} + 25792 T^{11} + 12136 T^{12} + 956 T^{13} + 221 T^{14} + 13 T^{15} + T^{16} \)
$41$ \( ( 256 + 1520 T + 784 T^{2} - 1504 T^{3} - 728 T^{4} + 272 T^{5} + 40 T^{6} - 16 T^{7} + T^{8} )^{2} \)
$43$ \( ( -937796 - 141104 T + 322712 T^{2} - 62812 T^{3} - 5446 T^{4} + 2214 T^{5} - 84 T^{6} - 15 T^{7} + T^{8} )^{2} \)
$47$ \( 1666159476804 - 1159855578486 T + 695209810887 T^{2} - 210204524799 T^{3} + 65736886302 T^{4} - 13380695991 T^{5} + 3332751201 T^{6} - 535493172 T^{7} + 106085199 T^{8} - 11793891 T^{9} + 1947597 T^{10} - 152700 T^{11} + 25423 T^{12} - 1117 T^{13} + 186 T^{14} - T^{15} + T^{16} \)
$53$ \( 886371984 + 422911260 T + 847090125 T^{2} - 135513495 T^{3} + 402231528 T^{4} - 43923213 T^{5} + 82194273 T^{6} - 16377144 T^{7} + 10649379 T^{8} - 1066539 T^{9} + 387699 T^{10} - 13614 T^{11} + 9415 T^{12} - 141 T^{13} + 122 T^{14} + 3 T^{15} + T^{16} \)
$59$ \( 1182003840000 + 1399878720000 T + 1191583238400 T^{2} + 557401881600 T^{3} + 208977801984 T^{4} + 48538940544 T^{5} + 11841928128 T^{6} + 2170762368 T^{7} + 425947680 T^{8} + 59226864 T^{9} + 8193072 T^{10} + 819264 T^{11} + 89800 T^{12} + 7184 T^{13} + 576 T^{14} + 26 T^{15} + T^{16} \)
$61$ \( 21489214464 + 50387774976 T + 127500334848 T^{2} + 1637860608 T^{3} + 30708284544 T^{4} + 796053888 T^{5} + 6130996416 T^{6} - 330979968 T^{7} + 193196256 T^{8} - 9169968 T^{9} + 4098640 T^{10} - 207704 T^{11} + 45480 T^{12} - 2032 T^{13} + 348 T^{14} - 14 T^{15} + T^{16} \)
$67$ \( 5849089043121 - 5613192044550 T + 6765308936676 T^{2} + 729078249696 T^{3} + 520660134382 T^{4} + 103343640722 T^{5} + 36233653556 T^{6} + 6404537314 T^{7} + 1196411301 T^{8} + 156577482 T^{9} + 21519852 T^{10} + 2313024 T^{11} + 234510 T^{12} + 17164 T^{13} + 1040 T^{14} + 38 T^{15} + T^{16} \)
$71$ \( ( 44639898 + 10497209 T - 2988199 T^{2} - 332779 T^{3} + 57905 T^{4} + 3195 T^{5} - 421 T^{6} - 9 T^{7} + T^{8} )^{2} \)
$73$ \( 91705403241 + 215413775202 T + 368981923930 T^{2} + 307987902624 T^{3} + 194594382860 T^{4} + 38999938666 T^{5} + 10550619996 T^{6} + 573000362 T^{7} + 328964253 T^{8} + 9605346 T^{9} + 5016924 T^{10} - 19198 T^{11} + 49860 T^{12} - 368 T^{13} + 290 T^{14} - 6 T^{15} + T^{16} \)
$79$ \( 219581777797264 + 61737864871776 T + 35534820882960 T^{2} + 4391428433760 T^{3} + 2579398273144 T^{4} + 344691162856 T^{5} + 102069667672 T^{6} + 11008729700 T^{7} + 2476161600 T^{8} + 254895776 T^{9} + 35847672 T^{10} + 2310288 T^{11} + 201696 T^{12} + 9158 T^{13} + 707 T^{14} + 23 T^{15} + T^{16} \)
$83$ \( ( 9331200 + 5526144 T + 105408 T^{2} - 285408 T^{3} - 13104 T^{4} + 4824 T^{5} + 36 T^{6} - 30 T^{7} + T^{8} )^{2} \)
$89$ \( 12483505303833664 - 8291899787356704 T + 6535970301064624 T^{2} + 377969485378960 T^{3} + 171193930529632 T^{4} + 6293999537824 T^{5} + 2399985852408 T^{6} + 95838581056 T^{7} + 19706822016 T^{8} + 598653508 T^{9} + 102934956 T^{10} + 2839996 T^{11} + 364372 T^{12} + 6868 T^{13} + 784 T^{14} + 12 T^{15} + T^{16} \)
$97$ \( ( 1304736 + 1800688 T - 354656 T^{2} - 190152 T^{3} + 18488 T^{4} + 3884 T^{5} - 300 T^{6} - 14 T^{7} + T^{8} )^{2} \)
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