# Properties

 Label 570.2.f.d Level $570$ Weight $2$ Character orbit 570.f Analytic conductor $4.551$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 570.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.55147291521$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.7278137344.1 Defining polynomial: $$x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81$$ x^8 - 2*x^7 + x^6 + 6*x^5 - 20*x^4 + 18*x^3 + 9*x^2 - 54*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + x^{6} + 6x^{5} - 20x^{4} + 18x^{3} + 9x^{2} - 54x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} + \nu^{6} - 5\nu^{5} + 9\nu^{4} - 2\nu^{3} - 15\nu^{2} + 36\nu - 27 ) / 54$$ (v^7 + v^6 - 5*v^5 + 9*v^4 - 2*v^3 - 15*v^2 + 36*v - 27) / 54 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} - \nu^{4} + 2\nu^{3} - 3\nu^{2} - \nu + 12 ) / 6$$ (-v^5 - v^4 + 2*v^3 - 3*v^2 - v + 12) / 6 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} - \nu^{6} - 4\nu^{5} + 9\nu^{4} - 7\nu^{3} - 12\nu^{2} + 36\nu - 54 ) / 54$$ (-v^7 - v^6 - 4*v^5 + 9*v^4 - 7*v^3 - 12*v^2 + 36*v - 54) / 54 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - \nu^{5} - 6\nu^{4} + 20\nu^{3} - 18\nu^{2} - 9\nu + 54 ) / 27$$ (-v^7 + 2*v^6 - v^5 - 6*v^4 + 20*v^3 - 18*v^2 - 9*v + 54) / 27 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + \nu^{5} + 6\nu^{4} - 11\nu^{3} + 9\nu^{2} + 9\nu - 27 ) / 18$$ (v^7 - 2*v^6 + v^5 + 6*v^4 - 11*v^3 + 9*v^2 + 9*v - 27) / 18 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + \nu^{6} + \nu^{5} - 7\nu^{4} + 14\nu^{3} + 11\nu^{2} - 18\nu + 45 ) / 18$$ (-v^7 + v^6 + v^5 - 7*v^4 + 14*v^3 + 11*v^2 - 18*v + 45) / 18
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{5} + \beta_{2}$$ b7 - b5 + b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{2} - 3$$ b7 + 2*b6 + 2*b5 + b2 - 3 $$\nu^{4}$$ $$=$$ $$\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 3\beta _1 + 4$$ b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 + 3*b2 - 3*b1 + 4 $$\nu^{5}$$ $$=$$ $$-2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 2\beta_{4} - 4\beta_{3} - 4\beta_{2} + 2\beta _1 + 2$$ -2*b7 + 2*b6 + 5*b5 - 2*b4 - 4*b3 - 4*b2 + 2*b1 + 2 $$\nu^{6}$$ $$=$$ $$-10\beta_{6} - 6\beta_{5} - 6\beta_{4} - 6\beta_{3} + 12\beta_{2} - 2\beta _1 + 9$$ -10*b6 - 6*b5 - 6*b4 - 6*b3 + 12*b2 - 2*b1 + 9 $$\nu^{7}$$ $$=$$ $$-2\beta_{7} + 6\beta_{6} + 2\beta_{5} - 22\beta_{4} + 4\beta_{3} + 12\beta_{2} + 3\beta _1 - 14$$ -2*b7 + 6*b6 + 2*b5 - 22*b4 + 4*b3 + 12*b2 + 3*b1 - 14

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/570\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$211$$ $$457$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 1.67936 + 0.423958i 1.67936 − 0.423958i 0.828750 + 1.52091i 0.828750 − 1.52091i 0.209196 + 1.71937i 0.209196 − 1.71937i −1.71731 + 0.225499i −1.71731 − 0.225499i
1.00000 −1.67936 0.423958i 1.00000 1.00000i −1.67936 0.423958i −2.47197 1.00000 2.64052 + 1.42396i 1.00000i
341.2 1.00000 −1.67936 + 0.423958i 1.00000 1.00000i −1.67936 + 0.423958i −2.47197 1.00000 2.64052 1.42396i 1.00000i
341.3 1.00000 −0.828750 1.52091i 1.00000 1.00000i −0.828750 1.52091i 4.83942 1.00000 −1.62635 + 2.52091i 1.00000i
341.4 1.00000 −0.828750 + 1.52091i 1.00000 1.00000i −0.828750 + 1.52091i 4.83942 1.00000 −1.62635 2.52091i 1.00000i
341.5 1.00000 −0.209196 1.71937i 1.00000 1.00000i −0.209196 1.71937i 0.264536 1.00000 −2.91247 + 0.719371i 1.00000i
341.6 1.00000 −0.209196 + 1.71937i 1.00000 1.00000i −0.209196 + 1.71937i 0.264536 1.00000 −2.91247 0.719371i 1.00000i
341.7 1.00000 1.71731 0.225499i 1.00000 1.00000i 1.71731 0.225499i −0.631989 1.00000 2.89830 0.774501i 1.00000i
341.8 1.00000 1.71731 + 0.225499i 1.00000 1.00000i 1.71731 + 0.225499i −0.631989 1.00000 2.89830 + 0.774501i 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.f.d yes 8
3.b odd 2 1 570.2.f.c 8
19.b odd 2 1 570.2.f.c 8
57.d even 2 1 inner 570.2.f.d yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.f.c 8 3.b odd 2 1
570.2.f.c 8 19.b odd 2 1
570.2.f.d yes 8 1.a even 1 1 trivial
570.2.f.d yes 8 57.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(570, [\chi])$$:

 $$T_{7}^{4} - 2T_{7}^{3} - 13T_{7}^{2} - 4T_{7} + 2$$ T7^4 - 2*T7^3 - 13*T7^2 - 4*T7 + 2 $$T_{29}^{4} + 2T_{29}^{3} - 11T_{29}^{2} - 24T_{29} - 8$$ T29^4 + 2*T29^3 - 11*T29^2 - 24*T29 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{8}$$
$3$ $$T^{8} + 2 T^{7} + T^{6} - 6 T^{5} + \cdots + 81$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$(T^{4} - 2 T^{3} - 13 T^{2} - 4 T + 2)^{2}$$
$11$ $$(T^{2} + 4)^{4}$$
$13$ $$T^{8} + 38 T^{6} + 265 T^{4} + \cdots + 256$$
$17$ $$T^{8} + 66 T^{6} + 1069 T^{4} + \cdots + 100$$
$19$ $$T^{8} - 12 T^{7} + 80 T^{6} + \cdots + 130321$$
$23$ $$T^{8} + 94 T^{6} + 2761 T^{4} + \cdots + 15376$$
$29$ $$(T^{4} + 2 T^{3} - 11 T^{2} - 24 T - 8)^{2}$$
$31$ $$T^{8} + 100 T^{6} + 3252 T^{4} + \cdots + 193600$$
$37$ $$T^{8} + 208 T^{6} + 11072 T^{4} + \cdots + 16384$$
$41$ $$(T^{4} - 8 T^{3} - 2 T^{2} + 24 T + 16)^{2}$$
$43$ $$(T^{4} + 20 T^{3} + 88 T^{2} - 112 T - 176)^{2}$$
$47$ $$T^{8} + 128 T^{6} + 5344 T^{4} + \cdots + 430336$$
$53$ $$(T^{4} - 14 T^{3} + 33 T^{2} + 168 T - 424)^{2}$$
$59$ $$(T^{4} + 2 T^{3} - 161 T^{2} - 280 T + 118)^{2}$$
$61$ $$(T^{4} - 8 T^{3} - 140 T^{2} + 848 T + 3104)^{2}$$
$67$ $$T^{8} + 510 T^{6} + 82801 T^{4} + \cdots + 1849600$$
$71$ $$(T^{4} - 12 T^{3} - 96 T^{2} + 1696 T - 5440)^{2}$$
$73$ $$(T^{4} + 2 T^{3} - 119 T^{2} - 764 T - 1228)^{2}$$
$79$ $$T^{8} + 220 T^{6} + 16132 T^{4} + \cdots + 4326400$$
$83$ $$T^{8} + 540 T^{6} + \cdots + 121000000$$
$89$ $$(T^{4} + 44 T^{3} + 698 T^{2} + \cdots + 11384)^{2}$$
$97$ $$T^{8} + 120 T^{6} + 4240 T^{4} + \cdots + 65536$$