Properties

Label 69.2.e.a
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.

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

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
−0.662317 1.45027i 0.959493 0.281733i −0.354905 + 0.409583i −0.438384 0.281733i −1.04408 1.20493i 0.188515 + 1.31115i −2.23047 0.654925i 0.841254 0.540641i −0.118239 + 0.822373i
13.1 1.44306 + 1.66538i −0.841254 0.540641i −0.406440 + 2.82685i 0.246902 0.540641i −0.313607 2.18119i −2.76921 0.813115i −1.58671 + 1.01971i 0.415415 + 0.909632i 1.25667 0.368991i
16.1 1.44306 1.66538i −0.841254 + 0.540641i −0.406440 2.82685i 0.246902 + 0.540641i −0.313607 + 2.18119i −2.76921 + 0.813115i −1.58671 1.01971i 0.415415 0.909632i 1.25667 + 0.368991i
25.1 −0.402869 0.258908i 0.142315 + 0.989821i −0.735560 1.61065i 3.37102 + 0.989821i 0.198939 0.435615i 0.527646 0.608936i −0.256983 + 1.78736i −0.959493 + 0.281733i −1.10181 1.27155i
31.1 0.0336545 0.234072i −0.415415 0.909632i 1.86533 + 0.547710i −0.788201 0.909632i −0.226900 + 0.0666238i 0.0566239 + 0.0363899i 0.387454 0.848406i −0.654861 + 0.755750i −0.239446 + 0.153882i
49.1 0.0336545 + 0.234072i −0.415415 + 0.909632i 1.86533 0.547710i −0.788201 + 0.909632i −0.226900 0.0666238i 0.0566239 0.0363899i 0.387454 + 0.848406i −0.654861 0.755750i −0.239446 0.153882i
52.1 −0.662317 + 1.45027i 0.959493 + 0.281733i −0.354905 0.409583i −0.438384 + 0.281733i −1.04408 + 1.20493i 0.188515 1.31115i −2.23047 + 0.654925i 0.841254 + 0.540641i −0.118239 0.822373i
55.1 −2.41153 + 0.708089i 0.654861 + 0.755750i 3.63158 2.33387i 0.108660 + 0.755750i −2.11435 1.35881i −2.00357 + 4.38721i −3.81329 + 4.40077i −0.142315 + 0.989821i −0.797176 1.74557i
58.1 −0.402869 + 0.258908i 0.142315 0.989821i −0.735560 + 1.61065i 3.37102 0.989821i 0.198939 + 0.435615i 0.527646 + 0.608936i −0.256983 1.78736i −0.959493 0.281733i −1.10181 + 1.27155i
64.1 −2.41153 0.708089i 0.654861 0.755750i 3.63158 + 2.33387i 0.108660 0.755750i −2.11435 + 1.35881i −2.00357 4.38721i −3.81329 4.40077i −0.142315 0.989821i −0.797176 + 1.74557i
\(n\): 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.a 10
3.b odd 2 1 207.2.i.b 10
23.c even 11 1 inner 69.2.e.a 10
23.c even 11 1 1587.2.a.o 5
23.d odd 22 1 1587.2.a.p 5
69.g even 22 1 4761.2.a.bq 5
69.h odd 22 1 207.2.i.b 10
69.h odd 22 1 4761.2.a.br 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.a 10 1.a even 1 1 trivial
69.2.e.a 10 23.c even 11 1 inner
207.2.i.b 10 3.b odd 2 1
207.2.i.b 10 69.h odd 22 1
1587.2.a.o 5 23.c even 11 1
1587.2.a.p 5 23.d odd 22 1
4761.2.a.bq 5 69.g even 22 1
4761.2.a.br 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

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 20 T^{2} + 71 T^{3} + 125 T^{4} + 78 T^{5} + 36 T^{6} + 9 T^{7} + 5 T^{8} + 4 T^{9} + T^{10} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \)
$5$ \( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} + 10 T^{5} + 20 T^{6} + 7 T^{7} + 3 T^{8} - 5 T^{9} + T^{10} \)
$7$ \( 1 - 26 T + 247 T^{2} - 251 T^{3} + 256 T^{4} + 65 T^{5} + 81 T^{6} + 105 T^{7} + 42 T^{8} + 8 T^{9} + T^{10} \)
$11$ \( 529 - 2208 T + 11317 T^{2} - 6111 T^{3} + 3282 T^{4} - 813 T^{5} - 74 T^{6} + 31 T^{7} + 16 T^{8} - 7 T^{9} + T^{10} \)
$13$ \( 2042041 + 2630789 T + 1984933 T^{2} + 1038776 T^{3} + 401294 T^{4} + 116610 T^{5} + 25359 T^{6} + 4010 T^{7} + 438 T^{8} + 30 T^{9} + T^{10} \)
$17$ \( 17161 - 46898 T + 69457 T^{2} - 57347 T^{3} + 27223 T^{4} - 2608 T^{5} + 720 T^{6} + 228 T^{7} - 18 T^{8} + 2 T^{9} + T^{10} \)
$19$ \( 1 + T + 67 T^{2} + 254 T^{3} + 56 T^{4} - 362 T^{5} + 375 T^{6} - 186 T^{7} + 56 T^{8} - 10 T^{9} + T^{10} \)
$23$ \( 6436343 + 279841 T + 12167 T^{2} - 11109 T^{3} - 5037 T^{4} + 1365 T^{5} - 219 T^{6} - 21 T^{7} + T^{8} + T^{9} + T^{10} \)
$29$ \( 734449 - 785869 T + 489428 T^{2} - 141572 T^{3} - 5046 T^{4} + 2509 T^{5} + 2061 T^{6} + 456 T^{7} + 108 T^{8} + 14 T^{9} + T^{10} \)
$31$ \( 4489 + 15611 T + 43586 T^{2} + 58462 T^{3} + 75355 T^{4} + 17302 T^{5} + 4519 T^{6} + 1514 T^{7} + 300 T^{8} + 28 T^{9} + T^{10} \)
$37$ \( 14645929 - 1530800 T + 4754029 T^{2} + 30677 T^{3} + 104954 T^{4} - 90947 T^{5} + 15426 T^{6} - 1293 T^{7} + 152 T^{8} - 19 T^{9} + T^{10} \)
$41$ \( 38809 - 130808 T + 192384 T^{2} - 143178 T^{3} + 70557 T^{4} - 25211 T^{5} + 6736 T^{6} - 1348 T^{7} + 196 T^{8} - 19 T^{9} + T^{10} \)
$43$ \( 1739761 - 4201015 T + 5725976 T^{2} - 1090225 T^{3} - 2290 T^{4} + 15971 T^{5} + 10202 T^{6} + 2417 T^{7} + 301 T^{8} + 24 T^{9} + T^{10} \)
$47$ \( ( -2507 - 249 T + 486 T^{2} - 27 T^{3} - 13 T^{4} + T^{5} )^{2} \)
$53$ \( 94249 - 868196 T + 11874435 T^{2} + 2059861 T^{3} + 277410 T^{4} + 10308 T^{5} + 309 T^{6} - 615 T^{7} - 54 T^{8} + T^{9} + T^{10} \)
$59$ \( 1515361 + 1347945 T + 776339 T^{2} - 486449 T^{3} + 76569 T^{4} - 7908 T^{5} + 390 T^{6} + 124 T^{7} + 37 T^{8} - 2 T^{9} + T^{10} \)
$61$ \( 519520849 - 470515899 T + 176708273 T^{2} - 37867403 T^{3} + 5860649 T^{4} - 734832 T^{5} + 87916 T^{6} - 8058 T^{7} + 581 T^{8} - 30 T^{9} + T^{10} \)
$67$ \( 24930049 - 12397619 T + 5237450 T^{2} - 1932518 T^{3} + 544350 T^{4} + 23407 T^{5} + 6559 T^{6} - 1582 T^{7} + 16 T^{8} - 4 T^{9} + T^{10} \)
$71$ \( 4663114369 + 1933204970 T + 476186716 T^{2} + 67817495 T^{3} + 5686486 T^{4} + 225182 T^{5} + 1291 T^{6} + 533 T^{7} + 130 T^{8} + 14 T^{9} + T^{10} \)
$73$ \( 12243001 + 10979862 T + 5751117 T^{2} + 1812453 T^{3} + 497842 T^{4} + 112157 T^{5} + 16591 T^{6} + 2077 T^{7} + 302 T^{8} + 26 T^{9} + T^{10} \)
$79$ \( 326041 + 1824345 T + 3339592 T^{2} + 1960009 T^{3} + 562142 T^{4} - 143375 T^{5} + 27494 T^{6} - 3215 T^{7} + 323 T^{8} - 20 T^{9} + T^{10} \)
$83$ \( 529 + 2346 T + 3804 T^{2} - 17 T^{3} + 224 T^{4} - 2914 T^{5} + 1455 T^{6} - 321 T^{7} + 192 T^{8} - 18 T^{9} + T^{10} \)
$89$ \( 3373402561 + 951831428 T + 245566468 T^{2} + 48380027 T^{3} + 6467928 T^{4} + 201279 T^{5} - 21254 T^{6} - 3923 T^{7} - 19 T^{8} + 5 T^{9} + T^{10} \)
$97$ \( 83740801 + 174070322 T + 69232216 T^{2} - 33910772 T^{3} + 6267309 T^{4} - 558724 T^{5} + 111642 T^{6} - 3782 T^{7} + 379 T^{8} - 15 T^{9} + T^{10} \)
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