# Properties

 Label 60.2.j.a Level $60$ Weight $2$ Character orbit 60.j Analytic conductor $0.479$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 60.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.479102412128$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.426337261060096.1 Defining polynomial: $$x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64$$ x^12 - 4*x^9 - 3*x^8 + 4*x^7 + 8*x^6 + 8*x^5 - 12*x^4 - 32*x^3 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} - \beta_{2} q^{3} + ( - \beta_{7} + \beta_{4} + \beta_{2}) q^{4} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4}) q^{5} - \beta_1 q^{6} + ( - \beta_{9} + \beta_{5} - \beta_{3} + \beta_1) q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 - 1) q^{8} + \beta_{8} q^{9}+O(q^{10})$$ q - b6 * q^2 - b2 * q^3 + (-b7 + b4 + b2) * q^4 + (-b8 + b6 - b5 - b4) * q^5 - b1 * q^6 + (-b9 + b5 - b3 + b1) * q^7 + (-b11 - b10 + b8 - b5 - b4 + 2*b3 + b1 - 1) * q^8 + b8 * q^9 $$q - \beta_{6} q^{2} - \beta_{2} q^{3} + ( - \beta_{7} + \beta_{4} + \beta_{2}) q^{4} + ( - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4}) q^{5} - \beta_1 q^{6} + ( - \beta_{9} + \beta_{5} - \beta_{3} + \beta_1) q^{7} + ( - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 - 1) q^{8} + \beta_{8} q^{9} + (\beta_{11} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{10} + (\beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{11} + (\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} - 1) q^{12} + (\beta_{11} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{13} + (\beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots - 2 \beta_{2}) q^{14}+ \cdots + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{99}+O(q^{100})$$ q - b6 * q^2 - b2 * q^3 + (-b7 + b4 + b2) * q^4 + (-b8 + b6 - b5 - b4) * q^5 - b1 * q^6 + (-b9 + b5 - b3 + b1) * q^7 + (-b11 - b10 + b8 - b5 - b4 + 2*b3 + b1 - 1) * q^8 + b8 * q^9 + (b11 - b8 + b7 + b5 - b3 + b2 - 1) * q^10 + (b10 + 2*b9 + b6 + b4 - b3 - 2*b1) * q^11 + (b9 - b8 + b6 - b5 - 1) * q^12 + (b11 + b8 + b5 + b4 + b1 - 1) * q^13 + (b11 + b10 - b9 + 2*b8 + 2*b7 - b6 + 2*b5 + b4 - b3 - 2*b2) * q^14 + (-b9 + b7 - b6 - b4 + b3 + b1) * q^15 + (-2*b11 + b7 - b4 + b3 + b2 - b1 + 2) * q^16 + (-b9 - b8 + b5 + b3 - b1 - 1) * q^17 + b10 * q^18 + (-b11 + b9 - 2*b5 - b4 + 3*b3) * q^19 + (b11 - b8 - 2*b7 + b6 + b5 + b4 - 2*b3 - 2*b2 + b1 + 1) * q^20 + (-b11 - b10 + b9 - b6) * q^21 + (b11 + b9 - 2*b7 + 2*b6 - 2*b5 - b4 + b1) * q^22 + (b11 - 2*b10 - b9 + b4 - b3 + 4*b2) * q^23 + (-b10 - b7 + b6 - b4 + b2) * q^24 + (-2*b11 + b9 - 2*b6 - b5 - b3 - b1 - 1) * q^25 + (-b11 - b9 - 2*b7 - b4 + b3 - 2*b2 + 2) * q^26 - b7 * q^27 + (b11 + 2*b10 - b9 + b4 - 2*b3 - 2*b2 - b1) * q^28 + (b10 - b6 + b4 - b3) * q^29 + (-b9 - b8 + b7 - b6 + b5 + b4 - b3 + b2 + 1) * q^30 + (2*b11 - 2*b10 - 2*b9 - 2*b6) * q^31 + (b9 + b8 + 4*b7 - b6 - b5 - b3 + b1 + 1) * q^32 + (2*b11 + 2*b10 - b9 + b5 + 2*b4 - b3 + b1) * q^33 + (-b11 - 2*b10 + b9 - 2*b8 + 2*b7 + 2*b6 - 2*b5 - b4 + b3 - 2*b2) * q^34 + (-2*b11 + b10 + 2*b9 - 4*b7 + b6 - 2*b5 - b4 + 3*b3) * q^35 + (b11 - b9 + b7 + b2 + b1) * q^36 + (-b9 + b8 + b5 + b3 - b1 + 1) * q^37 + (-2*b11 - 2*b10 - 2*b8 - 2*b5 - 2*b4 + 2*b3 + 4*b2 + 2) * q^38 + (-b10 - b7 + b6 + b4 - b3 + b2) * q^39 + (b11 - b9 + 2*b8 - b7 - 2*b6 + 2*b5 + b3 - 3*b2 - 2*b1) * q^40 + (b11 + b9 + b4 - b3 + 2*b1) * q^41 + (-b11 + b9 + 2*b8 - 2*b7 + b4 - b1 + 2) * q^42 + (4*b10 + 2*b9 + 2*b5 - 2*b3 - 4*b2 - 2*b1) * q^43 + (b11 + b10 - b9 - 2*b8 - 2*b7 - b6 + 2*b5 - b4 + b3 + 2*b2) * q^44 + (-b11 - b10 - b4 + b3 - b1 + 1) * q^45 + (-2*b11 + 2*b9 - 2*b7 - 2*b2 + 2*b1 - 4) * q^46 + (b11 - 3*b9 + 4*b7 - 2*b6 + 2*b5 - b4 - b3 + 2*b1) * q^47 + (b10 + b9 - b8 + b5 - 2*b2 + 1) * q^48 + (-b11 - 2*b10 + b9 - b8 + 2*b6 - 2*b5 - 3*b4 + b3) * q^49 + (b11 + b10 + b9 + 4*b8 + 2*b6 + 3*b4 - b3 + 4*b2 - 2) * q^50 + (-b11 + b10 + b9 + b7 + b6 + b2) * q^51 + (4*b7 - 2*b6 + 2*b3 - 2*b1) * q^52 + (-2*b11 - 2*b10 + b9 - b8 - b5 - 2*b4 + b3 - b1 + 1) * q^53 + b3 * q^54 + (-b11 - 2*b10 + 4*b7 + 2*b6 - b5 + b4 + b1) * q^55 + (3*b11 + b10 - b9 + 2*b7 + b6 + b4 - b3 + 2*b2 - 6) * q^56 + (b11 - b9 + 2*b6 - b4 + b3) * q^57 + (b11 - b9 + 2*b8 + b4 + 2*b2 + b1 - 2) * q^58 + (b11 + b10 - b9 + 4*b7 - b6 + 2*b5 - 2*b3 - 4*b2) * q^59 + (-b10 + 2*b8 + b7 - b6 + b4 - b3 - b2 + b1 - 2) * q^60 + (5*b11 + 4*b10 - 3*b9 + 4*b6 + b4 - b3 + 2*b1 - 4) * q^61 + (-2*b11 + 2*b9 - 4*b8 - 4*b7 + 2*b4 - 2*b1 - 4) * q^62 + (b11 + b5 + b4 - 2*b3 - b1) * q^63 + (b11 + 2*b10 - b9 + 2*b8 - 3*b7 - 2*b6 + 2*b5 + 2*b4 - 5*b3 + 3*b2) * q^64 + (2*b11 - b9 + 5*b8 + 2*b6 + b5 + b3 + b1 + 1) * q^65 + (b11 - b10 - 3*b9 - b6 - b4 + b3 + 2*b1 - 2) * q^66 + (-2*b11 + 4*b9 - 4*b7 + 4*b6 - 2*b5 + 2*b4 - 2*b1) * q^67 + (-2*b11 - 2*b10 + 2*b9 - 2*b4 - 2*b3 - 4*b1) * q^68 + (-b11 + b9 - 4*b8 - 2*b5 - b4 - b3) * q^69 + (-b11 - 2*b10 - b9 + 2*b7 - 2*b5 - 3*b4 + 6*b3 + 4*b2 + b1 + 4) * q^70 + (-2*b11 - 2*b9 - 4*b7 - 2*b4 + 2*b3 - 4*b2 + 4*b1) * q^71 + (-b9 - b8 - b6 + b5 - b3 + b1 - 1) * q^72 + (-4*b11 - 4*b10 + 2*b9 - 5*b8 - 2*b5 - 4*b4 + 2*b3 - 2*b1 + 5) * q^73 + (-b11 + b9 - 2*b8 + 2*b7 - 2*b5 - b4 + b3 - 2*b2) * q^74 + (b11 + b10 + b9 - b6 + b2 - 2*b1) * q^75 + (-2*b11 - 2*b10 + 2*b9 + 2*b7 - 2*b6 + 2*b2 + 2*b1) * q^76 + (-b11 + b9 + 4*b8 - 2*b6 + b4 - b3 + 4) * q^77 + (-b11 + b9 + 2*b8 - b4 + b3 - 2*b2 - 2) * q^78 + (2*b11 + 2*b10 - 2*b9 - 4*b7 - 2*b6 + 4*b5 - 4*b3 + 4*b2) * q^79 + (-b11 + b10 + 2*b9 - 5*b8 - 2*b7 + 2*b6 - 3*b5 + b4 + 2*b3 - 2*b2 - 3*b1 + 1) * q^80 - q^81 + (-2*b9 + 2*b8 - 4*b7 - 2*b6 + 2*b5 + 2) * q^82 + (-b11 - 2*b10 - b9 - 2*b5 - b4 + 3*b3 - 4*b2 + 2*b1) * q^83 + (-b11 + b10 + b9 + 2*b8 - b6 - 2*b5 - b4 + 3*b3) * q^84 + (3*b11 + 4*b10 - 2*b9 + b8 - 2*b6 + 3*b5 + 5*b4 - 2*b3 + b1 + 3) * q^85 + (6*b11 + 2*b10 - 2*b9 + 2*b6 + 2*b4 - 2*b3 + 4) * q^86 + (b9 - b5 + b3 - b1) * q^87 + (b11 - 2*b10 - b9 - 4*b8 + b4 + 2*b3 - 2*b2 + 3*b1 + 4) * q^88 + (b11 + 2*b10 - b9 - 2*b6 + 2*b5 + 3*b4 - b3) * q^89 + (-b11 + 2*b9 - b8 + b7 - b5 - b2 - b1 + 1) * q^90 + (2*b11 - 2*b10 - 2*b9 + 4*b7 - 2*b6 + 4*b2) * q^91 + (-2*b9 + 6*b8 + 2*b6 + 2*b5 + 2*b3 - 2*b1 + 6) * q^92 + (-2*b11 - 2*b5 - 2*b4 - 2*b1) * q^93 + (2*b11 + 2*b10 - 2*b9 + 2*b7 - 2*b6 + 4*b5 + 4*b4 - 6*b3 - 2*b2) * q^94 + (3*b11 - b9 + 4*b7 - 2*b6 + 2*b5 + b4 - 3*b3 - 4*b2 - 2*b1) * q^95 + (b11 - b10 - b9 - b7 - b6 - b2 - b1 + 4) * q^96 + (-2*b11 + 4*b9 - 3*b8 - 4*b6 - 2*b5 + 2*b4 - 4*b3 + 2*b1 - 3) * q^97 + (b10 + 2*b9 - 2*b8 + 2*b5 - 2*b3 - 2*b1 + 2) * q^98 + (-b11 - b10 + b9 + b6 - 2*b5 + 2*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{6} - 12 q^{8}+O(q^{10})$$ 12 * q - 4 * q^6 - 12 * q^8 $$12 q - 4 q^{6} - 12 q^{8} - 8 q^{10} - 8 q^{12} - 4 q^{13} + 12 q^{16} - 20 q^{17} + 20 q^{20} + 12 q^{22} - 20 q^{25} + 16 q^{26} - 4 q^{28} + 8 q^{30} + 20 q^{32} + 8 q^{33} + 4 q^{36} + 4 q^{37} + 16 q^{38} - 8 q^{40} + 16 q^{41} + 20 q^{42} + 4 q^{45} - 40 q^{46} + 16 q^{48} - 16 q^{50} - 8 q^{52} + 4 q^{53} - 64 q^{56} - 20 q^{58} - 20 q^{60} - 32 q^{61} - 56 q^{62} + 20 q^{65} - 24 q^{66} - 16 q^{68} + 44 q^{70} - 12 q^{72} + 44 q^{73} + 8 q^{76} + 48 q^{77} - 24 q^{78} + 4 q^{80} - 12 q^{81} + 16 q^{82} + 44 q^{85} + 64 q^{86} + 60 q^{88} + 12 q^{90} + 56 q^{92} - 16 q^{93} + 44 q^{96} - 20 q^{97} + 24 q^{98}+O(q^{100})$$ 12 * q - 4 * q^6 - 12 * q^8 - 8 * q^10 - 8 * q^12 - 4 * q^13 + 12 * q^16 - 20 * q^17 + 20 * q^20 + 12 * q^22 - 20 * q^25 + 16 * q^26 - 4 * q^28 + 8 * q^30 + 20 * q^32 + 8 * q^33 + 4 * q^36 + 4 * q^37 + 16 * q^38 - 8 * q^40 + 16 * q^41 + 20 * q^42 + 4 * q^45 - 40 * q^46 + 16 * q^48 - 16 * q^50 - 8 * q^52 + 4 * q^53 - 64 * q^56 - 20 * q^58 - 20 * q^60 - 32 * q^61 - 56 * q^62 + 20 * q^65 - 24 * q^66 - 16 * q^68 + 44 * q^70 - 12 * q^72 + 44 * q^73 + 8 * q^76 + 48 * q^77 - 24 * q^78 + 4 * q^80 - 12 * q^81 + 16 * q^82 + 44 * q^85 + 64 * q^86 + 60 * q^88 + 12 * q^90 + 56 * q^92 - 16 * q^93 + 44 * q^96 - 20 * q^97 + 24 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{10} - \nu^{9} - 4\nu^{8} + 2\nu^{7} + 3\nu^{6} - 5\nu^{5} - 24\nu^{4} - 2\nu^{3} + 8\nu^{2} + 24\nu + 48 ) / 80$$ (-v^10 - v^9 - 4*v^8 + 2*v^7 + 3*v^6 - 5*v^5 - 24*v^4 - 2*v^3 + 8*v^2 + 24*v + 48) / 80 $$\beta_{2}$$ $$=$$ $$( \nu^{11} + \nu^{10} + 4\nu^{9} - 2\nu^{8} - 3\nu^{7} + 5\nu^{6} + 24\nu^{5} + 2\nu^{4} - 8\nu^{3} - 24\nu^{2} - 48\nu ) / 160$$ (v^11 + v^10 + 4*v^9 - 2*v^8 - 3*v^7 + 5*v^6 + 24*v^5 + 2*v^4 - 8*v^3 - 24*v^2 - 48*v) / 160 $$\beta_{3}$$ $$=$$ $$( - \nu^{10} + 9 \nu^{9} + 6 \nu^{8} + 2 \nu^{7} - 17 \nu^{6} - 35 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} + 28 \nu^{2} - 56 \nu - 192 ) / 80$$ (-v^10 + 9*v^9 + 6*v^8 + 2*v^7 - 17*v^6 - 35*v^5 + 26*v^4 + 38*v^3 + 28*v^2 - 56*v - 192) / 80 $$\beta_{4}$$ $$=$$ $$( - \nu^{11} - \nu^{10} + \nu^{9} + 2 \nu^{8} + 13 \nu^{7} - 5 \nu^{6} - 19 \nu^{5} - 22 \nu^{4} - 22 \nu^{3} + 44 \nu^{2} + 88 \nu ) / 80$$ (-v^11 - v^10 + v^9 + 2*v^8 + 13*v^7 - 5*v^6 - 19*v^5 - 22*v^4 - 22*v^3 + 44*v^2 + 88*v) / 80 $$\beta_{5}$$ $$=$$ $$( \nu^{11} + 4\nu^{10} + 2\nu^{9} + \nu^{7} + 16\nu^{6} - 6\nu^{5} + 4\nu^{4} + 8\nu^{3} - 48\nu^{2} - 160\nu - 64 ) / 160$$ (v^11 + 4*v^10 + 2*v^9 + v^7 + 16*v^6 - 6*v^5 + 4*v^4 + 8*v^3 - 48*v^2 - 160*v - 64) / 160 $$\beta_{6}$$ $$=$$ $$( -\nu^{11} + 4\nu^{8} + 3\nu^{7} - 4\nu^{6} - 8\nu^{5} - 8\nu^{4} + 12\nu^{3} + 32\nu^{2} ) / 32$$ (-v^11 + 4*v^8 + 3*v^7 - 4*v^6 - 8*v^5 - 8*v^4 + 12*v^3 + 32*v^2) / 32 $$\beta_{7}$$ $$=$$ $$( - \nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + 38 \nu^{4} + 28 \nu^{3} - 56 \nu^{2} - 192 \nu ) / 160$$ (-v^11 + 9*v^10 + 6*v^9 + 2*v^8 - 17*v^7 - 35*v^6 + 26*v^5 + 38*v^4 + 28*v^3 - 56*v^2 - 192*v) / 160 $$\beta_{8}$$ $$=$$ $$( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80$$ (3*v^11 + v^10 - 4*v^8 - 5*v^7 + v^6 + 12*v^5 + 8*v^4 + 12*v^3 - 36*v^2 - 16*v - 64) / 80 $$\beta_{9}$$ $$=$$ $$( 9 \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 20 \nu^{8} - 31 \nu^{7} + 34 \nu^{6} + 46 \nu^{5} + 16 \nu^{4} - 88 \nu^{3} - 192 \nu^{2} - 160 \nu + 64 ) / 160$$ (9*v^11 + 6*v^10 - 2*v^9 - 20*v^8 - 31*v^7 + 34*v^6 + 46*v^5 + 16*v^4 - 88*v^3 - 192*v^2 - 160*v + 64) / 160 $$\beta_{10}$$ $$=$$ $$( - \nu^{11} - 3 \nu^{10} - \nu^{9} + 4 \nu^{8} + 7 \nu^{7} + \nu^{6} - 9 \nu^{5} - 20 \nu^{4} + 4 \nu^{3} + 20 \nu^{2} + 36 \nu + 16 ) / 40$$ (-v^11 - 3*v^10 - v^9 + 4*v^8 + 7*v^7 + v^6 - 9*v^5 - 20*v^4 + 4*v^3 + 20*v^2 + 36*v + 16) / 40 $$\beta_{11}$$ $$=$$ $$( 11 \nu^{11} + 6 \nu^{10} + 14 \nu^{9} - 12 \nu^{8} - 53 \nu^{7} + 10 \nu^{6} + 14 \nu^{5} + 112 \nu^{4} + 32 \nu^{3} - 224 \nu^{2} - 128 \nu - 320 ) / 160$$ (11*v^11 + 6*v^10 + 14*v^9 - 12*v^8 - 53*v^7 + 10*v^6 + 14*v^5 + 112*v^4 + 32*v^3 - 224*v^2 - 128*v - 320) / 160
 $$\nu$$ $$=$$ $$( \beta_{11} - \beta_{9} + \beta_{4} - \beta_{3} ) / 2$$ (b11 - b9 + b4 - b3) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{11} - 2\beta_{10} + \beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} ) / 2$$ (-b11 - 2*b10 + b9 - 2*b7 + 2*b6 - 2*b5 - b4 + b3 + 2*b2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{11} - 3\beta_{9} + 2\beta_{8} + 2\beta_{5} - \beta_{4} - \beta_{3} + 2\beta _1 + 2 ) / 2$$ (b11 - 3*b9 + 2*b8 + 2*b5 - b4 - b3 + 2*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{11} - 2\beta_{10} - \beta_{9} - 2\beta_{7} - 2\beta_{6} - \beta_{4} + \beta_{3} - 2\beta_{2} - 4\beta _1 + 4 ) / 2$$ (-b11 - 2*b10 - b9 - 2*b7 - 2*b6 - b4 + b3 - 2*b2 - 4*b1 + 4) / 2 $$\nu^{5}$$ $$=$$ $$( -\beta_{11} - \beta_{9} + 2\beta_{8} - 2\beta_{5} - \beta_{4} - \beta_{3} + 8\beta_{2} - 2\beta _1 - 2 ) / 2$$ (-b11 - b9 + 2*b8 - 2*b5 - b4 - b3 + 8*b2 - 2*b1 - 2) / 2 $$\nu^{6}$$ $$=$$ $$( 3 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} - \beta_{4} - 3 \beta_{3} + 6 \beta_{2} ) / 2$$ (3*b11 - 2*b10 - 3*b9 - 4*b8 - 6*b7 + 2*b6 + 6*b5 - b4 - 3*b3 + 6*b2) / 2 $$\nu^{7}$$ $$=$$ $$( -5\beta_{11} - 5\beta_{9} + 10\beta_{8} - 8\beta_{6} + 10\beta_{5} + 5\beta_{4} + \beta_{3} - 6\beta _1 + 10 ) / 2$$ (-5*b11 - 5*b9 + 10*b8 - 8*b6 + 10*b5 + 5*b4 + b3 - 6*b1 + 10) / 2 $$\nu^{8}$$ $$=$$ $$( 5 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 2 \beta_{7} + 10 \beta_{6} + 5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 20 \beta _1 + 4 ) / 2$$ (5*b11 + 10*b10 + 5*b9 + 2*b7 + 10*b6 + 5*b4 - 5*b3 + 2*b2 - 20*b1 + 4) / 2 $$\nu^{9}$$ $$=$$ $$( 9\beta_{11} - 7\beta_{9} - 10\beta_{8} + 2\beta_{5} + 9\beta_{4} + \beta_{3} + 40\beta_{2} + 10\beta _1 + 10 ) / 2$$ (9*b11 - 7*b9 - 10*b8 + 2*b5 + 9*b4 + b3 + 40*b2 + 10*b1 + 10) / 2 $$\nu^{10}$$ $$=$$ $$( 13 \beta_{11} - 14 \beta_{10} - 13 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 14 \beta_{6} + 26 \beta_{5} + 25 \beta_{4} - 21 \beta_{3} - 6 \beta_{2} ) / 2$$ (13*b11 - 14*b10 - 13*b9 + 4*b8 + 6*b7 + 14*b6 + 26*b5 + 25*b4 - 21*b3 - 6*b2) / 2 $$\nu^{11}$$ $$=$$ $$( - 11 \beta_{11} + 29 \beta_{9} + 54 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} - 18 \beta_{5} + 11 \beta_{4} + 15 \beta_{3} - 26 \beta _1 + 54 ) / 2$$ (-11*b11 + 29*b9 + 54*b8 - 16*b7 + 24*b6 - 18*b5 + 11*b4 + 15*b3 - 26*b1 + 54) / 2

## Character values

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

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$-1$$ $$\beta_{8}$$ $$1$$

## 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
 1.41127 + 0.0912546i 1.19252 + 0.760198i −0.0912546 − 1.41127i −0.394157 + 1.35818i −0.760198 − 1.19252i −1.35818 + 0.394157i 1.41127 − 0.0912546i 1.19252 − 0.760198i −0.0912546 + 1.41127i −0.394157 − 1.35818i −0.760198 + 1.19252i −1.35818 − 0.394157i
−1.41127 + 0.0912546i −0.707107 0.707107i 1.98335 0.257569i 1.32001 1.80487i 1.06244 + 0.933389i 1.86678 1.86678i −2.77552 + 0.544488i 1.00000i −1.69819 + 2.66761i
7.2 −1.19252 + 0.760198i 0.707107 + 0.707107i 0.844199 1.81310i 0.432320 + 2.19388i −1.38078 0.305697i −0.611393 + 0.611393i 0.371591 + 2.80391i 1.00000i −2.18333 2.28759i
7.3 0.0912546 1.41127i 0.707107 + 0.707107i −1.98335 0.257569i 1.32001 1.80487i 1.06244 0.933389i −1.86678 + 1.86678i −0.544488 + 2.77552i 1.00000i −2.42670 2.02759i
7.4 0.394157 + 1.35818i 0.707107 + 0.707107i −1.68928 + 1.07067i −1.75233 1.38900i −0.681664 + 1.23909i 2.47817 2.47817i −2.12000 1.87233i 1.00000i 1.19582 2.92746i
7.5 0.760198 1.19252i −0.707107 0.707107i −0.844199 1.81310i 0.432320 + 2.19388i −1.38078 + 0.305697i 0.611393 0.611393i −2.80391 0.371591i 1.00000i 2.94489 + 1.15223i
7.6 1.35818 + 0.394157i −0.707107 0.707107i 1.68928 + 1.07067i −1.75233 1.38900i −0.681664 1.23909i −2.47817 + 2.47817i 1.87233 + 2.12000i 1.00000i −1.83249 2.57720i
43.1 −1.41127 0.0912546i −0.707107 + 0.707107i 1.98335 + 0.257569i 1.32001 + 1.80487i 1.06244 0.933389i 1.86678 + 1.86678i −2.77552 0.544488i 1.00000i −1.69819 2.66761i
43.2 −1.19252 0.760198i 0.707107 0.707107i 0.844199 + 1.81310i 0.432320 2.19388i −1.38078 + 0.305697i −0.611393 0.611393i 0.371591 2.80391i 1.00000i −2.18333 + 2.28759i
43.3 0.0912546 + 1.41127i 0.707107 0.707107i −1.98335 + 0.257569i 1.32001 + 1.80487i 1.06244 + 0.933389i −1.86678 1.86678i −0.544488 2.77552i 1.00000i −2.42670 + 2.02759i
43.4 0.394157 1.35818i 0.707107 0.707107i −1.68928 1.07067i −1.75233 + 1.38900i −0.681664 1.23909i 2.47817 + 2.47817i −2.12000 + 1.87233i 1.00000i 1.19582 + 2.92746i
43.5 0.760198 + 1.19252i −0.707107 + 0.707107i −0.844199 + 1.81310i 0.432320 2.19388i −1.38078 0.305697i 0.611393 + 0.611393i −2.80391 + 0.371591i 1.00000i 2.94489 1.15223i
43.6 1.35818 0.394157i −0.707107 + 0.707107i 1.68928 1.07067i −1.75233 + 1.38900i −0.681664 + 1.23909i −2.47817 2.47817i 1.87233 2.12000i 1.00000i −1.83249 + 2.57720i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.6 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
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.j.a 12
3.b odd 2 1 180.2.k.e 12
4.b odd 2 1 inner 60.2.j.a 12
5.b even 2 1 300.2.j.d 12
5.c odd 4 1 inner 60.2.j.a 12
5.c odd 4 1 300.2.j.d 12
8.b even 2 1 960.2.w.g 12
8.d odd 2 1 960.2.w.g 12
12.b even 2 1 180.2.k.e 12
15.d odd 2 1 900.2.k.n 12
15.e even 4 1 180.2.k.e 12
15.e even 4 1 900.2.k.n 12
20.d odd 2 1 300.2.j.d 12
20.e even 4 1 inner 60.2.j.a 12
20.e even 4 1 300.2.j.d 12
40.i odd 4 1 960.2.w.g 12
40.k even 4 1 960.2.w.g 12
60.h even 2 1 900.2.k.n 12
60.l odd 4 1 180.2.k.e 12
60.l odd 4 1 900.2.k.n 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a 12 1.a even 1 1 trivial
60.2.j.a 12 4.b odd 2 1 inner
60.2.j.a 12 5.c odd 4 1 inner
60.2.j.a 12 20.e even 4 1 inner
180.2.k.e 12 3.b odd 2 1
180.2.k.e 12 12.b even 2 1
180.2.k.e 12 15.e even 4 1
180.2.k.e 12 60.l odd 4 1
300.2.j.d 12 5.b even 2 1
300.2.j.d 12 5.c odd 4 1
300.2.j.d 12 20.d odd 2 1
300.2.j.d 12 20.e even 4 1
900.2.k.n 12 15.d odd 2 1
900.2.k.n 12 15.e even 4 1
900.2.k.n 12 60.h even 2 1
900.2.k.n 12 60.l odd 4 1
960.2.w.g 12 8.b even 2 1
960.2.w.g 12 8.d odd 2 1
960.2.w.g 12 40.i odd 4 1
960.2.w.g 12 40.k even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 4 T^{9} - 3 T^{8} - 4 T^{7} + \cdots + 64$$
$3$ $$(T^{4} + 1)^{3}$$
$5$ $$(T^{6} + 5 T^{4} + 8 T^{3} + 25 T^{2} + \cdots + 125)^{2}$$
$7$ $$T^{12} + 200 T^{8} + 7440 T^{4} + \cdots + 4096$$
$11$ $$(T^{6} + 36 T^{4} + 260 T^{2} + 128)^{2}$$
$13$ $$(T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + \cdots + 32)^{2}$$
$17$ $$(T^{6} + 10 T^{5} + 50 T^{4} + 80 T^{3} + \cdots + 800)^{2}$$
$19$ $$(T^{6} - 40 T^{4} + 400 T^{2} - 512)^{2}$$
$23$ $$T^{12} + 4640 T^{8} + 37120 T^{4} + \cdots + 65536$$
$29$ $$(T^{6} + 20 T^{4} + 100 T^{2} + 64)^{2}$$
$31$ $$(T^{6} + 112 T^{4} + 3648 T^{2} + \cdots + 32768)^{2}$$
$37$ $$(T^{6} - 2 T^{5} + 2 T^{4} + 32 T^{3} + \cdots + 32)^{2}$$
$41$ $$(T^{3} - 4 T^{2} - 20 T + 64)^{4}$$
$43$ $$T^{12} + 9600 T^{8} + \cdots + 15352201216$$
$47$ $$T^{12} + 4896 T^{8} + \cdots + 40960000$$
$53$ $$(T^{6} - 2 T^{5} + 2 T^{4} + 16 T^{3} + \cdots + 128)^{2}$$
$59$ $$(T^{6} - 100 T^{4} + 1860 T^{2} + \cdots - 512)^{2}$$
$61$ $$(T^{3} + 8 T^{2} - 100 T + 176)^{4}$$
$67$ $$T^{12} + 9600 T^{8} + \cdots + 15352201216$$
$71$ $$(T^{6} + 256 T^{4} + 15616 T^{2} + \cdots + 204800)^{2}$$
$73$ $$(T^{6} - 22 T^{5} + 242 T^{4} + \cdots + 55112)^{2}$$
$79$ $$(T^{6} - 304 T^{4} + 12864 T^{2} + \cdots - 2048)^{2}$$
$83$ $$T^{12} + 16672 T^{8} + 10264832 T^{4} + \cdots + 65536$$
$89$ $$(T^{6} + 72 T^{4} + 1040 T^{2} + \cdots + 1024)^{2}$$
$97$ $$(T^{6} + 10 T^{5} + 50 T^{4} - 2048 T^{3} + \cdots + 35912)^{2}$$
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