# Properties

 Label 60.2.e.a Level $60$ Weight $2$ Character orbit 60.e Analytic conductor $0.479$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 60.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.479102412128$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16$$ x^8 + x^6 + 4*x^4 + 4*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{4} + \beta_1 q^{5} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{8} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b7 - b4) * q^3 + (b7 - b5 + b3) * q^4 + b1 * q^5 + (b6 + b5 - b2 - b1 - 1) * q^6 + (b5 - b3) * q^7 + (-b5 - b3 + b2 - 2*b1) * q^8 + (b7 - b6 + b3 + b2 + b1) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{4} + \beta_1 q^{5} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{8} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{9} + \beta_{7} q^{10} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{11} + (\beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1) q^{12} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{13} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{14} + (\beta_{3} - \beta_{2}) q^{15} + ( - 3 \beta_{7} - \beta_{6} - \beta_{4} - 2) q^{16} - 2 \beta_1 q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{18} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3}) q^{19} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{20} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{21} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4}) q^{22} + (\beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 4 \beta_{2}) q^{23} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{24} - q^{25} + (2 \beta_{6} - 2 \beta_{4} + 4 \beta_1) q^{26} + (\beta_{7} - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{27} + (2 \beta_{7} + 4) q^{28} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{29} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_1 + 1) q^{30} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{31} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{32} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{33} - 2 \beta_{7} q^{34} + (\beta_{6} - \beta_{4}) q^{35} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{36} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{37} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{38} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{39} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 2) q^{40} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{42} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{43} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{44} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{45} + (4 \beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 2) q^{46} + ( - \beta_{6} + \beta_{4}) q^{47} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{48} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{49} + \beta_{2} q^{50} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{51} + (4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4) q^{52} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{53} + ( - 2 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{54} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3}) q^{55} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2}) q^{56} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{57} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 4) q^{58} + (3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 6 \beta_{2}) q^{59} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 2) q^{60} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{62} + (2 \beta_{7} + 3 \beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{63} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2) q^{64} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{65} + (4 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 4) q^{66} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{67} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{68} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 1) q^{69} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{70} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{71} + (4 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 4) q^{72} + (4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2) q^{73} + ( - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{74} + (\beta_{7} + \beta_{4}) q^{75} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 4) q^{76} + 4 \beta_1 q^{77} + (2 \beta_{7} - 2 \beta_{6} + 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 6) q^{78} + ( - 6 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{79} + (\beta_{5} + \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{80} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 8 \beta_1 - 1) q^{81} + ( - 4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + 4) q^{82} + ( - 5 \beta_{6} + 5 \beta_{4}) q^{83} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_1 + 2) q^{84} + 2 q^{85} + (3 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{86} + (2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3} - 4 \beta_{2}) q^{87} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 4) q^{88} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{89} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{90} + ( - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3}) q^{91} + ( - 4 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{92} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{93} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{94} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{95} + (\beta_{7} - 3 \beta_{6} + \beta_{4} + 6 \beta_{2} + 6) q^{96} - 6 q^{97} + (2 \beta_{6} - 2 \beta_{4} - 3 \beta_{2} + 4 \beta_1) q^{98} + (4 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 3 \beta_{3} - 6 \beta_{2}) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b7 - b4) * q^3 + (b7 - b5 + b3) * q^4 + b1 * q^5 + (b6 + b5 - b2 - b1 - 1) * q^6 + (b5 - b3) * q^7 + (-b5 - b3 + b2 - 2*b1) * q^8 + (b7 - b6 + b3 + b2 + b1) * q^9 + b7 * q^10 + (-b6 - b5 + b4 - b3 + 2*b2) * q^11 + (b6 + b4 + b2 + 2*b1) * q^12 + (-2*b7 + b6 + b5 + b4 - b3 - 2) * q^13 + (-b6 + b5 + b4 + b3 + 2*b1) * q^14 + (b3 - b2) * q^15 + (-3*b7 - b6 - b4 - 2) * q^16 - 2*b1 * q^17 + (-b6 + b5 + b4 - b3 + b2 - 2*b1 + 2) * q^18 + (2*b7 + b6 - b5 + b4 + b3) * q^19 + (-b6 + b4 + b2) * q^20 + (b7 - b5 - b4 - b2 - 3*b1 + 1) * q^21 + (-2*b7 - 2*b6 - 2*b4) * q^22 + (b6 + 2*b5 - b4 + 2*b3 - 4*b2) * q^23 + (b7 + b6 - b4 - 2*b3 + 2*b1) * q^24 - q^25 + (2*b6 - 2*b4 + 4*b1) * q^26 + (b7 - 3*b5 + b4 + b3 + 2*b2) * q^27 + (2*b7 + 4) * q^28 + (b6 - b5 - b4 - b3 - 2*b2) * q^29 + (b7 + b6 - b5 - b1 + 1) * q^30 + (2*b7 + b6 + b5 + b4 - b3) * q^31 + (2*b6 + b5 - 2*b4 + b3 - b2 - 2*b1) * q^32 + (b6 - b5 - b4 - b3 - 2*b2 + 2*b1 - 2) * q^33 - 2*b7 * q^34 + (b6 - b4) * q^35 + (-3*b7 - 2*b6 + b5 + b3 - 2*b2 + 2*b1 + 2) * q^36 + (2*b7 - b6 - b5 - b4 + b3 + 2) * q^37 + (-2*b5 - 2*b3 + 2*b2) * q^38 + (-2*b7 - 3*b6 + 3*b5 + b4 - b3 - 2*b2) * q^39 + (-b7 - b6 - b4 + 2) * q^40 + (-b6 + b5 + b4 + b3 + 2*b2 - 2*b1) * q^41 + (-2*b7 - b6 - b5 + b4 + b3 - 2*b1 - 2) * q^42 + (-2*b7 - b6 - 2*b5 - b4 + 2*b3) * q^43 + (2*b5 + 2*b3 - 2*b2 - 4*b1) * q^44 + (-b7 + b5 + b4 + b2 - 1) * q^45 + (4*b7 + 3*b6 - b5 + 3*b4 + b3 + 2) * q^46 + (-b6 + b4) * q^47 + (2*b7 - 2*b6 + b5 + 2*b4 + b3 + b2 + 2*b1 - 4) * q^48 + (-2*b7 + b6 + b5 + b4 - b3 + 1) * q^49 + b2 * q^50 + (-2*b3 + 2*b2) * q^51 + (4*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 - 4) * q^52 + (-2*b6 + 2*b5 + 2*b4 + 2*b3 + 4*b2 + 6*b1) * q^53 + (-2*b7 - 2*b5 - 3*b4 - 3*b3 + b2 - 3*b1 - 1) * q^54 + (-2*b7 - b6 + b5 - b4 - b3) * q^55 + (-2*b6 + 2*b4 - 2*b2) * q^56 + (-2*b7 + b6 + b5 + b4 - b3 + 2*b1 + 2) * q^57 + (2*b7 - 2*b5 + 2*b3 - 4) * q^58 + (3*b6 - 3*b5 - 3*b4 - 3*b3 + 6*b2) * q^59 + (-b7 - b5 - b3 - 2) * q^60 + (2*b7 - b6 - b5 - b4 + b3) * q^61 + (-2*b6 + 2*b4 + 2*b2 + 4*b1) * q^62 + (2*b7 + 3*b6 - b4 - 2*b3 + 2*b2) * q^63 + (-b7 + 3*b6 + 2*b5 + 3*b4 - 2*b3 - 2) * q^64 + (b6 - b5 - b4 - b3 - 2*b2 - 2*b1) * q^65 + (4*b7 - 2*b5 + 2*b3 + 2*b2 - 4) * q^66 + (-2*b7 - b6 + 2*b5 - b4 - 2*b3) * q^67 + (2*b6 - 2*b4 - 2*b2) * q^68 + (-b7 - b6 + 2*b5 + 2*b4 + b3 + 3*b2 - 5*b1 + 1) * q^69 + (b6 + b5 + b4 - b3 - 2) * q^70 + (2*b5 + 2*b3 - 4*b2) * q^71 + (4*b7 + 2*b6 - b5 - 2*b4 + 3*b3 - 5*b2 - 2*b1 + 4) * q^72 + (4*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3 + 2) * q^73 + (-2*b6 + 2*b4 - 4*b1) * q^74 + (b7 + b4) * q^75 + (-2*b7 - 2*b6 - 2*b4 - 4) * q^76 + 4*b1 * q^77 + (2*b7 - 2*b6 + 6*b3 - 2*b2 + 2*b1 + 6) * q^78 + (-6*b7 - 3*b6 - b5 - 3*b4 + b3) * q^79 + (b5 + b3 - 3*b2 - 2*b1) * q^80 + (-2*b7 + b6 + b5 + b4 - b3 + 8*b1 - 1) * q^81 + (-4*b7 + 2*b5 - 2*b3 + 4) * q^82 + (-5*b6 + 5*b4) * q^83 + (-2*b7 + 2*b6 - 2*b5 - 4*b4 - 2*b1 + 2) * q^84 + 2 * q^85 + (3*b6 - b5 - 3*b4 - b3 - 2*b2 - 6*b1) * q^86 + (2*b7 + 3*b6 + 3*b5 - b4 + b3 - 4*b2) * q^87 + (-2*b7 + 2*b6 + 2*b4 + 4) * q^88 + (2*b6 - 2*b5 - 2*b4 - 2*b3 - 4*b2 - 4*b1) * q^89 + (-b7 + b6 + b5 - b4 - b3 + 2*b1 + 2) * q^90 + (-4*b7 - 2*b6 - 4*b5 - 2*b4 + 4*b3) * q^91 + (-4*b5 - 4*b3 + 2*b2 + 4*b1) * q^92 + (b6 - b5 - b4 - b3 - 2*b2 - 4*b1 + 4) * q^93 + (-b6 - b5 - b4 + b3 + 2) * q^94 + (-b6 - b5 + b4 - b3 + 2*b2) * q^95 + (b7 - 3*b6 + b4 + 6*b2 + 6) * q^96 - 6 * q^97 + (2*b6 - 2*b4 - 3*b2 + 4*b1) * q^98 + (4*b7 + 3*b6 + 3*b5 + b4 + 3*b3 - 6*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10})$$ 8 * q - 2 * q^4 - 6 * q^6 - 4 * q^9 $$8 q - 2 q^{4} - 6 q^{6} - 4 q^{9} - 2 q^{10} + 4 q^{12} - 8 q^{13} - 14 q^{16} + 16 q^{18} + 4 q^{21} - 4 q^{22} - 2 q^{24} - 8 q^{25} + 28 q^{28} + 8 q^{30} - 16 q^{33} + 4 q^{34} + 18 q^{36} + 8 q^{37} + 14 q^{40} - 12 q^{42} - 4 q^{45} + 20 q^{46} - 36 q^{48} + 16 q^{49} - 32 q^{52} - 10 q^{54} + 24 q^{57} - 36 q^{58} - 14 q^{60} - 8 q^{61} - 2 q^{64} - 40 q^{66} + 12 q^{69} - 12 q^{70} + 24 q^{72} - 36 q^{76} + 40 q^{78} + 40 q^{82} + 16 q^{84} + 16 q^{85} + 44 q^{88} + 18 q^{90} + 32 q^{93} + 12 q^{94} + 42 q^{96} - 48 q^{97}+O(q^{100})$$ 8 * q - 2 * q^4 - 6 * q^6 - 4 * q^9 - 2 * q^10 + 4 * q^12 - 8 * q^13 - 14 * q^16 + 16 * q^18 + 4 * q^21 - 4 * q^22 - 2 * q^24 - 8 * q^25 + 28 * q^28 + 8 * q^30 - 16 * q^33 + 4 * q^34 + 18 * q^36 + 8 * q^37 + 14 * q^40 - 12 * q^42 - 4 * q^45 + 20 * q^46 - 36 * q^48 + 16 * q^49 - 32 * q^52 - 10 * q^54 + 24 * q^57 - 36 * q^58 - 14 * q^60 - 8 * q^61 - 2 * q^64 - 40 * q^66 + 12 * q^69 - 12 * q^70 + 24 * q^72 - 36 * q^76 + 40 * q^78 + 40 * q^82 + 16 * q^84 + 16 * q^85 + 44 * q^88 + 18 * q^90 + 32 * q^93 + 12 * q^94 + 42 * q^96 - 48 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16$$ (-v^7 + v^5 + 2*v^3) / 16 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + \nu^{5} + 4\nu^{3} + 4\nu ) / 8$$ (v^7 + v^5 + 4*v^3 + 4*v) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16$$ (v^7 - v^6 + 3*v^5 - 3*v^4 + 2*v^3 - 10*v^2 + 16*v - 8) / 16 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + \nu^{6} - 3\nu^{5} + 3\nu^{4} - 2\nu^{3} - 6\nu^{2} + 8 ) / 16$$ (-v^7 + v^6 - 3*v^5 + 3*v^4 - 2*v^3 - 6*v^2 + 8) / 16 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} + 10\nu^{2} + 16\nu + 8 ) / 16$$ (v^7 + v^6 + 3*v^5 + 3*v^4 + 2*v^3 + 10*v^2 + 16*v + 8) / 16 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} - 6\nu^{2} + 8 ) / 16$$ (v^7 + v^6 + 3*v^5 + 3*v^4 + 2*v^3 - 6*v^2 + 8) / 16 $$\beta_{7}$$ $$=$$ $$( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8$$ (-v^6 + v^4 + 2*v^2) / 8
 $$\nu$$ $$=$$ $$( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2$$ (-b6 + b5 + b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} ) / 2$$ (-b6 + b5 - b4 - b3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 4\beta_{2} + 4\beta_1 ) / 2$$ (-b6 - b5 + b4 - b3 + 4*b2 + 4*b1) / 2 $$\nu^{4}$$ $$=$$ $$( 4\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - 4 ) / 2$$ (4*b7 + 3*b6 + b5 + 3*b4 - b3 - 4) / 2 $$\nu^{5}$$ $$=$$ $$( 5\beta_{6} + \beta_{5} - 5\beta_{4} + \beta_{3} - 4\beta_{2} + 4\beta_1 ) / 2$$ (5*b6 + b5 - 5*b4 + b3 - 4*b2 + 4*b1) / 2 $$\nu^{6}$$ $$=$$ $$( -12\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 4 ) / 2$$ (-12*b7 + b6 + 3*b5 + b4 - 3*b3 - 4) / 2 $$\nu^{7}$$ $$=$$ $$( 3\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} + 4\beta_{2} - 20\beta_1 ) / 2$$ (3*b6 - b5 - 3*b4 - b3 + 4*b2 - 20*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.17915 + 0.780776i −1.17915 − 0.780776i −0.599676 + 1.28078i −0.599676 − 1.28078i 0.599676 + 1.28078i 0.599676 − 1.28078i 1.17915 + 0.780776i 1.17915 − 0.780776i
−1.17915 0.780776i 1.51022 + 0.848071i 0.780776 + 1.84130i 1.00000i −1.11862 2.17915i 3.02045i 0.516994 2.78078i 1.56155 + 2.56155i 0.780776 1.17915i
11.2 −1.17915 + 0.780776i 1.51022 0.848071i 0.780776 1.84130i 1.00000i −1.11862 + 2.17915i 3.02045i 0.516994 + 2.78078i 1.56155 2.56155i 0.780776 + 1.17915i
11.3 −0.599676 1.28078i −0.468213 1.66757i −1.28078 + 1.53610i 1.00000i −1.85500 + 1.59968i 0.936426i 2.73546 + 0.719224i −2.56155 + 1.56155i −1.28078 + 0.599676i
11.4 −0.599676 + 1.28078i −0.468213 + 1.66757i −1.28078 1.53610i 1.00000i −1.85500 1.59968i 0.936426i 2.73546 0.719224i −2.56155 1.56155i −1.28078 0.599676i
11.5 0.599676 1.28078i 0.468213 + 1.66757i −1.28078 1.53610i 1.00000i 2.41656 + 0.400324i 0.936426i −2.73546 + 0.719224i −2.56155 + 1.56155i −1.28078 0.599676i
11.6 0.599676 + 1.28078i 0.468213 1.66757i −1.28078 + 1.53610i 1.00000i 2.41656 0.400324i 0.936426i −2.73546 0.719224i −2.56155 1.56155i −1.28078 + 0.599676i
11.7 1.17915 0.780776i −1.51022 0.848071i 0.780776 1.84130i 1.00000i −2.44293 + 0.179147i 3.02045i −0.516994 2.78078i 1.56155 + 2.56155i 0.780776 + 1.17915i
11.8 1.17915 + 0.780776i −1.51022 + 0.848071i 0.780776 + 1.84130i 1.00000i −2.44293 0.179147i 3.02045i −0.516994 + 2.78078i 1.56155 2.56155i 0.780776 1.17915i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.8 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.e.a 8
3.b odd 2 1 inner 60.2.e.a 8
4.b odd 2 1 inner 60.2.e.a 8
5.b even 2 1 300.2.e.c 8
5.c odd 4 1 300.2.h.a 8
5.c odd 4 1 300.2.h.b 8
8.b even 2 1 960.2.h.g 8
8.d odd 2 1 960.2.h.g 8
12.b even 2 1 inner 60.2.e.a 8
15.d odd 2 1 300.2.e.c 8
15.e even 4 1 300.2.h.a 8
15.e even 4 1 300.2.h.b 8
20.d odd 2 1 300.2.e.c 8
20.e even 4 1 300.2.h.a 8
20.e even 4 1 300.2.h.b 8
24.f even 2 1 960.2.h.g 8
24.h odd 2 1 960.2.h.g 8
60.h even 2 1 300.2.e.c 8
60.l odd 4 1 300.2.h.a 8
60.l odd 4 1 300.2.h.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.e.a 8 1.a even 1 1 trivial
60.2.e.a 8 3.b odd 2 1 inner
60.2.e.a 8 4.b odd 2 1 inner
60.2.e.a 8 12.b even 2 1 inner
300.2.e.c 8 5.b even 2 1
300.2.e.c 8 15.d odd 2 1
300.2.e.c 8 20.d odd 2 1
300.2.e.c 8 60.h even 2 1
300.2.h.a 8 5.c odd 4 1
300.2.h.a 8 15.e even 4 1
300.2.h.a 8 20.e even 4 1
300.2.h.a 8 60.l odd 4 1
300.2.h.b 8 5.c odd 4 1
300.2.h.b 8 15.e even 4 1
300.2.h.b 8 20.e even 4 1
300.2.h.b 8 60.l odd 4 1
960.2.h.g 8 8.b even 2 1
960.2.h.g 8 8.d odd 2 1
960.2.h.g 8 24.f even 2 1
960.2.h.g 8 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{6} + 4 T^{4} + 4 T^{2} + \cdots + 16$$
$3$ $$T^{8} + 2 T^{6} + 2 T^{4} + 18 T^{2} + \cdots + 81$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$(T^{4} + 10 T^{2} + 8)^{2}$$
$11$ $$(T^{4} - 20 T^{2} + 32)^{2}$$
$13$ $$(T^{2} + 2 T - 16)^{4}$$
$17$ $$(T^{2} + 4)^{4}$$
$19$ $$(T^{4} + 20 T^{2} + 32)^{2}$$
$23$ $$(T^{4} - 58 T^{2} + 8)^{2}$$
$29$ $$(T^{4} + 36 T^{2} + 256)^{2}$$
$31$ $$(T^{4} + 28 T^{2} + 128)^{2}$$
$37$ $$(T^{2} - 2 T - 16)^{4}$$
$41$ $$(T^{4} + 52 T^{2} + 64)^{2}$$
$43$ $$(T^{4} + 62 T^{2} + 128)^{2}$$
$47$ $$(T^{4} - 10 T^{2} + 8)^{2}$$
$53$ $$(T^{4} + 168 T^{2} + 2704)^{2}$$
$59$ $$(T^{4} - 252 T^{2} + 10368)^{2}$$
$61$ $$(T^{2} + 2 T - 16)^{4}$$
$67$ $$(T^{4} + 46 T^{2} + 512)^{2}$$
$71$ $$(T^{4} - 56 T^{2} + 512)^{2}$$
$73$ $$(T^{2} - 68)^{4}$$
$79$ $$(T^{4} + 148 T^{2} + 5408)^{2}$$
$83$ $$(T^{4} - 250 T^{2} + 5000)^{2}$$
$89$ $$(T^{4} + 144 T^{2} + 4096)^{2}$$
$97$ $$(T + 6)^{8}$$