Properties

Label 378.2.u.b
Level 378
Weight 2
Character orbit 378.u
Analytic conductor 3.018
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \)
\(\beta_{2}\)\(=\)\( 3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25 \)
\(\beta_{3}\)\(=\)\( -6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25 \)
\(\beta_{4}\)\(=\)\( -9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49 \)
\(\beta_{5}\)\(=\)\( 9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62 \)
\(\beta_{6}\)\(=\)\( 11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61 \)
\(\beta_{7}\)\(=\)\( -16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85 \)
\(\beta_{8}\)\(=\)\( -16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \)
\(\beta_{9}\)\(=\)\( 36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209 \)
\(\beta_{10}\)\(=\)\( -36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \)
\(\beta_{11}\)\(=\)\( -42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 3\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - \beta_{10} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_{1} - 6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + \beta_{8} - 8 \beta_{7} + 7 \beta_{6} + 4 \beta_{5} + 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 8 \beta_{1} - 18\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-11 \beta_{11} + 17 \beta_{10} - 5 \beta_{9} - 20 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} - 14 \beta_{5} + 4 \beta_{4} - 14 \beta_{3} - 8 \beta_{2} + \beta_{1} + 6\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(19 \beta_{11} - \beta_{10} + 31 \beta_{9} - 32 \beta_{8} + 43 \beta_{7} - 20 \beta_{6} - 41 \beta_{5} - 44 \beta_{4} + 10 \beta_{3} + 25 \beta_{2} + 40 \beta_{1} + 87\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(85 \beta_{11} - 97 \beta_{10} + 55 \beta_{9} + 64 \beta_{8} + 10 \beta_{7} - 101 \beta_{6} + 19 \beta_{5} - 62 \beta_{4} + 91 \beta_{3} + 70 \beta_{2} + 31 \beta_{1} + 60\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-20 \beta_{11} - 118 \beta_{10} - 134 \beta_{9} + 244 \beta_{8} - 218 \beta_{7} + \beta_{6} + 232 \beta_{5} + 157 \beta_{4} + 52 \beta_{3} - 74 \beta_{2} - 179 \beta_{1} - 357\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-503 \beta_{11} + 386 \beta_{10} - 440 \beta_{9} - 47 \beta_{8} - 233 \beta_{7} + 514 \beta_{6} + 163 \beta_{5} + 466 \beta_{4} - 431 \beta_{3} - 461 \beta_{2} - 329 \beta_{1} - 639\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-425 \beta_{11} + 1076 \beta_{10} + 319 \beta_{9} - 1313 \beta_{8} + 955 \beta_{7} + 502 \beta_{6} - 1013 \beta_{5} - 332 \beta_{4} - 743 \beta_{3} - 59 \beta_{2} + 631 \beta_{1} + 1164\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(2299 \beta_{11} - 862 \beta_{10} + 2725 \beta_{9} - 1193 \beta_{8} + 2104 \beta_{7} - 2135 \beta_{6} - 1907 \beta_{5} - 2705 \beta_{4} + 1495 \beta_{3} + 2425 \beta_{2} + 2344 \beta_{1} + 4356\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(4708 \beta_{11} - 6628 \beta_{10} + 985 \beta_{9} + 5506 \beta_{8} - 3107 \beta_{7} - 4679 \beta_{6} + 3238 \beta_{5} - 992 \beta_{4} + 5476 \beta_{3} + 2770 \beta_{2} - 1043 \beta_{1} - 1698\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(\beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 + 2.22827i
0.500000 0.258654i
0.500000 + 0.0126039i
0.500000 + 1.27297i
0.500000 0.0126039i
0.500000 1.27297i
0.500000 2.22827i
0.500000 + 0.258654i
0.500000 1.68614i
0.500000 + 1.00210i
0.500000 + 1.68614i
0.500000 1.00210i
0.766044 + 0.642788i −1.67071 0.456871i 0.173648 + 0.984808i −1.78124 0.648319i −0.986166 1.42389i 0.173648 0.984808i −0.500000 + 0.866025i 2.58254 + 1.52660i −0.947779 1.64160i
43.2 0.766044 + 0.642788i 1.23102 + 1.21844i 0.173648 + 0.984808i 0.667901 + 0.243096i 0.159815 + 1.72466i 0.173648 0.984808i −0.500000 + 0.866025i 0.0308031 + 2.99984i 0.355383 + 0.615541i
85.1 −0.939693 0.342020i −1.04508 + 1.38124i 0.766044 + 0.642788i 0.108876 0.617467i 1.45446 0.940501i 0.766044 0.642788i −0.500000 0.866025i −0.815631 2.88700i −0.313496 + 0.542992i
85.2 −0.939693 0.342020i 1.71872 0.214444i 0.766044 + 0.642788i −0.701272 + 3.97711i −1.68842 0.386327i 0.766044 0.642788i −0.500000 0.866025i 2.90803 0.737141i 2.01923 3.49741i
169.1 −0.939693 + 0.342020i −1.04508 1.38124i 0.766044 0.642788i 0.108876 + 0.617467i 1.45446 + 0.940501i 0.766044 + 0.642788i −0.500000 + 0.866025i −0.815631 + 2.88700i −0.313496 0.542992i
169.2 −0.939693 + 0.342020i 1.71872 + 0.214444i 0.766044 0.642788i −0.701272 3.97711i −1.68842 + 0.386327i 0.766044 + 0.642788i −0.500000 + 0.866025i 2.90803 + 0.737141i 2.01923 + 3.49741i
211.1 0.766044 0.642788i −1.67071 + 0.456871i 0.173648 0.984808i −1.78124 + 0.648319i −0.986166 + 1.42389i 0.173648 + 0.984808i −0.500000 0.866025i 2.58254 1.52660i −0.947779 + 1.64160i
211.2 0.766044 0.642788i 1.23102 1.21844i 0.173648 0.984808i 0.667901 0.243096i 0.159815 1.72466i 0.173648 + 0.984808i −0.500000 0.866025i 0.0308031 2.99984i 0.355383 0.615541i
295.1 0.173648 0.984808i −0.390623 1.68743i −0.939693 0.342020i 0.393151 0.329893i −1.72962 + 0.0916693i −0.939693 + 0.342020i −0.500000 + 0.866025i −2.69483 + 1.31830i −0.256611 0.444463i
295.2 0.173648 0.984808i 1.65667 0.505425i −0.939693 0.342020i 1.31259 1.10139i −0.210069 1.71926i −0.939693 + 0.342020i −0.500000 + 0.866025i 2.48909 1.67464i −0.856730 1.48390i
337.1 0.173648 + 0.984808i −0.390623 + 1.68743i −0.939693 + 0.342020i 0.393151 + 0.329893i −1.72962 0.0916693i −0.939693 0.342020i −0.500000 0.866025i −2.69483 1.31830i −0.256611 + 0.444463i
337.2 0.173648 + 0.984808i 1.65667 + 0.505425i −0.939693 + 0.342020i 1.31259 + 1.10139i −0.210069 + 1.71926i −0.939693 0.342020i −0.500000 0.866025i 2.48909 + 1.67464i −0.856730 + 1.48390i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.b 12
27.e even 9 1 inner 378.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.b 12 1.a even 1 1 trivial
378.2.u.b 12 27.e even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$3$ \( 1 - 3 T + 3 T^{3} + 9 T^{4} - 45 T^{6} + 81 T^{8} + 81 T^{9} - 729 T^{11} + 729 T^{12} \)
$5$ \( 1 + 12 T^{2} - 18 T^{3} + 78 T^{4} - 198 T^{5} + 467 T^{6} - 1233 T^{7} + 2205 T^{8} - 5832 T^{9} + 8847 T^{10} - 26739 T^{11} + 35469 T^{12} - 133695 T^{13} + 221175 T^{14} - 729000 T^{15} + 1378125 T^{16} - 3853125 T^{17} + 7296875 T^{18} - 15468750 T^{19} + 30468750 T^{20} - 35156250 T^{21} + 117187500 T^{22} + 244140625 T^{24} \)
$7$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$11$ \( 1 - 3 T - 15 T^{2} + 126 T^{3} - 201 T^{4} - 1488 T^{5} + 7145 T^{6} - 1530 T^{7} - 61974 T^{8} + 202716 T^{9} - 19692 T^{10} - 1304451 T^{11} + 4526883 T^{12} - 14348961 T^{13} - 2382732 T^{14} + 269814996 T^{15} - 907361334 T^{16} - 246408030 T^{17} + 12657803345 T^{18} - 28996910448 T^{19} - 43086135081 T^{20} + 297101409066 T^{21} - 389061369015 T^{22} - 855935011833 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 12 T + 93 T^{2} - 461 T^{3} + 1830 T^{4} - 6510 T^{5} + 27469 T^{6} - 129519 T^{7} + 566109 T^{8} - 2155844 T^{9} + 7183293 T^{10} - 24098388 T^{11} + 81886711 T^{12} - 313279044 T^{13} + 1213976517 T^{14} - 4736389268 T^{15} + 16168639149 T^{16} - 48089498067 T^{17} + 132587616421 T^{18} - 408492845670 T^{19} + 1492787219430 T^{20} - 4888674210953 T^{21} + 12820839741957 T^{22} - 21505924728444 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 45 T^{2} - 18 T^{3} + 810 T^{4} + 405 T^{5} - 4228 T^{6} + 10935 T^{7} - 8145 T^{8} - 541728 T^{9} - 2753640 T^{10} + 5149980 T^{11} + 87495483 T^{12} + 87549660 T^{13} - 795801960 T^{14} - 2661509664 T^{15} - 680278545 T^{16} + 15526136295 T^{17} - 102053641732 T^{18} + 166187162565 T^{19} + 5650363527210 T^{20} - 2134581776946 T^{21} - 90719725520205 T^{22} + 582622237229761 T^{24} \)
$19$ \( 1 - 12 T + 12 T^{2} + 52 T^{3} + 2649 T^{4} - 10209 T^{5} - 32138 T^{6} - 100341 T^{7} + 1360620 T^{8} + 1473505 T^{9} - 7164300 T^{10} - 30874461 T^{11} + 38067223 T^{12} - 586614759 T^{13} - 2586312300 T^{14} + 10106770795 T^{15} + 177317359020 T^{16} - 248454249759 T^{17} - 1511960523578 T^{18} - 9125536583451 T^{19} + 44989458495609 T^{20} + 16779760284508 T^{21} + 73572795093612 T^{22} - 1397883106778628 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 - 12 T + 57 T^{2} - 216 T^{3} + 870 T^{4} - 1974 T^{5} + 6398 T^{6} - 34596 T^{7} - 231615 T^{8} + 2798766 T^{9} - 5217057 T^{10} - 7068906 T^{11} - 3870063 T^{12} - 162584838 T^{13} - 2759823153 T^{14} + 34052585922 T^{15} - 64815373215 T^{16} - 222671722428 T^{17} + 947133617822 T^{18} - 6721125432378 T^{19} + 68130557194470 T^{20} - 389048974876008 T^{21} + 2361311139177993 T^{22} - 11433717094967124 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 + 3 T + 30 T^{2} + 207 T^{3} + 978 T^{4} + 4863 T^{5} + 23219 T^{6} + 31419 T^{7} + 849420 T^{8} - 4617 T^{9} - 2257902 T^{10} - 22744773 T^{11} - 94856679 T^{12} - 659598417 T^{13} - 1898895582 T^{14} - 112604013 T^{15} + 600778627020 T^{16} + 644439790431 T^{17} + 13811202690299 T^{18} + 83886148490667 T^{19} + 489240991875858 T^{20} + 3002979217004883 T^{21} + 12621216999006030 T^{22} + 36601529297117487 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 21 T + 255 T^{2} - 2414 T^{3} + 18624 T^{4} - 124887 T^{5} + 730225 T^{6} - 3464208 T^{7} + 11411442 T^{8} - 1810082 T^{9} - 324371289 T^{10} + 3091769544 T^{11} - 19862733551 T^{12} + 95844855864 T^{13} - 311720808729 T^{14} - 53924152862 T^{15} + 10538706327282 T^{16} - 99177333927408 T^{17} + 648077375458225 T^{18} - 3435967838480457 T^{19} + 15884242681301184 T^{20} - 63825247895859794 T^{21} + 209005213180104255 T^{22} - 533578014824501451 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 15 T + 69 T^{2} + 406 T^{3} - 7374 T^{4} + 38802 T^{5} + 15283 T^{6} - 1166850 T^{7} + 3628278 T^{8} + 22593355 T^{9} - 153725946 T^{10} - 917449914 T^{11} + 12989691655 T^{12} - 33945646818 T^{13} - 210450820074 T^{14} + 1144421210815 T^{15} + 6799977124758 T^{16} - 80913996225450 T^{17} + 39211996708747 T^{18} + 3683546696514666 T^{19} - 25901023493213454 T^{20} + 52764466356801262 T^{21} + 331792321696831581 T^{22} - 2668764326691906195 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 12 T + 39 T^{2} - 252 T^{3} + 6540 T^{4} - 62211 T^{5} + 254375 T^{6} - 1609425 T^{7} + 20028519 T^{8} - 143465580 T^{9} + 690787764 T^{10} - 4925423520 T^{11} + 40342142493 T^{12} - 201942364320 T^{13} + 1161214231284 T^{14} - 9887791239180 T^{15} + 56595807877959 T^{16} - 186461866294425 T^{17} + 1208307766304375 T^{18} - 12115858132410891 T^{19} + 52221410998451340 T^{20} - 82500247467278172 T^{21} + 523483713095943639 T^{22} - 6603948380594981292 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 21 T + 192 T^{2} - 560 T^{3} - 8403 T^{4} + 123036 T^{5} - 639908 T^{6} - 1742472 T^{7} + 55902231 T^{8} - 419028878 T^{9} + 874994820 T^{10} + 12129804183 T^{11} - 134937795011 T^{12} + 521581579869 T^{13} + 1617865422180 T^{14} - 33315729003146 T^{15} + 191118603245031 T^{16} - 256158095691096 T^{17} - 4045090785959492 T^{18} + 33443474636160852 T^{19} - 98215946932681203 T^{20} - 281451862684632080 T^{21} + 4149404604150575808 T^{22} - 19515168528895676847 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 12 T - 72 T^{2} + 1215 T^{3} + 6273 T^{4} - 85908 T^{5} - 561724 T^{6} + 5249610 T^{7} + 35240229 T^{8} - 201170169 T^{9} - 2024333352 T^{10} + 3145093866 T^{11} + 108872443875 T^{12} + 147819411702 T^{13} - 4471752374568 T^{14} - 20886090456087 T^{15} + 171961075886949 T^{16} + 1203971842197270 T^{17} - 6054943951467196 T^{18} - 43522979032735404 T^{19} + 149368201229226753 T^{20} + 1359743524819861905 T^{21} - 3787137520979763528 T^{22} - 29665910581008147636 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( ( 1 + 21 T + 450 T^{2} + 5868 T^{3} + 70182 T^{4} + 637455 T^{5} + 5208409 T^{6} + 33785115 T^{7} + 197141238 T^{8} + 873610236 T^{9} + 3550716450 T^{10} + 8782105353 T^{11} + 22164361129 T^{12} )^{2} \)
$59$ \( 1 - 18 T + 72 T^{2} + 3006 T^{4} - 28638 T^{5} + 390494 T^{6} - 4658796 T^{7} + 22409190 T^{8} - 176439384 T^{9} + 1983739302 T^{10} - 9795803364 T^{11} + 37789076343 T^{12} - 577952398476 T^{13} + 6905396510262 T^{14} - 36236944246536 T^{15} + 271540244947590 T^{16} - 3330686464484004 T^{17} + 16471245303608654 T^{18} - 71270001222246522 T^{19} + 441372295438588926 T^{20} + 36800406237646180872 T^{22} - \)\(54\!\cdots\!62\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 189 T^{2} + 130 T^{3} + 25974 T^{4} - 17334 T^{5} + 2536246 T^{6} - 3252339 T^{7} + 204923790 T^{8} - 455484935 T^{9} + 14770394064 T^{10} - 36254574540 T^{11} + 930935536957 T^{12} - 2211529046940 T^{13} + 54960636312144 T^{14} - 103386426031235 T^{15} + 2837342213457390 T^{16} - 2746913488998039 T^{17} + 130668343391588806 T^{18} - 54476304319588014 T^{19} + 4979405747791376694 T^{20} + 1520238992068438330 T^{21} + \)\(13\!\cdots\!89\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 33 T + 642 T^{2} + 8368 T^{3} + 69393 T^{4} + 186288 T^{5} - 4701338 T^{6} - 91813068 T^{7} - 902003391 T^{8} - 4727097308 T^{9} + 9492444366 T^{10} + 478832565675 T^{11} + 5293151069269 T^{12} + 32081781900225 T^{13} + 42611582758974 T^{14} - 1421735967646004 T^{15} - 18176379474451311 T^{16} - 123959128257498276 T^{17} - 425275429509642122 T^{18} + 1129037843532411024 T^{19} + 28178254348687988913 T^{20} + \)\(22\!\cdots\!96\)\( T^{21} + \)\(11\!\cdots\!58\)\( T^{22} + \)\(40\!\cdots\!39\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 24 T + 105 T^{2} + 2718 T^{3} - 33666 T^{4} - 35205 T^{5} + 2568722 T^{6} - 8972217 T^{7} - 55594071 T^{8} - 248099436 T^{9} + 7228158138 T^{10} + 28994592564 T^{11} - 876571769937 T^{12} + 2058616072044 T^{13} + 36437145173658 T^{14} - 88797517238196 T^{15} - 1412738797743351 T^{16} - 16187937254941167 T^{17} + 329054017514118962 T^{18} - 320193705176155155 T^{19} - 21739938382919789826 T^{20} + \)\(12\!\cdots\!58\)\( T^{21} + \)\(34\!\cdots\!05\)\( T^{22} - \)\(55\!\cdots\!04\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 9 T - 162 T^{2} + 1543 T^{3} + 6453 T^{4} - 30168 T^{5} - 1029257 T^{6} - 3127977 T^{7} + 217574586 T^{8} - 589643561 T^{9} - 12568109643 T^{10} + 46186633458 T^{11} + 321235971409 T^{12} + 3371624242434 T^{13} - 66975456287547 T^{14} - 229381369169537 T^{15} + 6178735528703226 T^{16} - 6484520262257361 T^{17} - 155761811747537273 T^{18} - 333277918524118296 T^{19} + 5204086972992504693 T^{20} + 90838858290857389759 T^{21} - \)\(69\!\cdots\!38\)\( T^{22} - \)\(28\!\cdots\!93\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 135 T^{2} + 1993 T^{3} - 2052 T^{4} - 232362 T^{5} + 3259333 T^{6} - 10002285 T^{7} - 260701254 T^{8} + 3944140684 T^{9} - 13669666056 T^{10} - 186122056770 T^{11} + 3171346188679 T^{12} - 14703642484830 T^{13} - 85312385855496 T^{14} + 1944615178698676 T^{15} - 10154334960101574 T^{16} - 30777595063871715 T^{17} + 792302965665627493 T^{18} - 4462258699841877558 T^{19} - 3113107277928263172 T^{20} + \)\(23\!\cdots\!67\)\( T^{21} - \)\(12\!\cdots\!35\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 - 15 T + 147 T^{2} - 1854 T^{3} + 25071 T^{4} - 149289 T^{5} + 1163042 T^{6} - 9519381 T^{7} + 3954789 T^{8} + 847137582 T^{9} - 4335015195 T^{10} + 98252949789 T^{11} - 1318582794483 T^{12} + 8154994832487 T^{13} - 29863919678355 T^{14} + 484382256599034 T^{15} + 187687645849269 T^{16} - 37497228655201983 T^{17} + 380245385723828498 T^{18} - 4051113916190425203 T^{19} + 56467218551957896911 T^{20} - \)\(34\!\cdots\!62\)\( T^{21} + \)\(22\!\cdots\!03\)\( T^{22} - \)\(19\!\cdots\!05\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 51 T + 1197 T^{2} + 18234 T^{3} + 216342 T^{4} + 2052636 T^{5} + 13370033 T^{6} + 34362828 T^{7} - 391121892 T^{8} - 7793030061 T^{9} - 89642793852 T^{10} - 757259381556 T^{11} - 6184239975177 T^{12} - 67396084958484 T^{13} - 710060570101692 T^{14} - 5493844609073109 T^{15} - 24539864008239972 T^{16} + 191884074387761772 T^{17} + 6644656260531171713 T^{18} + 90790830334619064444 T^{19} + \)\(85\!\cdots\!02\)\( T^{20} + \)\(63\!\cdots\!06\)\( T^{21} + \)\(37\!\cdots\!97\)\( T^{22} + \)\(14\!\cdots\!39\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 - 48 T + 1326 T^{2} - 27506 T^{3} + 457320 T^{4} - 6561246 T^{5} + 84857617 T^{6} - 1022695011 T^{7} + 11895196383 T^{8} - 134109734294 T^{9} + 1470887073393 T^{10} - 15554728611663 T^{11} + 156712969708897 T^{12} - 1508808675331311 T^{13} + 13839576473554737 T^{14} - 122398333527307862 T^{15} + 1053073183140790623 T^{16} - 8782230038593357827 T^{17} + 70684019365987194193 T^{18} - \)\(53\!\cdots\!98\)\( T^{19} + \)\(35\!\cdots\!20\)\( T^{20} - \)\(20\!\cdots\!02\)\( T^{21} + \)\(97\!\cdots\!74\)\( T^{22} - \)\(34\!\cdots\!44\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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