Properties

Label 210.2.u.b
Level 210
Weight 2
Character orbit 210.u
Analytic conductor 1.677
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 210.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + 2680829374033629 \nu^{11} - 1607350163227359 \nu^{10} - 900494454550979 \nu^{9} - 8792384561334166 \nu^{8} + 3536477625795241 \nu^{7} - 17495309067416083 \nu^{6} - 6622116382602757 \nu^{5} - 26263502656698172 \nu^{4} - 5327429661771194 \nu^{3} - 19428846797347660 \nu^{2} + 4742345058394342 \nu + 4680253552809794\)\()/ 5173992472766390 \)
\(\beta_{2}\)\(=\)\((\)\(-123434729989149 \nu^{15} + 492304781090243 \nu^{14} - 1406992399278677 \nu^{13} + 5707540667520213 \nu^{12} - 7621279603788347 \nu^{11} + 281604145705947 \nu^{10} - 12957947975492223 \nu^{9} + 22625006932641203 \nu^{8} - 34533205664857133 \nu^{7} + 18681773772251579 \nu^{6} - 26694324268699259 \nu^{5} + 48096185658956821 \nu^{4} - 21876042039069938 \nu^{3} + 47788976778038450 \nu^{2} - 814448920872776 \nu + 195044141171618\)\()/ 5173992472766390 \)
\(\beta_{3}\)\(=\)\((\)\(-137809854283897 \nu^{15} + 666808683532056 \nu^{14} - 2027562782554100 \nu^{13} + 7694802116157034 \nu^{12} - 13885376435295071 \nu^{11} + 7288708077624688 \nu^{10} - 15122432396762870 \nu^{9} + 37940274269902486 \nu^{8} - 58228159495736079 \nu^{7} + 55051784324946342 \nu^{6} - 44775296775045806 \nu^{5} + 85474895099571132 \nu^{4} - 62657114582598388 \nu^{3} + 77038886286558388 \nu^{2} - 37418608795158304 \nu + 6950822659807744\)\()/ 1034798494553278 \)
\(\beta_{4}\)\(=\)\((\)\(3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} - 162143292202288 \nu^{12} + 154868117754112 \nu^{11} + 18595969360728 \nu^{10} + 464520812336313 \nu^{9} - 395501573216208 \nu^{8} + 1073827229563468 \nu^{7} + 11545660383276 \nu^{6} + 1336974192211299 \nu^{5} - 488205468817916 \nu^{4} + 1087029641011808 \nu^{3} - 622728293466680 \nu^{2} + 162118602709476 \nu - 16575048540328\)\()/ 28585593772190 \)
\(\beta_{5}\)\(=\)\((\)\(-4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + 160755388928989 \nu^{12} - 115488770848206 \nu^{11} - 55417826512144 \nu^{10} - 507559901300404 \nu^{9} + 264781323438119 \nu^{8} - 1058958936197574 \nu^{7} - 327950045248708 \nu^{6} - 1495012504742632 \nu^{5} + 55329885176253 \nu^{4} - 1127861763864064 \nu^{3} + 338424533456400 \nu^{2} - 56848696686768 \nu + 3631882693804\)\()/ 28585593772190 \)
\(\beta_{6}\)\(=\)\((\)\(-1439070217869327 \nu^{15} + 5626426049162414 \nu^{14} - 16762149286150656 \nu^{13} + 67570735976932179 \nu^{12} - 90330349700000641 \nu^{11} + 26412913789425376 \nu^{10} - 167480504119187474 \nu^{9} + 237626790891811969 \nu^{8} - 482812364930441289 \nu^{7} + 215899963434715562 \nu^{6} - 494473747233645422 \nu^{5} + 437389184714757453 \nu^{4} - 517428552172387524 \nu^{3} + 446449271988400340 \nu^{2} - 192208665603330138 \nu + 35411484066617244\)\()/ 5173992472766390 \)
\(\beta_{7}\)\(=\)\((\)\(6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + 314950092835 \nu^{11} - 40514369326 \nu^{10} + 709289788986 \nu^{9} - 820031528032 \nu^{8} + 1863520118983 \nu^{7} - 458012963070 \nu^{6} + 2053184268060 \nu^{5} - 1324040796852 \nu^{4} + 1933409293532 \nu^{3} - 1441782411844 \nu^{2} + 520681411390 \nu - 77929949288\)\()/ 18265555126 \)
\(\beta_{8}\)\(=\)\((\)\(-6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + 264606089742658 \nu^{12} - 295959094070367 \nu^{11} + 17152993349497 \nu^{10} - 701312474278478 \nu^{9} + 777595182487398 \nu^{8} - 1797695843305223 \nu^{7} + 332786417052279 \nu^{6} - 1990877398734524 \nu^{5} + 1207350248071826 \nu^{4} - 1824887125776248 \nu^{3} + 1306467148621370 \nu^{2} - 411232893748306 \nu + 65633529574178\)\()/ 16530327389030 \)
\(\beta_{9}\)\(=\)\((\)\(2035054902038187 \nu^{15} - 7578221328661024 \nu^{14} + 22354204007810251 \nu^{13} - 91579556495364929 \nu^{12} + 111269852929486391 \nu^{11} - 19058949252444566 \nu^{10} + 235208674633202009 \nu^{9} - 292701695999447459 \nu^{8} + 636133145212162719 \nu^{7} - 188961643415479902 \nu^{6} + 683022296693330837 \nu^{5} - 490835043167996523 \nu^{4} + 666044503208027234 \nu^{3} - 504441981276855820 \nu^{2} + 198734847296148758 \nu - 28123111426177514\)\()/ 5173992472766390 \)
\(\beta_{10}\)\(=\)\((\)\(30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + 1575363211 \nu^{11} - 188871116 \nu^{10} + 3515753069 \nu^{9} - 4158557204 \nu^{8} + 9212296429 \nu^{7} - 2344575592 \nu^{6} + 10021703527 \nu^{5} - 6825888288 \nu^{4} + 9461086554 \nu^{3} - 7254602740 \nu^{2} + 2513736318 \nu - 330526984\)\()/63923990\)
\(\beta_{11}\)\(=\)\((\)\(567889621787826 \nu^{15} - 2163110819836755 \nu^{14} + 6376823282252541 \nu^{13} - 25955396623934578 \nu^{12} + 32832120300673190 \nu^{11} - 6275666808832427 \nu^{10} + 64640506440221435 \nu^{9} - 87228624024570764 \nu^{8} + 179418986889806458 \nu^{7} - 65452477393295353 \nu^{6} + 182753898461327581 \nu^{5} - 154497881200594756 \nu^{4} + 181717723715357082 \nu^{3} - 157676074754561598 \nu^{2} + 53424782974620022 \nu - 7740432548067096\)\()/ 1034798494553278 \)
\(\beta_{12}\)\(=\)\((\)\(5281395788857158 \nu^{15} - 19095159515314451 \nu^{14} + 56052725193796619 \nu^{13} - 232054361068567251 \nu^{12} + 264944456866733364 \nu^{11} - 25993531049108449 \nu^{10} + 615506034577358971 \nu^{9} - 694365286936506611 \nu^{8} + 1588081348228614396 \nu^{7} - 344839767257990343 \nu^{6} + 1765864153109925593 \nu^{5} - 1105265950762762657 \nu^{4} + 1638034897211160076 \nu^{3} - 1197019598040248580 \nu^{2} + 413425841726845612 \nu - 60388611162910706\)\()/ 5173992472766390 \)
\(\beta_{13}\)\(=\)\((\)\(-19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} - 1035106924390 \nu^{11} + 152629602893 \nu^{10} - 2258419134670 \nu^{9} + 2719627979686 \nu^{8} - 6018321521990 \nu^{7} + 1643327598977 \nu^{6} - 6516717498206 \nu^{5} + 4492931472132 \nu^{4} - 6274129258728 \nu^{3} + 4768566345236 \nu^{2} - 1753345508964 \nu + 258618081490\)\()/ 18265555126 \)
\(\beta_{14}\)\(=\)\((\)\(-82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} - 4191737118 \nu^{11} + 407798693 \nu^{10} - 9561674912 \nu^{9} + 11027434227 \nu^{8} - 24845185642 \nu^{7} + 5661417851 \nu^{6} - 27237589476 \nu^{5} + 17742563129 \nu^{4} - 25400492652 \nu^{3} + 18964234070 \nu^{2} - 6297003604 \nu + 791533522\)\()/63923990\)
\(\beta_{15}\)\(=\)\((\)\(6956104161848134 \nu^{15} - 25667361154231553 \nu^{14} + 75476948193017452 \nu^{13} - 310349129733580443 \nu^{12} + 369420728986424472 \nu^{11} - 50736196725114107 \nu^{10} + 803087087863156658 \nu^{9} - 975207215422026373 \nu^{8} + 2135058242437940538 \nu^{7} - 584968875881439509 \nu^{6} + 2300137181821316574 \nu^{5} - 1620912261036340171 \nu^{4} + 2200212896950517088 \nu^{3} - 1699651856766889730 \nu^{2} + 607998971285547366 \nu - 83437215315298168\)\()/ 5173992472766390 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{6} - \beta_{5} - \beta_{3}\)
\(\nu^{2}\)\(=\)\(\beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - 2 \beta_{8} + \beta_{6} - 5 \beta_{1} + 10\)
\(\nu^{4}\)\(=\)\(-2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} - 15 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 20 \beta_{2} - 9 \beta_{1} + 17\)
\(\nu^{5}\)\(=\)\(-23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + 22 \beta_{6} - 2 \beta_{5} + 39 \beta_{4} - 21 \beta_{3} + 14 \beta_{2} + 7 \beta_{1} - 37\)
\(\nu^{6}\)\(=\)\(28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} - 56 \beta_{9} + 12 \beta_{8} + 89 \beta_{7} + 84 \beta_{6} + 121 \beta_{5} - 12 \beta_{4} + 56 \beta_{3} + 173 \beta_{2} - 39 \beta_{1} + 93\)
\(\nu^{7}\)\(=\)\(163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} - 351 \beta_{8} - 35 \beta_{7} - 219 \beta_{6} - 128 \beta_{5} - 94 \beta_{4} + 35 \beta_{3} - 440 \beta_{2} - 400 \beta_{1} + 985\)
\(\nu^{8}\)\(=\)\(-645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + 914 \beta_{9} - 672 \beta_{8} - 663 \beta_{7} - 645 \beta_{6} - 1243 \beta_{5} + 914 \beta_{4} - 914 \beta_{3} - 2063 \beta_{2} - 251 \beta_{1} + 9\)
\(\nu^{9}\)\(=\)\(-1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} - 1299 \beta_{9} + 1776 \beta_{8} + 1915 \beta_{7} + 2296 \beta_{6} + 2491 \beta_{5} + 1915 \beta_{4} + 4267 \beta_{2} + 1776 \beta_{1} - 5566\)
\(\nu^{10}\)\(=\)\(6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} - 3465 \beta_{9} - 1493 \beta_{8} + 8974 \beta_{7} + 1665 \beta_{6} + 11899 \beta_{5} - 7481 \beta_{4} + 9507 \beta_{3} + 10946 \beta_{2} - 6042 \beta_{1} + 17408\)
\(\nu^{11}\)\(=\)\(8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} - 21849 \beta_{10} + 32416 \beta_{9} - 37414 \beta_{8} - 15992 \beta_{7} - 38812 \beta_{6} - 32352 \beta_{5} - 5857 \beta_{4} - 5857 \beta_{3} - 77618 \beta_{2} - 32416 \beta_{1} + 77085\)
\(\nu^{12}\)\(=\)\(-91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + 7240 \beta_{10} + 55995 \beta_{9} - 7240 \beta_{8} - 43472 \beta_{7} - 31804 \beta_{6} - 82510 \beta_{5} + 99467 \beta_{4} - 82510 \beta_{3} - 121181 \beta_{2} + 43472 \beta_{1} - 166538\)
\(\nu^{13}\)\(=\)\(-55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + 372602 \beta_{10} - 253893 \beta_{9} + 253893 \beta_{8} + 322709 \beta_{7} + 281000 \beta_{6} + 467745 \beta_{5} + 186301 \beta_{3} + 711477 \beta_{2} + 185857 \beta_{1} - 469625\)
\(\nu^{14}\)\(=\)\(880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + 13265 \beta_{10} + 13265 \beta_{9} - 533645 \beta_{8} + 533645 \beta_{7} - 400341 \beta_{6} + 656706 \beta_{5} - 1044347 \beta_{4} + 911043 \beta_{3} + 57243 \beta_{2} - 911043 \beta_{1} + 2650257\)
\(\nu^{15}\)\(=\)\(-533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} - 2855121 \beta_{10} + 3769764 \beta_{9} - 3264759 \beta_{8} - 2855121 \beta_{7} - 4161149 \beta_{6} - 4943250 \beta_{5} + 765039 \beta_{4} - 2090082 \beta_{3} - 9354046 \beta_{2} - 1884882 \beta_{1} + 3840272\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-\beta_{14}\) \(1\) \(\beta_{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
0.117630 0.893490i
−0.424637 + 3.22544i
2.69978 + 0.355433i
0.339278 + 0.0446668i
0.792206 1.03242i
−0.709944 + 0.925217i
−1.09227 0.838128i
0.277956 + 0.213283i
0.792206 + 1.03242i
−0.709944 0.925217i
−1.09227 + 0.838128i
0.277956 0.213283i
0.117630 + 0.893490i
−0.424637 3.22544i
2.69978 0.355433i
0.339278 0.0446668i
−0.965926 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i −0.575432 2.16076i 1.00000i 1.42843 2.22701i −0.707107 0.707107i −0.866025 + 0.500000i −0.00342112 + 2.23607i
73.2 −0.965926 0.258819i 0.258819 + 0.965926i 0.866025 + 0.500000i 2.23435 + 0.0876265i 1.00000i −0.703686 + 2.55046i −0.707107 0.707107i −0.866025 + 0.500000i −2.13554 0.662933i
73.3 0.965926 + 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 1.20307 1.88484i 1.00000i −0.781940 2.52756i 0.707107 + 0.707107i −0.866025 + 0.500000i 1.64991 1.50924i
73.4 0.965926 + 0.258819i −0.258819 0.965926i 0.866025 + 0.500000i 1.87006 + 1.22592i 1.00000i −0.942805 + 2.47207i 0.707107 + 0.707107i −0.866025 + 0.500000i 1.48905 + 1.66815i
103.1 −0.258819 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i −0.619385 + 2.14857i 1.00000i 2.25331 + 1.38658i 0.707107 + 0.707107i 0.866025 + 0.500000i 2.23567 + 0.0421887i
103.2 −0.258819 0.965926i 0.965926 + 0.258819i −0.866025 + 0.500000i 1.96047 1.07544i 1.00000i −1.52856 2.15951i 0.707107 + 0.707107i 0.866025 + 0.500000i −1.54620 1.61532i
103.3 0.258819 + 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i −2.21628 + 0.296818i 1.00000i −1.87796 + 1.86367i −0.707107 0.707107i 0.866025 + 0.500000i −0.860320 2.06394i
103.4 0.258819 + 0.965926i −0.965926 0.258819i −0.866025 + 0.500000i 2.14315 0.637899i 1.00000i 0.153213 + 2.64131i −0.707107 0.707107i 0.866025 + 0.500000i 1.17085 + 1.90502i
157.1 −0.258819 + 0.965926i 0.965926 0.258819i −0.866025 0.500000i −0.619385 2.14857i 1.00000i 2.25331 1.38658i 0.707107 0.707107i 0.866025 0.500000i 2.23567 0.0421887i
157.2 −0.258819 + 0.965926i 0.965926 0.258819i −0.866025 0.500000i 1.96047 + 1.07544i 1.00000i −1.52856 + 2.15951i 0.707107 0.707107i 0.866025 0.500000i −1.54620 + 1.61532i
157.3 0.258819 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i −2.21628 0.296818i 1.00000i −1.87796 1.86367i −0.707107 + 0.707107i 0.866025 0.500000i −0.860320 + 2.06394i
157.4 0.258819 0.965926i −0.965926 + 0.258819i −0.866025 0.500000i 2.14315 + 0.637899i 1.00000i 0.153213 2.64131i −0.707107 + 0.707107i 0.866025 0.500000i 1.17085 1.90502i
187.1 −0.965926 + 0.258819i 0.258819 0.965926i 0.866025 0.500000i −0.575432 + 2.16076i 1.00000i 1.42843 + 2.22701i −0.707107 + 0.707107i −0.866025 0.500000i −0.00342112 2.23607i
187.2 −0.965926 + 0.258819i 0.258819 0.965926i 0.866025 0.500000i 2.23435 0.0876265i 1.00000i −0.703686 2.55046i −0.707107 + 0.707107i −0.866025 0.500000i −2.13554 + 0.662933i
187.3 0.965926 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 1.20307 + 1.88484i 1.00000i −0.781940 + 2.52756i 0.707107 0.707107i −0.866025 0.500000i 1.64991 + 1.50924i
187.4 0.965926 0.258819i −0.258819 + 0.965926i 0.866025 0.500000i 1.87006 1.22592i 1.00000i −0.942805 2.47207i 0.707107 0.707107i −0.866025 0.500000i 1.48905 1.66815i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 187.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.u.b yes 16
3.b odd 2 1 630.2.bv.b 16
5.b even 2 1 1050.2.bc.g 16
5.c odd 4 1 210.2.u.a 16
5.c odd 4 1 1050.2.bc.h 16
7.c even 3 1 1470.2.m.e 16
7.d odd 6 1 210.2.u.a 16
7.d odd 6 1 1470.2.m.d 16
15.e even 4 1 630.2.bv.a 16
21.g even 6 1 630.2.bv.a 16
35.i odd 6 1 1050.2.bc.h 16
35.k even 12 1 inner 210.2.u.b yes 16
35.k even 12 1 1050.2.bc.g 16
35.k even 12 1 1470.2.m.e 16
35.l odd 12 1 1470.2.m.d 16
105.w odd 12 1 630.2.bv.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.u.a 16 5.c odd 4 1
210.2.u.a 16 7.d odd 6 1
210.2.u.b yes 16 1.a even 1 1 trivial
210.2.u.b yes 16 35.k even 12 1 inner
630.2.bv.a 16 15.e even 4 1
630.2.bv.a 16 21.g even 6 1
630.2.bv.b 16 3.b odd 2 1
630.2.bv.b 16 105.w odd 12 1
1050.2.bc.g 16 5.b even 2 1
1050.2.bc.g 16 35.k even 12 1
1050.2.bc.h 16 5.c odd 4 1
1050.2.bc.h 16 35.i odd 6 1
1470.2.m.d 16 7.d odd 6 1
1470.2.m.d 16 35.l odd 12 1
1470.2.m.e 16 7.c even 3 1
1470.2.m.e 16 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{13}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$5$ \( 1 - 12 T + 64 T^{2} - 200 T^{3} + 398 T^{4} - 380 T^{5} - 992 T^{6} + 6636 T^{7} - 19341 T^{8} + 33180 T^{9} - 24800 T^{10} - 47500 T^{11} + 248750 T^{12} - 625000 T^{13} + 1000000 T^{14} - 937500 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 4 T + 34 T^{2} + 80 T^{3} + 445 T^{4} + 704 T^{5} + 3926 T^{6} + 5044 T^{7} + 29688 T^{8} + 35308 T^{9} + 192374 T^{10} + 241472 T^{11} + 1068445 T^{12} + 1344560 T^{13} + 4000066 T^{14} + 3294172 T^{15} + 5764801 T^{16} \)
$11$ \( 1 - 4 T - 42 T^{2} + 72 T^{3} + 1354 T^{4} - 52 T^{5} - 25972 T^{6} - 39092 T^{7} + 336125 T^{8} + 1014028 T^{9} - 2401652 T^{10} - 15137644 T^{11} - 1718882 T^{12} + 141984000 T^{13} + 309093362 T^{14} - 600162532 T^{15} - 4715588236 T^{16} - 6601787852 T^{17} + 37400296802 T^{18} + 188980704000 T^{19} - 25166151362 T^{20} - 2437932703844 T^{21} - 4254673018772 T^{22} + 19760537034788 T^{23} + 72051378876125 T^{24} - 92176891136572 T^{25} - 673646791737172 T^{26} - 14836206871772 T^{27} + 4249432022080234 T^{28} + 2485635274363032 T^{29} - 15949493010496122 T^{30} - 16708992677662604 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 - 16 T + 128 T^{2} - 744 T^{3} + 3314 T^{4} - 10768 T^{5} + 24864 T^{6} - 28864 T^{7} - 4655 T^{8} - 572216 T^{9} + 6801728 T^{10} - 43610328 T^{11} + 198012786 T^{12} - 641713232 T^{13} + 1520861600 T^{14} - 2393411960 T^{15} + 3297590084 T^{16} - 31114355480 T^{17} + 257025610400 T^{18} - 1409843970704 T^{19} + 5655443180946 T^{20} - 16192209514104 T^{21} + 32830641925952 T^{22} - 35905705403672 T^{23} - 3797226506255 T^{24} - 306088269902272 T^{25} + 3427713541333536 T^{26} - 19297983122990416 T^{27} + 77209854095902034 T^{28} - 225339079304636232 T^{29} + 503984177369508992 T^{30} - 818974288225452112 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 - 12 T + 760 T^{3} - 4000 T^{4} - 9644 T^{5} + 164912 T^{6} - 417780 T^{7} - 2033646 T^{8} + 15406336 T^{9} - 21694928 T^{10} - 141136884 T^{11} + 768607008 T^{12} - 1405601268 T^{13} - 1667112976 T^{14} + 24205669888 T^{15} - 123458993805 T^{16} + 411496388096 T^{17} - 481795650064 T^{18} - 6905719029684 T^{19} + 64194825915168 T^{20} - 200394192705588 T^{21} - 523662821550032 T^{22} + 6321815470032128 T^{23} - 14186221216859886 T^{24} - 49543643042916660 T^{25} + 332461586110845488 T^{26} - 330518167990812652 T^{27} - 2330488948919044000 T^{28} + 7527479305008512120 T^{29} - 34349076618117789516 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 8 T - 22 T^{2} - 144 T^{3} + 3371 T^{4} + 4000 T^{5} - 7722 T^{6} - 652040 T^{7} - 370487 T^{8} + 2996624 T^{9} + 89400308 T^{10} + 59920656 T^{11} - 371517522 T^{12} - 10373644960 T^{13} - 8723924296 T^{14} + 39539357168 T^{15} + 977996314130 T^{16} + 751247786192 T^{17} - 3149336670856 T^{18} - 71152830780640 T^{19} - 48416534984562 T^{20} + 148369476400944 T^{21} + 4205916251531348 T^{22} + 2678597506009136 T^{23} - 6292189320370967 T^{24} - 210405286459819160 T^{25} - 47344093642739322 T^{26} + 465961035592876000 T^{27} + 7461084592172028731 T^{28} - 6055629618565016496 T^{29} - 17578147087223450662 T^{30} - \)\(12\!\cdots\!92\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 40 T + 776 T^{2} + 9784 T^{3} + 91799 T^{4} + 705368 T^{5} + 4787160 T^{6} + 29946728 T^{7} + 172895985 T^{8} + 902021104 T^{9} + 4178360768 T^{10} + 17091640336 T^{11} + 61289409778 T^{12} + 182133009408 T^{13} + 342869112800 T^{14} - 372347426016 T^{15} - 5577275049586 T^{16} - 8563990798368 T^{17} + 181377760671200 T^{18} + 2216012325467136 T^{19} + 17151289721685298 T^{20} + 110007659635131248 T^{21} + 618547350853602752 T^{22} + 3071224408630233488 T^{23} + 13539654936478996785 T^{24} + 53938628839308543064 T^{25} + \)\(19\!\cdots\!40\)\( T^{26} + \)\(67\!\cdots\!36\)\( T^{27} + \)\(20\!\cdots\!79\)\( T^{28} + \)\(49\!\cdots\!72\)\( T^{29} + \)\(89\!\cdots\!84\)\( T^{30} + \)\(10\!\cdots\!80\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 11228601281217568 T^{18} + 262138411756753378 T^{20} - 5206063149142513344 T^{22} + 85729661487925625265 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(11\!\cdots\!38\)\( T^{28} - \)\(73\!\cdots\!88\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( 1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 219348622553040 T^{17} + 1297120976901720 T^{18} + 7743746535986400 T^{19} + 44642930296011118 T^{20} + 244104359431513968 T^{21} + 1308878681740078284 T^{22} + 6899639338664300784 T^{23} + 37295035913646998185 T^{24} + \)\(20\!\cdots\!48\)\( T^{25} + \)\(10\!\cdots\!86\)\( T^{26} + \)\(50\!\cdots\!80\)\( T^{27} + \)\(21\!\cdots\!39\)\( T^{28} + \)\(83\!\cdots\!28\)\( T^{29} + \)\(25\!\cdots\!14\)\( T^{30} + \)\(56\!\cdots\!24\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 + 8 T + 248 T^{2} + 1752 T^{3} + 30199 T^{4} + 206976 T^{5} + 2540200 T^{6} + 17745904 T^{7} + 169033089 T^{8} + 1209867128 T^{9} + 9485737632 T^{10} + 68571970520 T^{11} + 465855584354 T^{12} + 3322795052152 T^{13} + 20345020109280 T^{14} + 139940970795928 T^{15} + 794780046531614 T^{16} + 5177815919449336 T^{17} + 27852332529604320 T^{18} + 168309537776655256 T^{19} + 873088367828476994 T^{20} + 4755051775144147640 T^{21} + 24337807551267523488 T^{22} + \)\(11\!\cdots\!24\)\( T^{23} + \)\(59\!\cdots\!69\)\( T^{24} + \)\(23\!\cdots\!08\)\( T^{25} + \)\(12\!\cdots\!00\)\( T^{26} + \)\(36\!\cdots\!88\)\( T^{27} + \)\(19\!\cdots\!19\)\( T^{28} + \)\(42\!\cdots\!44\)\( T^{29} + \)\(22\!\cdots\!72\)\( T^{30} + \)\(26\!\cdots\!44\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 - 172 T^{2} + 13974 T^{4} - 791464 T^{6} + 43993793 T^{8} - 2592908552 T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 9601764734939492 T^{18} + 379606521146518278 T^{20} - 12316585909380369032 T^{22} + \)\(35\!\cdots\!53\)\( T^{24} - \)\(10\!\cdots\!64\)\( T^{26} + \)\(31\!\cdots\!94\)\( T^{28} - \)\(65\!\cdots\!92\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 + 24 T + 288 T^{2} + 2456 T^{3} + 18144 T^{4} + 133752 T^{5} + 1000544 T^{6} + 6828856 T^{7} + 39963644 T^{8} + 227672888 T^{9} + 1517218592 T^{10} + 11117129912 T^{11} + 81274307360 T^{12} + 589957036568 T^{13} + 4310182316384 T^{14} + 31623166948824 T^{15} + 218467749390534 T^{16} + 1359796178799432 T^{17} + 7969527102994016 T^{18} + 46905714106411976 T^{19} + 277860683276675360 T^{20} + 1634311958991847016 T^{21} + 9590889544724607008 T^{22} + 61885728202879567016 T^{23} + \)\(46\!\cdots\!44\)\( T^{24} + \)\(34\!\cdots\!08\)\( T^{25} + \)\(21\!\cdots\!56\)\( T^{26} + \)\(12\!\cdots\!64\)\( T^{27} + \)\(72\!\cdots\!44\)\( T^{28} + \)\(42\!\cdots\!08\)\( T^{29} + \)\(21\!\cdots\!12\)\( T^{30} + \)\(76\!\cdots\!68\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 + 72 T^{2} + 616 T^{3} + 3239 T^{4} + 50872 T^{5} + 298520 T^{6} + 2523744 T^{7} + 15099153 T^{8} + 113209408 T^{9} + 814540096 T^{10} + 3210267840 T^{11} + 24971489874 T^{12} + 159883078560 T^{13} + 379015320032 T^{14} + 3041257505584 T^{15} + 39450787699278 T^{16} + 142939102762448 T^{17} + 837244841950688 T^{18} + 16599540865334880 T^{19} + 121852904679850194 T^{20} + 736258900236674880 T^{21} + 8780103088888331584 T^{22} + 57354503546728915904 T^{23} + \)\(35\!\cdots\!33\)\( T^{24} + \)\(28\!\cdots\!48\)\( T^{25} + \)\(15\!\cdots\!80\)\( T^{26} + \)\(12\!\cdots\!16\)\( T^{27} + \)\(37\!\cdots\!99\)\( T^{28} + \)\(33\!\cdots\!32\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 28 T + 344 T^{2} + 2912 T^{3} + 25298 T^{4} + 232644 T^{5} + 1917568 T^{6} + 14158268 T^{7} + 104375121 T^{8} + 770687892 T^{9} + 5270701040 T^{10} + 39047642852 T^{11} + 318443399442 T^{12} + 2209044273928 T^{13} + 14320519756920 T^{14} + 111315852463604 T^{15} + 873920983850628 T^{16} + 5899740180571012 T^{17} + 40226339997188280 T^{18} + 328875884369578856 T^{19} + 2512671592872511602 T^{20} + 16329548252980066036 T^{21} + \)\(11\!\cdots\!60\)\( T^{22} + \)\(90\!\cdots\!04\)\( T^{23} + \)\(64\!\cdots\!81\)\( T^{24} + \)\(46\!\cdots\!44\)\( T^{25} + \)\(33\!\cdots\!32\)\( T^{26} + \)\(21\!\cdots\!68\)\( T^{27} + \)\(12\!\cdots\!18\)\( T^{28} + \)\(75\!\cdots\!76\)\( T^{29} + \)\(47\!\cdots\!36\)\( T^{30} + \)\(20\!\cdots\!96\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 + 8 T - 236 T^{2} - 1632 T^{3} + 30563 T^{4} + 163648 T^{5} - 2682716 T^{6} - 9836488 T^{7} + 178469613 T^{8} + 332320192 T^{9} - 10946443072 T^{10} - 3817987168 T^{11} + 756189977838 T^{12} - 175777223888 T^{13} - 56160273686696 T^{14} + 6297587928448 T^{15} + 3675723794639094 T^{16} + 371557687778432 T^{17} - 195493912703388776 T^{18} - 36100950464893552 T^{19} + 9163026946045045518 T^{20} - 2729571799673395232 T^{21} - \)\(46\!\cdots\!52\)\( T^{22} + \)\(82\!\cdots\!48\)\( T^{23} + \)\(26\!\cdots\!73\)\( T^{24} - \)\(85\!\cdots\!32\)\( T^{25} - \)\(13\!\cdots\!16\)\( T^{26} + \)\(49\!\cdots\!32\)\( T^{27} + \)\(54\!\cdots\!03\)\( T^{28} - \)\(17\!\cdots\!28\)\( T^{29} - \)\(14\!\cdots\!96\)\( T^{30} + \)\(29\!\cdots\!92\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 89390037630613680 T^{17} - 596188052016653508 T^{18} + 3466669701018035640 T^{19} - 15772027818523273792 T^{20} + 30956710267288726104 T^{21} + \)\(43\!\cdots\!32\)\( T^{22} - \)\(77\!\cdots\!64\)\( T^{23} + \)\(82\!\cdots\!86\)\( T^{24} - \)\(72\!\cdots\!88\)\( T^{25} + \)\(54\!\cdots\!72\)\( T^{26} - \)\(36\!\cdots\!32\)\( T^{27} + \)\(21\!\cdots\!80\)\( T^{28} - \)\(10\!\cdots\!44\)\( T^{29} + \)\(46\!\cdots\!88\)\( T^{30} - \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 + 84 T + 3216 T^{2} + 71776 T^{3} + 957024 T^{4} + 5741436 T^{5} - 40997200 T^{6} - 1158667636 T^{7} - 7828301806 T^{8} + 48210145144 T^{9} + 1314996417536 T^{10} + 8000602781236 T^{11} - 45952509042976 T^{12} - 1085658642853028 T^{13} - 5561139754106368 T^{14} + 34677137718483048 T^{15} + 641528682100762995 T^{16} + 2323368227138364216 T^{17} - 24963956356183485952 T^{18} - \)\(32\!\cdots\!64\)\( T^{19} - \)\(92\!\cdots\!96\)\( T^{20} + \)\(10\!\cdots\!52\)\( T^{21} + \)\(11\!\cdots\!84\)\( T^{22} + \)\(29\!\cdots\!12\)\( T^{23} - \)\(31\!\cdots\!46\)\( T^{24} - \)\(31\!\cdots\!92\)\( T^{25} - \)\(74\!\cdots\!00\)\( T^{26} + \)\(70\!\cdots\!88\)\( T^{27} + \)\(78\!\cdots\!64\)\( T^{28} + \)\(39\!\cdots\!12\)\( T^{29} + \)\(11\!\cdots\!64\)\( T^{30} + \)\(20\!\cdots\!12\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 + 16 T + 392 T^{2} + 4432 T^{3} + 72412 T^{4} + 697872 T^{5} + 8835896 T^{6} + 71051216 T^{7} + 737387398 T^{8} + 5044636336 T^{9} + 44541751736 T^{10} + 249776065392 T^{11} + 1840110644572 T^{12} + 7996344483632 T^{13} + 50215311297032 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} )^{2} \)
$73$ \( 1 - 16 T + 416 T^{2} - 4440 T^{3} + 68404 T^{4} - 584824 T^{5} + 7229216 T^{6} - 53644504 T^{7} + 544685226 T^{8} - 2736110216 T^{9} + 18336160640 T^{10} + 72833715600 T^{11} - 1347659375280 T^{12} + 28574375923760 T^{13} - 284065502536896 T^{14} + 3320804755103832 T^{15} - 26931854637922413 T^{16} + 242418747122579736 T^{17} - 1513785063019118784 T^{18} + 11115917998733343920 T^{19} - 38271155725110882480 T^{20} + \)\(15\!\cdots\!00\)\( T^{21} + \)\(27\!\cdots\!60\)\( T^{22} - \)\(30\!\cdots\!52\)\( T^{23} + \)\(43\!\cdots\!06\)\( T^{24} - \)\(31\!\cdots\!52\)\( T^{25} + \)\(31\!\cdots\!84\)\( T^{26} - \)\(18\!\cdots\!48\)\( T^{27} + \)\(15\!\cdots\!84\)\( T^{28} - \)\(74\!\cdots\!20\)\( T^{29} + \)\(50\!\cdots\!44\)\( T^{30} - \)\(14\!\cdots\!12\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 12 T + 518 T^{2} + 5640 T^{3} + 137639 T^{4} + 1484376 T^{5} + 26147402 T^{6} + 279843540 T^{7} + 3950719429 T^{8} + 41115674928 T^{9} + 499791767516 T^{10} + 4982987471808 T^{11} + 54520483633690 T^{12} + 515559074594184 T^{13} + 5198103978036912 T^{14} + 46390749479828928 T^{15} + 436708099187124858 T^{16} + 3664869208906485312 T^{17} + 32441366926928367792 T^{18} + \)\(25\!\cdots\!76\)\( T^{19} + \)\(21\!\cdots\!90\)\( T^{20} + \)\(15\!\cdots\!92\)\( T^{21} + \)\(12\!\cdots\!36\)\( T^{22} + \)\(78\!\cdots\!52\)\( T^{23} + \)\(59\!\cdots\!69\)\( T^{24} + \)\(33\!\cdots\!60\)\( T^{25} + \)\(24\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!04\)\( T^{27} + \)\(81\!\cdots\!99\)\( T^{28} + \)\(26\!\cdots\!60\)\( T^{29} + \)\(19\!\cdots\!58\)\( T^{30} + \)\(34\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 16 T + 128 T^{2} - 48 T^{3} - 16022 T^{4} + 55056 T^{5} + 1171072 T^{6} - 29664624 T^{7} + 191234977 T^{8} + 1519529248 T^{9} - 27088145920 T^{10} + 210706506976 T^{11} + 1209312464722 T^{12} - 24316714882016 T^{13} + 89859490185984 T^{14} + 1324590918805248 T^{15} - 28693493560058396 T^{16} + 109941046260835584 T^{17} + 619042027891243776 T^{18} - 13903981452243282592 T^{19} + 57391939140077851762 T^{20} + \)\(82\!\cdots\!68\)\( T^{21} - \)\(88\!\cdots\!80\)\( T^{22} + \)\(41\!\cdots\!96\)\( T^{23} + \)\(43\!\cdots\!57\)\( T^{24} - \)\(55\!\cdots\!72\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} + \)\(70\!\cdots\!52\)\( T^{27} - \)\(17\!\cdots\!42\)\( T^{28} - \)\(42\!\cdots\!24\)\( T^{29} + \)\(94\!\cdots\!12\)\( T^{30} - \)\(97\!\cdots\!12\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 16 T - 440 T^{2} + 6624 T^{3} + 131636 T^{4} - 1629968 T^{5} - 29449232 T^{6} + 280328720 T^{7} + 5329291050 T^{8} - 36595065152 T^{9} - 800017753480 T^{10} + 3635275214288 T^{11} + 102057968083920 T^{12} - 258547916350832 T^{13} - 11209896420041672 T^{14} + 8808353629704256 T^{15} + 1069443829226298003 T^{16} + 783943473043678784 T^{17} - 88793589543150083912 T^{18} - \)\(18\!\cdots\!08\)\( T^{19} + \)\(64\!\cdots\!20\)\( T^{20} + \)\(20\!\cdots\!12\)\( T^{21} - \)\(39\!\cdots\!80\)\( T^{22} - \)\(16\!\cdots\!08\)\( T^{23} + \)\(20\!\cdots\!50\)\( T^{24} + \)\(98\!\cdots\!80\)\( T^{25} - \)\(91\!\cdots\!32\)\( T^{26} - \)\(45\!\cdots\!52\)\( T^{27} + \)\(32\!\cdots\!56\)\( T^{28} + \)\(14\!\cdots\!56\)\( T^{29} - \)\(86\!\cdots\!40\)\( T^{30} - \)\(27\!\cdots\!84\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 + 44 T + 968 T^{2} + 15000 T^{3} + 231654 T^{4} + 3860388 T^{5} + 58116000 T^{6} + 743419588 T^{7} + 9111376513 T^{8} + 116138978204 T^{9} + 1430090287664 T^{10} + 15964265880436 T^{11} + 171574947065342 T^{12} + 1888109943974960 T^{13} + 20331243756190952 T^{14} + 203613182755137748 T^{15} + 1985416913924716484 T^{16} + 19750478727248361556 T^{17} + \)\(19\!\cdots\!68\)\( T^{18} + \)\(17\!\cdots\!80\)\( T^{19} + \)\(15\!\cdots\!02\)\( T^{20} + \)\(13\!\cdots\!52\)\( T^{21} + \)\(11\!\cdots\!56\)\( T^{22} + \)\(93\!\cdots\!52\)\( T^{23} + \)\(71\!\cdots\!93\)\( T^{24} + \)\(56\!\cdots\!96\)\( T^{25} + \)\(42\!\cdots\!00\)\( T^{26} + \)\(27\!\cdots\!64\)\( T^{27} + \)\(16\!\cdots\!14\)\( T^{28} + \)\(10\!\cdots\!00\)\( T^{29} + \)\(63\!\cdots\!92\)\( T^{30} + \)\(27\!\cdots\!92\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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