# Properties

 Label 585.2.i.e Level $585$ Weight $2$ Character orbit 585.i Analytic conductor $4.671$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + \cdots + 268$$ x^16 - 5*x^15 + 20*x^14 - 44*x^13 + 96*x^12 - 107*x^11 + 178*x^10 - 19*x^9 + 231*x^8 + 326*x^7 + 551*x^6 + 859*x^5 + 1118*x^4 + 1215*x^3 + 1103*x^2 + 770*x + 268 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + (\beta_{14} - \beta_{7}) q^{3} + (\beta_{15} - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{4} - \beta_{3} q^{5} + (\beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{4} - \beta_1 + 1) q^{6} + (\beta_{15} - \beta_{14} + \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{7} + ( - \beta_{14} + \beta_{7} + \beta_{2} + \beta_1 - 1) q^{8} + ( - \beta_{10} - \beta_{7} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q - b6 * q^2 + (b14 - b7) * q^3 + (b15 - b13 - b11 + b10 + b9 - b8 + b7 - b5 + b4 + b3 + b2 - b1) * q^4 - b3 * q^5 + (b12 - b11 + b9 + b7 - b4 - b1 + 1) * q^6 + (b15 - b14 + b7 - b6 - b4 + 2*b3 - b1 + 2) * q^7 + (-b14 + b7 + b2 + b1 - 1) * q^8 + (-b10 - b7 - b4 - 2*b3 - b2 + b1) * q^9 $$q - \beta_{6} q^{2} + (\beta_{14} - \beta_{7}) q^{3} + (\beta_{15} - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{4} - \beta_{3} q^{5} + (\beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{4} - \beta_1 + 1) q^{6} + (\beta_{15} - \beta_{14} + \beta_{7} - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{7} + ( - \beta_{14} + \beta_{7} + \beta_{2} + \beta_1 - 1) q^{8} + ( - \beta_{10} - \beta_{7} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{9} - \beta_{2} q^{10} + ( - \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{3} + 2 \beta_1 - 2) q^{11} + ( - \beta_{15} - \beta_{13} - \beta_{11} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{12} - \beta_{3} q^{13} + ( - \beta_{14} - \beta_{13} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{6} + \beta_{4} + 2 \beta_{3} + \cdots + \beta_1) q^{14}+ \cdots + ( - 3 \beta_{15} - \beta_{14} + 3 \beta_{13} - \beta_{12} + 4 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} + \cdots - 3) q^{99}+O(q^{100})$$ q - b6 * q^2 + (b14 - b7) * q^3 + (b15 - b13 - b11 + b10 + b9 - b8 + b7 - b5 + b4 + b3 + b2 - b1) * q^4 - b3 * q^5 + (b12 - b11 + b9 + b7 - b4 - b1 + 1) * q^6 + (b15 - b14 + b7 - b6 - b4 + 2*b3 - b1 + 2) * q^7 + (-b14 + b7 + b2 + b1 - 1) * q^8 + (-b10 - b7 - b4 - 2*b3 - b2 + b1) * q^9 - b2 * q^10 + (-b15 + b14 - b12 + b11 - b9 - b8 - b7 - b3 + 2*b1 - 2) * q^11 + (-b15 - b13 - b11 + b9 - b7 - b6 + b4 - b3 + b2 + b1) * q^12 - b3 * q^13 + (-b14 - b13 + b10 + b9 - b8 - 2*b6 + b4 + 2*b3 + 3*b2 + b1) * q^14 + b14 * q^15 + (-b12 + b11 - b9 - b8 + b6 + b4 - b3 + b1 - 2) * q^16 + (-b15 + 2*b14 - 2*b9 - b7 + b5 - b4 - b2 - b1 - 2) * q^17 + (-b15 + b14 - b13 + b12 + b11 - b9 - b7 - b6 + 2*b3 - 2*b2 + 1) * q^18 + (b15 + b14 - b13 - b12 - 2*b7 + b5 + b2 - b1 - 2) * q^19 + (b15 - b13 + b12 - b11 + b9 - b8 + b7 - b5 + b3 - b1 + 2) * q^20 + (-b15 - b14 + b13 + 2*b10 + b9 - b6 - b3 + b2) * q^21 + (b15 + b14 + 2*b13 - b12 + 2*b7 + b6 + b5 - b4 + b3 - b2 - 2*b1) * q^22 + (-b14 + b13 - b12 - 2*b11 + b10 + b9 - b8 + b7 + b5 + b2 + b1) * q^23 + (-b13 + b10 - b9 + b7 - b5 + 2*b4 + 2*b3 + b2 - b1 - 4) * q^24 + (-b3 - 1) * q^25 - b2 * q^26 + (-b15 + 2*b13 + b11 + 2*b7 - b6 - b5 - b4 - 1) * q^27 + (b14 - 2*b12 + 2*b10 - 3*b9 - b7 + 2*b4 + 3*b2 - b1 - 8) * q^28 + (b12 + b8 + 2*b6 + b4 - 2*b3 - b1 - 2) * q^29 + (-b14 + b13 + b9 + b7 + b5 - b4 + 1) * q^30 + (-b15 + b14 + b12 + 3*b11 - 2*b10 - 2*b9 + 2*b8 - 2*b6 - b4 - b3 - 2*b1) * q^31 + (-b15 + 3*b13 - b12 + 2*b11 - b10 - b9 + b8 - b6 + b5 - 2*b4 + b3 + b1) * q^32 + (-b15 - b14 + b13 - b12 + 2*b9 + b7 + 2*b6 - b5 - b3 + b2 + 2*b1 - 2) * q^33 + (4*b15 - 3*b14 - b12 - 4*b11 + 4*b9 - b8 + 4*b7 + 2*b6 - 2*b1 + 4) * q^34 + (b15 - b14 + b5 - b4 - b2 + 2) * q^35 + (2*b14 - b13 + b12 - 3*b11 - b10 - b8 + b7 - b6 - b5 - b4 + 5*b3 - 2*b1 + 3) * q^36 + (-b14 - b12 + b10 - b9 + b7 - b5 + 2*b4 - b2 + b1) * q^37 + (-b15 + 2*b14 - b13 + 3*b12 + b11 - b9 + b8 - b7 + 2*b6 - b5 - 2*b3 - b1 - 3) * q^38 + b14 * q^39 + (-b15 - b14 - b6 + b3 + b2 + b1) * q^40 + (-b14 + 3*b13 - 2*b12 + b10 + b9 - b8 + b7 - b6 - b4 - 2*b3 + 2*b2 + 2*b1) * q^41 + (b15 - 3*b13 + b12 - 6*b11 + b10 + 3*b9 - 2*b8 + 2*b7 + 2*b6 - 3*b5 + 2*b4 + b3 - 3*b1 + 1) * q^42 + (-b13 + b12 - 3*b11 + 3*b9 - b8 - b5 + 3*b4 - 2*b3 - b1 + 1) * q^43 + (b15 + b14 - b13 + 2*b12 - 3*b10 + 3*b9 - 2*b7 + 2*b5 - 4*b4 - 2*b2 - b1 + 5) * q^44 + (-b15 + b14 + b11 - b10 - b9 + b8 - 2*b7 + b5 - b4 - b3 - b2 + b1 - 2) * q^45 + (2*b15 - 2*b13 - 2*b12 + b9 - 2*b7 - b5 + 3*b4 + 3*b2 - 4) * q^46 + (2*b15 - 2*b14 + 2*b13 - 2*b12 + 2*b8 + 2*b7 - 2*b6 + 2*b5 - 2*b4 + 3*b3 + 3) * q^47 + (-b12 + b9 + b7 + 3*b6 - b5 + 2*b4 - 2*b3 + 2*b2 - 2) * q^48 + (b14 - b13 + b12 - b11 - b10 - b9 + b8 + 2*b7 - 2*b6 - b5 + 4*b3 + b2 - 4*b1) * q^49 + (b6 - b2) * q^50 + (-2*b14 - b13 - b12 + b11 + b10 + 2*b9 - 2*b8 + b6 - b4 + b2 + 3*b1 + 2) * q^51 + (b15 - b13 + b12 - b11 + b9 - b8 + b7 - b5 + b3 - b1 + 2) * q^52 + (-2*b15 - b14 + b13 - b12 + 2*b10 - b9 + 3*b7 - 2*b5 + 3*b4 + 2*b1 - 1) * q^53 + (2*b15 - 3*b14 + b13 - b12 - 2*b11 + b10 + 5*b9 - b8 + 2*b7 + b6 + 2*b5 - 2*b4 + 2*b3 - b1 + 4) * q^54 + (-b15 + 2*b14 - b13 - b10 - 2*b9 - b7 - b2 - 2) * q^55 + (3*b15 - b14 + b13 - 3*b12 - b11 + b9 - b8 + 3*b7 + 6*b6 + b5 + b4 - 4*b3 + 2*b1 - 3) * q^56 + (-2*b15 - b14 + b13 - 3*b12 - b11 - b9 - b8 - b4 - 4*b3 + 3*b1) * q^57 + (-2*b15 + 2*b13 + b11 - 2*b10 - 2*b9 + 2*b8 - 2*b7 + 3*b6 + b5 - 2*b4 - 7*b3 - 5*b2 + 2*b1) * q^58 + (b14 + b13 + 2*b11 - b10 - b9 + b8 + b7 + b6 + 2*b5 - b4 + 5*b3 - 2*b2 - 2*b1) * q^59 + (-b15 + b14 - b13 + b12 + b9 - b7 - b6 - b5 + b4 - b1) * q^60 + (-4*b15 + b13 - 2*b12 + 5*b11 - 5*b9 - 4*b7 + b5 - 2*b4 + 2*b3 + 2*b1 - 3) * q^61 + (-b14 + 2*b13 + 2*b10 + b9 + b7 + b5 - b4 - 4*b2 - b1 - 1) * q^62 + (3*b15 - b14 - b13 + b12 - 4*b11 + b10 + 4*b9 - 2*b8 + 2*b7 + b6 - 2*b2 + b1 + 7) * q^63 + (b15 + 2*b14 - b13 - b10 - 3*b7 - b2 - 2*b1 - 3) * q^64 + (-b3 - 1) * q^65 + (-4*b15 + b14 + b13 + b12 + 5*b11 - 2*b10 - 4*b9 + 3*b8 - 5*b7 + 2*b6 + 3*b5 - 2*b4 - 6*b3 - 3*b2 + 3*b1 - 2) * q^66 + (-3*b14 - b13 - b12 - 4*b11 + 3*b10 + 3*b9 - 3*b8 + 2*b7 + b6 - 5*b5 + 2*b4 - 6*b3 + 2*b2 + 2*b1) * q^67 + (2*b14 - 4*b13 + 3*b12 + 4*b11 - 2*b10 - 2*b9 + 2*b8 - 7*b7 - 3*b6 + 2*b5 + b4 + b2 + 2*b1) * q^68 + (-b15 + b14 - 3*b13 + 3*b12 - 2*b11 + b9 - 3*b8 + 2*b6 - 3*b5 + b4) * q^69 + (-b15 + b14 - b13 + b12 - b8 - b7 - 2*b6 - b5 + b4 + 2*b3 + 2) * q^70 + (2*b15 - 2*b14 + b13 - 2*b12 + 3*b10 + 2*b9 + 2*b4 + b2 - b1 + 2) * q^71 + (2*b15 - 4*b14 - 2*b13 - b12 - b11 + b10 + b9 + 2*b7 - b6 + b5 + 2*b4 + 2*b3 + 4*b2) * q^72 + (-b14 + 3*b13 + b12 + 2*b10 + 2*b9 + b7 - b4 - 2*b1 + 2) * q^73 + (4*b15 - b14 - 2*b12 - 2*b8 + 4*b7 + b6 + 5*b3 + b1 + 5) * q^74 + b7 * q^75 + (-b15 + b14 + 2*b13 - 2*b11 - 2*b10 - 2*b9 + 2*b8 + 2*b6 + 4*b5 - 2*b4 - 7*b3 - 4*b2 - b1) * q^76 + (b15 + 3*b13 - 2*b12 + 2*b11 + b10 + b9 - b8 - 3*b7 + 4*b5 - b4 - 5*b3 + b2 + 5*b1) * q^77 + (-b14 + b13 + b9 + b7 + b5 - b4 + 1) * q^78 + (2*b15 - 2*b14 - b13 + b12 - b8 + 2*b7 + 6*b6 - b5 + b4 - 3*b1) * q^79 + (b14 - b13 - b10 - 2*b9 - b7 - b5 + b4 - 2) * q^80 + (-b15 - b14 + b13 + 3*b12 + 3*b11 - b10 - b9 + 4*b8 + b5 - b4 - 2*b2 + 1) * q^81 + (b15 - b12 + b10 + 4*b9 - b7 + 2*b5 - b4 - b1 - 3) * q^82 + (-b15 - 2*b12 + b11 - b9 - 2*b8 - b7 - b6 - 2*b4 + b3 + 2*b1) * q^83 + (3*b15 - 5*b14 - 4*b13 - 2*b12 - 2*b11 + 2*b10 + 2*b9 - 3*b8 + 2*b7 - 2*b5 + 3*b4 - 4*b3 + 7*b2 + 5*b1 - 1) * q^84 + (b15 + b14 - 2*b11 + b7 + b6 + b5 - b2 - 2*b1) * q^85 + (3*b15 + 3*b14 - 6*b13 + 3*b12 - 2*b11 - 2*b7 - 2*b6 - 7*b5 + 3*b4 + 6*b3 + 2*b2 - 4*b1) * q^86 + (3*b15 + b14 - 2*b13 - b12 + b11 - b10 - 3*b9 + b5 + 2*b4 + b3 - b2 - b1) * q^87 + (-2*b15 + b14 - 2*b13 + 5*b12 + b11 - b9 + b8 - 2*b7 - 6*b6 - 2*b5 + b4 + 5*b3 - 4*b1 + 4) * q^88 + (-3*b15 + b14 + b13 + 3*b12 - 2*b10 + 2*b7 - 2*b5 - b4 - 3*b2 + b1) * q^89 + (-b15 + b14 + b12 - b9 + 2*b6 - 3*b2 - b1 + 1) * q^90 + (b15 - b14 + b5 - b4 - b2 + 2) * q^91 + (-3*b15 + 5*b14 + b12 + 3*b11 - 3*b9 + b8 - 3*b7 + 4*b6 + b4 - 3*b3 + 4*b1 - 6) * q^92 + (3*b14 + 2*b13 - b12 + 3*b11 - 4*b9 + 5*b8 - b7 - b6 + 3*b5 - 2*b4 + 3*b3 - b2 - 3*b1 + 2) * q^93 + (-2*b14 - 2*b13 + 4*b11 + 2*b10 + 2*b9 - 2*b8 - 2*b7 - b6 + 2*b5 + 2*b4 + 4*b3 + 3*b2 + 4*b1) * q^94 + (2*b15 + 2*b14 - 2*b13 + b12 - b7 - b6 + b4 + 2*b3 + b2 - 2*b1) * q^95 + (-b15 - b14 + 2*b13 + 2*b12 + 3*b11 - b10 - 3*b9 + 3*b8 - b7 + 2*b6 - 2*b4 - 3*b3 - 3*b2 - 1) * q^96 + (-3*b15 + b14 + 2*b13 + b12 + 2*b11 - 2*b9 + 5*b8 - 3*b7 - 3*b6 + 2*b5 - b4 + 5*b3 + 3) * q^97 + (4*b15 - 6*b14 - b13 - 3*b12 + 2*b10 + 2*b9 + 2*b7 + b5 + 2*b4 + 6*b2 + 3*b1 - 4) * q^98 + (-3*b15 - b14 + 3*b13 - b12 + 4*b11 - 2*b10 - 6*b9 + 4*b8 - b7 - 2*b6 + b5 + b4 + 3*b3 - 2*b2 + 3*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 3 q^{2} + q^{3} - 9 q^{4} + 8 q^{5} + 8 q^{6} + 11 q^{7} - 12 q^{8} + q^{9}+O(q^{10})$$ 16 * q - 3 * q^2 + q^3 - 9 * q^4 + 8 * q^5 + 8 * q^6 + 11 * q^7 - 12 * q^8 + q^9 $$16 q - 3 q^{2} + q^{3} - 9 q^{4} + 8 q^{5} + 8 q^{6} + 11 q^{7} - 12 q^{8} + q^{9} - 6 q^{10} - 6 q^{11} + 2 q^{12} + 8 q^{13} - 10 q^{14} + 5 q^{15} - 11 q^{16} - 4 q^{17} + 5 q^{18} - 20 q^{19} + 9 q^{20} + 11 q^{21} - 3 q^{22} - 6 q^{23} - 57 q^{24} - 8 q^{25} - 6 q^{26} - 14 q^{27} - 68 q^{28} - 14 q^{29} + 7 q^{30} + 31 q^{31} - q^{32} - 31 q^{33} + 7 q^{34} + 22 q^{35} + 2 q^{36} + 2 q^{37} - 9 q^{38} + 5 q^{39} - 6 q^{40} + 12 q^{41} - 26 q^{42} - 15 q^{43} + 32 q^{44} - 16 q^{45} - 64 q^{46} + 18 q^{47} - 4 q^{48} - 17 q^{49} - 3 q^{50} + 32 q^{51} + 9 q^{52} + 4 q^{53} + 2 q^{54} - 12 q^{55} - 16 q^{56} + 45 q^{57} + 42 q^{58} - 24 q^{59} - 8 q^{60} + 9 q^{61} - 40 q^{62} + 47 q^{63} - 60 q^{64} - 8 q^{65} + 55 q^{66} + 18 q^{67} + 14 q^{68} - 12 q^{69} + 10 q^{70} + 20 q^{71} - 9 q^{72} + 12 q^{73} + 37 q^{74} + 4 q^{75} + 53 q^{76} + 34 q^{77} + 7 q^{78} + 3 q^{79} - 22 q^{80} + 13 q^{81} - 68 q^{82} + 10 q^{83} + 22 q^{84} - 2 q^{85} - 60 q^{86} + 11 q^{87} + 14 q^{88} - 26 q^{89} + 19 q^{90} + 22 q^{91} - 5 q^{92} + 74 q^{93} - 17 q^{94} - 10 q^{95} + 13 q^{96} + 34 q^{97} - 60 q^{98} - 43 q^{99}+O(q^{100})$$ 16 * q - 3 * q^2 + q^3 - 9 * q^4 + 8 * q^5 + 8 * q^6 + 11 * q^7 - 12 * q^8 + q^9 - 6 * q^10 - 6 * q^11 + 2 * q^12 + 8 * q^13 - 10 * q^14 + 5 * q^15 - 11 * q^16 - 4 * q^17 + 5 * q^18 - 20 * q^19 + 9 * q^20 + 11 * q^21 - 3 * q^22 - 6 * q^23 - 57 * q^24 - 8 * q^25 - 6 * q^26 - 14 * q^27 - 68 * q^28 - 14 * q^29 + 7 * q^30 + 31 * q^31 - q^32 - 31 * q^33 + 7 * q^34 + 22 * q^35 + 2 * q^36 + 2 * q^37 - 9 * q^38 + 5 * q^39 - 6 * q^40 + 12 * q^41 - 26 * q^42 - 15 * q^43 + 32 * q^44 - 16 * q^45 - 64 * q^46 + 18 * q^47 - 4 * q^48 - 17 * q^49 - 3 * q^50 + 32 * q^51 + 9 * q^52 + 4 * q^53 + 2 * q^54 - 12 * q^55 - 16 * q^56 + 45 * q^57 + 42 * q^58 - 24 * q^59 - 8 * q^60 + 9 * q^61 - 40 * q^62 + 47 * q^63 - 60 * q^64 - 8 * q^65 + 55 * q^66 + 18 * q^67 + 14 * q^68 - 12 * q^69 + 10 * q^70 + 20 * q^71 - 9 * q^72 + 12 * q^73 + 37 * q^74 + 4 * q^75 + 53 * q^76 + 34 * q^77 + 7 * q^78 + 3 * q^79 - 22 * q^80 + 13 * q^81 - 68 * q^82 + 10 * q^83 + 22 * q^84 - 2 * q^85 - 60 * q^86 + 11 * q^87 + 14 * q^88 - 26 * q^89 + 19 * q^90 + 22 * q^91 - 5 * q^92 + 74 * q^93 - 17 * q^94 - 10 * q^95 + 13 * q^96 + 34 * q^97 - 60 * q^98 - 43 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 5 x^{15} + 20 x^{14} - 44 x^{13} + 96 x^{12} - 107 x^{11} + 178 x^{10} - 19 x^{9} + 231 x^{8} + 326 x^{7} + 551 x^{6} + 859 x^{5} + 1118 x^{4} + 1215 x^{3} + 1103 x^{2} + \cdots + 268$$ :

 $$\beta_{1}$$ $$=$$ $$( - 89 \nu^{15} - 1135 \nu^{14} + 7576 \nu^{13} - 31264 \nu^{12} + 72320 \nu^{11} - 147579 \nu^{10} + 190360 \nu^{9} - 259593 \nu^{8} + 151587 \nu^{7} - 267712 \nu^{6} + \cdots + 194918 ) / 205578$$ (-89*v^15 - 1135*v^14 + 7576*v^13 - 31264*v^12 + 72320*v^11 - 147579*v^10 + 190360*v^9 - 259593*v^8 + 151587*v^7 - 267712*v^6 - 48005*v^5 - 224641*v^4 - 207308*v^3 + 176473*v^2 - 45495*v + 194918) / 205578 $$\beta_{2}$$ $$=$$ $$( \nu^{15} - 4 \nu^{14} + 16 \nu^{13} - 28 \nu^{12} + 68 \nu^{11} - 39 \nu^{10} + 139 \nu^{9} + 120 \nu^{8} + 351 \nu^{7} + 677 \nu^{6} + 1228 \nu^{5} + 2087 \nu^{4} + 3205 \nu^{3} + 4420 \nu^{2} + \cdots + 6293 ) / 2187$$ (v^15 - 4*v^14 + 16*v^13 - 28*v^12 + 68*v^11 - 39*v^10 + 139*v^9 + 120*v^8 + 351*v^7 + 677*v^6 + 1228*v^5 + 2087*v^4 + 3205*v^3 + 4420*v^2 + 3336*v + 6293) / 2187 $$\beta_{3}$$ $$=$$ $$( - 53 \nu^{15} + 1221 \nu^{14} - 4880 \nu^{13} + 14280 \nu^{12} - 18634 \nu^{11} + 24185 \nu^{10} + 23394 \nu^{9} - 18231 \nu^{8} + 152463 \nu^{7} + 42206 \nu^{6} + \cdots + 125592 ) / 68526$$ (-53*v^15 + 1221*v^14 - 4880*v^13 + 14280*v^12 - 18634*v^11 + 24185*v^10 + 23394*v^9 - 18231*v^8 + 152463*v^7 + 42206*v^6 + 312111*v^5 + 354893*v^4 + 509676*v^3 + 567013*v^2 + 436375*v + 125592) / 68526 $$\beta_{4}$$ $$=$$ $$( - 265 \nu^{15} - 851 \nu^{14} - 2404 \nu^{13} + 18478 \nu^{12} - 122780 \nu^{11} + 267189 \nu^{10} - 630706 \nu^{9} + 600027 \nu^{8} - 1096911 \nu^{7} - 12878 \nu^{6} + \cdots - 1138394 ) / 205578$$ (-265*v^15 - 851*v^14 - 2404*v^13 + 18478*v^12 - 122780*v^11 + 267189*v^10 - 630706*v^9 + 600027*v^8 - 1096911*v^7 - 12878*v^6 - 1628113*v^5 - 1542137*v^4 - 2561554*v^3 - 2836237*v^2 - 2196081*v - 1138394) / 205578 $$\beta_{5}$$ $$=$$ $$( - 547 \nu^{15} - 2825 \nu^{14} + 11696 \nu^{13} - 42716 \nu^{12} + 9478 \nu^{11} + 20721 \nu^{10} - 417232 \nu^{9} + 363147 \nu^{8} - 1222965 \nu^{7} - 271472 \nu^{6} + \cdots - 568472 ) / 205578$$ (-547*v^15 - 2825*v^14 + 11696*v^13 - 42716*v^12 + 9478*v^11 + 20721*v^10 - 417232*v^9 + 363147*v^8 - 1222965*v^7 - 271472*v^6 - 2073673*v^5 - 2132363*v^4 - 3222844*v^3 - 3007411*v^2 - 2176341*v - 568472) / 205578 $$\beta_{6}$$ $$=$$ $$( 685 \nu^{15} - 2275 \nu^{14} + 10492 \nu^{13} - 25438 \nu^{12} + 90950 \nu^{11} - 164457 \nu^{10} + 406510 \nu^{9} - 423375 \nu^{8} + 877113 \nu^{7} - 182266 \nu^{6} + \cdots + 752246 ) / 205578$$ (685*v^15 - 2275*v^14 + 10492*v^13 - 25438*v^12 + 90950*v^11 - 164457*v^10 + 406510*v^9 - 423375*v^8 + 877113*v^7 - 182266*v^6 + 1218265*v^5 + 955763*v^4 + 1546546*v^3 + 1686079*v^2 + 1281639*v + 752246) / 205578 $$\beta_{7}$$ $$=$$ $$( - 556 \nu^{15} + 910 \nu^{14} + 1274 \nu^{13} - 22988 \nu^{12} + 57808 \nu^{11} - 136806 \nu^{10} + 137726 \nu^{9} - 222348 \nu^{8} + 10989 \nu^{7} - 315041 \nu^{6} + \cdots + 188851 ) / 102789$$ (-556*v^15 + 910*v^14 + 1274*v^13 - 22988*v^12 + 57808*v^11 - 136806*v^10 + 137726*v^9 - 222348*v^8 + 10989*v^7 - 315041*v^6 - 213625*v^5 - 501563*v^4 - 330217*v^3 - 274471*v^2 - 28941*v + 188851) / 102789 $$\beta_{8}$$ $$=$$ $$( 1099 \nu^{15} - 4885 \nu^{14} + 28834 \nu^{13} - 92848 \nu^{12} + 290714 \nu^{11} - 524601 \nu^{10} + 991078 \nu^{9} - 865473 \nu^{8} + 1367433 \nu^{7} + 197588 \nu^{6} + \cdots + 1087838 ) / 205578$$ (1099*v^15 - 4885*v^14 + 28834*v^13 - 92848*v^12 + 290714*v^11 - 524601*v^10 + 991078*v^9 - 865473*v^8 + 1367433*v^7 + 197588*v^6 + 1766167*v^5 + 2184713*v^4 + 3032698*v^3 + 3272689*v^2 + 2521281*v + 1087838) / 205578 $$\beta_{9}$$ $$=$$ $$( - 14 \nu^{15} + 44 \nu^{14} - 53 \nu^{13} - 331 \nu^{12} + 1253 \nu^{11} - 3357 \nu^{10} + 4663 \nu^{9} - 6243 \nu^{8} + 2448 \nu^{7} - 3007 \nu^{6} - 4508 \nu^{5} + 1643 \nu^{4} + \cdots + 3587 ) / 2187$$ (-14*v^15 + 44*v^14 - 53*v^13 - 331*v^12 + 1253*v^11 - 3357*v^10 + 4663*v^9 - 6243*v^8 + 2448*v^7 - 3007*v^6 - 4508*v^5 + 1643*v^4 - 1982*v^3 + 5971*v^2 + 8634*v + 3587) / 2187 $$\beta_{10}$$ $$=$$ $$( 195 \nu^{15} - 931 \nu^{14} + 2748 \nu^{13} - 3688 \nu^{12} + 3582 \nu^{11} + 2953 \nu^{10} - 3464 \nu^{9} + 5595 \nu^{8} + 21477 \nu^{7} - 29046 \nu^{6} + 43807 \nu^{5} + \cdots - 138346 ) / 22842$$ (195*v^15 - 931*v^14 + 2748*v^13 - 3688*v^12 + 3582*v^11 + 2953*v^10 - 3464*v^9 + 5595*v^8 + 21477*v^7 - 29046*v^6 + 43807*v^5 - 79347*v^4 - 86552*v^3 - 151197*v^2 - 216913*v - 138346) / 22842 $$\beta_{11}$$ $$=$$ $$( - 938 \nu^{15} + 5045 \nu^{14} - 19022 \nu^{13} + 40454 \nu^{12} - 78373 \nu^{11} + 83322 \nu^{10} - 119705 \nu^{9} + 16041 \nu^{8} - 150696 \nu^{7} - 141799 \nu^{6} + \cdots - 220840 ) / 102789$$ (-938*v^15 + 5045*v^14 - 19022*v^13 + 40454*v^12 - 78373*v^11 + 83322*v^10 - 119705*v^9 + 16041*v^8 - 150696*v^7 - 141799*v^6 - 375791*v^5 - 287764*v^4 - 648728*v^3 - 371333*v^2 - 454875*v - 220840) / 102789 $$\beta_{12}$$ $$=$$ $$( 405 \nu^{15} - 2212 \nu^{14} + 6369 \nu^{13} - 7450 \nu^{12} - 3732 \nu^{11} + 40360 \nu^{10} - 89195 \nu^{9} + 120006 \nu^{8} - 108444 \nu^{7} + 31020 \nu^{6} - 74402 \nu^{5} + \cdots - 177730 ) / 34263$$ (405*v^15 - 2212*v^14 + 6369*v^13 - 7450*v^12 - 3732*v^11 + 40360*v^10 - 89195*v^9 + 120006*v^8 - 108444*v^7 + 31020*v^6 - 74402*v^5 - 167943*v^4 - 271589*v^3 - 340359*v^2 - 435703*v - 177730) / 34263 $$\beta_{13}$$ $$=$$ $$( 523 \nu^{15} - 1973 \nu^{14} + 5068 \nu^{13} - 650 \nu^{12} - 10177 \nu^{11} + 56326 \nu^{10} - 68467 \nu^{9} + 139773 \nu^{8} - 14142 \nu^{7} + 172772 \nu^{6} + 180167 \nu^{5} + \cdots - 53165 ) / 34263$$ (523*v^15 - 1973*v^14 + 5068*v^13 - 650*v^12 - 10177*v^11 + 56326*v^10 - 68467*v^9 + 139773*v^8 - 14142*v^7 + 172772*v^6 + 180167*v^5 + 203561*v^4 + 277010*v^3 + 101233*v^2 + 19619*v - 53165) / 34263 $$\beta_{14}$$ $$=$$ $$( - 4021 \nu^{15} + 14221 \nu^{14} - 42610 \nu^{13} + 32740 \nu^{12} - 46970 \nu^{11} - 188259 \nu^{10} + 63962 \nu^{9} - 750819 \nu^{8} - 383103 \nu^{7} - 1812884 \nu^{6} + \cdots - 512798 ) / 205578$$ (-4021*v^15 + 14221*v^14 - 42610*v^13 + 32740*v^12 - 46970*v^11 - 188259*v^10 + 63962*v^9 - 750819*v^8 - 383103*v^7 - 1812884*v^6 - 1944757*v^5 - 3449927*v^4 - 3468346*v^3 - 3279607*v^2 - 1943709*v - 512798) / 205578 $$\beta_{15}$$ $$=$$ $$( 4721 \nu^{15} - 27797 \nu^{14} + 107414 \nu^{13} - 251234 \nu^{12} + 501802 \nu^{11} - 668505 \nu^{10} + 915602 \nu^{9} - 643377 \nu^{8} + 951039 \nu^{7} + \cdots - 191036 ) / 205578$$ (4721*v^15 - 27797*v^14 + 107414*v^13 - 251234*v^12 + 501802*v^11 - 668505*v^10 + 915602*v^9 - 643377*v^8 + 951039*v^7 + 92026*v^6 + 839147*v^5 + 597901*v^4 + 482192*v^3 + 7385*v^2 - 493101*v - 191036) / 205578
 $$\nu$$ $$=$$ $$( - \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} + 2 \beta _1 + 1 ) / 3$$ (-b14 + b13 - 2*b12 + b10 + b9 - 2*b8 + b7 + b6 - b3 - b2 + 2*b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{15} + 2 \beta_{13} - \beta_{12} + 3 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{2} + 2 \beta _1 - 3 ) / 3$$ (b15 + 2*b13 - b12 + 3*b11 - b10 - b9 - b8 + b7 + b5 - 2*b4 - b2 + 2*b1 - 3) / 3 $$\nu^{3}$$ $$=$$ $$( 2 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + 4 \beta_{8} - \beta_{7} + 5 \beta_{5} - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 2 \beta _1 - 12 ) / 3$$ (2*b15 + 3*b14 + 4*b13 - 2*b12 + 6*b11 - 2*b10 - 5*b9 + 4*b8 - b7 + 5*b5 - b4 + 3*b3 + 4*b2 - 2*b1 - 12) / 3 $$\nu^{4}$$ $$=$$ $$( - 2 \beta_{15} + \beta_{14} + 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 7 \beta_{10} - 2 \beta_{9} + 7 \beta_{8} - 3 \beta_{7} + 5 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} + 21 \beta_{2} - 31 ) / 3$$ (-2*b15 + b14 + 4*b13 - 8*b12 + 3*b11 + 7*b10 - 2*b9 + 7*b8 - 3*b7 + 5*b6 + 7*b5 + 7*b4 - 2*b3 + 21*b2 - 31) / 3 $$\nu^{5}$$ $$=$$ $$( - 14 \beta_{15} - 10 \beta_{14} - 12 \beta_{13} - 6 \beta_{12} - 21 \beta_{11} + 21 \beta_{10} + 12 \beta_{9} - \beta_{7} + 10 \beta_{6} - 14 \beta_{5} + 22 \beta_{4} - 28 \beta_{3} + 43 \beta_{2} + 7 \beta _1 - 56 ) / 3$$ (-14*b15 - 10*b14 - 12*b13 - 6*b12 - 21*b11 + 21*b10 + 12*b9 - b7 + 10*b6 - 14*b5 + 22*b4 - 28*b3 + 43*b2 + 7*b1 - 56) / 3 $$\nu^{6}$$ $$=$$ $$( - 21 \beta_{15} - 23 \beta_{14} - 76 \beta_{13} + 41 \beta_{12} - 78 \beta_{11} + 8 \beta_{10} + 23 \beta_{9} - 16 \beta_{8} - \beta_{7} - 10 \beta_{6} - 72 \beta_{5} + 36 \beta_{4} - 35 \beta_{3} + 25 \beta_{2} - 11 \beta _1 - 13 ) / 3$$ (-21*b15 - 23*b14 - 76*b13 + 41*b12 - 78*b11 + 8*b10 + 23*b9 - 16*b8 - b7 - 10*b6 - 72*b5 + 36*b4 - 35*b3 + 25*b2 - 11*b1 - 13) / 3 $$\nu^{7}$$ $$=$$ $$( 8 \beta_{15} - 14 \beta_{14} - 183 \beta_{13} + 156 \beta_{12} - 153 \beta_{11} - 60 \beta_{10} + 18 \beta_{9} - 36 \beta_{8} - 5 \beta_{7} - 106 \beta_{6} - 133 \beta_{5} + 14 \beta_{4} + 103 \beta_{3} - 115 \beta_{2} - 88 \beta _1 + 299 ) / 3$$ (8*b15 - 14*b14 - 183*b13 + 156*b12 - 153*b11 - 60*b10 + 18*b9 - 36*b8 - 5*b7 - 106*b6 - 133*b5 + 14*b4 + 103*b3 - 115*b2 - 88*b1 + 299) / 3 $$\nu^{8}$$ $$=$$ $$36 \beta_{15} + 16 \beta_{14} - 56 \beta_{13} + 89 \beta_{12} - 45 \beta_{11} - 52 \beta_{10} + 19 \beta_{9} - 28 \beta_{8} + 7 \beta_{7} - 103 \beta_{6} - 32 \beta_{5} - 42 \beta_{4} + 163 \beta_{3} - 143 \beta_{2} - 62 \beta _1 + 368$$ 36*b15 + 16*b14 - 56*b13 + 89*b12 - 45*b11 - 52*b10 + 19*b9 - 28*b8 + 7*b7 - 103*b6 - 32*b5 - 42*b4 + 163*b3 - 143*b2 - 62*b1 + 368 $$\nu^{9}$$ $$=$$ $$( 267 \beta_{15} + 200 \beta_{14} + 415 \beta_{13} + 136 \beta_{12} + 309 \beta_{11} - 194 \beta_{10} + 256 \beta_{9} - 179 \beta_{8} + 139 \beta_{7} - 407 \beta_{6} + 258 \beta_{5} - 534 \beta_{4} + 875 \beta_{3} - 835 \beta_{2} + \cdots + 2065 ) / 3$$ (267*b15 + 200*b14 + 415*b13 + 136*b12 + 309*b11 - 194*b10 + 256*b9 - 179*b8 + 139*b7 - 407*b6 + 258*b5 - 534*b4 + 875*b3 - 835*b2 - 172*b1 + 2065) / 3 $$\nu^{10}$$ $$=$$ $$( 271 \beta_{15} + 495 \beta_{14} + 2219 \beta_{13} - 718 \beta_{12} + 1797 \beta_{11} - 94 \beta_{10} + 401 \beta_{9} - 25 \beta_{8} + 187 \beta_{7} + 483 \beta_{6} + 1432 \beta_{5} - 1343 \beta_{4} + 15 \beta_{3} - 973 \beta_{2} + \cdots + 966 ) / 3$$ (271*b15 + 495*b14 + 2219*b13 - 718*b12 + 1797*b11 - 94*b10 + 401*b9 - 25*b8 + 187*b7 + 483*b6 + 1432*b5 - 1343*b4 + 15*b3 - 973*b2 + 200*b1 + 966) / 3 $$\nu^{11}$$ $$=$$ $$( - 562 \beta_{15} + 819 \beta_{14} + 5284 \beta_{13} - 3086 \beta_{12} + 4665 \beta_{11} + 250 \beta_{10} - 785 \beta_{9} + 1402 \beta_{8} - 643 \beta_{7} + 4206 \beta_{6} + 3965 \beta_{5} - 1933 \beta_{4} + \cdots - 7629 ) / 3$$ (-562*b15 + 819*b14 + 5284*b13 - 3086*b12 + 4665*b11 + 250*b10 - 785*b9 + 1402*b8 - 643*b7 + 4206*b6 + 3965*b5 - 1933*b4 - 5007*b3 + 145*b2 + 1642*b1 - 7629) / 3 $$\nu^{12}$$ $$=$$ $$( - 3293 \beta_{15} + 13 \beta_{14} + 6391 \beta_{13} - 7562 \beta_{12} + 6747 \beta_{11} + 1306 \beta_{10} - 5876 \beta_{9} + 5200 \beta_{8} - 3246 \beta_{7} + 12074 \beta_{6} + 6283 \beta_{5} + 1096 \beta_{4} + \cdots - 30547 ) / 3$$ (-3293*b15 + 13*b14 + 6391*b13 - 7562*b12 + 6747*b11 + 1306*b10 - 5876*b9 + 5200*b8 - 3246*b7 + 12074*b6 + 6283*b5 + 1096*b4 - 17279*b3 + 4887*b2 + 5874*b1 - 30547) / 3 $$\nu^{13}$$ $$=$$ $$( - 7271 \beta_{15} - 5467 \beta_{14} - 6057 \beta_{13} - 11703 \beta_{12} - 993 \beta_{11} + 4242 \beta_{10} - 16398 \beta_{9} + 8862 \beta_{8} - 5269 \beta_{7} + 17608 \beta_{6} - 410 \beta_{5} + 15937 \beta_{4} + \cdots - 63437 ) / 3$$ (-7271*b15 - 5467*b14 - 6057*b13 - 11703*b12 - 993*b11 + 4242*b10 - 16398*b9 + 8862*b8 - 5269*b7 + 17608*b6 - 410*b5 + 15937*b4 - 30142*b3 + 17758*b2 + 13723*b1 - 63437) / 3 $$\nu^{14}$$ $$=$$ $$( - 4191 \beta_{15} - 20177 \beta_{14} - 55156 \beta_{13} - 2431 \beta_{12} - 38424 \beta_{11} + 10259 \beta_{10} - 23467 \beta_{9} - 1204 \beta_{8} + 4397 \beta_{7} - 9829 \beta_{6} - 36678 \beta_{5} + \cdots - 56845 ) / 3$$ (-4191*b15 - 20177*b14 - 55156*b13 - 2431*b12 - 38424*b11 + 10259*b10 - 23467*b9 - 1204*b8 + 4397*b7 - 9829*b6 - 36678*b5 + 50877*b4 - 1727*b3 + 43735*b2 + 15172*b1 - 56845) / 3 $$\nu^{15}$$ $$=$$ $$( 28025 \beta_{15} - 38420 \beta_{14} - 157983 \beta_{13} + 55743 \beta_{12} - 129234 \beta_{11} + 17493 \beta_{10} + 15225 \beta_{9} - 55227 \beta_{8} + 45550 \beta_{7} - 138565 \beta_{6} + \cdots + 136049 ) / 3$$ (28025*b15 - 38420*b14 - 157983*b13 + 55743*b12 - 129234*b11 + 17493*b10 + 15225*b9 - 55227*b8 + 45550*b7 - 138565*b6 - 124777*b5 + 88751*b4 + 172129*b3 + 75692*b2 - 31399*b1 + 136049) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
196.1
 −0.724143 − 0.165319i −0.628312 + 0.590424i −0.317019 − 1.12493i 0.172467 − 1.52157i 0.252952 + 1.56266i 0.466399 + 1.64781i 1.48460 − 1.66288i 1.79305 + 1.53983i −0.724143 + 0.165319i −0.628312 − 0.590424i −0.317019 + 1.12493i 0.172467 + 1.52157i 0.252952 − 1.56266i 0.466399 − 1.64781i 1.48460 + 1.66288i 1.79305 − 1.53983i
−1.22414 + 2.12028i −1.71693 0.228383i −1.99705 3.45900i 0.500000 + 0.866025i 2.58600 3.36079i 1.96726 3.40740i 4.88214 2.89568 + 0.784233i −2.44829
196.2 −1.12831 + 1.95429i −0.604199 1.62325i −1.54617 2.67805i 0.500000 + 0.866025i 3.85403 + 0.650751i −0.353285 + 0.611908i 2.46502 −2.26989 + 1.96153i −2.25662
196.3 −0.817019 + 1.41512i 1.40359 + 1.01486i −0.335039 0.580304i 0.500000 + 0.866025i −2.58290 + 1.15709i −1.06506 + 1.84473i −2.17314 0.940129 + 2.84889i −1.63404
196.4 −0.327533 + 0.567303i 0.997116 1.41625i 0.785445 + 1.36043i 0.500000 + 0.866025i 0.476854 + 1.02954i 0.388592 0.673061i −2.33917 −1.01152 2.82433i −0.655066
196.5 −0.247048 + 0.427900i 0.674928 + 1.59514i 0.877935 + 1.52063i 0.500000 + 0.866025i −0.849299 0.105275i 2.14269 3.71125i −1.85576 −2.08895 + 2.15321i −0.494096
196.6 −0.0336011 + 0.0581988i −0.332146 1.69991i 0.997742 + 1.72814i 0.500000 + 0.866025i 0.110093 + 0.0377882i −1.23179 + 2.13352i −0.268505 −2.77936 + 1.12923i −0.0672022
196.7 0.984603 1.70538i −1.60495 0.651266i −0.938888 1.62620i 0.500000 + 0.866025i −2.69089 + 2.09581i 1.51414 2.62256i 0.240686 2.15171 + 2.09049i 1.96921
196.8 1.29305 2.23963i 1.68259 + 0.410979i −2.34397 4.05987i 0.500000 + 0.866025i 3.09611 3.23696i 2.13745 3.70217i −6.95128 2.66219 + 1.38302i 2.58610
391.1 −1.22414 2.12028i −1.71693 + 0.228383i −1.99705 + 3.45900i 0.500000 0.866025i 2.58600 + 3.36079i 1.96726 + 3.40740i 4.88214 2.89568 0.784233i −2.44829
391.2 −1.12831 1.95429i −0.604199 + 1.62325i −1.54617 + 2.67805i 0.500000 0.866025i 3.85403 0.650751i −0.353285 0.611908i 2.46502 −2.26989 1.96153i −2.25662
391.3 −0.817019 1.41512i 1.40359 1.01486i −0.335039 + 0.580304i 0.500000 0.866025i −2.58290 1.15709i −1.06506 1.84473i −2.17314 0.940129 2.84889i −1.63404
391.4 −0.327533 0.567303i 0.997116 + 1.41625i 0.785445 1.36043i 0.500000 0.866025i 0.476854 1.02954i 0.388592 + 0.673061i −2.33917 −1.01152 + 2.82433i −0.655066
391.5 −0.247048 0.427900i 0.674928 1.59514i 0.877935 1.52063i 0.500000 0.866025i −0.849299 + 0.105275i 2.14269 + 3.71125i −1.85576 −2.08895 2.15321i −0.494096
391.6 −0.0336011 0.0581988i −0.332146 + 1.69991i 0.997742 1.72814i 0.500000 0.866025i 0.110093 0.0377882i −1.23179 2.13352i −0.268505 −2.77936 1.12923i −0.0672022
391.7 0.984603 + 1.70538i −1.60495 + 0.651266i −0.938888 + 1.62620i 0.500000 0.866025i −2.69089 2.09581i 1.51414 + 2.62256i 0.240686 2.15171 2.09049i 1.96921
391.8 1.29305 + 2.23963i 1.68259 0.410979i −2.34397 + 4.05987i 0.500000 0.866025i 3.09611 + 3.23696i 2.13745 + 3.70217i −6.95128 2.66219 1.38302i 2.58610
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.i.e 16
3.b odd 2 1 1755.2.i.f 16
9.c even 3 1 inner 585.2.i.e 16
9.c even 3 1 5265.2.a.bf 8
9.d odd 6 1 1755.2.i.f 16
9.d odd 6 1 5265.2.a.ba 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.i.e 16 1.a even 1 1 trivial
585.2.i.e 16 9.c even 3 1 inner
1755.2.i.f 16 3.b odd 2 1
1755.2.i.f 16 9.d odd 6 1
5265.2.a.ba 8 9.d odd 6 1
5265.2.a.bf 8 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$:

 $$T_{2}^{16} + 3 T_{2}^{15} + 17 T_{2}^{14} + 38 T_{2}^{13} + 158 T_{2}^{12} + 324 T_{2}^{11} + 875 T_{2}^{10} + 1368 T_{2}^{9} + 2812 T_{2}^{8} + 3873 T_{2}^{7} + 5539 T_{2}^{6} + 4627 T_{2}^{5} + 3027 T_{2}^{4} + 1114 T_{2}^{3} + \cdots + 1$$ T2^16 + 3*T2^15 + 17*T2^14 + 38*T2^13 + 158*T2^12 + 324*T2^11 + 875*T2^10 + 1368*T2^9 + 2812*T2^8 + 3873*T2^7 + 5539*T2^6 + 4627*T2^5 + 3027*T2^4 + 1114*T2^3 + 295*T2^2 + 19*T2 + 1 $$T_{7}^{16} - 11 T_{7}^{15} + 97 T_{7}^{14} - 476 T_{7}^{13} + 2127 T_{7}^{12} - 6158 T_{7}^{11} + 20644 T_{7}^{10} - 44604 T_{7}^{9} + 140146 T_{7}^{8} - 191361 T_{7}^{7} + 544052 T_{7}^{6} - 359094 T_{7}^{5} + \cdots + 395641$$ T7^16 - 11*T7^15 + 97*T7^14 - 476*T7^13 + 2127*T7^12 - 6158*T7^11 + 20644*T7^10 - 44604*T7^9 + 140146*T7^8 - 191361*T7^7 + 544052*T7^6 - 359094*T7^5 + 1614395*T7^4 - 251056*T7^3 + 856340*T7^2 - 18870*T7 + 395641

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 3 T^{15} + 17 T^{14} + 38 T^{13} + \cdots + 1$$
$3$ $$T^{16} - T^{15} + 5 T^{13} - 8 T^{12} + \cdots + 6561$$
$5$ $$(T^{2} - T + 1)^{8}$$
$7$ $$T^{16} - 11 T^{15} + 97 T^{14} + \cdots + 395641$$
$11$ $$T^{16} + 6 T^{15} + 74 T^{14} + \cdots + 401956$$
$13$ $$(T^{2} - T + 1)^{8}$$
$17$ $$(T^{8} + 2 T^{7} - 78 T^{6} - 73 T^{5} + \cdots + 892)^{2}$$
$19$ $$(T^{8} + 10 T^{7} - 37 T^{6} - 541 T^{5} + \cdots - 1584)^{2}$$
$23$ $$T^{16} + 6 T^{15} + 115 T^{14} + \cdots + 144360225$$
$29$ $$T^{16} + 14 T^{15} + 191 T^{14} + \cdots + 1243225$$
$31$ $$T^{16} - 31 T^{15} + \cdots + 2547422784$$
$37$ $$(T^{8} - T^{7} - 120 T^{6} - 211 T^{5} + \cdots - 33660)^{2}$$
$41$ $$T^{16} - 12 T^{15} + \cdots + 371525625$$
$43$ $$T^{16} + 15 T^{15} + \cdots + 62415940941376$$
$47$ $$T^{16} - 18 T^{15} + \cdots + 424482219529$$
$53$ $$(T^{8} - 2 T^{7} - 282 T^{6} + \cdots + 111438)^{2}$$
$59$ $$T^{16} + 24 T^{15} + \cdots + 3524698346724$$
$61$ $$T^{16} + \cdots + 106354483234561$$
$67$ $$T^{16} + \cdots + 974523752673769$$
$71$ $$(T^{8} - 10 T^{7} - 211 T^{6} + \cdots - 127950)^{2}$$
$73$ $$(T^{8} - 6 T^{7} - 300 T^{6} + \cdots + 1196532)^{2}$$
$79$ $$T^{16} - 3 T^{15} + \cdots + 17\!\cdots\!04$$
$83$ $$T^{16} - 10 T^{15} + 215 T^{14} + \cdots + 469225$$
$89$ $$(T^{8} + 13 T^{7} - 228 T^{6} + \cdots + 1032219)^{2}$$
$97$ $$T^{16} + \cdots + 122483314259524$$