# Properties

 Label 462.2.k.e Level $462$ Weight $2$ Character orbit 462.k 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.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 0.965926 − 0.258819i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i
−0.866025 0.500000i −1.33195 1.10721i 0.500000 + 0.866025i 0.465926 0.807007i 0.599900 + 1.62484i 2.19067 + 1.48356i 1.00000i 0.548188 + 2.94949i −0.807007 + 0.465926i
89.2 −0.866025 0.500000i 0.599900 1.62484i 0.500000 + 0.866025i −1.46593 + 2.53906i −1.33195 + 1.10721i −2.19067 1.48356i 1.00000i −2.28024 1.94949i 2.53906 1.46593i
89.3 0.866025 + 0.500000i 1.10721 1.33195i 0.500000 + 0.866025i −0.241181 + 0.417738i 1.62484 0.599900i 1.48356 2.19067i 1.00000i −0.548188 2.94949i −0.417738 + 0.241181i
89.4 0.866025 + 0.500000i 1.62484 + 0.599900i 0.500000 + 0.866025i −0.758819 + 1.31431i 1.10721 + 1.33195i −1.48356 + 2.19067i 1.00000i 2.28024 + 1.94949i −1.31431 + 0.758819i
353.1 −0.866025 + 0.500000i −1.33195 + 1.10721i 0.500000 0.866025i 0.465926 + 0.807007i 0.599900 1.62484i 2.19067 1.48356i 1.00000i 0.548188 2.94949i −0.807007 0.465926i
353.2 −0.866025 + 0.500000i 0.599900 + 1.62484i 0.500000 0.866025i −1.46593 2.53906i −1.33195 1.10721i −2.19067 + 1.48356i 1.00000i −2.28024 + 1.94949i 2.53906 + 1.46593i
353.3 0.866025 0.500000i 1.10721 + 1.33195i 0.500000 0.866025i −0.241181 0.417738i 1.62484 + 0.599900i 1.48356 + 2.19067i 1.00000i −0.548188 + 2.94949i −0.417738 0.241181i
353.4 0.866025 0.500000i 1.62484 0.599900i 0.500000 0.866025i −0.758819 1.31431i 1.10721 1.33195i −1.48356 2.19067i 1.00000i 2.28024 1.94949i −1.31431 0.758819i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.k.e 8
3.b odd 2 1 462.2.k.f yes 8
7.d odd 6 1 462.2.k.f yes 8
21.g even 6 1 inner 462.2.k.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.e 8 1.a even 1 1 trivial
462.2.k.e 8 21.g even 6 1 inner
462.2.k.f yes 8 3.b odd 2 1
462.2.k.f yes 8 7.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 $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}^{8} + 4T_{5}^{7} + 14T_{5}^{6} + 16T_{5}^{5} + 22T_{5}^{4} + 8T_{5}^{3} + 20T_{5}^{2} + 8T_{5} + 4$$ T5^8 + 4*T5^7 + 14*T5^6 + 16*T5^5 + 22*T5^4 + 8*T5^3 + 20*T5^2 + 8*T5 + 4 $$T_{17}^{8} - 20 T_{17}^{7} + 260 T_{17}^{6} - 2000 T_{17}^{5} + 11224 T_{17}^{4} - 40960 T_{17}^{3} + 107360 T_{17}^{2} - 150400 T_{17} + 141376$$ T17^8 - 20*T17^7 + 260*T17^6 - 2000*T17^5 + 11224*T17^4 - 40960*T17^3 + 107360*T17^2 - 150400*T17 + 141376

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{2}$$
$3$ $$T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81$$
$5$ $$T^{8} + 4 T^{7} + 14 T^{6} + 16 T^{5} + \cdots + 4$$
$7$ $$T^{8} + 71T^{4} + 2401$$
$11$ $$(T^{4} - T^{2} + 1)^{2}$$
$13$ $$(T^{4} + 30 T^{2} + 9)^{2}$$
$17$ $$T^{8} - 20 T^{7} + 260 T^{6} + \cdots + 141376$$
$19$ $$T^{8} + 12 T^{7} + 46 T^{6} - 24 T^{5} + \cdots + 4$$
$23$ $$T^{8} + 12 T^{7} + 54 T^{6} + 72 T^{5} + \cdots + 4$$
$29$ $$T^{8} + 108 T^{6} + 3782 T^{4} + \cdots + 187489$$
$31$ $$T^{8} - 12 T^{7} - 4 T^{6} + \cdots + 153664$$
$37$ $$T^{8} + 58 T^{6} + 168 T^{5} + \cdots + 145924$$
$41$ $$(T^{4} - 20 T^{3} + 74 T^{2} + 620 T - 3554)^{2}$$
$43$ $$(T^{4} + 4 T^{3} - 4 T^{2} - 16 T - 8)^{2}$$
$47$ $$(T^{4} - 12 T^{3} + 116 T^{2} - 336 T + 784)^{2}$$
$53$ $$T^{8} + 60 T^{7} + 1630 T^{6} + \cdots + 19909444$$
$59$ $$T^{8} - 4 T^{7} + 44 T^{6} - 88 T^{5} + \cdots + 625$$
$61$ $$T^{8} + 36 T^{7} + 534 T^{6} + \cdots + 4977361$$
$67$ $$T^{8} - 12 T^{7} + 148 T^{6} + \cdots + 5329$$
$71$ $$T^{8} + 60 T^{6} + 1088 T^{4} + \cdots + 8836$$
$73$ $$T^{8} - 48 T^{7} + 986 T^{6} + \cdots + 3694084$$
$79$ $$T^{8} + 8 T^{7} + 236 T^{6} + \cdots + 12794929$$
$83$ $$(T^{4} + 36 T^{3} + 444 T^{2} + 2160 T + 3384)^{2}$$
$89$ $$T^{8} + 4 T^{7} + 128 T^{6} + \cdots + 583696$$
$97$ $$T^{8} + 1020 T^{6} + \cdots + 2618266561$$