# Properties

 Label 462.2.j.h Level $462$ Weight $2$ Character orbit 462.j Analytic conductor $3.689$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.j (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.20164000000.8 Defining polynomial: $$x^{8} + 6x^{6} + 76x^{4} + 781x^{2} + 5041$$ x^8 + 6*x^6 + 76*x^4 + 781*x^2 + 5041 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6x^{6} + 76x^{4} + 781x^{2} + 5041$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 17\nu^{6} - 537\nu^{4} + 44034\nu^{2} - 40328 ) / 383471$$ (17*v^6 - 537*v^4 + 44034*v^2 - 40328) / 383471 $$\beta_{3}$$ $$=$$ $$( -153\nu^{6} + 4833\nu^{4} - 12835\nu^{2} - 20519 ) / 383471$$ (-153*v^6 + 4833*v^4 - 12835*v^2 - 20519) / 383471 $$\beta_{4}$$ $$=$$ $$( -153\nu^{7} + 4833\nu^{5} - 12835\nu^{3} - 20519\nu ) / 383471$$ (-153*v^7 + 4833*v^5 - 12835*v^3 - 20519*v) / 383471 $$\beta_{5}$$ $$=$$ $$( -585\nu^{7} - 4078\nu^{5} - 49075\nu^{3} - 461926\nu ) / 383471$$ (-585*v^7 - 4078*v^5 - 49075*v^3 - 461926*v) / 383471 $$\beta_{6}$$ $$=$$ $$( -721\nu^{6} + 218\nu^{4} - 17876\nu^{2} - 139302 ) / 383471$$ (-721*v^6 + 218*v^4 - 17876*v^2 - 139302) / 383471 $$\beta_{7}$$ $$=$$ $$( -721\nu^{7} + 218\nu^{5} - 17876\nu^{3} - 139302\nu ) / 383471$$ (-721*v^7 + 218*v^5 - 17876*v^3 - 139302*v) / 383471
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 9\beta_{2} + 1$$ b3 + 9*b2 + 1 $$\nu^{3}$$ $$=$$ $$9\beta_{7} - 9\beta_{5} - 8\beta_{4} - 8\beta_1$$ 9*b7 - 9*b5 - 8*b4 - 8*b1 $$\nu^{4}$$ $$=$$ $$-17\beta_{6} + 82\beta_{3} + 17\beta_{2}$$ -17*b6 + 82*b3 + 17*b2 $$\nu^{5}$$ $$=$$ $$-17\beta_{5} + 65\beta_{4} - 17\beta_1$$ -17*b5 + 65*b4 - 17*b1 $$\nu^{6}$$ $$=$$ $$-537\beta_{6} - 218\beta_{2} - 218$$ -537*b6 - 218*b2 - 218 $$\nu^{7}$$ $$=$$ $$-755\beta_{7} + 218\beta_{5} + 218\beta_{4}$$ -755*b7 + 218*b5 + 218*b4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{2} - \beta_{3} + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 −2.50900 + 1.82290i 2.50900 − 1.82290i −0.839592 − 2.58400i 0.839592 + 2.58400i −0.839592 + 2.58400i 0.839592 − 2.58400i −2.50900 − 1.82290i 2.50900 + 1.82290i
0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −3.31802 + 2.41068i −0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i −4.10130
169.2 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 1.69998 1.23511i −0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i 2.10130
295.1 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.530575 1.63294i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i 1.71698
295.2 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 1.14861 + 3.53506i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i −3.71698
379.1 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.530575 + 1.63294i 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.71698
379.2 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 1.14861 3.53506i 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i −3.71698
421.1 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −3.31802 2.41068i −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i −4.10130
421.2 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i 1.69998 + 1.23511i −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i 2.10130
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.j.h 8
11.c even 5 1 inner 462.2.j.h 8
11.c even 5 1 5082.2.a.by 4
11.d odd 10 1 5082.2.a.cd 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.h 8 1.a even 1 1 trivial
462.2.j.h 8 11.c even 5 1 inner
5082.2.a.by 4 11.c even 5 1
5082.2.a.cd 4 11.d odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 2T_{5}^{7} + 9T_{5}^{6} + 28T_{5}^{5} + 156T_{5}^{4} - 370T_{5}^{3} + 790T_{5}^{2} - 550T_{5} + 3025$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$3$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$5$ $$T^{8} + 2 T^{7} + 9 T^{6} + 28 T^{5} + \cdots + 3025$$
$7$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$11$ $$T^{8} - 10 T^{7} + 51 T^{6} + \cdots + 14641$$
$13$ $$T^{8} - 10 T^{7} + 74 T^{6} + \cdots + 26896$$
$17$ $$(T^{4} + 6 T^{3} + 36 T^{2} + 81 T + 81)^{2}$$
$19$ $$T^{8} - 2 T^{7} - 3 T^{6} + 20 T^{5} + \cdots + 2401$$
$23$ $$(T^{4} - 57 T^{2} + 20 T + 131)^{2}$$
$29$ $$T^{8} + 2 T^{7} - 26 T^{6} - 112 T^{5} + \cdots + 400$$
$31$ $$T^{8} + 4 T^{7} - 27 T^{6} + \cdots + 32761$$
$37$ $$T^{8} - 30 T^{7} + 506 T^{6} + \cdots + 1771561$$
$41$ $$T^{8} + 24 T^{7} + 408 T^{6} + \cdots + 25110121$$
$43$ $$(T^{4} + 10 T^{3} - 48 T^{2} - 540 T - 604)^{2}$$
$47$ $$T^{8} + 6 T^{7} - 2 T^{6} + \cdots + 99856$$
$53$ $$T^{8} - 24 T^{7} + 466 T^{6} + \cdots + 64320400$$
$59$ $$T^{8} - 14 T^{7} + 158 T^{6} + \cdots + 6948496$$
$61$ $$T^{8} + 12 T^{7} + 138 T^{6} + \cdots + 355216$$
$67$ $$(T^{4} + 6 T^{3} - 74 T^{2} - 224 T + 1516)^{2}$$
$71$ $$T^{8} - 4 T^{7} + 156 T^{6} + \cdots + 774400$$
$73$ $$T^{8} + 34 T^{7} + 758 T^{6} + \cdots + 4648336$$
$79$ $$T^{8} + 22 T^{7} + \cdots + 383376400$$
$83$ $$T^{8} - 12 T^{7} + 98 T^{6} + \cdots + 698896$$
$89$ $$(T^{4} - 22 T^{3} - 9 T^{2} + 1570 T - 4895)^{2}$$
$97$ $$T^{8} + 8 T^{7} + 34 T^{6} + 72 T^{5} + \cdots + 400$$