Properties

 Label 44.2.g.a Level $44$ Weight $2$ Character orbit 44.g Analytic conductor $0.351$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$44 = 2^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 44.g (of order $$10$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.351341768894$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + 120 x^{7} - 8 x^{6} - 240 x^{5} + 560 x^{4} - 800 x^{3} + 832 x^{2} - 640 x + 256$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + 120 x^{7} - 8 x^{6} - 240 x^{5} + 560 x^{4} - 800 x^{3} + 832 x^{2} - 640 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} - 5 \nu^{14} + 13 \nu^{13} - 25 \nu^{12} + 35 \nu^{11} - 30 \nu^{10} - 2 \nu^{9} + 60 \nu^{8} - 116 \nu^{7} + 120 \nu^{6} - 8 \nu^{5} - 240 \nu^{4} + 560 \nu^{3} - 800 \nu^{2} + 832 \nu - 640$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$-35 \nu^{15} + 94 \nu^{14} - 210 \nu^{13} + 334 \nu^{12} - 328 \nu^{11} + 111 \nu^{10} + 476 \nu^{9} - 1018 \nu^{8} + 1368 \nu^{7} - 500 \nu^{6} - 1600 \nu^{5} + 4744 \nu^{4} - 7520 \nu^{3} + 7760 \nu^{2} - 7072 \nu + 2304$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-30 \nu^{15} + 141 \nu^{14} - 309 \nu^{13} + 573 \nu^{12} - 693 \nu^{11} + 429 \nu^{10} + 398 \nu^{9} - 1670 \nu^{8} + 2452 \nu^{7} - 2036 \nu^{6} - 1384 \nu^{5} + 6728 \nu^{4} - 13008 \nu^{3} + 15856 \nu^{2} - 14368 \nu + 9664$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$33 \nu^{15} - 147 \nu^{14} + 323 \nu^{13} - 591 \nu^{12} + 705 \nu^{11} - 424 \nu^{10} - 446 \nu^{9} + 1736 \nu^{8} - 2524 \nu^{7} + 2000 \nu^{6} + 1544 \nu^{5} - 7072 \nu^{4} + 13424 \nu^{3} - 16128 \nu^{2} + 14592 \nu - 9472$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$-37 \nu^{15} + 152 \nu^{14} - 334 \nu^{13} + 602 \nu^{12} - 704 \nu^{11} + 405 \nu^{10} + 498 \nu^{9} - 1774 \nu^{8} + 2556 \nu^{7} - 1916 \nu^{6} - 1704 \nu^{5} + 7336 \nu^{4} - 13648 \nu^{3} + 16176 \nu^{2} - 14656 \nu + 9088$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$35 \nu^{15} - 120 \nu^{14} + 265 \nu^{13} - 455 \nu^{12} + 501 \nu^{11} - 248 \nu^{10} - 475 \nu^{9} + 1358 \nu^{8} - 1902 \nu^{7} + 1168 \nu^{6} + 1612 \nu^{5} - 5880 \nu^{4} + 10280 \nu^{3} - 11632 \nu^{2} + 10544 \nu - 5536$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-47 \nu^{15} + 153 \nu^{14} - 338 \nu^{13} + 572 \nu^{12} - 618 \nu^{11} + 291 \nu^{10} + 633 \nu^{9} - 1712 \nu^{8} + 2382 \nu^{7} - 1356 \nu^{6} - 2148 \nu^{5} + 7520 \nu^{4} - 12920 \nu^{3} + 14400 \nu^{2} - 13072 \nu + 6464$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$71 \nu^{15} - 281 \nu^{14} + 619 \nu^{13} - 1107 \nu^{12} + 1281 \nu^{11} - 722 \nu^{10} - 952 \nu^{9} + 3264 \nu^{8} - 4688 \nu^{7} + 3408 \nu^{6} + 3264 \nu^{5} - 13632 \nu^{4} + 25088 \nu^{3} - 29504 \nu^{2} + 26720 \nu - 16128$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$-279 \nu^{15} + 987 \nu^{14} - 2179 \nu^{13} + 3775 \nu^{12} - 4213 \nu^{11} + 2162 \nu^{10} + 3766 \nu^{9} - 11244 \nu^{8} + 15852 \nu^{7} - 10168 \nu^{6} - 12808 \nu^{5} + 48240 \nu^{4} - 85456 \nu^{3} + 97632 \nu^{2} - 88384 \nu + 48384$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$307 \nu^{15} - 1097 \nu^{14} + 2421 \nu^{13} - 4209 \nu^{12} + 4711 \nu^{11} - 2444 \nu^{10} - 4142 \nu^{9} + 12520 \nu^{8} - 17676 \nu^{7} + 11488 \nu^{6} + 14088 \nu^{5} - 53600 \nu^{4} + 95248 \nu^{3} - 109056 \nu^{2} + 98880 \nu - 54528$$$$)/128$$ $$\beta_{12}$$ $$=$$ $$($$$$-293 \nu^{15} + 1079 \nu^{14} - 2383 \nu^{13} + 4179 \nu^{12} - 4733 \nu^{11} + 2528 \nu^{10} + 3942 \nu^{9} - 12400 \nu^{8} + 17612 \nu^{7} - 11888 \nu^{6} - 13448 \nu^{5} + 52672 \nu^{4} - 94608 \nu^{3} + 109376 \nu^{2} - 99072 \nu + 56320$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$162 \nu^{15} - 591 \nu^{14} + 1304 \nu^{13} - 2280 \nu^{12} + 2572 \nu^{11} - 1360 \nu^{10} - 2183 \nu^{9} + 6772 \nu^{8} - 9594 \nu^{7} + 6392 \nu^{6} + 7444 \nu^{5} - 28824 \nu^{4} + 51608 \nu^{3} - 59456 \nu^{2} + 53872 \nu - 30400$$$$)/32$$ $$\beta_{14}$$ $$=$$ $$($$$$-361 \nu^{15} + 1262 \nu^{14} - 2786 \nu^{13} + 4814 \nu^{12} - 5348 \nu^{11} + 2721 \nu^{10} + 4868 \nu^{9} - 14342 \nu^{8} + 20176 \nu^{7} - 12748 \nu^{6} - 16560 \nu^{5} + 61736 \nu^{4} - 108864 \nu^{3} + 123984 \nu^{2} - 112352 \nu + 60672$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$-785 \nu^{15} + 2815 \nu^{14} - 6215 \nu^{13} + 10819 \nu^{12} - 12133 \nu^{11} + 6324 \nu^{10} + 10574 \nu^{9} - 32160 \nu^{8} + 45468 \nu^{7} - 29696 \nu^{6} - 36008 \nu^{5} + 137504 \nu^{4} - 244752 \nu^{3} + 280832 \nu^{2} - 254464 \nu + 141056$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$-\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{15} + \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} + 4$$ $$\nu^{5}$$ $$=$$ $$\beta_{13} - 2 \beta_{11} + 3 \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{14} - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} + 10$$ $$\nu^{8}$$ $$=$$ $$-5 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} - 8 \beta_{9} + 10 \beta_{8} - 12 \beta_{7} + \beta_{6} - 6 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 9 \beta_{2} - 10 \beta_{1} + 6$$ $$\nu^{9}$$ $$=$$ $$-7 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} - 4 \beta_{8} - 11 \beta_{7} + 17 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} - 5 \beta_{1} + 4$$ $$\nu^{10}$$ $$=$$ $$-2 \beta_{15} + 14 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} - 14 \beta_{8} + 10 \beta_{7} + \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 2 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$26 \beta_{15} + 21 \beta_{14} + 40 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} + 19 \beta_{10} + 3 \beta_{9} - 51 \beta_{8} + 39 \beta_{7} - 31 \beta_{6} + \beta_{5} + 40 \beta_{4} + 20 \beta_{3} + 27 \beta_{2} + 42 \beta_{1} + 28$$ $$\nu^{12}$$ $$=$$ $$20 \beta_{15} - \beta_{14} - \beta_{13} - 30 \beta_{12} + \beta_{11} - 23 \beta_{10} - 21 \beta_{9} - 15 \beta_{8} - 12 \beta_{7} - 16 \beta_{6} + 8 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 9 \beta_{1} + 10$$ $$\nu^{13}$$ $$=$$ $$-21 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 31 \beta_{12} + 24 \beta_{11} - 11 \beta_{10} - 38 \beta_{9} + 48 \beta_{8} - 8 \beta_{7} + 95 \beta_{6} - 2 \beta_{5} - 49 \beta_{4} - 8 \beta_{3} + 11 \beta_{2} + 50$$ $$\nu^{14}$$ $$=$$ $$7 \beta_{15} + 45 \beta_{13} + 19 \beta_{12} - 33 \beta_{11} - 9 \beta_{10} + 52 \beta_{9} + 62 \beta_{8} + 23 \beta_{7} + 85 \beta_{6} - 33 \beta_{5} + 27 \beta_{4} - 9 \beta_{3} + 32 \beta_{2} - 29 \beta_{1} + 32$$ $$\nu^{15}$$ $$=$$ $$20 \beta_{15} + 20 \beta_{14} + 61 \beta_{13} - 56 \beta_{12} + 22 \beta_{11} + 40 \beta_{10} - 88 \beta_{9} - 24 \beta_{8} - 104 \beta_{7} - 73 \beta_{6} - 144 \beta_{5} - 66 \beta_{4} - 21 \beta_{3} - 46 \beta_{2} + 62 \beta_{1} + 68$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/44\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$-\beta_{12}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.40958 − 0.114404i 0.0737040 − 1.41229i −0.544389 + 1.30524i −1.36594 − 0.366325i 1.40958 + 0.114404i 0.0737040 + 1.41229i −0.544389 − 1.30524i −1.36594 + 0.366325i 1.40874 + 0.124276i 1.06665 − 0.928579i 0.656642 + 1.25253i −0.204982 − 1.39928i 1.40874 − 0.124276i 1.06665 + 0.928579i 0.656642 − 1.25253i −0.204982 + 1.39928i
−1.40958 0.114404i 0.704424 + 0.228881i 1.97382 + 0.322523i 1.09089 0.792578i −0.966756 0.403215i 0.503194 + 1.54867i −2.74536 0.680436i −1.98322 1.44090i −1.62837 + 0.992398i
7.2 −0.0737040 1.41229i 1.70537 + 0.554109i −1.98914 + 0.208183i −2.39991 + 1.74363i 0.656871 2.44932i −0.815620 2.51022i 0.440622 + 2.79390i 0.174207 + 0.126569i 2.63940 + 3.26086i
7.3 0.544389 + 1.30524i −0.704424 0.228881i −1.40728 + 1.42111i 1.09089 0.792578i −0.0847364 1.04404i −0.503194 1.54867i −2.62099 1.06320i −1.98322 1.44090i 1.62837 + 0.992398i
7.4 1.36594 0.366325i −1.70537 0.554109i 1.73161 1.00076i −2.39991 + 1.74363i −2.53243 0.132161i 0.815620 + 2.51022i 1.99868 2.00132i 0.174207 + 0.126569i −2.63940 + 3.26086i
19.1 −1.40958 + 0.114404i 0.704424 0.228881i 1.97382 0.322523i 1.09089 + 0.792578i −0.966756 + 0.403215i 0.503194 1.54867i −2.74536 + 0.680436i −1.98322 + 1.44090i −1.62837 0.992398i
19.2 −0.0737040 + 1.41229i 1.70537 0.554109i −1.98914 0.208183i −2.39991 1.74363i 0.656871 + 2.44932i −0.815620 + 2.51022i 0.440622 2.79390i 0.174207 0.126569i 2.63940 3.26086i
19.3 0.544389 1.30524i −0.704424 + 0.228881i −1.40728 1.42111i 1.09089 + 0.792578i −0.0847364 + 1.04404i −0.503194 + 1.54867i −2.62099 + 1.06320i −1.98322 + 1.44090i 1.62837 0.992398i
19.4 1.36594 + 0.366325i −1.70537 + 0.554109i 1.73161 + 1.00076i −2.39991 1.74363i −2.53243 + 0.132161i 0.815620 2.51022i 1.99868 + 2.00132i 0.174207 0.126569i −2.63940 3.26086i
35.1 −1.40874 + 0.124276i −1.59814 2.19965i 1.96911 0.350146i −0.720859 2.21858i 2.52473 + 2.90013i 1.04462 + 0.758960i −2.73046 + 0.737979i −1.35736 + 4.17752i 1.29122 + 3.03582i
35.2 −1.06665 0.928579i 1.59814 + 2.19965i 0.275480 + 1.98094i −0.720859 2.21858i 0.337896 3.83025i −1.04462 0.758960i 1.54562 2.36877i −1.35736 + 4.17752i −1.29122 + 3.03582i
35.3 −0.656642 + 1.25253i 0.539857 + 0.743049i −1.13764 1.64492i 0.529876 + 1.63079i −1.28518 + 0.188268i −1.93399 1.40513i 2.80733 0.344804i 0.666375 2.05089i −2.39055 0.407162i
35.4 0.204982 1.39928i −0.539857 0.743049i −1.91596 0.573655i 0.529876 + 1.63079i −1.15039 + 0.603098i 1.93399 + 1.40513i −1.19544 + 2.56338i 0.666375 2.05089i 2.39055 0.407162i
39.1 −1.40874 0.124276i −1.59814 + 2.19965i 1.96911 + 0.350146i −0.720859 + 2.21858i 2.52473 2.90013i 1.04462 0.758960i −2.73046 0.737979i −1.35736 4.17752i 1.29122 3.03582i
39.2 −1.06665 + 0.928579i 1.59814 2.19965i 0.275480 1.98094i −0.720859 + 2.21858i 0.337896 + 3.83025i −1.04462 + 0.758960i 1.54562 + 2.36877i −1.35736 4.17752i −1.29122 3.03582i
39.3 −0.656642 1.25253i 0.539857 0.743049i −1.13764 + 1.64492i 0.529876 1.63079i −1.28518 0.188268i −1.93399 + 1.40513i 2.80733 + 0.344804i 0.666375 + 2.05089i −2.39055 + 0.407162i
39.4 0.204982 + 1.39928i −0.539857 + 0.743049i −1.91596 + 0.573655i 0.529876 1.63079i −1.15039 0.603098i 1.93399 1.40513i −1.19544 2.56338i 0.666375 + 2.05089i 2.39055 + 0.407162i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 39.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.2.g.a 16
3.b odd 2 1 396.2.r.a 16
4.b odd 2 1 inner 44.2.g.a 16
8.b even 2 1 704.2.u.c 16
8.d odd 2 1 704.2.u.c 16
11.b odd 2 1 484.2.g.i 16
11.c even 5 1 484.2.c.d 16
11.c even 5 1 484.2.g.f 16
11.c even 5 1 484.2.g.i 16
11.c even 5 1 484.2.g.j 16
11.d odd 10 1 inner 44.2.g.a 16
11.d odd 10 1 484.2.c.d 16
11.d odd 10 1 484.2.g.f 16
11.d odd 10 1 484.2.g.j 16
12.b even 2 1 396.2.r.a 16
33.f even 10 1 396.2.r.a 16
44.c even 2 1 484.2.g.i 16
44.g even 10 1 inner 44.2.g.a 16
44.g even 10 1 484.2.c.d 16
44.g even 10 1 484.2.g.f 16
44.g even 10 1 484.2.g.j 16
44.h odd 10 1 484.2.c.d 16
44.h odd 10 1 484.2.g.f 16
44.h odd 10 1 484.2.g.i 16
44.h odd 10 1 484.2.g.j 16
88.k even 10 1 704.2.u.c 16
88.p odd 10 1 704.2.u.c 16
132.n odd 10 1 396.2.r.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.g.a 16 1.a even 1 1 trivial
44.2.g.a 16 4.b odd 2 1 inner
44.2.g.a 16 11.d odd 10 1 inner
44.2.g.a 16 44.g even 10 1 inner
396.2.r.a 16 3.b odd 2 1
396.2.r.a 16 12.b even 2 1
396.2.r.a 16 33.f even 10 1
396.2.r.a 16 132.n odd 10 1
484.2.c.d 16 11.c even 5 1
484.2.c.d 16 11.d odd 10 1
484.2.c.d 16 44.g even 10 1
484.2.c.d 16 44.h odd 10 1
484.2.g.f 16 11.c even 5 1
484.2.g.f 16 11.d odd 10 1
484.2.g.f 16 44.g even 10 1
484.2.g.f 16 44.h odd 10 1
484.2.g.i 16 11.b odd 2 1
484.2.g.i 16 11.c even 5 1
484.2.g.i 16 44.c even 2 1
484.2.g.i 16 44.h odd 10 1
484.2.g.j 16 11.c even 5 1
484.2.g.j 16 11.d odd 10 1
484.2.g.j 16 44.g even 10 1
484.2.g.j 16 44.h odd 10 1
704.2.u.c 16 8.b even 2 1
704.2.u.c 16 8.d odd 2 1
704.2.u.c 16 88.k even 10 1
704.2.u.c 16 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(44, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 640 T + 832 T^{2} + 800 T^{3} + 560 T^{4} + 240 T^{5} - 8 T^{6} - 120 T^{7} - 116 T^{8} - 60 T^{9} - 2 T^{10} + 30 T^{11} + 35 T^{12} + 25 T^{13} + 13 T^{14} + 5 T^{15} + T^{16}$$
$3$ $$121 - 319 T^{2} + 432 T^{4} - 357 T^{6} + 675 T^{8} - 253 T^{10} + 42 T^{12} - T^{14} + T^{16}$$
$5$ $$( 256 - 192 T + 128 T^{2} - 24 T^{3} + 25 T^{4} + 6 T^{5} + 8 T^{6} + 3 T^{7} + T^{8} )^{2}$$
$7$ $$30976 + 11264 T^{2} + 10272 T^{4} + 6572 T^{6} + 2505 T^{8} + 378 T^{10} + 72 T^{12} + 11 T^{14} + T^{16}$$
$11$ $$214358881 - 62004635 T^{2} + 7598679 T^{4} - 649165 T^{6} + 56496 T^{8} - 5365 T^{10} + 519 T^{12} - 35 T^{14} + T^{16}$$
$13$ $$( 16 + 120 T + 296 T^{2} + 220 T^{3} + 61 T^{4} + 20 T^{5} + 16 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$17$ $$( 1 + 15 T + 176 T^{2} - 335 T^{3} + 401 T^{4} - 85 T^{5} - 24 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$19$ $$1771561 - 247566 T^{2} + 614542 T^{4} + 124182 T^{6} + 47600 T^{8} + 15948 T^{10} + 4387 T^{12} + 106 T^{14} + T^{16}$$
$23$ $$( 2816 + 10752 T^{2} + 2064 T^{4} + 88 T^{6} + T^{8} )^{2}$$
$29$ $$( 400 + 1400 T + 2600 T^{2} + 1900 T^{3} + 85 T^{4} - 320 T^{5} + 20 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$31$ $$428888770816 - 18536176384 T^{2} + 2166038752 T^{4} - 166420932 T^{6} + 8183825 T^{8} - 149838 T^{10} + 5812 T^{12} - 111 T^{14} + T^{16}$$
$37$ $$( 13456 + 54984 T + 82132 T^{2} - 25278 T^{3} + 5305 T^{4} - 732 T^{5} + 132 T^{6} - 9 T^{7} + T^{8} )^{2}$$
$41$ $$( 2588881 - 1488325 T + 399696 T^{2} - 48155 T^{3} - 639 T^{4} + 895 T^{5} - 74 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$43$ $$( 148016 - 58912 T^{2} + 5429 T^{4} - 143 T^{6} + T^{8} )^{2}$$
$47$ $$1206517709056 - 122161433856 T^{2} + 7676727392 T^{4} - 380460228 T^{6} + 35709505 T^{8} - 297282 T^{10} + 8412 T^{12} - 129 T^{14} + T^{16}$$
$53$ $$( 15376 - 11656 T + 11112 T^{2} - 4528 T^{3} + 1905 T^{4} - 672 T^{5} + 162 T^{6} - 19 T^{7} + T^{8} )^{2}$$
$59$ $$17080137481 - 50132283454 T^{2} + 56821159822 T^{4} - 2926349122 T^{6} + 64872880 T^{8} - 181268 T^{10} + 40067 T^{12} + 114 T^{14} + T^{16}$$
$61$ $$( 633616 - 429840 T + 247784 T^{2} + 61030 T^{3} + 4781 T^{4} - 880 T^{5} - 126 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$67$ $$( 1760000 + 226000 T^{2} + 10225 T^{4} + 185 T^{6} + T^{8} )^{2}$$
$71$ $$21908736256 - 4629348416 T^{2} + 460206432 T^{4} - 25650768 T^{6} + 1299225 T^{8} - 62942 T^{10} + 2112 T^{12} + T^{14} + T^{16}$$
$73$ $$( 6561 - 10935 T + 2916 T^{2} + 1215 T^{3} + 1701 T^{4} - 135 T^{5} + 36 T^{6} + 15 T^{7} + T^{8} )^{2}$$
$79$ $$28606986496 + 4499694144 T^{2} + 610810832 T^{4} + 79698372 T^{6} + 12605305 T^{8} + 46638 T^{10} + 8112 T^{12} + 141 T^{14} + T^{16}$$
$83$ $$435963075625 + 72795318750 T^{2} + 6940690750 T^{4} + 533610000 T^{6} + 40502900 T^{8} + 1975500 T^{10} + 46705 T^{12} - 100 T^{14} + T^{16}$$
$89$ $$( 116 - 186 T - 19 T^{2} + 9 T^{3} + T^{4} )^{4}$$
$97$ $$( 3721 + 1098 T + 184 T^{2} + 17 T^{3} + T^{4} )^{4}$$