Properties

Label 349.2.a.a
Level 349
Weight 2
Character orbit 349.a
Self dual yes
Analytic conductor 2.787
Analytic rank 1
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.78677903054\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 5 x^{10} - x^{9} + 35 x^{8} - 24 x^{7} - 80 x^{6} + 66 x^{5} + 77 x^{4} - 56 x^{3} - 31 x^{2} + 15 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{10} + 6 \nu^{9} - 4 \nu^{8} - 33 \nu^{7} + 49 \nu^{6} + 42 \nu^{5} - 85 \nu^{4} - 2 \nu^{3} + 33 \nu^{2} - 12 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\( -\nu^{10} + 5 \nu^{9} - 30 \nu^{7} + 24 \nu^{6} + 50 \nu^{5} - 42 \nu^{4} - 27 \nu^{3} + 13 \nu^{2} + 5 \nu + 2 \)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{10} - 9 \nu^{9} - 6 \nu^{8} + 64 \nu^{7} - 15 \nu^{6} - 147 \nu^{5} + 36 \nu^{4} + 131 \nu^{3} + 4 \nu^{2} - 39 \nu - 14 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{10} - 9 \nu^{9} - 6 \nu^{8} + 64 \nu^{7} - 15 \nu^{6} - 149 \nu^{5} + 40 \nu^{4} + 141 \nu^{3} - 14 \nu^{2} - 47 \nu - 2 \)\()/2\)
\(\beta_{7}\)\(=\)\( -\nu^{10} + 5 \nu^{9} - 31 \nu^{7} + 27 \nu^{6} + 55 \nu^{5} - 58 \nu^{4} - 33 \nu^{3} + 31 \nu^{2} + 6 \nu - 1 \)
\(\beta_{8}\)\(=\)\( \nu^{10} - 5 \nu^{9} + 31 \nu^{7} - 27 \nu^{6} - 55 \nu^{5} + 59 \nu^{4} + 32 \nu^{3} - 36 \nu^{2} - 4 \nu + 5 \)
\(\beta_{9}\)\(=\)\((\)\( \nu^{10} - 2 \nu^{9} - 16 \nu^{8} + 33 \nu^{7} + 73 \nu^{6} - 144 \nu^{5} - 125 \nu^{4} + 202 \nu^{3} + 87 \nu^{2} - 78 \nu - 20 \)\()/2\)
\(\beta_{10}\)\(=\)\((\)\( -2 \nu^{10} + 7 \nu^{9} + 16 \nu^{8} - 64 \nu^{7} - 47 \nu^{6} + 201 \nu^{5} + 72 \nu^{4} - 243 \nu^{3} - 60 \nu^{2} + 87 \nu + 16 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 6 \beta_{2} + 8 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(-7 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 6 \beta_{6} + 8 \beta_{5} + 7 \beta_{3} + 8 \beta_{2} + 28 \beta_{1} + 7\)
\(\nu^{6}\)\(=\)\(-2 \beta_{10} - 13 \beta_{9} + 9 \beta_{8} + 21 \beta_{7} + 9 \beta_{6} + 13 \beta_{5} + 11 \beta_{3} + 34 \beta_{2} + 55 \beta_{1} + 30\)
\(\nu^{7}\)\(=\)\(-6 \beta_{10} - 52 \beta_{9} + 21 \beta_{8} + 69 \beta_{7} + 35 \beta_{6} + 57 \beta_{5} + \beta_{4} + 46 \beta_{3} + 58 \beta_{2} + 166 \beta_{1} + 40\)
\(\nu^{8}\)\(=\)\(-29 \beta_{10} - 121 \beta_{9} + 68 \beta_{8} + 171 \beta_{7} + 65 \beta_{6} + 118 \beta_{5} + 2 \beta_{4} + 93 \beta_{3} + 199 \beta_{2} + 364 \beta_{1} + 138\)
\(\nu^{9}\)\(=\)\(-84 \beta_{10} - 389 \beta_{9} + 169 \beta_{8} + 499 \beta_{7} + 206 \beta_{6} + 400 \beta_{5} + 10 \beta_{4} + 311 \beta_{3} + 397 \beta_{2} + 1019 \beta_{1} + 216\)
\(\nu^{10}\)\(=\)\(-288 \beta_{10} - 978 \beta_{9} + 489 \beta_{8} + 1268 \beta_{7} + 427 \beta_{6} + 933 \beta_{5} + 19 \beta_{4} + 720 \beta_{3} + 1195 \beta_{2} + 2382 \beta_{1} + 657\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.60070
2.42122
2.17018
1.31062
0.831463
0.734924
−0.216390
−0.767986
−0.916892
−1.25205
−1.91579
−2.60070 −0.209864 4.76365 1.72230 0.545795 −2.16571 −7.18744 −2.95596 −4.47919
1.2 −2.42122 2.15404 3.86232 −2.99362 −5.21540 0.559596 −4.50909 1.63987 7.24821
1.3 −2.17018 −2.21125 2.70967 −2.88560 4.79881 3.08193 −1.54011 1.88963 6.26228
1.4 −1.31062 0.477277 −0.282282 −1.24067 −0.625527 0.925788 2.99120 −2.77221 1.62604
1.5 −0.831463 −3.29989 −1.30867 1.20940 2.74374 3.20765 2.75104 7.88930 −1.00557
1.6 −0.734924 0.314167 −1.45989 2.34643 −0.230889 −4.45589 2.54275 −2.90130 −1.72445
1.7 0.216390 2.05625 −1.95318 −4.07164 0.444951 −1.74178 −0.855428 1.22815 −0.881063
1.8 0.767986 0.193628 −1.41020 −0.953573 0.148703 −2.43623 −2.61898 −2.96251 −0.732330
1.9 0.916892 −2.19816 −1.15931 2.45207 −2.01547 −1.41213 −2.89675 1.83189 2.24828
1.10 1.25205 −1.02718 −0.432368 −3.34750 −1.28608 4.81118 −3.04545 −1.94490 −4.19124
1.11 1.91579 −2.24901 1.67025 −1.23759 −4.30862 −3.37442 −0.631741 2.05803 −2.37097
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.2.a.a 11
3.b odd 2 1 3141.2.a.b 11
4.b odd 2 1 5584.2.a.j 11
5.b even 2 1 8725.2.a.l 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.2.a.a 11 1.a even 1 1 trivial
3141.2.a.b 11 3.b odd 2 1
5584.2.a.j 11 4.b odd 2 1
8725.2.a.l 11 5.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(349\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{11} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(349))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 21 T^{2} + 65 T^{3} + 178 T^{4} + 420 T^{5} + 906 T^{6} + 1763 T^{7} + 3196 T^{8} + 5335 T^{9} + 8367 T^{10} + 12192 T^{11} + 16734 T^{12} + 21340 T^{13} + 25568 T^{14} + 28208 T^{15} + 28992 T^{16} + 26880 T^{17} + 22784 T^{18} + 16640 T^{19} + 10752 T^{20} + 5120 T^{21} + 2048 T^{22} \)
$3$ \( 1 + 6 T + 33 T^{2} + 125 T^{3} + 431 T^{4} + 1245 T^{5} + 3329 T^{6} + 7963 T^{7} + 17806 T^{8} + 36586 T^{9} + 70528 T^{10} + 126077 T^{11} + 211584 T^{12} + 329274 T^{13} + 480762 T^{14} + 645003 T^{15} + 808947 T^{16} + 907605 T^{17} + 942597 T^{18} + 820125 T^{19} + 649539 T^{20} + 354294 T^{21} + 177147 T^{22} \)
$5$ \( 1 + 9 T + 63 T^{2} + 321 T^{3} + 1440 T^{4} + 5535 T^{5} + 19431 T^{6} + 61267 T^{7} + 179102 T^{8} + 479850 T^{9} + 1201056 T^{10} + 2775966 T^{11} + 6005280 T^{12} + 11996250 T^{13} + 22387750 T^{14} + 38291875 T^{15} + 60721875 T^{16} + 86484375 T^{17} + 112500000 T^{18} + 125390625 T^{19} + 123046875 T^{20} + 87890625 T^{21} + 48828125 T^{22} \)
$7$ \( 1 + 3 T + 36 T^{2} + 73 T^{3} + 599 T^{4} + 886 T^{5} + 6706 T^{6} + 7260 T^{7} + 58319 T^{8} + 44843 T^{9} + 431862 T^{10} + 269606 T^{11} + 3023034 T^{12} + 2197307 T^{13} + 20003417 T^{14} + 17431260 T^{15} + 112707742 T^{16} + 104237014 T^{17} + 493302257 T^{18} + 420830473 T^{19} + 1452729852 T^{20} + 847425747 T^{21} + 1977326743 T^{22} \)
$11$ \( 1 + 31 T + 531 T^{2} + 6413 T^{3} + 60422 T^{4} + 467688 T^{5} + 3069703 T^{6} + 17433416 T^{7} + 86846022 T^{8} + 382944129 T^{9} + 1503494100 T^{10} + 5273070684 T^{11} + 16538435100 T^{12} + 46336239609 T^{13} + 115592055282 T^{14} + 255242643656 T^{15} + 494378737853 T^{16} + 828537820968 T^{17} + 1177453846162 T^{18} + 1374683503853 T^{19} + 1252070223921 T^{20} + 804060162631 T^{21} + 285311670611 T^{22} \)
$13$ \( 1 + 4 T + 70 T^{2} + 205 T^{3} + 2502 T^{4} + 6277 T^{5} + 64760 T^{6} + 145370 T^{7} + 1294441 T^{8} + 2599706 T^{9} + 20644875 T^{10} + 37409596 T^{11} + 268383375 T^{12} + 439350314 T^{13} + 2843886877 T^{14} + 4151912570 T^{15} + 24044934680 T^{16} + 30297880093 T^{17} + 156996789534 T^{18} + 167224797805 T^{19} + 742314956110 T^{20} + 551433967396 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 + T + 84 T^{2} + 63 T^{3} + 3698 T^{4} + 1926 T^{5} + 110912 T^{6} + 30427 T^{7} + 2571395 T^{8} + 181737 T^{9} + 49922464 T^{10} - 452741 T^{11} + 848681888 T^{12} + 52521993 T^{13} + 12633263635 T^{14} + 2541293467 T^{15} + 157479179584 T^{16} + 46488957894 T^{17} + 1517432412754 T^{18} + 439472718783 T^{19} + 9961381625748 T^{20} + 2015993900449 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 + 17 T + 226 T^{2} + 2068 T^{3} + 16001 T^{4} + 99620 T^{5} + 537641 T^{6} + 2402987 T^{7} + 9332226 T^{8} + 29918227 T^{9} + 91775769 T^{10} + 313894131 T^{11} + 1743739611 T^{12} + 10800479947 T^{13} + 64009738134 T^{14} + 313159668827 T^{15} + 1331252342459 T^{16} + 4686710665220 T^{17} + 14302841695739 T^{18} + 35122008368788 T^{19} + 72927419698054 T^{20} + 104228126382617 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 + 24 T + 456 T^{2} + 6060 T^{3} + 69390 T^{4} + 658653 T^{5} + 5584079 T^{6} + 41423274 T^{7} + 279404880 T^{8} + 1686913894 T^{9} + 9341224951 T^{10} + 46717149029 T^{11} + 214848173873 T^{12} + 892377449926 T^{13} + 3399519174960 T^{14} + 11591930419434 T^{15} + 35941047783097 T^{16} + 97504282397517 T^{17} + 236260837767330 T^{18} + 474564570802860 T^{19} + 821325613627128 T^{20} + 994236269127576 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 + 17 T + 312 T^{2} + 3279 T^{3} + 35609 T^{4} + 278714 T^{5} + 2291527 T^{6} + 14722036 T^{7} + 102293462 T^{8} + 575540002 T^{9} + 3582194714 T^{10} + 18273808685 T^{11} + 103883646706 T^{12} + 484029141682 T^{13} + 2494835244718 T^{14} + 10412616344116 T^{15} + 47001851734523 T^{16} + 165785587089194 T^{17} + 614250845487181 T^{18} + 1640307988099119 T^{19} + 4526229544471128 T^{20} + 7152022966103417 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 + 10 T + 209 T^{2} + 1511 T^{3} + 18966 T^{4} + 113255 T^{5} + 1109705 T^{6} + 5858249 T^{7} + 49248292 T^{8} + 238704003 T^{9} + 1803457161 T^{10} + 8093669384 T^{11} + 55907171991 T^{12} + 229394546883 T^{13} + 1467155866972 T^{14} + 5410215974729 T^{15} + 31769912010455 T^{16} + 100514229391655 T^{17} + 521804239229226 T^{18} + 1288718357573351 T^{19} + 5525881031580239 T^{20} + 8196282869808010 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 + T + 148 T^{2} + 671 T^{3} + 11897 T^{4} + 84685 T^{5} + 812968 T^{6} + 5833528 T^{7} + 47735175 T^{8} + 297515886 T^{9} + 2211783831 T^{10} + 12276997715 T^{11} + 81836001747 T^{12} + 407299247934 T^{13} + 2417929819275 T^{14} + 10932970670008 T^{15} + 56374418034376 T^{16} + 217278540946165 T^{17} + 1129404542251301 T^{18} + 2356873713580991 T^{19} + 19234337489671396 T^{20} + 4808584372417849 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 + 15 T + 464 T^{2} + 5462 T^{3} + 94566 T^{4} + 913668 T^{5} + 11425824 T^{6} + 92868482 T^{7} + 920707511 T^{8} + 6370944600 T^{9} + 52310068897 T^{10} + 308742520868 T^{11} + 2144712824777 T^{12} + 10709557872600 T^{13} + 63456082365631 T^{14} + 262424134564802 T^{15} + 1323752561934624 T^{16} + 4340018241665988 T^{17} + 18417132663830646 T^{18} + 43613661601458902 T^{19} + 151905217558797904 T^{20} + 201339889652286015 T^{21} + 550329031716248441 T^{22} \)
$43$ \( 1 + 5 T + 397 T^{2} + 1835 T^{3} + 74045 T^{4} + 314401 T^{5} + 8578800 T^{6} + 33083766 T^{7} + 686577253 T^{8} + 2365470392 T^{9} + 39868889363 T^{10} + 120003944224 T^{11} + 1714362242609 T^{12} + 4373754754808 T^{13} + 54587697654271 T^{14} + 113106812284566 T^{15} + 1261156030808400 T^{16} + 1987442863968649 T^{17} + 20126809059417815 T^{18} + 21447847509397835 T^{19} + 199529266938926671 T^{20} + 108057411566421245 T^{21} + 929293739471222707 T^{22} \)
$47$ \( 1 - 4 T + 133 T^{2} - 1165 T^{3} + 15951 T^{4} - 122352 T^{5} + 1476135 T^{6} - 10679138 T^{7} + 99342939 T^{8} - 714139231 T^{9} + 5858599654 T^{10} - 35823804252 T^{11} + 275354183738 T^{12} - 1577533561279 T^{13} + 10314081955797 T^{14} - 52110786794978 T^{15} + 338544191907945 T^{16} - 1318858553933808 T^{17} + 8081145394505313 T^{18} - 27740148960951565 T^{19} + 148844352922668011 T^{20} - 210396528943320196 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 + 3 T + 131 T^{2} - 113 T^{3} + 11900 T^{4} - 35620 T^{5} + 973068 T^{6} - 2905877 T^{7} + 73723127 T^{8} - 217817831 T^{9} + 4472676520 T^{10} - 12277722876 T^{11} + 237051855560 T^{12} - 611850287279 T^{13} + 10975677978379 T^{14} - 22928767256837 T^{15} + 406932651982524 T^{16} - 789494543414980 T^{17} + 13979062564060300 T^{18} - 7035345016483793 T^{19} + 432269030526079423 T^{20} + 524662411096539147 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( 1 + 52 T + 1591 T^{2} + 35655 T^{3} + 646035 T^{4} + 9906888 T^{5} + 132455588 T^{6} + 1571731651 T^{7} + 16762500494 T^{8} + 161934343201 T^{9} + 1424678579860 T^{10} + 11445471606214 T^{11} + 84056036211740 T^{12} + 563693448682681 T^{13} + 3442665588957226 T^{14} + 19045239810293011 T^{15} + 94695718399532812 T^{16} + 417877822561619208 T^{17} + 1607755961995042665 T^{18} + 5235239252782065255 T^{19} + 13782826347480007949 T^{20} + 26578071171633352852 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( 1 + 305 T^{2} - 1022 T^{3} + 46959 T^{4} - 238047 T^{5} + 5591294 T^{6} - 27036499 T^{7} + 525969946 T^{8} - 2368610576 T^{9} + 38193372810 T^{10} - 167452201806 T^{11} + 2329795741410 T^{12} - 8813599953296 T^{13} + 119385184313026 T^{14} - 374343066350659 T^{15} + 4722386230203494 T^{16} - 12264270555512967 T^{17} + 147580060836710139 T^{18} - 195924873883221182 T^{19} + 3566714558314413005 T^{20} + 43513917611435838661 T^{22} \)
$67$ \( 1 + 23 T + 810 T^{2} + 13243 T^{3} + 269718 T^{4} + 3466456 T^{5} + 52096518 T^{6} + 553620189 T^{7} + 6706250203 T^{8} + 60455149013 T^{9} + 614420394552 T^{10} + 4744114263579 T^{11} + 41166166434984 T^{12} + 271383163919357 T^{13} + 2016991929804889 T^{14} + 11156067416581869 T^{15} + 70336816939077426 T^{16} + 313570001620023064 T^{17} + 1634683012764508914 T^{18} + 5377554253882596763 T^{19} + 22037292860998907070 T^{20} + 41925269504690513327 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 + 30 T + 879 T^{2} + 16593 T^{3} + 295747 T^{4} + 4228076 T^{5} + 57278317 T^{6} + 671008140 T^{7} + 7489732329 T^{8} + 74677531487 T^{9} + 711427669644 T^{10} + 6130590425082 T^{11} + 50511364544724 T^{12} + 376449436225967 T^{13} + 2680657587604719 T^{14} + 17051444802083340 T^{15} + 103343220707282267 T^{16} + 541617736039565996 T^{17} + 2689854501483663077 T^{18} + 10714988343960912273 T^{19} + 40300832131516698249 T^{20} + 97657306530296436030 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 - 12 T + 379 T^{2} - 2866 T^{3} + 64346 T^{4} - 309225 T^{5} + 6811163 T^{6} - 12873028 T^{7} + 506136815 T^{8} + 687792516 T^{9} + 32137115719 T^{10} + 114959140581 T^{11} + 2346009447487 T^{12} + 3665246317764 T^{13} + 196895825360855 T^{14} - 365571351543748 T^{15} + 14120028530592659 T^{16} - 46796326124216025 T^{17} + 710855905109815562 T^{18} - 2311314623368436146 T^{19} + 22312331362433539027 T^{20} - 51571509956442691788 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 - 11 T + 561 T^{2} - 4855 T^{3} + 148594 T^{4} - 1083574 T^{5} + 25875251 T^{6} - 165665964 T^{7} + 3338743854 T^{8} - 19013446671 T^{9} + 333875716292 T^{10} - 1694281754028 T^{11} + 26376181587068 T^{12} - 118662920673711 T^{13} + 1646130931032306 T^{14} - 6452702716743084 T^{15} + 79619606665281149 T^{16} - 263403246528712054 T^{17} + 2853585651889310446 T^{18} - 7365563272096353655 T^{19} + 67236745346248876959 T^{20} - \)\(10\!\cdots\!11\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 + 13 T + 601 T^{2} + 7481 T^{3} + 184200 T^{4} + 2103509 T^{5} + 36953980 T^{6} + 380316811 T^{7} + 5333867435 T^{8} + 48916596181 T^{9} + 578558064924 T^{10} + 4678349738759 T^{11} + 48020319388692 T^{12} + 336986431090909 T^{13} + 3049836059056345 T^{14} + 18049197298134331 T^{15} + 145563229140609140 T^{16} + 687722017845051821 T^{17} + 4998460592289293400 T^{18} + 16849398188632165721 T^{19} + \)\(11\!\cdots\!03\)\( T^{20} + \)\(20\!\cdots\!37\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 + 19 T + 466 T^{2} + 5953 T^{3} + 93394 T^{4} + 896288 T^{5} + 10921543 T^{6} + 77200192 T^{7} + 815530713 T^{8} + 3671172923 T^{9} + 44897516730 T^{10} + 145517283008 T^{11} + 3995878988970 T^{12} + 29079360723083 T^{13} + 574923871212897 T^{14} + 4843713051710272 T^{15} + 60986545386809807 T^{16} + 445438367312852768 T^{17} + 4130941291233035426 T^{18} + 23434513160344488193 T^{19} + \)\(16\!\cdots\!94\)\( T^{20} + \)\(59\!\cdots\!19\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 - 26 T + 836 T^{2} - 14101 T^{3} + 278601 T^{4} - 3720808 T^{5} + 58309271 T^{6} - 670846910 T^{7} + 9018538123 T^{8} - 91787845546 T^{9} + 1089314495233 T^{10} - 9927024051904 T^{11} + 105663506037601 T^{12} - 863631838742314 T^{13} + 8230976244332779 T^{14} - 59389594603371710 T^{15} + 500721550214622647 T^{16} - 3099328899715862632 T^{17} + 22510482853886759913 T^{18} - \)\(11\!\cdots\!61\)\( T^{19} + \)\(63\!\cdots\!12\)\( T^{20} - \)\(19\!\cdots\!74\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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