Properties

Label 29.2.e.a
Level $29$
Weight $2$
Character orbit 29.e
Analytic conductor $0.232$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: 12.0.7877952219361.1
Defining polynomial: \(x^{12} - 3 x^{11} + 13 x^{9} - 18 x^{8} - 14 x^{7} + 57 x^{6} - 28 x^{5} - 72 x^{4} + 104 x^{3} - 96 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} + 13 x^{9} - 18 x^{8} - 14 x^{7} + 57 x^{6} - 28 x^{5} - 72 x^{4} + 104 x^{3} - 96 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + 68 \nu^{3} - 176 \nu^{2} + 32 \nu + 96 \)\()/128\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} - 196 \nu^{3} + 96 \nu^{2} + 256 \nu - 288 \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} - 100 \nu^{3} - 128 \nu^{2} + 192 \nu - 32 \)\()/128\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + 44 \nu^{3} + 40 \nu^{2} - 80 \nu + 64 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + 220 \nu^{3} - 288 \nu^{2} + 224 \)\()/128\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + 12 \nu^{3} - 128 \nu^{2} + 96 \)\()/128\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{10} + 3 \nu^{9} - 9 \nu^{7} + 10 \nu^{6} + 6 \nu^{5} - 29 \nu^{4} + 16 \nu^{3} + 20 \nu^{2} - 40 \nu + 16 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 5 \nu^{10} + 10 \nu^{9} + 9 \nu^{8} - 36 \nu^{7} + 26 \nu^{6} + 53 \nu^{5} - 86 \nu^{4} - 12 \nu^{3} + 96 \nu^{2} - 64 \nu + 32 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{11} - 5 \nu^{9} + 5 \nu^{8} + 9 \nu^{7} - 16 \nu^{6} - 5 \nu^{5} + 31 \nu^{4} - 12 \nu^{2} + 8 \nu + 16 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + 82 \nu^{4} - 420 \nu^{3} + 176 \nu^{2} + 352 \nu - 352 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( -15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} - 146 \nu^{4} + 740 \nu^{3} - 368 \nu^{2} - 416 \nu + 480 \)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{8} + \beta_{5} + \beta_{2}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{6} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(-\beta_{10} - \beta_{8} - \beta_{7} - \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 2 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} - 4\)
\(\nu^{7}\)\(=\)\(-5 \beta_{10} + \beta_{9} + 6 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} + 5 \beta_{3} + \beta_{2} - 10 \beta_{1} - 5\)
\(\nu^{8}\)\(=\)\(4 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} + 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} - 6 \beta_{4} + 8 \beta_{3} + \beta_{2} - 4 \beta_{1} - 10\)
\(\nu^{9}\)\(=\)\(-6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} + 4 \beta_{8} - 11 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} - 5 \beta_{3} + 6 \beta_{1} + 5\)
\(\nu^{10}\)\(=\)\(2 \beta_{11} + 9 \beta_{9} + 9 \beta_{7} - 7 \beta_{6} - 9 \beta_{5} + \beta_{4} + 9 \beta_{3} + 5 \beta_{2} + 18 \beta_{1} - 1\)
\(\nu^{11}\)\(=\)\(-18 \beta_{11} - 29 \beta_{10} - 29 \beta_{9} + 8 \beta_{8} - 28 \beta_{7} - 7 \beta_{6} + \beta_{5} - 39 \beta_{3} + 8 \beta_{2} + 36 \beta_{1} + 11\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.639551 + 1.26134i
1.38491 0.286410i
1.23295 + 0.692694i
−1.41140 + 0.0891373i
1.23295 0.692694i
−1.41140 0.0891373i
−1.25719 0.647667i
0.911180 + 1.08155i
−1.25719 + 0.647667i
0.911180 1.08155i
0.639551 1.26134i
1.38491 + 0.286410i
−2.21089 0.504621i −1.23248 2.55926i 2.83146 + 1.36356i 0.0128801 0.0564316i 1.43341 + 6.28018i 1.40728 0.677709i −2.02596 1.61565i −3.16036 + 3.96297i −0.0569531 + 0.118264i
4.2 −0.536089 0.122359i 0.855966 + 1.77743i −1.52952 0.736577i 0.610610 2.67526i −0.241390 1.05760i −4.03077 + 1.94112i 1.58965 + 1.26771i −0.556117 + 0.697349i −0.654683 + 1.35946i
5.1 −0.909335 + 0.725171i 0.960118 0.219141i −0.144024 + 0.631009i −1.18424 1.48499i −0.714155 + 0.895521i −0.339509 1.48749i −1.33591 2.77404i −1.82910 + 0.880850i 2.15374 + 0.491577i
5.2 1.21127 0.965958i −2.86109 + 0.653024i 0.0890656 0.390222i 0.283269 + 0.355208i −2.83476 + 3.55468i −0.759522 3.32768i 1.07536 + 2.23300i 5.05647 2.43507i 0.686232 + 0.156628i
6.1 −0.909335 0.725171i 0.960118 + 0.219141i −0.144024 0.631009i −1.18424 + 1.48499i −0.714155 0.895521i −0.339509 + 1.48749i −1.33591 + 2.77404i −1.82910 0.880850i 2.15374 0.491577i
6.2 1.21127 + 0.965958i −2.86109 0.653024i 0.0890656 + 0.390222i 0.283269 0.355208i −2.83476 3.55468i −0.759522 + 3.32768i 1.07536 2.23300i 5.05647 + 2.43507i 0.686232 0.156628i
9.1 −1.12916 2.34472i −0.343489 + 0.273923i −2.97573 + 3.73144i 2.32488 1.11960i 1.03013 + 0.496082i 0.0468435 + 0.0587399i 7.03485 + 1.60566i −0.624612 + 2.73660i −5.25031 4.18698i
9.2 0.0741982 + 0.154074i −0.879032 + 0.701005i 1.22875 1.54080i −2.54740 + 1.22676i −0.173229 0.0834229i −1.82432 2.28763i 0.662012 + 0.151100i −0.386273 + 1.69237i −0.378025 0.301465i
13.1 −1.12916 + 2.34472i −0.343489 0.273923i −2.97573 3.73144i 2.32488 + 1.11960i 1.03013 0.496082i 0.0468435 0.0587399i 7.03485 1.60566i −0.624612 2.73660i −5.25031 + 4.18698i
13.2 0.0741982 0.154074i −0.879032 0.701005i 1.22875 + 1.54080i −2.54740 1.22676i −0.173229 + 0.0834229i −1.82432 + 2.28763i 0.662012 0.151100i −0.386273 1.69237i −0.378025 + 0.301465i
22.1 −2.21089 + 0.504621i −1.23248 + 2.55926i 2.83146 1.36356i 0.0128801 + 0.0564316i 1.43341 6.28018i 1.40728 + 0.677709i −2.02596 + 1.61565i −3.16036 3.96297i −0.0569531 0.118264i
22.2 −0.536089 + 0.122359i 0.855966 1.77743i −1.52952 + 0.736577i 0.610610 + 2.67526i −0.241390 + 1.05760i −4.03077 1.94112i 1.58965 1.26771i −0.556117 0.697349i −0.654683 1.35946i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.2.e.a 12
3.b odd 2 1 261.2.o.a 12
4.b odd 2 1 464.2.y.d 12
5.b even 2 1 725.2.q.a 12
5.c odd 4 2 725.2.p.a 24
29.b even 2 1 841.2.e.i 12
29.c odd 4 2 841.2.d.m 24
29.d even 7 1 841.2.b.e 12
29.d even 7 1 841.2.e.a 12
29.d even 7 1 841.2.e.e 12
29.d even 7 1 841.2.e.f 12
29.d even 7 1 841.2.e.h 12
29.d even 7 1 841.2.e.i 12
29.e even 14 1 inner 29.2.e.a 12
29.e even 14 1 841.2.b.e 12
29.e even 14 1 841.2.e.a 12
29.e even 14 1 841.2.e.e 12
29.e even 14 1 841.2.e.f 12
29.e even 14 1 841.2.e.h 12
29.f odd 28 2 841.2.a.k 12
29.f odd 28 4 841.2.d.k 24
29.f odd 28 4 841.2.d.l 24
29.f odd 28 2 841.2.d.m 24
87.h odd 14 1 261.2.o.a 12
87.k even 28 2 7569.2.a.bp 12
116.h odd 14 1 464.2.y.d 12
145.l even 14 1 725.2.q.a 12
145.q odd 28 2 725.2.p.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a 12 1.a even 1 1 trivial
29.2.e.a 12 29.e even 14 1 inner
261.2.o.a 12 3.b odd 2 1
261.2.o.a 12 87.h odd 14 1
464.2.y.d 12 4.b odd 2 1
464.2.y.d 12 116.h odd 14 1
725.2.p.a 24 5.c odd 4 2
725.2.p.a 24 145.q odd 28 2
725.2.q.a 12 5.b even 2 1
725.2.q.a 12 145.l even 14 1
841.2.a.k 12 29.f odd 28 2
841.2.b.e 12 29.d even 7 1
841.2.b.e 12 29.e even 14 1
841.2.d.k 24 29.f odd 28 4
841.2.d.l 24 29.f odd 28 4
841.2.d.m 24 29.c odd 4 2
841.2.d.m 24 29.f odd 28 2
841.2.e.a 12 29.d even 7 1
841.2.e.a 12 29.e even 14 1
841.2.e.e 12 29.d even 7 1
841.2.e.e 12 29.e even 14 1
841.2.e.f 12 29.d even 7 1
841.2.e.f 12 29.e even 14 1
841.2.e.h 12 29.d even 7 1
841.2.e.h 12 29.e even 14 1
841.2.e.i 12 29.b even 2 1
841.2.e.i 12 29.d even 7 1
7569.2.a.bp 12 87.k even 28 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(29, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 18 T^{2} + 133 T^{3} + 289 T^{4} + 259 T^{5} + 92 T^{6} + 7 T^{7} + 32 T^{8} + 42 T^{9} + 23 T^{10} + 7 T^{11} + T^{12} \)
$3$ \( 64 + 224 T + 256 T^{2} - 280 T^{3} - 432 T^{4} - 84 T^{5} + 211 T^{6} + 105 T^{7} + 67 T^{8} + 49 T^{9} + 23 T^{10} + 7 T^{11} + T^{12} \)
$5$ \( 1 - 10 T + 320 T^{2} - 717 T^{3} + 1109 T^{4} + 521 T^{5} + 196 T^{6} - 79 T^{7} + 6 T^{8} + 4 T^{9} - T^{10} + T^{11} + T^{12} \)
$7$ \( 64 - 1056 T + 11248 T^{2} + 872 T^{3} + 2540 T^{4} - 1878 T^{5} - 385 T^{6} + 297 T^{7} + 451 T^{8} + 207 T^{9} + 61 T^{10} + 11 T^{11} + T^{12} \)
$11$ \( 10816 + 2912 T + 11680 T^{2} - 12152 T^{3} + 6456 T^{4} - 1932 T^{5} + 3403 T^{6} - 1015 T^{7} - 297 T^{8} + 161 T^{9} - 5 T^{10} - 7 T^{11} + T^{12} \)
$13$ \( 841 + 1566 T + 3692 T^{2} - 1845 T^{3} + 8125 T^{4} - 4115 T^{5} + 2646 T^{6} + 449 T^{7} - 2 T^{8} + 66 T^{9} + 17 T^{10} - 9 T^{11} + T^{12} \)
$17$ \( 53824 + 259120 T^{2} + 125672 T^{4} + 22695 T^{6} + 1870 T^{8} + 71 T^{10} + T^{12} \)
$19$ \( 3136 + 9408 T + 18816 T^{2} + 12936 T^{3} - 1176 T^{4} - 11466 T^{5} + 3297 T^{6} + 315 T^{7} + 77 T^{8} - 77 T^{9} - 7 T^{10} + 7 T^{11} + T^{12} \)
$23$ \( 64 + 96 T + 1200 T^{2} - 200 T^{3} + 1580 T^{4} + 5090 T^{5} + 5299 T^{6} + 2899 T^{7} + 1279 T^{8} + 115 T^{9} + 3 T^{10} + 5 T^{11} + T^{12} \)
$29$ \( 594823321 + 307667235 T + 89117406 T^{2} + 15169958 T^{3} + 1400265 T^{4} - 119161 T^{5} - 52800 T^{6} - 4109 T^{7} + 1665 T^{8} + 622 T^{9} + 126 T^{10} + 15 T^{11} + T^{12} \)
$31$ \( 817216 - 911232 T + 17376 T^{2} + 1297800 T^{3} + 1169872 T^{4} + 551362 T^{5} + 172733 T^{6} + 40551 T^{7} + 7933 T^{8} + 1393 T^{9} + 207 T^{10} + 21 T^{11} + T^{12} \)
$37$ \( 110103049 + 51709504 T + 9550100 T^{2} - 7248521 T^{3} - 3193771 T^{4} - 343441 T^{5} + 221690 T^{6} + 57799 T^{7} + 5712 T^{8} - 462 T^{9} - 77 T^{10} - 7 T^{11} + T^{12} \)
$41$ \( 107584 + 383328 T^{2} + 240836 T^{4} + 46551 T^{6} + 3354 T^{8} + 99 T^{10} + T^{12} \)
$43$ \( 24364096 - 156036832 T + 297070944 T^{2} - 123964456 T^{3} + 23501608 T^{4} - 1972348 T^{5} + 245659 T^{6} - 54047 T^{7} + 5735 T^{8} - 525 T^{9} + 53 T^{10} - 7 T^{11} + T^{12} \)
$47$ \( 11343424 - 63183680 T + 393855872 T^{2} - 264630072 T^{3} + 39023448 T^{4} + 4670190 T^{5} - 350475 T^{6} + 67669 T^{7} + 19989 T^{8} + 161 T^{9} - 103 T^{10} + 7 T^{11} + T^{12} \)
$53$ \( 9409 - 23668 T + 7461 T^{2} - 928 T^{3} + 97201 T^{4} - 26542 T^{5} + 33131 T^{6} + 578 T^{7} + 339 T^{8} + 570 T^{9} + 131 T^{10} + 10 T^{11} + T^{12} \)
$59$ \( ( 1856 - 288 T - 1616 T^{2} + 440 T^{3} + 92 T^{4} - 22 T^{5} + T^{6} )^{2} \)
$61$ \( 325694209 - 454531742 T + 245176194 T^{2} - 63413315 T^{3} + 5400761 T^{4} + 1785091 T^{5} - 347290 T^{6} - 16723 T^{7} + 12576 T^{8} + 1386 T^{9} + 37 T^{10} + 7 T^{11} + T^{12} \)
$67$ \( 415833664 + 140215392 T + 442353712 T^{2} - 61163224 T^{3} + 3141164 T^{4} - 155226 T^{5} + 101339 T^{6} + 74717 T^{7} + 33787 T^{8} + 6241 T^{9} + 647 T^{10} + 37 T^{11} + T^{12} \)
$71$ \( 2671649344 + 638243424 T + 1782816 T^{2} + 8854888 T^{3} + 6120688 T^{4} - 1767332 T^{5} + 411173 T^{6} + 22785 T^{7} + 12901 T^{8} + 1715 T^{9} + 315 T^{10} + 21 T^{11} + T^{12} \)
$73$ \( 625681 - 2070838 T + 4875059 T^{2} - 4728990 T^{3} + 1687021 T^{4} + 1002050 T^{5} - 180061 T^{6} - 31234 T^{7} + 8113 T^{8} - 616 T^{9} + 77 T^{10} - 14 T^{11} + T^{12} \)
$79$ \( 30056463424 - 51708046208 T + 36243227456 T^{2} - 13718782952 T^{3} + 3236678008 T^{4} - 508283370 T^{5} + 56898577 T^{6} - 4897893 T^{7} + 357513 T^{8} - 23023 T^{9} + 1243 T^{10} - 49 T^{11} + T^{12} \)
$83$ \( 419758144 + 590546112 T + 249611008 T^{2} + 28644144 T^{3} + 65368364 T^{4} - 3763292 T^{5} + 1456973 T^{6} + 418137 T^{7} + 36097 T^{8} + 1369 T^{9} + 185 T^{10} - 5 T^{11} + T^{12} \)
$89$ \( 203946961 - 274309448 T + 22907116 T^{2} + 51450665 T^{3} + 26790541 T^{4} - 18842803 T^{5} + 3818480 T^{6} - 404103 T^{7} + 18096 T^{8} + 1008 T^{9} + 141 T^{10} - 7 T^{11} + T^{12} \)
$97$ \( 1697809 + 25484074 T + 86045131 T^{2} - 118114444 T^{3} + 66938013 T^{4} - 19681494 T^{5} + 3340051 T^{6} - 302162 T^{7} + 3945 T^{8} + 364 T^{9} + 247 T^{10} - 14 T^{11} + T^{12} \)
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