# Properties

 Label 99.2.e.d Level $99$ Weight $2$ Character orbit 99.e Analytic conductor $0.791$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ -v^4 + v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ -v^5 + v^4 $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$ -v^5 - v^4 + v
 $$\zeta_{18}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$$ (b5 + b4 + 2*b2) / 3 $$\zeta_{18}^{2}$$ $$=$$ $$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$$ (-2*b5 + b4 + 3*b3 - b2) / 3 $$\zeta_{18}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{18}^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$$ (-b5 + 2*b4 + b2) / 3 $$\zeta_{18}^{5}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$$ (-b5 - b4 + b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i
−0.766044 + 1.32683i 0.592396 + 1.62760i −0.173648 0.300767i 0.266044 + 0.460802i −2.61334 0.460802i 1.43969 2.49362i −2.53209 −2.29813 + 1.92836i −0.815207
34.2 −0.173648 + 0.300767i 1.11334 1.32683i 0.939693 + 1.62760i −0.326352 0.565258i 0.205737 + 0.565258i −0.266044 + 0.460802i −1.34730 −0.520945 2.95442i 0.226682
34.3 0.939693 1.62760i −1.70574 0.300767i −0.766044 1.32683i −1.43969 2.49362i −2.09240 + 2.49362i 0.326352 0.565258i 0.879385 2.81908 + 1.02606i −5.41147
67.1 −0.766044 1.32683i 0.592396 1.62760i −0.173648 + 0.300767i 0.266044 0.460802i −2.61334 + 0.460802i 1.43969 + 2.49362i −2.53209 −2.29813 1.92836i −0.815207
67.2 −0.173648 0.300767i 1.11334 + 1.32683i 0.939693 1.62760i −0.326352 + 0.565258i 0.205737 0.565258i −0.266044 0.460802i −1.34730 −0.520945 + 2.95442i 0.226682
67.3 0.939693 + 1.62760i −1.70574 + 0.300767i −0.766044 + 1.32683i −1.43969 + 2.49362i −2.09240 2.49362i 0.326352 + 0.565258i 0.879385 2.81908 1.02606i −5.41147
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.3 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 99.2.e.d 6
3.b odd 2 1 297.2.e.d 6
9.c even 3 1 inner 99.2.e.d 6
9.c even 3 1 891.2.a.l 3
9.d odd 6 1 297.2.e.d 6
9.d odd 6 1 891.2.a.k 3
11.b odd 2 1 1089.2.e.h 6
99.g even 6 1 9801.2.a.bd 3
99.h odd 6 1 1089.2.e.h 6
99.h odd 6 1 9801.2.a.be 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.d 6 1.a even 1 1 trivial
99.2.e.d 6 9.c even 3 1 inner
297.2.e.d 6 3.b odd 2 1
297.2.e.d 6 9.d odd 6 1
891.2.a.k 3 9.d odd 6 1
891.2.a.l 3 9.c even 3 1
1089.2.e.h 6 11.b odd 2 1
1089.2.e.h 6 99.h odd 6 1
9801.2.a.bd 3 99.g even 6 1
9801.2.a.be 3 99.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3T_{2}^{4} + 2T_{2}^{3} + 9T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$3$ $$T^{6} + 9T^{3} + 27$$
$5$ $$T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1$$
$7$ $$T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1$$
$11$ $$(T^{2} - T + 1)^{3}$$
$13$ $$T^{6} - 9 T^{5} + 66 T^{4} - 137 T^{3} + \cdots + 1$$
$17$ $$(T^{3} + 3 T^{2} - 6 T + 1)^{2}$$
$19$ $$(T^{3} + 9 T^{2} + 18 T - 9)^{2}$$
$23$ $$T^{6} - 6 T^{5} + 33 T^{4} - 56 T^{3} + \cdots + 361$$
$29$ $$T^{6} - 6 T^{5} + 81 T^{4} + \cdots + 45369$$
$31$ $$T^{6} - 9 T^{5} + 75 T^{4} - 92 T^{3} + \cdots + 361$$
$37$ $$(T^{3} + 12 T^{2} + 27 T - 3)^{2}$$
$41$ $$T^{6} - 6 T^{5} + 117 T^{4} + \cdots + 218089$$
$43$ $$T^{6} + 3 T^{5} + 153 T^{4} + \cdots + 294849$$
$47$ $$T^{6} + 12 T^{5} + 159 T^{4} + \cdots + 32041$$
$53$ $$(T^{3} - 3 T^{2} - 126 T + 57)^{2}$$
$59$ $$T^{6} + 21 T^{5} + 375 T^{4} + \cdots + 218089$$
$61$ $$T^{6} - 21 T^{5} + 297 T^{4} + \cdots + 103041$$
$67$ $$T^{6} + 3 T^{5} + 135 T^{4} + \cdots + 332929$$
$71$ $$(T^{3} - 12 T^{2} + 9 T + 111)^{2}$$
$73$ $$(T^{3} + 6 T^{2} - 105 T - 703)^{2}$$
$79$ $$T^{6} - 9 T^{5} + 57 T^{4} - 178 T^{3} + \cdots + 361$$
$83$ $$T^{6} + 27 T^{4} - 54 T^{3} + \cdots + 729$$
$89$ $$(T^{3} + 12 T^{2} - 108 T - 408)^{2}$$
$97$ $$T^{6} - 3 T^{5} + 150 T^{4} + \cdots + 516961$$