# Properties

 Label 61.2.g.a Level $61$ Weight $2$ Character orbit 61.g Analytic conductor $0.487$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 61.g (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.487087452330$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 17 x^{14} + 111 x^{12} + 361 x^{10} + 624 x^{8} + 558 x^{6} + 229 x^{4} + 34 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 17 x^{14} + 111 x^{12} + 361 x^{10} + 624 x^{8} + 558 x^{6} + 229 x^{4} + 34 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$15 \nu^{15} - 9 \nu^{14} + 226 \nu^{13} - 162 \nu^{12} + 1231 \nu^{11} - 1117 \nu^{10} + 3082 \nu^{9} - 3750 \nu^{8} + 3686 \nu^{7} - 6374 \nu^{6} + 2112 \nu^{5} - 5104 \nu^{4} + 839 \nu^{3} - 1533 \nu^{2} + 293 \nu - 123$$$$)/88$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{15} - 3 \nu^{14} + 162 \nu^{13} - 54 \nu^{12} + 1117 \nu^{11} - 387 \nu^{10} + 3750 \nu^{9} - 1426 \nu^{8} + 6374 \nu^{7} - 2770 \nu^{6} + 5104 \nu^{5} - 2508 \nu^{4} + 1533 \nu^{3} - 731 \nu^{2} + 79 \nu - 41$$$$)/88$$ $$\beta_{3}$$ $$=$$ $$($$$$9 \nu^{15} - 29 \nu^{14} + 162 \nu^{13} - 434 \nu^{12} + 1117 \nu^{11} - 2333 \nu^{10} + 3750 \nu^{9} - 5674 \nu^{8} + 6374 \nu^{7} - 6258 \nu^{6} + 5104 \nu^{5} - 2596 \nu^{4} + 1533 \nu^{3} - 217 \nu^{2} + 123 \nu - 15$$$$)/88$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{15} - 3 \nu^{14} - 162 \nu^{13} - 54 \nu^{12} - 1117 \nu^{11} - 387 \nu^{10} - 3750 \nu^{9} - 1426 \nu^{8} - 6374 \nu^{7} - 2770 \nu^{6} - 5104 \nu^{5} - 2508 \nu^{4} - 1533 \nu^{3} - 731 \nu^{2} - 79 \nu - 41$$$$)/88$$ $$\beta_{5}$$ $$=$$ $$($$$$9 \nu^{15} + 29 \nu^{14} + 162 \nu^{13} + 434 \nu^{12} + 1117 \nu^{11} + 2333 \nu^{10} + 3750 \nu^{9} + 5674 \nu^{8} + 6374 \nu^{7} + 6258 \nu^{6} + 5104 \nu^{5} + 2596 \nu^{4} + 1533 \nu^{3} + 217 \nu^{2} + 123 \nu + 15$$$$)/88$$ $$\beta_{6}$$ $$=$$ $$($$$$-41 \nu^{15} - 9 \nu^{14} - 650 \nu^{13} - 118 \nu^{12} - 3837 \nu^{11} - 501 \nu^{10} - 10850 \nu^{9} - 758 \nu^{8} - 15402 \nu^{7} - 82 \nu^{6} - 10164 \nu^{5} + 484 \nu^{4} - 2217 \nu^{3} - 37 \nu^{2} + 129 \nu - 123$$$$)/88$$ $$\beta_{7}$$ $$=$$ $$($$$$-18 \nu^{15} + 21 \nu^{14} - 302 \nu^{13} + 334 \nu^{12} - 1926 \nu^{11} + 1983 \nu^{10} - 6004 \nu^{9} + 5670 \nu^{8} - 9602 \nu^{7} + 8258 \nu^{6} - 7392 \nu^{5} + 5896 \nu^{4} - 2186 \nu^{3} + 1817 \nu^{2} - 114 \nu + 133$$$$)/44$$ $$\beta_{8}$$ $$=$$ $$($$$$-25 \nu^{15} + 5 \nu^{14} - 406 \nu^{13} + 68 \nu^{12} - 2477 \nu^{11} + 293 \nu^{10} - 7300 \nu^{9} + 338 \nu^{8} - 10888 \nu^{7} - 634 \nu^{6} - 7634 \nu^{5} - 1606 \nu^{4} - 1897 \nu^{3} - 835 \nu^{2} - 85 \nu - 49$$$$)/44$$ $$\beta_{9}$$ $$=$$ $$($$$$25 \nu^{15} - 13 \nu^{14} + 406 \nu^{13} - 212 \nu^{12} + 2477 \nu^{11} - 1303 \nu^{10} + 7300 \nu^{9} - 3884 \nu^{8} + 10888 \nu^{7} - 5858 \nu^{6} + 7634 \nu^{5} - 4048 \nu^{4} + 1875 \nu^{3} - 865 \nu^{2} - 3 \nu - 31$$$$)/44$$ $$\beta_{10}$$ $$=$$ $$($$$$-41 \nu^{15} + 9 \nu^{14} - 694 \nu^{13} + 162 \nu^{12} - 4497 \nu^{11} + 1117 \nu^{10} - 14414 \nu^{9} + 3750 \nu^{8} - 24158 \nu^{7} + 6374 \nu^{6} - 20108 \nu^{5} + 5104 \nu^{4} - 6881 \nu^{3} + 1533 \nu^{2} - 663 \nu + 79$$$$)/88$$ $$\beta_{11}$$ $$=$$ $$($$$$25 \nu^{15} + 5 \nu^{14} + 406 \nu^{13} + 68 \nu^{12} + 2477 \nu^{11} + 293 \nu^{10} + 7300 \nu^{9} + 338 \nu^{8} + 10888 \nu^{7} - 634 \nu^{6} + 7634 \nu^{5} - 1606 \nu^{4} + 1897 \nu^{3} - 835 \nu^{2} + 85 \nu - 49$$$$)/44$$ $$\beta_{12}$$ $$=$$ $$($$$$43 \nu^{15} - 15 \nu^{14} + 730 \nu^{13} - 226 \nu^{12} + 4755 \nu^{11} - 1231 \nu^{10} + 15394 \nu^{9} - 3082 \nu^{8} + 26342 \nu^{7} - 3686 \nu^{6} + 22924 \nu^{5} - 2112 \nu^{4} + 8659 \nu^{3} - 795 \nu^{2} + 1057 \nu - 117$$$$)/88$$ $$\beta_{13}$$ $$=$$ $$($$$$41 \nu^{15} + 9 \nu^{14} + 694 \nu^{13} + 162 \nu^{12} + 4497 \nu^{11} + 1117 \nu^{10} + 14414 \nu^{9} + 3750 \nu^{8} + 24158 \nu^{7} + 6374 \nu^{6} + 20108 \nu^{5} + 5104 \nu^{4} + 6881 \nu^{3} + 1533 \nu^{2} + 663 \nu + 79$$$$)/88$$ $$\beta_{14}$$ $$=$$ $$($$$$-43 \nu^{15} - 15 \nu^{14} - 730 \nu^{13} - 226 \nu^{12} - 4755 \nu^{11} - 1231 \nu^{10} - 15394 \nu^{9} - 3082 \nu^{8} - 26342 \nu^{7} - 3686 \nu^{6} - 22924 \nu^{5} - 2112 \nu^{4} - 8659 \nu^{3} - 795 \nu^{2} - 1057 \nu - 117$$$$)/88$$ $$\beta_{15}$$ $$=$$ $$($$$$-18 \nu^{15} - 21 \nu^{14} - 302 \nu^{13} - 334 \nu^{12} - 1926 \nu^{11} - 1983 \nu^{10} - 6004 \nu^{9} - 5670 \nu^{8} - 9602 \nu^{7} - 8258 \nu^{6} - 7392 \nu^{5} - 5896 \nu^{4} - 2186 \nu^{3} - 1817 \nu^{2} - 158 \nu - 133$$$$)/44$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} - 5 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$\beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} + 5 \beta_{9} - \beta_{7} + 5 \beta_{6} + \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 7 \beta_{2} + 3$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{15} + 7 \beta_{14} + 9 \beta_{13} + 5 \beta_{12} - 6 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 26 \beta_{5} + 18 \beta_{4} + 20 \beta_{3} - 18 \beta_{2} + 2 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{15} + 17 \beta_{14} - 10 \beta_{13} + 17 \beta_{12} + \beta_{11} - 10 \beta_{10} - 27 \beta_{9} + \beta_{8} + 10 \beta_{7} - 27 \beta_{6} - 12 \beta_{5} + 26 \beta_{4} + 19 \beta_{3} + 46 \beta_{2} - 13$$ $$\nu^{7}$$ $$=$$ $$-22 \beta_{15} - 45 \beta_{14} - 63 \beta_{13} - 27 \beta_{12} + 34 \beta_{11} - 25 \beta_{10} - 36 \beta_{9} - 34 \beta_{8} - 22 \beta_{7} + 36 \beta_{6} - 149 \beta_{5} - 93 \beta_{4} - 113 \beta_{3} + 93 \beta_{2} - 16 \beta_{1} - 8$$ $$\nu^{8}$$ $$=$$ $$76 \beta_{15} - 119 \beta_{14} + 76 \beta_{13} - 119 \beta_{12} - 13 \beta_{11} + 76 \beta_{10} + 158 \beta_{9} - 13 \beta_{8} - 76 \beta_{7} + 158 \beta_{6} + 99 \beta_{5} - 144 \beta_{4} - 105 \beta_{3} - 296 \beta_{2} + 70$$ $$\nu^{9}$$ $$=$$ $$174 \beta_{15} + 286 \beta_{14} + 414 \beta_{13} + 162 \beta_{12} - 201 \beta_{11} + 140 \beta_{10} + 224 \beta_{9} + 201 \beta_{8} + 174 \beta_{7} - 224 \beta_{6} + 906 \beta_{5} + 530 \beta_{4} + 682 \beta_{3} - 530 \beta_{2} + 106 \beta_{1} + 53$$ $$\nu^{10}$$ $$=$$ $$-527 \beta_{15} + 790 \beta_{14} - 528 \beta_{13} + 790 \beta_{12} + 112 \beta_{11} - 528 \beta_{10} - 968 \beta_{9} + 112 \beta_{8} + 527 \beta_{7} - 968 \beta_{6} - 715 \beta_{5} + 841 \beta_{4} + 629 \beta_{3} + 1895 \beta_{2} - 419$$ $$\nu^{11}$$ $$=$$ $$-1227 \beta_{15} - 1819 \beta_{14} - 2670 \beta_{13} - 1019 \beta_{12} + 1231 \beta_{11} - 846 \beta_{10} - 1419 \beta_{9} - 1231 \beta_{8} - 1227 \beta_{7} + 1419 \beta_{6} - 5669 \beta_{5} - 3195 \beta_{4} - 4250 \beta_{3} + 3195 \beta_{2} - 678 \beta_{1} - 339$$ $$\nu^{12}$$ $$=$$ $$3513 \beta_{15} - 5144 \beta_{14} + 3529 \beta_{13} - 5144 \beta_{12} - 827 \beta_{11} + 3529 \beta_{10} + 6069 \beta_{9} - 827 \beta_{8} - 3513 \beta_{7} + 6069 \beta_{6} + 4867 \beta_{5} - 5099 \beta_{4} - 3910 \beta_{3} - 12125 \beta_{2} + 2619$$ $$\nu^{13}$$ $$=$$ $$8237 \beta_{15} + 11597 \beta_{14} + 17125 \beta_{13} + 6511 \beta_{12} - 7700 \beta_{11} + 5299 \beta_{10} + 9054 \beta_{9} + 7700 \beta_{8} + 8237 \beta_{7} - 9054 \beta_{6} + 35936 \beta_{5} + 19840 \beta_{4} + 26882 \beta_{3} - 19840 \beta_{2} + 4316 \beta_{1} + 2158$$ $$\nu^{14}$$ $$=$$ $$-22979 \beta_{15} + 33222 \beta_{14} - 23138 \beta_{13} + 33222 \beta_{12} + 5694 \beta_{11} - 23138 \beta_{10} - 38479 \beta_{9} + 5694 \beta_{8} + 22979 \beta_{7} - 38479 \beta_{6} - 32185 \beta_{5} + 31656 \beta_{4} + 24706 \beta_{3} + 77614 \beta_{2} - 16639$$ $$\nu^{15}$$ $$=$$ $$-54035 \beta_{15} - 74084 \beta_{14} - 109689 \beta_{13} - 41772 \beta_{12} + 48722 \beta_{11} - 33677 \beta_{10} - 57928 \beta_{9} - 48722 \beta_{8} - 54035 \beta_{7} + 57928 \beta_{6} - 229139 \beta_{5} - 125094 \beta_{4} - 171211 \beta_{3} + 125094 \beta_{2} - 27510 \beta_{1} - 13755$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/61\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 2.53165i − 0.776536i − 0.196205i 1.60228i 1.85647i 1.25523i − 0.475317i − 1.46081i 2.53165i 0.776536i 0.196205i − 1.60228i − 1.85647i − 1.25523i 0.475317i 1.46081i
−2.40774 0.782322i −0.277331 + 0.853536i 3.56715 + 2.59169i 0.429180 + 0.311818i 1.33548 1.83813i 3.57843 1.16270i −3.58511 4.93448i 1.77544 + 1.28993i −0.789413 1.08653i
3.2 −0.738530 0.239963i 0.554995 1.70810i −1.13019 0.821131i −0.515969 0.374873i −0.819760 + 1.12830i 1.31473 0.427183i 1.55051 + 2.13410i −0.182527 0.132614i 0.291103 + 0.400668i
3.3 −0.186602 0.0606307i −0.865057 + 2.66237i −1.58689 1.15294i 2.93210 + 2.13030i 0.322843 0.444355i −0.222647 + 0.0723423i 0.456866 + 0.628822i −3.91286 2.84286i −0.417975 0.575293i
3.4 1.52385 + 0.495130i −0.221623 + 0.682087i 0.458946 + 0.333444i −1.72728 1.25494i −0.675444 + 0.929669i −1.05248 + 0.341973i −1.34932 1.85718i 2.01093 + 1.46102i −2.01077 2.76758i
27.1 −1.09120 1.50191i 2.38911 1.73579i −0.446986 + 1.37568i −1.04459 + 3.21492i −5.21402 1.69414i −0.380995 + 0.524395i −0.977305 + 0.317546i 1.76783 5.44082i 5.96841 1.93925i
27.2 −0.737805 1.01550i −0.446713 + 0.324556i 0.131147 0.403630i 0.620635 1.91012i 0.659173 + 0.214178i 1.40505 1.93389i −2.89423 + 0.940394i −0.832835 + 2.56320i −2.39763 + 0.779038i
27.3 0.279384 + 0.384540i 0.261794 0.190204i 0.548219 1.68724i 0.00765597 0.0235626i 0.146282 + 0.0475300i −2.52527 + 3.47573i 1.70608 0.554340i −0.894693 + 2.75358i 0.0111997 0.00363901i
27.4 0.858642 + 1.18182i −1.89518 + 1.37693i −0.0413973 + 0.127408i −0.701732 + 2.15971i −3.25455 1.05747i 2.88318 3.96835i 2.59251 0.842356i 0.768714 2.36586i −3.15492 + 1.02510i
41.1 −2.40774 + 0.782322i −0.277331 0.853536i 3.56715 2.59169i 0.429180 0.311818i 1.33548 + 1.83813i 3.57843 + 1.16270i −3.58511 + 4.93448i 1.77544 1.28993i −0.789413 + 1.08653i
41.2 −0.738530 + 0.239963i 0.554995 + 1.70810i −1.13019 + 0.821131i −0.515969 + 0.374873i −0.819760 1.12830i 1.31473 + 0.427183i 1.55051 2.13410i −0.182527 + 0.132614i 0.291103 0.400668i
41.3 −0.186602 + 0.0606307i −0.865057 2.66237i −1.58689 + 1.15294i 2.93210 2.13030i 0.322843 + 0.444355i −0.222647 0.0723423i 0.456866 0.628822i −3.91286 + 2.84286i −0.417975 + 0.575293i
41.4 1.52385 0.495130i −0.221623 0.682087i 0.458946 0.333444i −1.72728 + 1.25494i −0.675444 0.929669i −1.05248 0.341973i −1.34932 + 1.85718i 2.01093 1.46102i −2.01077 + 2.76758i
52.1 −1.09120 + 1.50191i 2.38911 + 1.73579i −0.446986 1.37568i −1.04459 3.21492i −5.21402 + 1.69414i −0.380995 0.524395i −0.977305 0.317546i 1.76783 + 5.44082i 5.96841 + 1.93925i
52.2 −0.737805 + 1.01550i −0.446713 0.324556i 0.131147 + 0.403630i 0.620635 + 1.91012i 0.659173 0.214178i 1.40505 + 1.93389i −2.89423 0.940394i −0.832835 2.56320i −2.39763 0.779038i
52.3 0.279384 0.384540i 0.261794 + 0.190204i 0.548219 + 1.68724i 0.00765597 + 0.0235626i 0.146282 0.0475300i −2.52527 3.47573i 1.70608 + 0.554340i −0.894693 2.75358i 0.0111997 + 0.00363901i
52.4 0.858642 1.18182i −1.89518 1.37693i −0.0413973 0.127408i −0.701732 2.15971i −3.25455 + 1.05747i 2.88318 + 3.96835i 2.59251 + 0.842356i 0.768714 + 2.36586i −3.15492 1.02510i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 52.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.g even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.2.g.a 16
3.b odd 2 1 549.2.y.b 16
4.b odd 2 1 976.2.bd.b 16
61.g even 10 1 inner 61.2.g.a 16
61.j odd 20 2 3721.2.a.k 16
183.l odd 10 1 549.2.y.b 16
244.m odd 10 1 976.2.bd.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.g.a 16 1.a even 1 1 trivial
61.2.g.a 16 61.g even 10 1 inner
549.2.y.b 16 3.b odd 2 1
549.2.y.b 16 183.l odd 10 1
976.2.bd.b 16 4.b odd 2 1
976.2.bd.b 16 244.m odd 10 1
3721.2.a.k 16 61.j odd 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(61, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 10 T + 29 T^{2} + 15 T^{3} + 74 T^{4} + 265 T^{5} + 278 T^{6} + 55 T^{7} - 51 T^{8} - 10 T^{9} + 46 T^{10} - 19 T^{12} - 5 T^{13} + 7 T^{14} + 5 T^{15} + T^{16}$$
$3$ $$16 - 8 T - 16 T^{2} + 52 T^{3} + 633 T^{4} + 1352 T^{5} + 2466 T^{6} + 2091 T^{7} + 1948 T^{8} + 652 T^{9} + 504 T^{10} + 131 T^{11} + 43 T^{12} - T^{13} + 6 T^{14} + T^{15} + T^{16}$$
$5$ $$1 - 25 T + 1629 T^{2} - 50 T^{3} - 2483 T^{4} + 860 T^{5} + 14037 T^{6} + 7020 T^{7} + 6385 T^{8} + 2440 T^{9} + 1363 T^{10} + 255 T^{11} + 112 T^{12} - 15 T^{13} + 11 T^{14} + T^{16}$$
$7$ $$1936 + 18040 T + 54848 T^{2} + 43920 T^{3} - 22619 T^{4} - 84060 T^{5} + 9194 T^{6} + 57050 T^{7} - 14751 T^{8} - 13920 T^{9} + 12657 T^{10} - 3945 T^{11} + 704 T^{12} - 165 T^{13} + 51 T^{14} - 10 T^{15} + T^{16}$$
$11$ $$1936 + 78724 T^{2} + 477629 T^{4} + 661998 T^{6} + 255109 T^{8} + 39221 T^{10} + 2756 T^{12} + 87 T^{14} + T^{16}$$
$13$ $$( 10261 - 9881 T - 2638 T^{2} + 2772 T^{3} + 430 T^{4} - 242 T^{5} - 38 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$17$ $$4879681 + 32185130 T + 46720350 T^{2} - 37873305 T^{3} + 76201662 T^{4} + 60532010 T^{5} - 3305715 T^{6} - 2114915 T^{7} + 3004179 T^{8} + 93460 T^{9} - 105200 T^{10} + 5250 T^{11} + 2208 T^{12} - 210 T^{13} - 25 T^{14} + T^{16}$$
$19$ $$7311616 + 23762752 T + 42179696 T^{2} + 38471108 T^{3} + 35081121 T^{4} + 4969806 T^{5} + 14007439 T^{6} - 966106 T^{7} + 1051517 T^{8} - 261136 T^{9} + 70329 T^{10} + 13786 T^{11} + 2696 T^{12} - 327 T^{13} + 36 T^{14} - 3 T^{15} + T^{16}$$
$23$ $$30976 + 239360 T + 893760 T^{2} + 2090900 T^{3} + 3359253 T^{4} + 3868080 T^{5} + 3267730 T^{6} + 2054040 T^{7} + 968974 T^{8} + 344880 T^{9} + 93360 T^{10} + 19830 T^{11} + 3637 T^{12} + 660 T^{13} + 115 T^{14} + 15 T^{15} + T^{16}$$
$29$ $$383161 + 22311765 T^{2} + 45705437 T^{4} + 22386825 T^{6} + 3899509 T^{8} + 304345 T^{10} + 11273 T^{12} + 185 T^{14} + T^{16}$$
$31$ $$1008016 + 11274920 T + 41931520 T^{2} + 49968300 T^{3} + 27726233 T^{4} + 9349335 T^{5} + 3667805 T^{6} + 1846210 T^{7} + 666689 T^{8} + 98620 T^{9} - 18875 T^{10} - 10545 T^{11} - 1258 T^{12} + 240 T^{13} + 105 T^{14} + 15 T^{15} + T^{16}$$
$37$ $$46787420416 + 33925119360 T - 4662352160 T^{2} + 21744165380 T^{3} + 29232507733 T^{4} + 3607503380 T^{5} - 364807040 T^{6} + 164222510 T^{7} + 54558584 T^{8} - 560520 T^{9} - 720670 T^{10} - 31290 T^{11} + 11927 T^{12} + 70 T^{13} - 115 T^{14} + 5 T^{15} + T^{16}$$
$41$ $$1437601 - 10831766 T + 27151796 T^{2} + 47844491 T^{3} + 29631578 T^{4} - 28184306 T^{5} + 29581129 T^{6} - 14094927 T^{7} + 5755703 T^{8} - 1709906 T^{9} + 444796 T^{10} - 91142 T^{11} + 16218 T^{12} - 1812 T^{13} + 159 T^{14} - 12 T^{15} + T^{16}$$
$43$ $$591267856 - 4995965360 T + 35723310576 T^{2} + 91683240430 T^{3} + 86510714899 T^{4} + 42328629475 T^{5} + 11969319757 T^{6} + 2077327020 T^{7} + 254995504 T^{8} + 31929235 T^{9} + 4588559 T^{10} + 525840 T^{11} + 42551 T^{12} + 3180 T^{13} + 303 T^{14} + 25 T^{15} + T^{16}$$
$47$ $$( -3524 - 49326 T + 68369 T^{2} - 27367 T^{3} + 2409 T^{4} + 716 T^{5} - 124 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$53$ $$162537001 - 1014820400 T + 2087323354 T^{2} - 1309557500 T^{3} + 337276979 T^{4} - 113980200 T^{5} + 99124723 T^{6} - 42058290 T^{7} + 3961189 T^{8} + 2488705 T^{9} + 1245631 T^{10} + 64765 T^{11} - 9469 T^{12} - 2425 T^{13} - 13 T^{14} + 20 T^{15} + T^{16}$$
$59$ $$138529862416 - 66202502520 T + 29476272760 T^{2} - 75581348920 T^{3} + 70357909053 T^{4} - 31973478985 T^{5} + 8574249190 T^{6} - 1421450690 T^{7} + 142003914 T^{8} - 9220145 T^{9} + 1027865 T^{10} - 173485 T^{11} + 9607 T^{12} + 1515 T^{13} - 180 T^{14} - 5 T^{15} + T^{16}$$
$61$ $$191707312997281 + 166565370309113 T + 65121753192304 T^{2} + 14949354527700 T^{3} + 2266813396838 T^{4} + 272157028430 T^{5} + 37745827721 T^{6} + 6488253227 T^{7} + 963246013 T^{8} + 106364807 T^{9} + 10144001 T^{10} + 1199030 T^{11} + 163718 T^{12} + 17700 T^{13} + 1264 T^{14} + 53 T^{15} + T^{16}$$
$67$ $$395837272336 + 899246379240 T + 1000050118920 T^{2} + 713102272280 T^{3} + 360845437077 T^{4} + 136043784405 T^{5} + 39178639680 T^{6} + 8725004425 T^{7} + 1513828394 T^{8} + 207458760 T^{9} + 23395915 T^{10} + 2369540 T^{11} + 232053 T^{12} + 20475 T^{13} + 1355 T^{14} + 55 T^{15} + T^{16}$$
$71$ $$121101216016 - 70872865360 T - 5005860392 T^{2} - 7842296090 T^{3} + 12074523441 T^{4} - 1269000190 T^{5} - 558306726 T^{6} - 526455 T^{7} + 22086204 T^{8} + 1005760 T^{9} - 293448 T^{10} + 122800 T^{11} + 61464 T^{12} + 10900 T^{13} + 1026 T^{14} + 50 T^{15} + T^{16}$$
$73$ $$687101735056 + 406677794424 T + 238029559164 T^{2} + 81283937854 T^{3} + 26600133711 T^{4} + 6784966787 T^{5} + 1656349196 T^{6} + 241017122 T^{7} + 45173152 T^{8} + 5912003 T^{9} + 672131 T^{10} + 18513 T^{11} + 14331 T^{12} + 491 T^{13} + 74 T^{14} + 11 T^{15} + T^{16}$$
$79$ $$2515192996096 - 3152460143360 T + 663609099456 T^{2} + 103531566980 T^{3} + 318703374109 T^{4} - 66003267600 T^{5} - 207839798 T^{6} + 491020160 T^{7} + 95303404 T^{8} - 16499840 T^{9} - 1194136 T^{10} + 377760 T^{11} - 6454 T^{12} - 5195 T^{13} + 688 T^{14} - 40 T^{15} + T^{16}$$
$83$ $$5678526736 - 16357074784 T + 23659165664 T^{2} - 21652351954 T^{3} + 14127820391 T^{4} - 6792072547 T^{5} + 2469560586 T^{6} - 691901457 T^{7} + 157205207 T^{8} - 29607513 T^{9} + 4786831 T^{10} - 648143 T^{11} + 75326 T^{12} - 6096 T^{13} + 524 T^{14} - 31 T^{15} + T^{16}$$
$89$ $$10571397384924601 - 1360843528250550 T + 49109649674910 T^{2} - 141132542698590 T^{3} + 38426648249173 T^{4} - 3264182062050 T^{5} + 122099493905 T^{6} - 29624245130 T^{7} + 5298727629 T^{8} - 234436195 T^{9} - 27789715 T^{10} + 3656505 T^{11} - 63203 T^{12} - 16245 T^{13} + 1515 T^{14} - 60 T^{15} + T^{16}$$
$97$ $$394856641 - 3991984545 T + 17322855371 T^{2} - 28230829385 T^{3} + 32766093282 T^{4} - 24138326165 T^{5} + 11873488188 T^{6} - 4106238865 T^{7} + 1048592185 T^{8} - 203926145 T^{9} + 30813742 T^{10} - 3624925 T^{11} + 332912 T^{12} - 23500 T^{13} + 1244 T^{14} - 45 T^{15} + T^{16}$$