# Properties

 Label 585.2.h.g Level $585$ Weight $2$ Character orbit 585.h Analytic conductor $4.671$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.2593100598870016.1 Defining polynomial: $$x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64$$ x^12 - 4*x^10 + 9*x^8 - 16*x^6 + 36*x^4 - 64*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{2} + ( - \beta_{11} + \beta_{5} + 1) q^{4} + ( - \beta_{10} - \beta_1) q^{5} + (\beta_{6} - \beta_{5}) q^{7} + (\beta_{10} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + b10 * q^2 + (-b11 + b5 + 1) * q^4 + (-b10 - b1) * q^5 + (b6 - b5) * q^7 + (b10 + b6 - b5 + b2 - b1) * q^8 $$q + \beta_{10} q^{2} + ( - \beta_{11} + \beta_{5} + 1) q^{4} + ( - \beta_{10} - \beta_1) q^{5} + (\beta_{6} - \beta_{5}) q^{7} + (\beta_{10} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{8} + (\beta_{11} - \beta_{8} - 2) q^{10} + \beta_{3} q^{11} + (\beta_{10} + \beta_{9} + \beta_{6}) q^{13} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - 2) q^{14} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + 1) q^{16} + ( - 2 \beta_{8} - \beta_{6} + \beta_{5}) q^{17} + (\beta_{3} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + \beta_1) q^{20} + ( - \beta_{9} + \beta_{8} - \beta_{5}) q^{22} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5}) q^{23} + (\beta_{8} - \beta_{7} - 2 \beta_{5}) q^{25} + ( - \beta_{11} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 2) q^{26} + (\beta_{6} - \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{28} + (2 \beta_{11} - 2 \beta_{5} + 2) q^{29} + (3 \beta_{3} - \beta_{2} - \beta_1) q^{31} + (\beta_{10} + \beta_{6} - \beta_{5}) q^{32} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{34} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5}) q^{35} + ( - 2 \beta_{10} - \beta_{6} + \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{11} - \beta_{9} + 3 \beta_{8} - 2 \beta_{5}) q^{38} + (\beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 1) q^{40} + (2 \beta_{3} - \beta_{2} - \beta_1) q^{41} + (\beta_{11} - \beta_{9} - \beta_{8}) q^{43} + (\beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{44} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + (4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 5) q^{49} + ( - \beta_{10} + \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{50} + ( - \beta_{11} + 3 \beta_{10} + \beta_{9} - 2 \beta_{8} + 2 \beta_{2} - 2 \beta_1) q^{52} + (2 \beta_{9} + \beta_{6} + \beta_{5}) q^{53} + (\beta_{11} + \beta_{9} - \beta_{7} + 2 \beta_{6} + 1) q^{55} + ( - 2 \beta_{11} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} + 6) q^{56} + ( - 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{58} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 2 \beta_{11} - 2 \beta_{7} + \beta_{6} + \beta_{5}) q^{61} + (\beta_{11} - 3 \beta_{9} + \beta_{8} - 2 \beta_{5}) q^{62} + ( - \beta_{11} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 1) q^{64} + (\beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} - \beta_{2} - 2) q^{65} + ( - 4 \beta_{10} + \beta_{6} - \beta_{5} + 4 \beta_{2} - 4 \beta_1) q^{67} + (2 \beta_{11} + 4 \beta_{9} - 6 \beta_{8} + \beta_{6} + 5 \beta_{5}) q^{68} + ( - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{70} + (2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{71} + (2 \beta_{10} + \beta_{6} - \beta_{5}) q^{73} + (4 \beta_{11} + 2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 8) q^{74} + (2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 5 \beta_{2} + \beta_1) q^{76} + ( - 2 \beta_{11} - 2 \beta_{6}) q^{77} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - 6) q^{79} + ( - 3 \beta_{10} - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{80} + (\beta_{11} - 2 \beta_{9} - \beta_{5}) q^{82} + ( - 4 \beta_{10} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{2} - 3 \beta_1) q^{83} + ( - 4 \beta_{10} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{85} + (\beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{86} + ( - \beta_{9} + 3 \beta_{8} + \beta_{6} - 2 \beta_{5}) q^{88} + ( - 2 \beta_{3} - \beta_{2} - \beta_1) q^{89} + (\beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 4) q^{91} + ( - \beta_{11} - 3 \beta_{9} + 3 \beta_{8} - \beta_{6} - 3 \beta_{5}) q^{92} + (\beta_{11} - \beta_{5} - 2) q^{94} + (\beta_{11} + \beta_{9} - \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + 6) q^{95} + (6 \beta_{10} - \beta_{6} + \beta_{5} - 4 \beta_{2} + 4 \beta_1) q^{97} + (\beta_{10} - 6 \beta_{6} + 6 \beta_{5} - 4 \beta_{2} + 4 \beta_1) q^{98}+O(q^{100})$$ q + b10 * q^2 + (-b11 + b5 + 1) * q^4 + (-b10 - b1) * q^5 + (b6 - b5) * q^7 + (b10 + b6 - b5 + b2 - b1) * q^8 + (b11 - b8 - 2) * q^10 + b3 * q^11 + (b10 + b9 + b6) * q^13 + (-2*b7 + b6 - b5 - 2) * q^14 + (-2*b7 + b6 - b5 + 1) * q^16 + (-2*b8 - b6 + b5) * q^17 + (b3 + b2 + b1) * q^19 + (-b10 - b6 + b5 - b4 - 3*b2 + b1) * q^20 + (-b9 + b8 - b5) * q^22 + (b11 - b9 + b8 + b6 - b5) * q^23 + (b8 - b7 - 2*b5) * q^25 + (-b11 - b7 + b5 - b4 - b3 + b1 + 2) * q^26 + (b6 - b5 + 2*b2 - 2*b1) * q^28 + (2*b11 - 2*b5 + 2) * q^29 + (3*b3 - b2 - b1) * q^31 + (b10 + b6 - b5) * q^32 + (-b6 + b5 - 2*b4 - 4*b2 - 2*b1) * q^34 + (-2*b11 + 2*b9 + 2*b7 - b6 + 3*b5) * q^35 + (-2*b10 - b6 + b5 - 2*b2 + 2*b1) * q^37 + (-b11 - b9 + 3*b8 - 2*b5) * q^38 + (b11 + 2*b9 - 2*b8 + b7 - b6 + b5 - 1) * q^40 + (2*b3 - b2 - b1) * q^41 + (b11 - b9 - b8) * q^43 + (b6 - b5 + 2*b4 - b3 + 2*b2) * q^44 + (b6 - b5 + 2*b4 + b3 + b2 - b1) * q^46 + (-b2 + b1) * q^47 + (4*b7 - 2*b6 + 2*b5 + 5) * q^49 + (-b10 + b6 - b5 + b4 + 3*b2 + 2*b1) * q^50 + (-b11 + 3*b10 + b9 - 2*b8 + 2*b2 - 2*b1) * q^52 + (2*b9 + b6 + b5) * q^53 + (b11 + b9 - b7 + 2*b6 + 1) * q^55 + (-2*b11 + 2*b7 - b6 + 3*b5 + 6) * q^56 + (-2*b10 - 2*b6 + 2*b5 - 2*b2 + 2*b1) * q^58 + (b3 + 2*b2 + 2*b1) * q^59 + (-2*b11 - 2*b7 + b6 + b5) * q^61 + (b11 - 3*b9 + b8 - 2*b5) * q^62 + (-b11 + 2*b7 - b6 + 2*b5 - 1) * q^64 + (b9 - b8 + b7 - 2*b6 + 3*b5 + 2*b3 - b2 - 2) * q^65 + (-4*b10 + b6 - b5 + 4*b2 - 4*b1) * q^67 + (2*b11 + 4*b9 - 6*b8 + b6 + 5*b5) * q^68 + (-3*b6 + 3*b5 - 2*b4 - 2*b3 + 4*b1) * q^70 + (2*b6 - 2*b5 + 4*b4 - b3 + 2*b2 - 2*b1) * q^71 + (2*b10 + b6 - b5) * q^73 + (4*b11 + 2*b7 - b6 - 3*b5 - 8) * q^74 + (2*b6 - 2*b5 + 4*b4 - b3 + 5*b2 + b1) * q^76 + (-2*b11 - 2*b6) * q^77 + (-2*b7 + b6 - b5 - 6) * q^79 + (-3*b10 - 3*b6 + 3*b5 - 2*b4 - 2*b3 + b1) * q^80 + (b11 - 2*b9 - b5) * q^82 + (-4*b10 + 2*b6 - 2*b5 + 3*b2 - 3*b1) * q^83 + (-4*b10 + b6 - b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1) * q^85 + (b3 - 3*b2 - 3*b1) * q^86 + (-b9 + 3*b8 + b6 - 2*b5) * q^88 + (-2*b3 - b2 - b1) * q^89 + (b6 - b5 + 2*b4 - 2*b2 - 4*b1 + 4) * q^91 + (-b11 - 3*b9 + 3*b8 - b6 - 3*b5) * q^92 + (b11 - b5 - 2) * q^94 + (b11 + b9 - b8 + 2*b6 + 2*b5 + 6) * q^95 + (6*b10 - b6 + b5 - 4*b2 + 4*b1) * q^97 + (b10 - 6*b6 + 6*b5 - 4*b2 + 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 12 q^{4}+O(q^{10})$$ 12 * q + 12 * q^4 $$12 q + 12 q^{4} - 24 q^{10} - 16 q^{14} + 20 q^{16} + 4 q^{25} + 28 q^{26} + 24 q^{29} - 8 q^{35} - 16 q^{40} + 44 q^{49} + 16 q^{55} + 64 q^{56} + 8 q^{61} - 20 q^{64} - 28 q^{65} - 104 q^{74} - 64 q^{79} + 48 q^{91} - 24 q^{94} + 72 q^{95}+O(q^{100})$$ 12 * q + 12 * q^4 - 24 * q^10 - 16 * q^14 + 20 * q^16 + 4 * q^25 + 28 * q^26 + 24 * q^29 - 8 * q^35 - 16 * q^40 + 44 * q^49 + 16 * q^55 + 64 * q^56 + 8 * q^61 - 20 * q^64 - 28 * q^65 - 104 * q^74 - 64 * q^79 + 48 * q^91 - 24 * q^94 + 72 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4x^{10} + 9x^{8} - 16x^{6} + 36x^{4} - 64x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} - 4\nu^{9} - \nu^{7} + 8\nu^{5} - 12\nu^{3} - 112\nu ) / 80$$ (-v^11 - 4*v^9 - v^7 + 8*v^5 - 12*v^3 - 112*v) / 80 $$\beta_{2}$$ $$=$$ $$( -3\nu^{11} + 8\nu^{9} - 43\nu^{7} + 44\nu^{5} - 76\nu^{3} + 144\nu ) / 160$$ (-3*v^11 + 8*v^9 - 43*v^7 + 44*v^5 - 76*v^3 + 144*v) / 160 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - 8\nu^{9} + 9\nu^{7} - 20\nu^{5} + 52\nu^{3} - 80\nu ) / 32$$ (v^11 - 8*v^9 + 9*v^7 - 20*v^5 + 52*v^3 - 80*v) / 32 $$\beta_{4}$$ $$=$$ $$( -\nu^{11} + \nu^{9} - \nu^{7} - 7\nu^{5} - 12\nu^{3} - 12\nu ) / 20$$ (-v^11 + v^9 - v^7 - 7*v^5 - 12*v^3 - 12*v) / 20 $$\beta_{5}$$ $$=$$ $$( \nu^{11} - 2 \nu^{10} - \nu^{9} + 2 \nu^{8} + \nu^{7} - 2 \nu^{6} - 13 \nu^{5} + 26 \nu^{4} + 32 \nu^{3} - 24 \nu^{2} - 28 \nu + 16 ) / 40$$ (v^11 - 2*v^10 - v^9 + 2*v^8 + v^7 - 2*v^6 - 13*v^5 + 26*v^4 + 32*v^3 - 24*v^2 - 28*v + 16) / 40 $$\beta_{6}$$ $$=$$ $$( - \nu^{11} - 2 \nu^{10} + \nu^{9} + 2 \nu^{8} - \nu^{7} - 2 \nu^{6} + 13 \nu^{5} + 26 \nu^{4} - 32 \nu^{3} - 24 \nu^{2} + 28 \nu + 16 ) / 40$$ (-v^11 - 2*v^10 + v^9 + 2*v^8 - v^7 - 2*v^6 + 13*v^5 + 26*v^4 - 32*v^3 - 24*v^2 + 28*v + 16) / 40 $$\beta_{7}$$ $$=$$ $$( - 2 \nu^{11} + 5 \nu^{10} + 2 \nu^{9} - 20 \nu^{8} - 2 \nu^{7} + 45 \nu^{6} + 26 \nu^{5} - 80 \nu^{4} - 64 \nu^{3} + 100 \nu^{2} + 56 \nu - 240 ) / 80$$ (-2*v^11 + 5*v^10 + 2*v^9 - 20*v^8 - 2*v^7 + 45*v^6 + 26*v^5 - 80*v^4 - 64*v^3 + 100*v^2 + 56*v - 240) / 80 $$\beta_{8}$$ $$=$$ $$( \nu^{11} - 3 \nu^{10} - \nu^{9} - 2 \nu^{8} + \nu^{7} - 3 \nu^{6} - 13 \nu^{5} - 6 \nu^{4} + 32 \nu^{3} + 4 \nu^{2} - 28 \nu - 56 ) / 40$$ (v^11 - 3*v^10 - v^9 - 2*v^8 + v^7 - 3*v^6 - 13*v^5 - 6*v^4 + 32*v^3 + 4*v^2 - 28*v - 56) / 40 $$\beta_{9}$$ $$=$$ $$( 9\nu^{10} - 24\nu^{8} + 49\nu^{6} - 132\nu^{4} + 308\nu^{2} - 352 ) / 80$$ (9*v^10 - 24*v^8 + 49*v^6 - 132*v^4 + 308*v^2 - 352) / 80 $$\beta_{10}$$ $$=$$ $$( -3\nu^{11} + 8\nu^{9} - 13\nu^{7} + 24\nu^{5} - 46\nu^{3} + 84\nu ) / 40$$ (-3*v^11 + 8*v^9 - 13*v^7 + 24*v^5 - 46*v^3 + 84*v) / 40 $$\beta_{11}$$ $$=$$ $$( 2 \nu^{11} + 11 \nu^{10} - 2 \nu^{9} - 36 \nu^{8} + 2 \nu^{7} + 51 \nu^{6} - 26 \nu^{5} - 88 \nu^{4} + 64 \nu^{3} + 252 \nu^{2} - 56 \nu - 368 ) / 80$$ (2*v^11 + 11*v^10 - 2*v^9 - 36*v^8 + 2*v^7 + 51*v^6 - 26*v^5 - 88*v^4 + 64*v^3 + 252*v^2 - 56*v - 368) / 80
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 3\beta_1 ) / 4$$ (b6 - b5 + b3 + b2 - 3*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{9} - \beta_{7} + \beta_{6} + 1 ) / 2$$ (b9 - b7 + b6 + 1) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{10} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} ) / 2$$ (2*b10 - b6 + b5 - b4 + b3 - b2) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{11} - \beta_{8} - \beta_{7} + 2\beta_{6} + \beta_{5} - 1 ) / 2$$ (b11 - b8 - b7 + 2*b6 + b5 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( 2\beta_{10} - 3\beta_{4} - 2\beta_{2} + 3\beta_1 ) / 2$$ (2*b10 - 3*b4 - 2*b2 + 3*b1) / 2 $$\nu^{6}$$ $$=$$ $$( -3\beta_{11} + 2\beta_{9} - 3\beta_{8} + 5\beta_{7} - 2\beta_{6} + 9\beta_{5} + 3 ) / 2$$ (-3*b11 + 2*b9 - 3*b8 + 5*b7 - 2*b6 + 9*b5 + 3) / 2 $$\nu^{7}$$ $$=$$ $$( 2\beta_{10} + 2\beta_{6} - 2\beta_{5} - \beta_{4} - 10\beta_{2} - \beta_1 ) / 2$$ (2*b10 + 2*b6 - 2*b5 - b4 - 10*b2 - b1) / 2 $$\nu^{8}$$ $$=$$ $$( -9\beta_{11} + 8\beta_{9} - 7\beta_{8} + \beta_{7} - 6\beta_{6} + 11\beta_{5} - 15 ) / 2$$ (-9*b11 + 8*b9 - 7*b8 + b7 - 6*b6 + 11*b5 - 15) / 2 $$\nu^{9}$$ $$=$$ $$( 6\beta_{10} - 10\beta_{6} + 10\beta_{5} - \beta_{4} - 10\beta_{3} - 16\beta_{2} + 7\beta_1 ) / 2$$ (6*b10 - 10*b6 + 10*b5 - b4 - 10*b3 - 16*b2 + 7*b1) / 2 $$\nu^{10}$$ $$=$$ $$( 7\beta_{11} - 6\beta_{9} - 17\beta_{8} - 5\beta_{7} - 10\beta_{6} - 5\beta_{5} - 27 ) / 2$$ (7*b11 - 6*b9 - 17*b8 - 5*b7 - 10*b6 - 5*b5 - 27) / 2 $$\nu^{11}$$ $$=$$ $$( -34\beta_{10} - 6\beta_{6} + 6\beta_{5} - 7\beta_{4} - 28\beta_{3} + 14\beta_{2} + 5\beta_1 ) / 2$$ (-34*b10 - 6*b6 + 6*b5 - 7*b4 - 28*b3 + 14*b2 + 5*b1) / 2

## Character values

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

 $$n$$ $$326$$ $$352$$ $$496$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −1.37820 − 0.317122i −1.37820 + 0.317122i 0.721581 − 1.21627i 0.721581 + 1.21627i 1.25694 + 0.648161i 1.25694 − 0.648161i −1.25694 + 0.648161i −1.25694 − 0.648161i −0.721581 − 1.21627i −0.721581 + 1.21627i 1.37820 − 0.317122i 1.37820 + 0.317122i
−2.51912 0 4.34596 1.45804 1.69532i 0 −1.26849 −5.90976 0 −3.67298 + 4.27072i
64.2 −2.51912 0 4.34596 1.45804 + 1.69532i 0 −1.26849 −5.90976 0 −3.67298 4.27072i
64.3 −1.61036 0 0.593272 1.11567 1.93785i 0 4.86509 2.26534 0 −1.79664 + 3.12065i
64.4 −1.61036 0 0.593272 1.11567 + 1.93785i 0 4.86509 2.26534 0 −1.79664 3.12065i
64.5 −0.246506 0 −1.93923 2.15160 0.608775i 0 −2.59264 0.971044 0 −0.530383 + 0.150067i
64.6 −0.246506 0 −1.93923 2.15160 + 0.608775i 0 −2.59264 0.971044 0 −0.530383 0.150067i
64.7 0.246506 0 −1.93923 −2.15160 0.608775i 0 2.59264 −0.971044 0 −0.530383 0.150067i
64.8 0.246506 0 −1.93923 −2.15160 + 0.608775i 0 2.59264 −0.971044 0 −0.530383 + 0.150067i
64.9 1.61036 0 0.593272 −1.11567 1.93785i 0 −4.86509 −2.26534 0 −1.79664 3.12065i
64.10 1.61036 0 0.593272 −1.11567 + 1.93785i 0 −4.86509 −2.26534 0 −1.79664 + 3.12065i
64.11 2.51912 0 4.34596 −1.45804 1.69532i 0 1.26849 5.90976 0 −3.67298 4.27072i
64.12 2.51912 0 4.34596 −1.45804 + 1.69532i 0 1.26849 5.90976 0 −3.67298 + 4.27072i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
65.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.h.g 12
3.b odd 2 1 195.2.h.c 12
5.b even 2 1 inner 585.2.h.g 12
12.b even 2 1 3120.2.r.n 12
13.b even 2 1 inner 585.2.h.g 12
15.d odd 2 1 195.2.h.c 12
15.e even 4 1 975.2.b.j 6
15.e even 4 1 975.2.b.l 6
39.d odd 2 1 195.2.h.c 12
60.h even 2 1 3120.2.r.n 12
65.d even 2 1 inner 585.2.h.g 12
156.h even 2 1 3120.2.r.n 12
195.e odd 2 1 195.2.h.c 12
195.s even 4 1 975.2.b.j 6
195.s even 4 1 975.2.b.l 6
780.d even 2 1 3120.2.r.n 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.h.c 12 3.b odd 2 1
195.2.h.c 12 15.d odd 2 1
195.2.h.c 12 39.d odd 2 1
195.2.h.c 12 195.e odd 2 1
585.2.h.g 12 1.a even 1 1 trivial
585.2.h.g 12 5.b even 2 1 inner
585.2.h.g 12 13.b even 2 1 inner
585.2.h.g 12 65.d even 2 1 inner
975.2.b.j 6 15.e even 4 1
975.2.b.j 6 195.s even 4 1
975.2.b.l 6 15.e even 4 1
975.2.b.l 6 195.s even 4 1
3120.2.r.n 12 12.b even 2 1
3120.2.r.n 12 60.h even 2 1
3120.2.r.n 12 156.h even 2 1
3120.2.r.n 12 780.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 9T_{2}^{4} + 17T_{2}^{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(585, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - 9 T^{4} + 17 T^{2} - 1)^{2}$$
$3$ $$T^{12}$$
$5$ $$T^{12} - 2 T^{10} + 27 T^{8} + \cdots + 15625$$
$7$ $$(T^{6} - 32 T^{4} + 208 T^{2} - 256)^{2}$$
$11$ $$(T^{6} + 20 T^{4} + 84 T^{2} + 64)^{2}$$
$13$ $$T^{12} + 26 T^{10} + 343 T^{8} + \cdots + 4826809$$
$17$ $$(T^{6} + 88 T^{4} + 2192 T^{2} + \cdots + 16384)^{2}$$
$19$ $$(T^{6} + 64 T^{4} + 1232 T^{2} + \cdots + 6400)^{2}$$
$23$ $$(T^{2} + 16)^{6}$$
$29$ $$(T^{3} - 6 T^{2} - 28 T + 104)^{4}$$
$31$ $$(T^{6} + 160 T^{4} + 6224 T^{2} + \cdots + 43264)^{2}$$
$37$ $$(T^{6} - 132 T^{4} + 2304 T^{2} + \cdots - 256)^{2}$$
$41$ $$(T^{6} + 76 T^{4} + 1460 T^{2} + \cdots + 1024)^{2}$$
$43$ $$(T^{6} + 72 T^{4} + 656 T^{2} + 256)^{2}$$
$47$ $$(T^{6} - 20 T^{4} + 84 T^{2} - 64)^{2}$$
$53$ $$(T^{6} + 104 T^{4} + 3216 T^{2} + \cdots + 25600)^{2}$$
$59$ $$(T^{6} + 164 T^{4} + 8500 T^{2} + \cdots + 141376)^{2}$$
$61$ $$(T^{3} - 2 T^{2} - 96 T - 256)^{4}$$
$67$ $$(T^{6} - 240 T^{4} + 10064 T^{2} + \cdots - 1024)^{2}$$
$71$ $$(T^{6} + 468 T^{4} + 70484 T^{2} + \cdots + 3356224)^{2}$$
$73$ $$(T^{6} - 52 T^{4} + 512 T^{2} - 1024)^{2}$$
$79$ $$(T^{3} + 16 T^{2} + 52 T + 32)^{4}$$
$83$ $$(T^{6} - 228 T^{4} + 9428 T^{2} + \cdots - 12544)^{2}$$
$89$ $$(T^{6} + 140 T^{4} + 5044 T^{2} + \cdots + 16384)^{2}$$
$97$ $$(T^{6} - 340 T^{4} + 31232 T^{2} + \cdots - 640000)^{2}$$