Properties

Label 69.2.e.b
Level $69$
Weight $2$
Character orbit 69.e
Analytic conductor $0.551$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.550967773947\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \(x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-\zeta_{22}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−0.415415 0.909632i
0.654861 + 0.755750i
0.654861 0.755750i
−0.841254 0.540641i
0.142315 0.989821i
0.142315 + 0.989821i
−0.415415 + 0.909632i
0.959493 0.281733i
−0.841254 + 0.540641i
0.959493 + 0.281733i
0.915415 + 2.00448i 0.959493 0.281733i −1.87023 + 2.15836i −2.61903 1.68315i 1.44306 + 1.66538i −0.427961 2.97653i −1.80972 0.531382i 0.841254 0.540641i 0.976337 6.79057i
13.1 −0.154861 0.178719i −0.841254 0.540641i 0.276671 1.92429i 0.418382 0.916128i 0.0336545 + 0.234072i 1.97204 + 0.579043i −0.784630 + 0.504251i 0.415415 + 0.909632i −0.228520 + 0.0670996i
16.1 −0.154861 + 0.178719i −0.841254 + 0.540641i 0.276671 + 1.92429i 0.418382 + 0.916128i 0.0336545 0.234072i 1.97204 0.579043i −0.784630 0.504251i 0.415415 0.909632i −0.228520 0.0670996i
25.1 1.34125 + 0.861971i 0.142315 + 0.989821i 0.225136 + 0.492980i −2.43560 0.715158i −0.662317 + 1.45027i 0.729022 0.841336i 0.330830 2.30097i −0.959493 + 0.281733i −2.65032 3.05863i
31.1 0.357685 2.48775i −0.415415 0.909632i −4.14200 1.21620i 2.65565 + 3.06479i −2.41153 + 0.708089i −1.15843 0.744479i −2.41899 + 5.29684i −0.654861 + 0.755750i 8.57432 5.51038i
49.1 0.357685 + 2.48775i −0.415415 + 0.909632i −4.14200 + 1.21620i 2.65565 3.06479i −2.41153 0.708089i −1.15843 + 0.744479i −2.41899 5.29684i −0.654861 0.755750i 8.57432 + 5.51038i
52.1 0.915415 2.00448i 0.959493 + 0.281733i −1.87023 2.15836i −2.61903 + 1.68315i 1.44306 1.66538i −0.427961 + 2.97653i −1.80972 + 0.531382i 0.841254 + 0.540641i 0.976337 + 6.79057i
55.1 −0.459493 + 0.134919i 0.654861 + 0.755750i −1.48958 + 0.957293i 0.480602 + 3.34266i −0.402869 0.258908i 1.88533 4.12830i 1.18251 1.36469i −0.142315 + 0.989821i −0.671822 1.47109i
58.1 1.34125 0.861971i 0.142315 0.989821i 0.225136 0.492980i −2.43560 + 0.715158i −0.662317 1.45027i 0.729022 + 0.841336i 0.330830 + 2.30097i −0.959493 0.281733i −2.65032 + 3.05863i
64.1 −0.459493 0.134919i 0.654861 0.755750i −1.48958 0.957293i 0.480602 3.34266i −0.402869 + 0.258908i 1.88533 + 4.12830i 1.18251 + 1.36469i −0.142315 0.989821i −0.671822 + 1.47109i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.2.e.b 10
3.b odd 2 1 207.2.i.a 10
23.c even 11 1 inner 69.2.e.b 10
23.c even 11 1 1587.2.a.q 5
23.d odd 22 1 1587.2.a.r 5
69.g even 22 1 4761.2.a.bm 5
69.h odd 22 1 207.2.i.a 10
69.h odd 22 1 4761.2.a.bp 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.b 10 1.a even 1 1 trivial
69.2.e.b 10 23.c even 11 1 inner
207.2.i.a 10 3.b odd 2 1
207.2.i.a 10 69.h odd 22 1
1587.2.a.q 5 23.c even 11 1
1587.2.a.r 5 23.d odd 22 1
4761.2.a.bm 5 69.g even 22 1
4761.2.a.bp 5 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(69, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T + 31 T^{2} + 39 T^{3} - 18 T^{4} - 23 T^{5} + 47 T^{6} - 31 T^{7} + 16 T^{8} - 4 T^{9} + T^{10} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$5$ \( 11881 + 763 T + 6737 T^{2} + 8336 T^{3} + 3204 T^{4} + 782 T^{5} + 290 T^{6} + 71 T^{7} + 9 T^{8} + 3 T^{9} + T^{10} \)
$7$ \( 1849 - 1806 T + 576 T^{2} + 597 T^{3} + 93 T^{4} - 472 T^{5} + 240 T^{6} - 95 T^{7} + 36 T^{8} - 6 T^{9} + T^{10} \)
$11$ \( 1 + 14 T + 1384 T^{2} + 4856 T^{3} + 7209 T^{4} + 5754 T^{5} + 2676 T^{6} + 746 T^{7} + 137 T^{8} + 15 T^{9} + T^{10} \)
$13$ \( 1 + 4 T + 16 T^{2} + 31 T^{3} + 47 T^{4} + 23 T^{5} - 18 T^{6} - 39 T^{7} + 31 T^{8} - 8 T^{9} + T^{10} \)
$17$ \( 39601 - 35223 T + 24740 T^{2} - 2465 T^{3} + 2190 T^{4} + 615 T^{5} + 111 T^{6} - 111 T^{7} + T^{8} - T^{9} + T^{10} \)
$19$ \( 109561 - 96652 T + 42056 T^{2} - 7718 T^{3} + 3573 T^{4} + 529 T^{5} - 28 T^{6} + 124 T^{7} + 48 T^{8} + 9 T^{9} + T^{10} \)
$23$ \( 6436343 - 5876661 T + 2688907 T^{2} - 860683 T^{3} + 218615 T^{4} - 47937 T^{5} + 9505 T^{6} - 1627 T^{7} + 221 T^{8} - 21 T^{9} + T^{10} \)
$29$ \( 109561 + 293597 T + 375303 T^{2} + 270569 T^{3} + 115052 T^{4} + 27103 T^{5} + 4272 T^{6} + 743 T^{7} + 108 T^{8} + 8 T^{9} + T^{10} \)
$31$ \( 11881 - 12099 T + 12035 T^{2} - 99824 T^{3} + 95085 T^{4} + 52383 T^{5} + 14697 T^{6} + 2487 T^{7} + 298 T^{8} + 23 T^{9} + T^{10} \)
$37$ \( 4190209 - 2034718 T + 1033785 T^{2} - 144516 T^{3} + 88102 T^{4} - 3268 T^{5} + 2919 T^{6} - 137 T^{7} - 57 T^{8} - 3 T^{9} + T^{10} \)
$41$ \( 203889841 - 34826481 T + 3215309 T^{2} - 1042685 T^{3} + 13094 T^{4} + 4038 T^{5} + 2896 T^{6} + 372 T^{7} + 137 T^{8} + 15 T^{9} + T^{10} \)
$43$ \( 1437601 + 1147443 T + 735317 T^{2} + 254947 T^{3} + 55176 T^{4} - 6633 T^{5} - 660 T^{6} - 759 T^{7} + 220 T^{8} - 22 T^{9} + T^{10} \)
$47$ \( ( -13133 + 5064 T + 365 T^{2} - 148 T^{3} - 2 T^{4} + T^{5} )^{2} \)
$53$ \( 55965361 - 69678034 T + 43238776 T^{2} - 16284428 T^{3} + 4803407 T^{4} - 747889 T^{5} + 73296 T^{6} - 5634 T^{7} + 434 T^{8} - 29 T^{9} + T^{10} \)
$59$ \( 474411961 + 314321611 T + 133825286 T^{2} + 45244121 T^{3} + 11903068 T^{4} + 2150587 T^{5} + 270832 T^{6} + 23847 T^{7} + 1431 T^{8} + 54 T^{9} + T^{10} \)
$61$ \( 591361 - 5842093 T + 15157235 T^{2} + 2908798 T^{3} + 921198 T^{4} + 186856 T^{5} + 46193 T^{6} + 5440 T^{7} + 504 T^{8} + 30 T^{9} + T^{10} \)
$67$ \( 6285049 + 10757537 T + 11220727 T^{2} - 1085206 T^{3} + 496189 T^{4} - 103357 T^{5} + 5479 T^{6} + 131 T^{7} - 32 T^{8} - T^{9} + T^{10} \)
$71$ \( 21743569 - 1916493 T - 1903248 T^{2} + 748405 T^{3} + 40406 T^{4} - 41381 T^{5} + 10443 T^{6} - 941 T^{7} + 163 T^{8} + 3 T^{9} + T^{10} \)
$73$ \( 58081 - 228950 T + 16654951 T^{2} + 15303594 T^{3} + 6224023 T^{4} + 1422433 T^{5} + 201700 T^{6} + 18892 T^{7} + 1197 T^{8} + 47 T^{9} + T^{10} \)
$79$ \( 34774609 - 29237326 T + 7610920 T^{2} + 1218145 T^{3} + 965980 T^{4} + 100464 T^{5} + 21189 T^{6} - 783 T^{7} + 16 T^{8} - 18 T^{9} + T^{10} \)
$83$ \( 141681409 - 222026659 T + 127482474 T^{2} - 33988191 T^{3} + 6243582 T^{4} - 810501 T^{5} + 79346 T^{6} - 5535 T^{7} + 335 T^{8} - 18 T^{9} + T^{10} \)
$89$ \( 69169 + 238278 T + 110555 T^{2} - 189693 T^{3} + 151704 T^{4} - 73646 T^{5} + 23621 T^{6} - 4009 T^{7} + 438 T^{8} - 25 T^{9} + T^{10} \)
$97$ \( 39601 - 203378 T + 359844 T^{2} + 57432 T^{3} + 43055 T^{4} - 17940 T^{5} + 6755 T^{6} - 1440 T^{7} + 210 T^{8} - 21 T^{9} + T^{10} \)
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