# Properties

 Label 585.2.i.f 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: $$x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} - 9 x^{7} + 1221 x^{6} + 133 x^{5} + 1245 x^{4} - 14 x^{3} + 391 x^{2} + 63 x + 81$$ x^16 - x^15 + 11*x^14 - 4*x^13 + 74*x^12 - 18*x^11 + 289*x^10 - 4*x^9 + 784*x^8 - 9*x^7 + 1221*x^6 + 133*x^5 + 1245*x^4 - 14*x^3 + 391*x^2 + 63*x + 81 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + 11 x^{14} - 4 x^{13} + 74 x^{12} - 18 x^{11} + 289 x^{10} - 4 x^{9} + 784 x^{8} - 9 x^{7} + 1221 x^{6} + 133 x^{5} + 1245 x^{4} - 14 x^{3} + 391 x^{2} + 63 x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 482086147342 \nu^{15} - 82999536459994 \nu^{14} + 51742196981879 \nu^{13} - 899262551344468 \nu^{12} + \cdots - 15\!\cdots\!53 ) / 91\!\cdots\!73$$ (-482086147342*v^15 - 82999536459994*v^14 + 51742196981879*v^13 - 899262551344468*v^12 + 29888665074896*v^11 - 6008483088362072*v^10 - 655850999572883*v^9 - 22854640199651823*v^8 - 6835814771962126*v^7 - 61053441921082007*v^6 - 16621010179418725*v^5 - 88736525406721413*v^4 - 21804108527677176*v^3 - 85814814008979952*v^2 - 8427487924594911*v - 15267025981692153) / 9139478381478873 $$\beta_{3}$$ $$=$$ $$( - 3469225716849 \nu^{15} + 7812059681008 \nu^{14} - 8290755936386 \nu^{13} - 2697108934580 \nu^{12} + \cdots + 12\!\cdots\!15 ) / 91\!\cdots\!73$$ (-3469225716849*v^15 + 7812059681008*v^14 - 8290755936386*v^13 - 2697108934580*v^12 + 189377660393089*v^11 + 16556478346708*v^10 + 1498667977954772*v^9 + 543181503433067*v^8 + 6528933517925721*v^7 + 5162807733352603*v^6 + 13241752875048308*v^5 + 18517847455679833*v^4 + 17630686105303827*v^3 + 28007951530950219*v^2 + 82246147650769*v + 12833642664201015) / 9139478381478873 $$\beta_{4}$$ $$=$$ $$( 6246753964615 \nu^{15} - 59652178722282 \nu^{14} + 157853267854272 \nu^{13} - 554550721812852 \nu^{12} + \cdots + 188168453461782 ) / 91\!\cdots\!73$$ (6246753964615*v^15 - 59652178722282*v^14 + 157853267854272*v^13 - 554550721812852*v^12 + 939963055735038*v^11 - 3319089373556393*v^10 + 5043192392654460*v^9 - 11225648306363066*v^8 + 13576539103281376*v^7 - 26605135287110714*v^6 + 36230288966505302*v^5 - 33863989763810635*v^4 + 36951118425472407*v^3 - 29817520384644188*v^2 + 47347580732583266*v + 188168453461782) / 9139478381478873 $$\beta_{5}$$ $$=$$ $$( - 10150138176959 \nu^{15} + 69644913861 \nu^{14} - 83292291525414 \nu^{13} - 118768769872326 \nu^{12} + \cdots + 10\!\cdots\!11 ) / 91\!\cdots\!73$$ (-10150138176959*v^15 + 69644913861*v^14 - 83292291525414*v^13 - 118768769872326*v^12 - 568340771522781*v^11 - 634932225031991*v^10 - 2850156002058894*v^9 - 2287548272381948*v^8 - 10896063663200009*v^7 - 3020771374915802*v^6 - 28231418467650946*v^5 + 346318987982825*v^4 - 43280949748322271*v^3 + 1711574395907992*v^2 - 26296054994954299*v + 10022852458122111) / 9139478381478873 $$\beta_{6}$$ $$=$$ $$( 16744116205687 \nu^{15} - 45341940014915 \nu^{14} + 176286820995922 \nu^{13} - 394886776051364 \nu^{12} + \cdots + 34\!\cdots\!58 ) / 91\!\cdots\!73$$ (16744116205687*v^15 - 45341940014915*v^14 + 176286820995922*v^13 - 394886776051364*v^12 + 1026293635905532*v^11 - 2740194677046448*v^10 + 2842579273044968*v^9 - 9946106370147720*v^8 + 3410762945368948*v^7 - 29202452252119336*v^6 - 7293677405066531*v^5 - 42465191302472856*v^4 - 21257885787172131*v^3 - 45244610063730803*v^2 - 14181352848452430*v + 3465152003612358) / 9139478381478873 $$\beta_{7}$$ $$=$$ $$( 26595513341993 \nu^{15} + 73952194191366 \nu^{14} + 112052967333795 \nu^{13} + \cdots + 79\!\cdots\!68 ) / 91\!\cdots\!73$$ (26595513341993*v^15 + 73952194191366*v^14 + 112052967333795*v^13 + 1032516115451622*v^12 + 899822933969919*v^11 + 6370987760378243*v^10 + 2174607198860307*v^9 + 24965334872155991*v^8 + 8193067138457009*v^7 + 58933039323116513*v^6 + 3455848364026450*v^5 + 86744344986601537*v^4 + 8345422433200593*v^3 + 66520663704511511*v^2 - 15105298015423202*v + 7952721863900868) / 9139478381478873 $$\beta_{8}$$ $$=$$ $$( 10058702437115 \nu^{15} - 6203109654385 \nu^{14} + 103630645608798 \nu^{13} + 1974780746302 \nu^{12} + \cdots + 845713476880083 ) / 30\!\cdots\!91$$ (10058702437115*v^15 - 6203109654385*v^14 + 103630645608798*v^13 + 1974780746302*v^12 + 701826108448870*v^11 + 80541385038920*v^10 + 2653504647339595*v^9 + 914336412587371*v^8 + 7278933995715497*v^7 + 2179877825131728*v^6 + 10645673810520968*v^5 + 4368639176864571*v^4 + 12190149830375740*v^3 + 2099036328412212*v^2 + 993128556611358*v + 845713476880083) / 3046492793826291 $$\beta_{9}$$ $$=$$ $$( - 31322721365929 \nu^{15} + 61498828677274 \nu^{14} - 363159263988374 \nu^{13} + 436182822290110 \nu^{12} + \cdots + 10\!\cdots\!47 ) / 91\!\cdots\!73$$ (-31322721365929*v^15 + 61498828677274*v^14 - 363159263988374*v^13 + 436182822290110*v^12 - 2311957038839840*v^11 + 2669287309933332*v^10 - 8810642319636721*v^9 + 8085804827482501*v^8 - 21814004313126223*v^7 + 22118706479439852*v^6 - 31705409312404125*v^5 + 27771099489894347*v^4 - 25890870569987892*v^3 + 37008967590250226*v^2 - 5950075068841603*v + 1006054223780547) / 9139478381478873 $$\beta_{10}$$ $$=$$ $$( 35353445168842 \nu^{15} - 42906025379669 \nu^{14} + 360549986338030 \nu^{13} - 261408818119538 \nu^{12} + \cdots - 18\!\cdots\!67 ) / 91\!\cdots\!73$$ (35353445168842*v^15 - 42906025379669*v^14 + 360549986338030*v^13 - 261408818119538*v^12 + 2346977874531172*v^11 - 1738189450969768*v^10 + 7939379548056260*v^9 - 5381830987437618*v^8 + 18436346491318150*v^7 - 17754813181134811*v^6 + 15550848768311545*v^5 - 28360469660381838*v^4 + 8592989216359623*v^3 - 39267507171071219*v^2 - 11838012848643870*v - 18971878017965067) / 9139478381478873 $$\beta_{11}$$ $$=$$ $$( 47477396037032 \nu^{15} - 55102847496393 \nu^{14} + 436243278002799 \nu^{13} - 112029436571388 \nu^{12} + \cdots - 18\!\cdots\!59 ) / 91\!\cdots\!73$$ (47477396037032*v^15 - 55102847496393*v^14 + 436243278002799*v^13 - 112029436571388*v^12 + 2518026227560113*v^11 - 864033044630458*v^10 + 8674336787121810*v^9 - 1319639334852235*v^8 + 19687039264476329*v^7 - 11054480958405496*v^6 + 24635102803156015*v^5 - 19746766936098476*v^4 + 10463030136046236*v^3 - 33052810323229990*v^2 - 10729277200479152*v - 18277046259484959) / 9139478381478873 $$\beta_{12}$$ $$=$$ $$( 67174987872150 \nu^{15} - 20896415688563 \nu^{14} + 620697587330554 \nu^{13} + 489098601157807 \nu^{12} + \cdots + 27\!\cdots\!48 ) / 91\!\cdots\!73$$ (67174987872150*v^15 - 20896415688563*v^14 + 620697587330554*v^13 + 489098601157807*v^12 + 3837661637756542*v^11 + 4030094128429255*v^10 + 13772024529051905*v^9 + 22166122749372422*v^8 + 36094915770719934*v^7 + 59485031448387424*v^6 + 53253791588998406*v^5 + 109067758838777617*v^4 + 52649488556200071*v^3 + 91314609178178160*v^2 + 5618911102245064*v + 27855550205743248) / 9139478381478873 $$\beta_{13}$$ $$=$$ $$( - 88820209793387 \nu^{15} + 39343350153959 \nu^{14} - 891308516903932 \nu^{13} - 287803296009742 \nu^{12} + \cdots - 71\!\cdots\!75 ) / 91\!\cdots\!73$$ (-88820209793387*v^15 + 39343350153959*v^14 - 891308516903932*v^13 - 287803296009742*v^12 - 5885557766462179*v^11 - 2688129593090100*v^10 - 22404008242064108*v^9 - 15810686435053003*v^8 - 62381927939342093*v^7 - 38750539349362944*v^6 - 98955484082891763*v^5 - 62266292327076857*v^4 - 110918644261366557*v^3 - 36447991581067544*v^2 - 23673309380329397*v - 7147318694846175) / 9139478381478873 $$\beta_{14}$$ $$=$$ $$( - 31322721365929 \nu^{15} + 61498828677274 \nu^{14} - 363159263988374 \nu^{13} + 436182822290110 \nu^{12} + \cdots + 10\!\cdots\!47 ) / 30\!\cdots\!91$$ (-31322721365929*v^15 + 61498828677274*v^14 - 363159263988374*v^13 + 436182822290110*v^12 - 2311957038839840*v^11 + 2669287309933332*v^10 - 8810642319636721*v^9 + 8085804827482501*v^8 - 21814004313126223*v^7 + 22118706479439852*v^6 - 31705409312404125*v^5 + 27771099489894347*v^4 - 25890870569987892*v^3 + 33962474796423935*v^2 - 5950075068841603*v + 1006054223780547) / 3046492793826291 $$\beta_{15}$$ $$=$$ $$( 241239132189497 \nu^{15} - 269626419785329 \nu^{14} + \cdots - 19\!\cdots\!11 ) / 91\!\cdots\!73$$ (241239132189497*v^15 - 269626419785329*v^14 + 2707842518071070*v^13 - 1384847430059254*v^12 + 18188750599418894*v^11 - 7008331821372974*v^10 + 70587911675040145*v^9 - 12238829708683665*v^8 + 187862907036005150*v^7 - 30755186597641787*v^6 + 280378078725409877*v^5 - 17879932341420474*v^4 + 262801827503435319*v^3 - 43331342630160394*v^2 + 47651869820136828*v - 1957620602448711) / 9139478381478873
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{14} + 3\beta_{9}$$ -b14 + 3*b9 $$\nu^{3}$$ $$=$$ $$\beta_{13} - \beta_{10} + 4\beta_{8} - \beta_{3} - 1$$ b13 - b10 + 4*b8 - b3 - 1 $$\nu^{4}$$ $$=$$ $$5\beta_{14} - 4\beta_{10} - 12\beta_{9} + 5\beta_{8} - \beta_{7} + 6\beta_{6} - \beta_{5} - \beta_{2} - \beta _1 - 12$$ 5*b14 - 4*b10 - 12*b9 + 5*b8 - b7 + 6*b6 - b5 - b2 - b1 - 12 $$\nu^{5}$$ $$=$$ $$- \beta_{15} + 8 \beta_{14} - 7 \beta_{13} + \beta_{11} + 8 \beta_{10} - 9 \beta_{9} - 11 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} + 6 \beta_{4} + 8 \beta_{3} - \beta_{2} - 11 \beta_1$$ -b15 + 8*b14 - 7*b13 + b11 + 8*b10 - 9*b9 - 11*b8 - 2*b7 + 8*b6 + 6*b4 + 8*b3 - b2 - 11*b1 $$\nu^{6}$$ $$=$$ $$- 2 \beta_{15} + 9 \beta_{14} - 13 \beta_{13} - 2 \beta_{12} + 10 \beta_{11} + 28 \beta_{10} - 25 \beta_{8} - 2 \beta_{7} - 14 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} + 21 \beta_{3} + 6 \beta_{2} + 9 \beta _1 + 54$$ -2*b15 + 9*b14 - 13*b13 - 2*b12 + 10*b11 + 28*b10 - 25*b8 - 2*b7 - 14*b6 + 10*b5 + 10*b4 + 21*b3 + 6*b2 + 9*b1 + 54 $$\nu^{7}$$ $$=$$ $$- 32 \beta_{14} - 11 \beta_{13} - 10 \beta_{12} + 11 \beta_{11} + \beta_{10} + 63 \beta_{9} - 32 \beta_{8} + 11 \beta_{7} - 52 \beta_{6} + 23 \beta_{5} - 12 \beta_{4} + 10 \beta_{3} + 20 \beta_{2} + 65 \beta _1 + 63$$ -32*b14 - 11*b13 - 10*b12 + 11*b11 + b10 + 63*b9 - 32*b8 + 11*b7 - 52*b6 + 23*b5 - 12*b4 + 10*b3 + 20*b2 + 65*b1 + 63 $$\nu^{8}$$ $$=$$ $$23 \beta_{15} - 170 \beta_{14} + 86 \beta_{13} - 51 \beta_{11} - 110 \beta_{10} + 263 \beta_{9} + 23 \beta_{8} + 75 \beta_{7} - 125 \beta_{6} - 63 \beta_{4} - 138 \beta_{3} + 36 \beta_{2} + 23 \beta_1$$ 23*b15 - 170*b14 + 86*b13 - 51*b11 - 110*b10 + 263*b9 + 23*b8 + 75*b7 - 125*b6 - 63*b4 - 138*b3 + 36*b2 + 23*b1 $$\nu^{9}$$ $$=$$ $$76 \beta_{15} - 148 \beta_{14} + 357 \beta_{13} + 76 \beta_{12} - 173 \beta_{11} - 368 \beta_{10} + 370 \beta_{8} + 87 \beta_{7} - 14 \beta_{6} - 184 \beta_{5} - 173 \beta_{4} - 443 \beta_{3} - 36 \beta_{2} - 148 \beta _1 - 403$$ 76*b15 - 148*b14 + 357*b13 + 76*b12 - 173*b11 - 368*b10 + 370*b8 + 87*b7 - 14*b6 - 184*b5 - 173*b4 - 443*b3 - 36*b2 - 148*b1 - 403 $$\nu^{10}$$ $$=$$ $$514 \beta_{14} + 198 \beta_{13} + 184 \beta_{12} - 198 \beta_{11} - 222 \beta_{10} - 1358 \beta_{9} + 514 \beta_{8} - 307 \beta_{7} + 921 \beta_{6} - 519 \beta_{5} - 101 \beta_{4} - 184 \beta_{3} - 407 \beta_{2} - 590 \beta _1 - 1358$$ 514*b14 + 198*b13 + 184*b12 - 198*b11 - 222*b10 - 1358*b9 + 514*b8 - 307*b7 + 921*b6 - 519*b5 - 101*b4 - 184*b3 - 407*b2 - 590*b1 - 1358 $$\nu^{11}$$ $$=$$ $$- 519 \beta_{15} + 1886 \beta_{14} - 1638 \beta_{13} + 590 \beta_{11} + 2241 \beta_{10} - 2473 \beta_{9} - 966 \beta_{8} - 1193 \beta_{7} + 2016 \beta_{6} + 1119 \beta_{4} + 2312 \beta_{3} - 815 \beta_{2} - 966 \beta_1$$ -519*b15 + 1886*b14 - 1638*b13 + 590*b11 + 2241*b10 - 2473*b9 - 966*b8 - 1193*b7 + 2016*b6 + 1119*b4 + 2312*b3 - 815*b2 - 966*b1 $$\nu^{12}$$ $$=$$ $$- 1277 \beta_{15} + 2535 \beta_{14} - 5171 \beta_{13} - 1277 \beta_{12} + 3222 \beta_{11} + 6063 \beta_{10} - 3825 \beta_{8} - 1405 \beta_{7} - 205 \beta_{6} + 3350 \beta_{5} + 3222 \beta_{4} + 6988 \beta_{3} + \cdots + 7316$$ -1277*b15 + 2535*b14 - 5171*b13 - 1277*b12 + 3222*b11 + 6063*b10 - 3825*b8 - 1405*b7 - 205*b6 + 3350*b5 + 3222*b4 + 6988*b3 + 443*b2 + 2535*b1 + 7316 $$\nu^{13}$$ $$=$$ $$- 4940 \beta_{14} - 3909 \beta_{13} - 3350 \beta_{12} + 3909 \beta_{11} + 526 \beta_{10} + 14858 \beta_{9} - 4940 \beta_{8} + 3797 \beta_{7} - 11113 \beta_{6} + 8265 \beta_{5} + 1200 \beta_{4} + 3350 \beta_{3} + \cdots + 14858$$ -4940*b14 - 3909*b13 - 3350*b12 + 3909*b11 + 526*b10 + 14858*b9 - 4940*b8 + 3797*b7 - 11113*b6 + 8265*b5 + 1200*b4 + 3350*b3 + 6173*b2 + 11097*b1 + 14858 $$\nu^{14}$$ $$=$$ $$8265 \beta_{15} - 28774 \beta_{14} + 23287 \beta_{13} - 10699 \beta_{11} - 32526 \beta_{10} + 40602 \beta_{9} + 8068 \beta_{8} + 19938 \beta_{7} - 29518 \beta_{6} - 15022 \beta_{4} - 34960 \beta_{3} + \cdots + 8068 \beta_1$$ 8265*b15 - 28774*b14 + 23287*b13 - 10699*b11 - 32526*b10 + 40602*b9 + 8068*b8 + 19938*b7 - 29518*b6 - 15022*b4 - 34960*b3 + 13707*b2 + 8068*b1 $$\nu^{15}$$ $$=$$ $$20912 \beta_{15} - 37783 \beta_{14} + 83331 \beta_{13} + 20912 \beta_{12} - 47996 \beta_{11} - 86483 \beta_{10} + 53709 \beta_{8} + 24406 \beta_{7} - 7061 \beta_{6} - 51490 \beta_{5} - 47996 \beta_{4} + \cdots - 88279$$ 20912*b15 - 37783*b14 + 83331*b13 + 20912*b12 - 47996*b11 - 86483*b10 + 53709*b8 + 24406*b7 - 7061*b6 - 51490*b5 - 47996*b4 - 106921*b3 - 3164*b2 - 37783*b1 - 88279

## Character values

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

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$\beta_{9}$$ $$1$$ $$1$$

## 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
 −0.947115 + 1.64045i −0.921972 + 1.59690i −0.624867 + 1.08230i −0.237622 + 0.411573i 0.316605 − 0.548376i 0.715744 − 1.23970i 0.987387 − 1.71020i 1.21184 − 2.09897i −0.947115 − 1.64045i −0.921972 − 1.59690i −0.624867 − 1.08230i −0.237622 − 0.411573i 0.316605 + 0.548376i 0.715744 + 1.23970i 0.987387 + 1.71020i 1.21184 + 2.09897i
−0.947115 + 1.64045i 1.31847 1.12323i −0.794055 1.37534i −0.500000 0.866025i 0.593868 + 3.22671i −0.837925 + 1.45133i −0.780216 0.476703 2.96188i 1.89423
196.2 −0.921972 + 1.59690i −0.495399 + 1.65969i −0.700064 1.21255i −0.500000 0.866025i −2.19362 2.32129i 2.38510 4.13111i −1.10613 −2.50916 1.64442i 1.84394
196.3 −0.624867 + 1.08230i 1.28189 + 1.16480i 0.219082 + 0.379462i −0.500000 0.866025i −2.06168 + 0.659540i 0.148724 0.257597i −3.04706 0.286466 + 2.98629i 1.24973
196.4 −0.237622 + 0.411573i −1.28679 1.15939i 0.887072 + 1.53645i −0.500000 0.866025i 0.782942 0.254110i −1.24774 + 2.16115i −1.79364 0.311632 + 2.98377i 0.475244
196.5 0.316605 0.548376i −1.15302 + 1.29250i 0.799523 + 1.38481i −0.500000 0.866025i 0.343725 + 1.04150i 0.896323 1.55248i 2.27895 −0.341107 2.98054i −0.633210
196.6 0.715744 1.23970i −1.70726 0.291973i −0.0245786 0.0425715i −0.500000 0.866025i −1.58392 + 1.90753i 0.625355 1.08315i 2.79261 2.82950 + 0.996950i −1.43149
196.7 0.987387 1.71020i 1.73142 + 0.0465761i −0.949865 1.64522i −0.500000 0.866025i 1.78924 2.91510i 0.124489 0.215622i 0.198009 2.99566 + 0.161286i −1.97477
196.8 1.21184 2.09897i −0.689312 1.58898i −1.93711 3.35518i −0.500000 0.866025i −4.17055 0.478743i 0.905675 1.56867i −4.54253 −2.04970 + 2.19060i −2.42368
391.1 −0.947115 1.64045i 1.31847 + 1.12323i −0.794055 + 1.37534i −0.500000 + 0.866025i 0.593868 3.22671i −0.837925 1.45133i −0.780216 0.476703 + 2.96188i 1.89423
391.2 −0.921972 1.59690i −0.495399 1.65969i −0.700064 + 1.21255i −0.500000 + 0.866025i −2.19362 + 2.32129i 2.38510 + 4.13111i −1.10613 −2.50916 + 1.64442i 1.84394
391.3 −0.624867 1.08230i 1.28189 1.16480i 0.219082 0.379462i −0.500000 + 0.866025i −2.06168 0.659540i 0.148724 + 0.257597i −3.04706 0.286466 2.98629i 1.24973
391.4 −0.237622 0.411573i −1.28679 + 1.15939i 0.887072 1.53645i −0.500000 + 0.866025i 0.782942 + 0.254110i −1.24774 2.16115i −1.79364 0.311632 2.98377i 0.475244
391.5 0.316605 + 0.548376i −1.15302 1.29250i 0.799523 1.38481i −0.500000 + 0.866025i 0.343725 1.04150i 0.896323 + 1.55248i 2.27895 −0.341107 + 2.98054i −0.633210
391.6 0.715744 + 1.23970i −1.70726 + 0.291973i −0.0245786 + 0.0425715i −0.500000 + 0.866025i −1.58392 1.90753i 0.625355 + 1.08315i 2.79261 2.82950 0.996950i −1.43149
391.7 0.987387 + 1.71020i 1.73142 0.0465761i −0.949865 + 1.64522i −0.500000 + 0.866025i 1.78924 + 2.91510i 0.124489 + 0.215622i 0.198009 2.99566 0.161286i −1.97477
391.8 1.21184 + 2.09897i −0.689312 + 1.58898i −1.93711 + 3.35518i −0.500000 + 0.866025i −4.17055 + 0.478743i 0.905675 + 1.56867i −4.54253 −2.04970 2.19060i −2.42368
 $$n$$: 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.f 16
3.b odd 2 1 1755.2.i.e 16
9.c even 3 1 inner 585.2.i.f 16
9.c even 3 1 5265.2.a.bb 8
9.d odd 6 1 1755.2.i.e 16
9.d odd 6 1 5265.2.a.be 8

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

## Hecke kernels

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

 $$T_{2}^{16} - T_{2}^{15} + 11 T_{2}^{14} - 4 T_{2}^{13} + 74 T_{2}^{12} - 18 T_{2}^{11} + 289 T_{2}^{10} - 4 T_{2}^{9} + 784 T_{2}^{8} - 9 T_{2}^{7} + 1221 T_{2}^{6} + 133 T_{2}^{5} + 1245 T_{2}^{4} - 14 T_{2}^{3} + 391 T_{2}^{2} + 63 T_{2} + 81$$ T2^16 - T2^15 + 11*T2^14 - 4*T2^13 + 74*T2^12 - 18*T2^11 + 289*T2^10 - 4*T2^9 + 784*T2^8 - 9*T2^7 + 1221*T2^6 + 133*T2^5 + 1245*T2^4 - 14*T2^3 + 391*T2^2 + 63*T2 + 81 $$T_{7}^{16} - 6 T_{7}^{15} + 38 T_{7}^{14} - 88 T_{7}^{13} + 353 T_{7}^{12} - 763 T_{7}^{11} + 2210 T_{7}^{10} - 3575 T_{7}^{9} + 7015 T_{7}^{8} - 9159 T_{7}^{7} + 14472 T_{7}^{6} - 14373 T_{7}^{5} + 13110 T_{7}^{4} - 5679 T_{7}^{3} + \cdots + 36$$ T7^16 - 6*T7^15 + 38*T7^14 - 88*T7^13 + 353*T7^12 - 763*T7^11 + 2210*T7^10 - 3575*T7^9 + 7015*T7^8 - 9159*T7^7 + 14472*T7^6 - 14373*T7^5 + 13110*T7^4 - 5679*T7^3 + 1827*T7^2 - 306*T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - T^{15} + 11 T^{14} - 4 T^{13} + \cdots + 81$$
$3$ $$T^{16} + 2 T^{15} - 10 T^{13} + \cdots + 6561$$
$5$ $$(T^{2} + T + 1)^{8}$$
$7$ $$T^{16} - 6 T^{15} + 38 T^{14} - 88 T^{13} + \cdots + 36$$
$11$ $$T^{16} - 9 T^{15} + 83 T^{14} + \cdots + 6561$$
$13$ $$(T^{2} + T + 1)^{8}$$
$17$ $$(T^{8} - 6 T^{7} - 36 T^{6} + 159 T^{5} + \cdots + 822)^{2}$$
$19$ $$(T^{8} + 11 T^{7} - 21 T^{6} - 648 T^{5} + \cdots + 51733)^{2}$$
$23$ $$T^{16} - 3 T^{15} + 125 T^{14} + \cdots + 898560576$$
$29$ $$T^{16} - 8 T^{15} + \cdots + 310831895529$$
$31$ $$T^{16} - 18 T^{15} + 269 T^{14} + \cdots + 18344089$$
$37$ $$(T^{8} + 18 T^{7} - 61 T^{6} + \cdots + 236259)^{2}$$
$41$ $$T^{16} + 17 T^{15} + \cdots + 3109623696$$
$43$ $$T^{16} - 17 T^{15} + \cdots + 16841810176$$
$47$ $$T^{16} - 11 T^{15} + 218 T^{14} + \cdots + 10850436$$
$53$ $$(T^{8} - 10 T^{7} - 166 T^{6} + \cdots - 405738)^{2}$$
$59$ $$T^{16} - 7 T^{15} + \cdots + 1642567767129$$
$61$ $$T^{16} - 21 T^{15} + \cdots + 28213508866321$$
$67$ $$T^{16} - 13 T^{15} + \cdots + 9767162560516$$
$71$ $$(T^{8} + 34 T^{7} + 297 T^{6} + \cdots + 1374696)^{2}$$
$73$ $$(T^{8} + 16 T^{7} - 196 T^{6} + \cdots + 373998)^{2}$$
$79$ $$T^{16} - 37 T^{15} + \cdots + 5500298896$$
$83$ $$T^{16} + \cdots + 568679695164036$$
$89$ $$(T^{8} - 14 T^{7} - 96 T^{6} + \cdots + 1052154)^{2}$$
$97$ $$T^{16} - 17 T^{15} + \cdots + 71550365121$$