# Properties

 Label 69.3.b.a Level $69$ Weight $3$ Character orbit 69.b Analytic conductor $1.880$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 69.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.88011382409$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 40 x^{12} + 598 x^{10} + 4207 x^{8} + 14465 x^{6} + 23786 x^{4} + 17144 x^{2} + 3887$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{8} q^{3} + ( -2 + \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( 1 + \beta_{4} ) q^{6} + \beta_{5} q^{7} + ( -2 \beta_{1} + \beta_{9} + \beta_{12} + \beta_{13} ) q^{8} + ( -1 + \beta_{1} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{8} q^{3} + ( -2 + \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( 1 + \beta_{4} ) q^{6} + \beta_{5} q^{7} + ( -2 \beta_{1} + \beta_{9} + \beta_{12} + \beta_{13} ) q^{8} + ( -1 + \beta_{1} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{9} + ( \beta_{2} + \beta_{6} - \beta_{7} ) q^{10} + ( \beta_{1} - \beta_{3} + \beta_{9} - \beta_{11} - \beta_{13} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{12} ) q^{12} + ( -\beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} ) q^{13} + ( -\beta_{1} + \beta_{9} + \beta_{11} + \beta_{13} ) q^{14} + ( -2 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - \beta_{13} ) q^{15} + ( 4 - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} - 3 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{16} + ( \beta_{1} - 2 \beta_{3} + \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{17} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{18} + ( 2 + \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{19} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{9} ) q^{20} + ( -\beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{12} - 2 \beta_{13} ) q^{21} + ( -6 + \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + 4 \beta_{8} - 2 \beta_{10} ) q^{22} -\beta_{9} q^{23} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{24} + ( 1 - \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{25} + ( 1 - \beta_{2} - 2 \beta_{3} - \beta_{6} + 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{26} + ( 5 - 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{27} + ( 6 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{10} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} - 5 \beta_{13} ) q^{29} + ( 4 - \beta_{1} + 3 \beta_{3} - \beta_{5} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{13} ) q^{30} + ( -12 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{31} + ( -1 + 9 \beta_{1} + \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{32} + ( 8 - 4 \beta_{1} - \beta_{3} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} ) q^{33} + ( -6 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{34} + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} + \beta_{12} + 5 \beta_{13} ) q^{35} + ( -9 \beta_{1} - \beta_{2} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} ) q^{36} + ( 4 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{37} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} ) q^{38} + ( -5 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{39} + ( 12 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{40} + ( -2 + \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 6 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + \beta_{12} - \beta_{13} ) q^{41} + ( -6 + 5 \beta_{1} + 5 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{42} + ( -2 + 4 \beta_{1} + \beta_{2} + 6 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{10} ) q^{43} + ( 2 - 11 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 6 \beta_{8} + 7 \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{44} + ( -4 + 6 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{12} ) q^{45} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{46} + ( -1 - 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} + \beta_{10} + 4 \beta_{12} + 8 \beta_{13} ) q^{47} + ( -24 - 11 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 4 \beta_{12} + 3 \beta_{13} ) q^{48} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} - 8 \beta_{8} + 3 \beta_{10} ) q^{49} + ( 2 + 12 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} + 6 \beta_{8} - 5 \beta_{9} + 2 \beta_{10} + \beta_{11} - 3 \beta_{12} - 2 \beta_{13} ) q^{50} + ( -3 \beta_{1} + 5 \beta_{2} + \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + \beta_{6} + \beta_{7} + 7 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - \beta_{13} ) q^{51} + ( 3 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{52} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} ) q^{53} + ( 23 + 11 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} + 2 \beta_{11} - \beta_{13} ) q^{54} + ( 23 + 4 \beta_{1} + 5 \beta_{2} + 5 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{55} + ( -4 + 19 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} - 12 \beta_{8} - 7 \beta_{9} - 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 3 \beta_{13} ) q^{56} + ( 10 + 3 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} + 4 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} ) q^{57} + ( 5 - \beta_{1} + 5 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} ) q^{58} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} + 9 \beta_{8} - 8 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{59} + ( -4 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{60} + ( -12 - 4 \beta_{1} - \beta_{2} - 10 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 6 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{61} + ( 3 - 21 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{6} + 9 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + \beta_{11} + \beta_{13} ) q^{62} + ( -16 + 2 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 6 \beta_{9} + 4 \beta_{12} + 4 \beta_{13} ) q^{63} + ( -24 - 2 \beta_{1} + 7 \beta_{2} - 11 \beta_{4} + 8 \beta_{5} - \beta_{6} - \beta_{7} + 16 \beta_{8} - 9 \beta_{10} ) q^{64} + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} + 6 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} ) q^{65} + ( 28 + 19 \beta_{1} - 7 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{9} - \beta_{11} - 4 \beta_{12} - 3 \beta_{13} ) q^{66} + ( 12 - 4 \beta_{1} - \beta_{2} - 2 \beta_{4} + 6 \beta_{5} - 5 \beta_{6} + \beta_{7} - 8 \beta_{8} + 2 \beta_{10} ) q^{67} + ( -4 - 17 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 12 \beta_{8} - 5 \beta_{9} - 6 \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{68} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{69} + ( 7 - \beta_{1} - \beta_{2} - \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{70} + ( 2 \beta_{1} + 4 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} - 2 \beta_{12} - 7 \beta_{13} ) q^{71} + ( 30 - 5 \beta_{1} - 14 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} - \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} - \beta_{12} ) q^{72} + ( -22 - 3 \beta_{1} + 6 \beta_{2} + \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + \beta_{7} - 9 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{73} + ( 2 + 21 \beta_{1} - 2 \beta_{2} - 11 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + 6 \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} - 8 \beta_{12} - 5 \beta_{13} ) q^{74} + ( -6 + 12 \beta_{1} + 2 \beta_{2} - 10 \beta_{3} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + 6 \beta_{8} + 9 \beta_{9} - 3 \beta_{10} + \beta_{12} + \beta_{13} ) q^{75} + ( 6 - 6 \beta_{1} - 15 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 22 \beta_{8} + 8 \beta_{10} ) q^{76} + ( -19 \beta_{1} - 10 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} ) q^{77} + ( -25 + 7 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 7 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} ) q^{78} + ( -24 + 6 \beta_{1} + 8 \beta_{2} + 2 \beta_{4} - \beta_{5} + 6 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{79} + ( -2 + 12 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 2 \beta_{6} - 6 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 4 \beta_{13} ) q^{80} + ( -10 + 3 \beta_{1} + \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 5 \beta_{7} + 6 \beta_{8} + 5 \beta_{9} - 6 \beta_{10} - \beta_{11} - 6 \beta_{12} ) q^{81} + ( -13 - 14 \beta_{1} - 9 \beta_{2} - 14 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 20 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} + 6 \beta_{11} + 6 \beta_{12} ) q^{82} + ( -2 + 9 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{6} - 6 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} + \beta_{11} + 6 \beta_{12} + 7 \beta_{13} ) q^{83} + ( -24 - 28 \beta_{1} + 9 \beta_{2} - \beta_{3} - 4 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} + 4 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{84} + ( 37 - \beta_{1} + 3 \beta_{2} + \beta_{4} - 3 \beta_{6} - 3 \beta_{7} + 5 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} - 5 \beta_{12} ) q^{85} + ( -6 + 2 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} - 18 \beta_{8} - 10 \beta_{9} - 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} ) q^{86} + ( 1 - 3 \beta_{1} + \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - \beta_{5} + 5 \beta_{6} - 4 \beta_{8} - \beta_{9} - 3 \beta_{10} + 8 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{87} + ( 28 + 4 \beta_{1} - 27 \beta_{2} + 10 \beta_{4} - 5 \beta_{5} - \beta_{6} + 11 \beta_{7} - 14 \beta_{8} - 6 \beta_{9} + 12 \beta_{10} + 6 \beta_{11} + 6 \beta_{12} ) q^{88} + ( -2 - 9 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + \beta_{9} - 3 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{89} + ( -40 - 10 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} - 2 \beta_{13} ) q^{90} + ( -8 - 4 \beta_{2} - 4 \beta_{4} + 6 \beta_{6} - 4 \beta_{7} + 6 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} ) q^{91} + ( 1 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{6} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{92} + ( 1 + 18 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 12 \beta_{8} - 4 \beta_{9} + 7 \beta_{10} + 3 \beta_{11} + 5 \beta_{12} + 3 \beta_{13} ) q^{93} + ( 44 - 5 \beta_{1} - 6 \beta_{2} - 9 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} + 4 \beta_{8} + \beta_{9} - 5 \beta_{10} - \beta_{11} - \beta_{12} ) q^{94} + ( 6 - 5 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} - 6 \beta_{6} + 18 \beta_{8} - 9 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} + 5 \beta_{12} + 2 \beta_{13} ) q^{95} + ( 49 - 41 \beta_{1} - 22 \beta_{2} - 10 \beta_{3} + \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} - 5 \beta_{8} + 2 \beta_{9} + 9 \beta_{10} + 6 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} ) q^{96} + ( 8 + 6 \beta_{1} - 6 \beta_{2} + 8 \beta_{4} + 2 \beta_{5} + 8 \beta_{8} + 6 \beta_{9} - 4 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} ) q^{97} + ( -1 - 14 \beta_{1} + \beta_{2} + 8 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} + 7 \beta_{13} ) q^{98} + ( 16 - 7 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 6 \beta_{6} + 6 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 3 \beta_{11} - 4 \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 4q^{3} - 24q^{4} + 11q^{6} - 4q^{7} - 8q^{9} + O(q^{10})$$ $$14q - 4q^{3} - 24q^{4} + 11q^{6} - 4q^{7} - 8q^{9} - 8q^{10} + 19q^{12} - 14q^{15} + 72q^{16} - 31q^{18} + 8q^{19} - 2q^{21} - 84q^{22} - 44q^{24} + 38q^{25} + 62q^{27} + 76q^{28} + 62q^{30} - 144q^{31} + 90q^{33} - 68q^{34} + 3q^{36} + 48q^{37} - 78q^{39} + 120q^{40} - 76q^{42} - 48q^{43} - 18q^{45} - 317q^{48} - 30q^{49} + 18q^{51} - 6q^{52} + 312q^{54} + 232q^{55} + 76q^{57} + 66q^{58} - 36q^{60} - 140q^{61} - 206q^{63} - 346q^{64} + 398q^{66} + 204q^{67} + 80q^{70} + 384q^{72} - 224q^{73} - 80q^{75} + 100q^{76} - 341q^{78} - 344q^{79} - 232q^{81} - 62q^{82} - 330q^{84} + 480q^{85} + 86q^{87} + 436q^{88} - 514q^{90} - 172q^{91} + 62q^{93} + 514q^{94} + 609q^{96} - 24q^{97} + 234q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 40 x^{12} + 598 x^{10} + 4207 x^{8} + 14465 x^{6} + 23786 x^{4} + 17144 x^{2} + 3887$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 6$$ $$\beta_{3}$$ $$=$$ $$($$$$-32 \nu^{13} - 1839 \nu^{11} - 39104 \nu^{9} - 384237 \nu^{7} - 1758889 \nu^{5} - 3225120 \nu^{3} - 1623643 \nu$$$$)/88920$$ $$\beta_{4}$$ $$=$$ $$($$$$9 \nu^{13} + 40 \nu^{12} + 348 \nu^{11} + 1515 \nu^{10} + 5013 \nu^{9} + 20950 \nu^{8} + 34029 \nu^{7} + 130815 \nu^{6} + 111888 \nu^{5} + 362285 \nu^{4} + 158655 \nu^{3} + 369150 \nu^{2} + 61506 \nu + 81365$$$$)/6840$$ $$\beta_{5}$$ $$=$$ $$($$$$409 \nu^{12} + 15783 \nu^{10} + 221923 \nu^{8} + 1396044 \nu^{6} + 3838193 \nu^{4} + 3932805 \nu^{2} + 990011$$$$)/13680$$ $$\beta_{6}$$ $$=$$ $$($$$$178 \nu^{13} + 1820 \nu^{12} + 6756 \nu^{11} + 70785 \nu^{10} + 91546 \nu^{9} + 1008800 \nu^{8} + 526338 \nu^{7} + 6502275 \nu^{6} + 1112036 \nu^{5} + 18734755 \nu^{4} + 244650 \nu^{3} + 20957040 \nu^{2} - 629968 \nu + 5975125$$$$)/88920$$ $$\beta_{7}$$ $$=$$ $$($$$$178 \nu^{13} + 2379 \nu^{12} + 6756 \nu^{11} + 90753 \nu^{10} + 91546 \nu^{9} + 1258413 \nu^{8} + 526338 \nu^{7} + 7798284 \nu^{6} + 1112036 \nu^{5} + 21198723 \nu^{4} + 244650 \nu^{3} + 22120995 \nu^{2} - 629968 \nu + 6384261$$$$)/88920$$ $$\beta_{8}$$ $$=$$ $$($$$$-295 \nu^{13} + 117 \nu^{12} - 11280 \nu^{11} + 4524 \nu^{10} - 156715 \nu^{9} + 65169 \nu^{8} - 968715 \nu^{7} + 442377 \nu^{6} - 2566580 \nu^{5} + 1454544 \nu^{4} - 2307165 \nu^{3} + 2062515 \nu^{2} - 258530 \nu + 799578$$$$)/88920$$ $$\beta_{9}$$ $$=$$ $$($$$$-131 \nu^{13} - 5097 \nu^{11} - 73385 \nu^{9} - 489744 \nu^{7} - 1548127 \nu^{5} - 2150079 \nu^{3} - 913741 \nu$$$$)/35568$$ $$\beta_{10}$$ $$=$$ $$($$$$-707 \nu^{13} + 1469 \nu^{12} - 27084 \nu^{11} + 57213 \nu^{10} - 378599 \nu^{9} + 813293 \nu^{8} - 2379807 \nu^{7} + 5175144 \nu^{6} - 6587704 \nu^{5} + 14371123 \nu^{4} - 6676845 \nu^{3} + 14858415 \nu^{2} - 1316638 \nu + 4020991$$$$)/88920$$ $$\beta_{11}$$ $$=$$ $$($$$$2565 \nu^{13} - 169 \nu^{12} + 98895 \nu^{11} - 6123 \nu^{10} + 1389375 \nu^{9} - 76843 \nu^{8} + 8734680 \nu^{7} - 373464 \nu^{6} + 23987025 \nu^{5} - 301613 \nu^{4} + 24224145 \nu^{3} + 2011035 \nu^{2} + 5038515 \nu + 2452489$$$$)/177840$$ $$\beta_{12}$$ $$=$$ $$($$$$-2752 \nu^{13} - 169 \nu^{12} - 106284 \nu^{11} - 6123 \nu^{10} - 1495624 \nu^{9} - 76843 \nu^{8} - 9413892 \nu^{7} - 373464 \nu^{6} - 25909484 \nu^{5} - 301613 \nu^{4} - 26724480 \nu^{3} + 2011035 \nu^{2} - 6231068 \nu + 2452489$$$$)/177840$$ $$\beta_{13}$$ $$=$$ $$($$$$3407 \nu^{13} + 169 \nu^{12} + 131769 \nu^{11} + 6123 \nu^{10} + 1862549 \nu^{9} + 76843 \nu^{8} + 11862612 \nu^{7} + 373464 \nu^{6} + 33650119 \nu^{5} + 301613 \nu^{4} + 37652715 \nu^{3} - 2011035 \nu^{2} + 12578173 \nu - 2452489$$$$)/177840$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{13} + \beta_{12} + \beta_{9} - 10 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{12} + \beta_{11} + 2 \beta_{10} - \beta_{9} - 3 \beta_{8} + \beta_{7} - 2 \beta_{5} + \beta_{4} - 15 \beta_{2} + 60$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{13} - 15 \beta_{12} - 3 \beta_{11} + \beta_{10} - 19 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + \beta_{2} + 121 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$-20 \beta_{12} - 20 \beta_{11} - 49 \beta_{10} + 20 \beta_{9} + 76 \beta_{8} - 21 \beta_{7} - \beta_{6} + 48 \beta_{5} - 31 \beta_{4} + 211 \beta_{2} - 2 \beta_{1} - 712$$ $$\nu^{7}$$ $$=$$ $$189 \beta_{13} + 210 \beta_{12} + 77 \beta_{11} - 18 \beta_{10} + 304 \beta_{9} + 93 \beta_{8} - 49 \beta_{7} - 80 \beta_{6} + 98 \beta_{5} + 49 \beta_{4} + 8 \beta_{3} - 31 \beta_{2} - 1586 \beta_{1} + 31$$ $$\nu^{8}$$ $$=$$ $$326 \beta_{12} + 326 \beta_{11} + 899 \beta_{10} - 326 \beta_{9} - 1404 \beta_{8} + 355 \beta_{7} + 39 \beta_{6} - 884 \beta_{5} + 641 \beta_{4} - 2992 \beta_{2} + 68 \beta_{1} + 9173$$ $$\nu^{9}$$ $$=$$ $$-2622 \beta_{13} - 2963 \beta_{12} - 1457 \beta_{11} + 258 \beta_{10} - 4693 \beta_{9} - 1923 \beta_{8} + 899 \beta_{7} + 1540 \beta_{6} - 1798 \beta_{5} - 899 \beta_{4} - 228 \beta_{3} + 641 \beta_{2} + 21777 \beta_{1} - 641$$ $$\nu^{10}$$ $$=$$ $$-5019 \beta_{12} - 5019 \beta_{11} - 14875 \beta_{10} + 5019 \beta_{9} + 23217 \beta_{8} - 5664 \beta_{7} - 869 \beta_{6} + 14766 \beta_{5} - 11370 \beta_{4} + 43064 \beta_{2} - 1514 \beta_{1} - 124369$$ $$\nu^{11}$$ $$=$$ $$37291 \beta_{13} + 42419 \beta_{12} + 24622 \beta_{11} - 3505 \beta_{10} + 71541 \beta_{9} + 34110 \beta_{8} - 14875 \beta_{7} - 26245 \beta_{6} + 29750 \beta_{5} + 14875 \beta_{4} + 4416 \beta_{3} - 11370 \beta_{2} - 308110 \beta_{1} + 11370$$ $$\nu^{12}$$ $$=$$ $$75674 \beta_{12} + 75674 \beta_{11} + 234702 \beta_{10} - 75674 \beta_{9} - 365376 \beta_{8} + 88242 \beta_{7} + 15786 \beta_{6} - 235188 \beta_{5} + 187382 \beta_{4} - 627410 \beta_{2} + 28354 \beta_{1} + 1744535$$ $$\nu^{13}$$ $$=$$ $$-539654 \beta_{13} - 614842 \beta_{12} - 394216 \beta_{11} + 47320 \beta_{10} - 1083222 \beta_{9} - 562146 \beta_{8} + 234702 \beta_{7} + 422084 \beta_{6} - 469404 \beta_{5} - 234702 \beta_{4} - 74004 \beta_{3} + 187382 \beta_{2} + 4445261 \beta_{1} - 187382$$

## Character values

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

 $$n$$ $$28$$ $$47$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 − 3.86310i − 3.07980i − 2.92693i − 1.90763i − 1.29748i − 1.13976i − 0.634638i 0.634638i 1.13976i 1.29748i 1.90763i 2.92693i 3.07980i 3.86310i
3.86310i −2.26727 + 1.96456i −10.9236 1.45241i 7.58928 + 8.75871i −6.75781 26.7464i 1.28104 8.90836i −5.61080
47.2 3.07980i −0.479940 2.96136i −5.48516 0.529218i −9.12040 + 1.47812i 5.42850 4.57400i −8.53932 + 2.84255i 1.62989
47.3 2.92693i 2.91955 + 0.690107i −4.56694 2.77451i 2.01990 8.54532i −1.17249 1.65938i 8.04750 + 4.02960i −8.12081
47.4 1.90763i 0.570904 + 2.94518i 0.360941 9.03892i 5.61831 1.08907i 1.87935 8.31907i −8.34814 + 3.36283i 17.2429
47.5 1.29748i −2.20436 + 2.03489i 2.31655 5.31444i 2.64023 + 2.86012i 12.0394 8.19559i 0.718425 8.97128i −6.89538
47.6 1.13976i −2.53605 1.60264i 2.70094 4.29888i −1.82663 + 2.89049i −9.25811 7.63748i 3.86309 + 8.12875i −4.89970
47.7 0.634638i 1.99717 2.23859i 3.59723 4.18173i −1.42070 1.26748i −4.15888 4.82149i −1.02261 8.94172i 2.65389
47.8 0.634638i 1.99717 + 2.23859i 3.59723 4.18173i −1.42070 + 1.26748i −4.15888 4.82149i −1.02261 + 8.94172i 2.65389
47.9 1.13976i −2.53605 + 1.60264i 2.70094 4.29888i −1.82663 2.89049i −9.25811 7.63748i 3.86309 8.12875i −4.89970
47.10 1.29748i −2.20436 2.03489i 2.31655 5.31444i 2.64023 2.86012i 12.0394 8.19559i 0.718425 + 8.97128i −6.89538
47.11 1.90763i 0.570904 2.94518i 0.360941 9.03892i 5.61831 + 1.08907i 1.87935 8.31907i −8.34814 3.36283i 17.2429
47.12 2.92693i 2.91955 0.690107i −4.56694 2.77451i 2.01990 + 8.54532i −1.17249 1.65938i 8.04750 4.02960i −8.12081
47.13 3.07980i −0.479940 + 2.96136i −5.48516 0.529218i −9.12040 1.47812i 5.42850 4.57400i −8.53932 2.84255i 1.62989
47.14 3.86310i −2.26727 1.96456i −10.9236 1.45241i 7.58928 8.75871i −6.75781 26.7464i 1.28104 + 8.90836i −5.61080
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.3.b.a 14
3.b odd 2 1 inner 69.3.b.a 14
4.b odd 2 1 1104.3.g.b 14
12.b even 2 1 1104.3.g.b 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.3.b.a 14 1.a even 1 1 trivial
69.3.b.a 14 3.b odd 2 1 inner
1104.3.g.b 14 4.b odd 2 1
1104.3.g.b 14 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(69, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3887 + 17144 T^{2} + 23786 T^{4} + 14465 T^{6} + 4207 T^{8} + 598 T^{10} + 40 T^{12} + T^{14}$$
$3$ $$4782969 + 2125764 T + 708588 T^{2} + 39366 T^{3} + 18954 T^{4} - 15876 T^{5} - 567 T^{6} - 1260 T^{7} - 63 T^{8} - 196 T^{9} + 26 T^{10} + 6 T^{11} + 12 T^{12} + 4 T^{13} + T^{14}$$
$5$ $$3391488 + 14696704 T^{2} + 9802560 T^{4} + 2067024 T^{6} + 187728 T^{8} + 8076 T^{10} + 156 T^{12} + T^{14}$$
$7$ $$( -37472 - 20608 T + 15744 T^{2} + 4724 T^{3} - 572 T^{4} - 162 T^{5} + 2 T^{6} + T^{7} )^{2}$$
$11$ $$44251280863232 + 8246528920832 T^{2} + 375181206336 T^{4} + 7286044176 T^{6} + 71366880 T^{8} + 369336 T^{10} + 964 T^{12} + T^{14}$$
$13$ $$( -839808 - 326592 T + 41472 T^{2} + 19584 T^{3} - 330 T^{4} - 263 T^{5} + T^{7} )^{2}$$
$17$ $$7651346734641152 + 680348693286144 T^{2} + 18219163861056 T^{4} + 198018435920 T^{6} + 983868304 T^{8} + 2318796 T^{10} + 2508 T^{12} + T^{14}$$
$19$ $$( -369654688 - 167081776 T - 1379704 T^{2} + 993620 T^{3} + 6816 T^{4} - 1826 T^{5} - 4 T^{6} + T^{7} )^{2}$$
$23$ $$( 23 + T^{2} )^{7}$$
$29$ $$1999193468185608192 + 52898033644943616 T^{2} + 528363013499136 T^{4} + 2584688672736 T^{6} + 6644147504 T^{8} + 8753825 T^{10} + 5182 T^{12} + T^{14}$$
$31$ $$( -1909922976 + 116096112 T + 16061976 T^{2} - 595644 T^{3} - 55310 T^{4} + 457 T^{5} + 72 T^{6} + T^{7} )^{2}$$
$37$ $$( 31114889632 - 1899741664 T - 106729328 T^{2} + 6292308 T^{3} + 94416 T^{4} - 4672 T^{5} - 24 T^{6} + T^{7} )^{2}$$
$41$ $$2430391354376192 + 4960056125242835200 T^{2} + 172181270974330752 T^{4} + 349977504468144 T^{6} + 265036408008 T^{8} + 93787761 T^{10} + 15686 T^{12} + T^{14}$$
$43$ $$( 8823945600 - 779652000 T - 82637496 T^{2} + 7562076 T^{3} - 19984 T^{4} - 5310 T^{5} + 24 T^{6} + T^{7} )^{2}$$
$47$ $$13\!\cdots\!08$$$$+ 7416147882206890240 T^{2} + 104145401596556672 T^{4} + 288985999258544 T^{6} + 250523310872 T^{8} + 93828713 T^{10} + 15902 T^{12} + T^{14}$$
$53$ $$85\!\cdots\!48$$$$+ 67740377911622142208 T^{2} + 215176998049632128 T^{4} + 347383019904720 T^{6} + 295734419008 T^{8} + 123098220 T^{10} + 19904 T^{12} + T^{14}$$
$59$ $$32\!\cdots\!12$$$$+$$$$22\!\cdots\!28$$$$T^{2} + 4243699752990855168 T^{4} + 3468876816386304 T^{6} + 1386408923520 T^{8} + 280363056 T^{10} + 27192 T^{12} + T^{14}$$
$61$ $$( 1148734281696 - 105718252496 T + 1515991304 T^{2} + 59094756 T^{3} - 811776 T^{4} - 14244 T^{5} + 70 T^{6} + T^{7} )^{2}$$
$67$ $$( -173963774816 - 16531619488 T - 177395264 T^{2} + 19197540 T^{3} + 350532 T^{4} - 6898 T^{5} - 102 T^{6} + T^{7} )^{2}$$
$71$ $$41984638580586119168 + 3526845269035434240 T^{2} + 61139835975655936 T^{4} + 288410890837984 T^{6} + 391720442624 T^{8} + 175228097 T^{10} + 27342 T^{12} + T^{14}$$
$73$ $$( 921220683136 + 54066679488 T - 730747328 T^{2} - 107003216 T^{3} - 2345810 T^{4} - 12543 T^{5} + 112 T^{6} + T^{7} )^{2}$$
$79$ $$( 607693691232 + 72942469008 T + 432105704 T^{2} - 85671804 T^{3} - 1796872 T^{4} - 2466 T^{5} + 172 T^{6} + T^{7} )^{2}$$
$83$ $$16\!\cdots\!32$$$$+$$$$54\!\cdots\!80$$$$T^{2} + 1733960036345289280 T^{4} + 1936006067227920 T^{6} + 992381445344 T^{8} + 245595384 T^{10} + 27220 T^{12} + T^{14}$$
$89$ $$14\!\cdots\!00$$$$+$$$$16\!\cdots\!00$$$$T^{2} +$$$$69\!\cdots\!32$$$$T^{4} + 156695438150447888 T^{6} + 20161524037984 T^{8} + 1498180668 T^{10} + 59928 T^{12} + T^{14}$$
$97$ $$( 144273643904 - 197517606784 T - 913842112 T^{2} + 139529840 T^{3} + 185736 T^{4} - 25240 T^{5} + 12 T^{6} + T^{7} )^{2}$$