# Properties

 Label 99.2.e.e Level $99$ Weight $2$ Character orbit 99.e Analytic conductor $0.791$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.508277025.1 Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19$$ x^8 - 3*x^7 + 5*x^6 - 15*x^5 + 21*x^4 + 3*x^3 - 22*x^2 + 3*x + 19 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -4\nu^{7} + 79\nu^{6} - 177\nu^{5} + 459\nu^{4} - 1008\nu^{3} + 1011\nu^{2} - 752\nu - 478 ) / 933$$ (-4*v^7 + 79*v^6 - 177*v^5 + 459*v^4 - 1008*v^3 + 1011*v^2 - 752*v - 478) / 933 $$\beta_{3}$$ $$=$$ $$( 35\nu^{7} + 164\nu^{6} - 395\nu^{5} + 260\nu^{4} - 2687\nu^{3} + 2894\nu^{2} + 1604\nu - 2193 ) / 1866$$ (35*v^7 + 164*v^6 - 395*v^5 + 260*v^4 - 2687*v^3 + 2894*v^2 + 1604*v - 2193) / 1866 $$\beta_{4}$$ $$=$$ $$( 217\nu^{7} + 146\nu^{6} + 39\nu^{5} - 876\nu^{4} - 4095\nu^{3} + 6498\nu^{2} + 1610\nu - 2525 ) / 5598$$ (217*v^7 + 146*v^6 + 39*v^5 - 876*v^4 - 4095*v^3 + 6498*v^2 + 1610*v - 2525) / 5598 $$\beta_{5}$$ $$=$$ $$( 241\nu^{7} - 328\nu^{6} + 1101\nu^{5} - 3630\nu^{4} + 1953\nu^{3} - 5166\nu^{2} + 6122\nu + 343 ) / 5598$$ (241*v^7 - 328*v^6 + 1101*v^5 - 3630*v^4 + 1953*v^3 - 5166*v^2 + 6122*v + 343) / 5598 $$\beta_{6}$$ $$=$$ $$( -145\nu^{7} + 298\nu^{6} - 585\nu^{5} + 1944\nu^{4} - 2019\nu^{3} - 438\nu^{2} + 730\nu + 1799 ) / 1866$$ (-145*v^7 + 298*v^6 - 585*v^5 + 1944*v^4 - 2019*v^3 - 438*v^2 + 730*v + 1799) / 1866 $$\beta_{7}$$ $$=$$ $$( 701\nu^{7} - 1016\nu^{6} + 1863\nu^{5} - 6966\nu^{4} + 2181\nu^{3} + 10122\nu^{2} - 6296\nu - 6265 ) / 5598$$ (701*v^7 - 1016*v^6 + 1863*v^5 - 6966*v^4 + 2181*v^3 + 10122*v^2 - 6296*v - 6265) / 5598
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} - \beta_{2}$$ -b5 + b4 - b2 $$\nu^{3}$$ $$=$$ $$-3\beta_{7} - 5\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{2} + \beta _1 + 3$$ -3*b7 - 5*b6 - b5 + b4 + 2*b2 + b1 + 3 $$\nu^{4}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} + 6\beta_{2} + 7\beta_1$$ -b6 + b5 - b4 - 3*b3 + 6*b2 + 7*b1 $$\nu^{5}$$ $$=$$ $$5\beta_{7} + 12\beta_{6} - \beta_{5} + 12\beta_{4} - 8\beta_{3} - 8\beta_{2} - \beta _1 - 14$$ 5*b7 + 12*b6 - b5 + 12*b4 - 8*b3 - 8*b2 - b1 - 14 $$\nu^{6}$$ $$=$$ $$-29\beta_{7} - 35\beta_{6} - 7\beta_{5} + 32\beta_{4} - \beta_{3} + 2\beta_{2} - 18\beta _1 + 16$$ -29*b7 - 35*b6 - 7*b5 + 32*b4 - b3 + 2*b2 - 18*b1 + 16 $$\nu^{7}$$ $$=$$ $$-38\beta_{7} - 77\beta_{6} + 20\beta_{5} - 13\beta_{4} - 10\beta_{3} + 92\beta_{2} + 52\beta _1 + 60$$ -38*b7 - 77*b6 + 20*b5 - 13*b4 - 10*b3 + 92*b2 + 52*b1 + 60

## Character values

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

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
34.1
 −0.734668 + 0.348716i −0.577806 − 2.22188i 0.947217 + 0.807294i 1.86526 + 0.199842i −0.734668 − 0.348716i −0.577806 + 2.22188i 0.947217 − 0.807294i 1.86526 − 0.199842i
−1.23467 + 2.13851i 1.66933 0.461883i −2.04881 3.54864i 1.21814 + 2.10988i −1.07333 + 4.14015i −1.16933 + 2.02534i 5.17972 2.57333 1.54207i −6.01598
34.2 −1.07781 + 1.86682i −0.635299 1.61133i −1.32333 2.29208i −1.81197 3.13842i 3.69279 + 0.550720i 1.13530 1.96640i 1.39396 −2.19279 + 2.04736i 7.81179
34.3 0.447217 0.774602i 1.22553 + 1.22396i 0.599994 + 1.03922i −1.87447 3.24667i 1.49616 0.401921i −0.725528 + 1.25665i 2.86218 0.00384004 + 3.00000i −3.35317
34.4 1.36526 2.36469i 0.240440 + 1.71528i −2.72785 4.72478i 0.468293 + 0.811107i 4.38438 + 1.77323i 0.259560 0.449571i −9.43585 −2.88438 + 0.824844i 2.55736
67.1 −1.23467 2.13851i 1.66933 + 0.461883i −2.04881 + 3.54864i 1.21814 2.10988i −1.07333 4.14015i −1.16933 2.02534i 5.17972 2.57333 + 1.54207i −6.01598
67.2 −1.07781 1.86682i −0.635299 + 1.61133i −1.32333 + 2.29208i −1.81197 + 3.13842i 3.69279 0.550720i 1.13530 + 1.96640i 1.39396 −2.19279 2.04736i 7.81179
67.3 0.447217 + 0.774602i 1.22553 1.22396i 0.599994 1.03922i −1.87447 + 3.24667i 1.49616 + 0.401921i −0.725528 1.25665i 2.86218 0.00384004 3.00000i −3.35317
67.4 1.36526 + 2.36469i 0.240440 1.71528i −2.72785 + 4.72478i 0.468293 0.811107i 4.38438 1.77323i 0.259560 + 0.449571i −9.43585 −2.88438 0.824844i 2.55736
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.e 8
3.b odd 2 1 297.2.e.e 8
9.c even 3 1 inner 99.2.e.e 8
9.c even 3 1 891.2.a.q 4
9.d odd 6 1 297.2.e.e 8
9.d odd 6 1 891.2.a.p 4
11.b odd 2 1 1089.2.e.i 8
99.g even 6 1 9801.2.a.bl 4
99.h odd 6 1 1089.2.e.i 8
99.h odd 6 1 9801.2.a.bi 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.e 8 1.a even 1 1 trivial
99.2.e.e 8 9.c even 3 1 inner
297.2.e.e 8 3.b odd 2 1
297.2.e.e 8 9.d odd 6 1
891.2.a.p 4 9.d odd 6 1
891.2.a.q 4 9.c even 3 1
1089.2.e.i 8 11.b odd 2 1
1089.2.e.i 8 99.h odd 6 1
9801.2.a.bi 4 99.h odd 6 1
9801.2.a.bl 4 99.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + T_{2}^{7} + 10T_{2}^{6} + 7T_{2}^{5} + 76T_{2}^{4} + 46T_{2}^{3} + 181T_{2}^{2} - 104T_{2} + 169$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{7} + 10 T^{6} + 7 T^{5} + \cdots + 169$$
$3$ $$T^{8} - 5 T^{7} + 15 T^{6} - 35 T^{5} + \cdots + 81$$
$5$ $$T^{8} + 4 T^{7} + 25 T^{6} + 22 T^{5} + \cdots + 961$$
$7$ $$T^{8} + T^{7} + 7 T^{6} + 4 T^{5} + \cdots + 16$$
$11$ $$(T^{2} + T + 1)^{4}$$
$13$ $$T^{8} + 7 T^{7} + 64 T^{6} + \cdots + 24964$$
$17$ $$(T^{4} + 5 T^{3} - 24 T^{2} - 169 T - 236)^{2}$$
$19$ $$(T^{4} - 9 T^{3} + 81 T - 54)^{2}$$
$23$ $$T^{8} + 14 T^{7} + 145 T^{6} + \cdots + 35344$$
$29$ $$T^{8} - 6 T^{7} + 45 T^{6} + \cdots + 22500$$
$31$ $$T^{8} - 2 T^{7} + 25 T^{6} + \cdots + 3481$$
$37$ $$(T^{4} - 3 T^{3} - 81 T^{2} + 144 T + 57)^{2}$$
$41$ $$T^{8} - 2 T^{7} + 67 T^{6} + 448 T^{5} + \cdots + 676$$
$43$ $$T^{8} - 21 T^{7} + 285 T^{6} + \cdots + 248004$$
$47$ $$T^{8} - 7 T^{7} + 64 T^{6} + 101 T^{5} + \cdots + 1$$
$53$ $$(T^{4} + 6 T^{3} - 45 T^{2} - 165 T - 123)^{2}$$
$59$ $$T^{8} + 2 T^{7} + 25 T^{6} + 56 T^{5} + \cdots + 169$$
$61$ $$T^{8} + 15 T^{7} + 147 T^{6} + \cdots + 14400$$
$67$ $$T^{8} + 14 T^{7} + 217 T^{6} + \cdots + 4713241$$
$71$ $$(T^{4} + 3 T^{3} - 87 T^{2} + 270 T - 195)^{2}$$
$73$ $$(T^{4} - 22 T^{3} + 129 T^{2} + 11 T - 1028)^{2}$$
$79$ $$T^{8} + 11 T^{7} + 145 T^{6} + \cdots + 1600$$
$83$ $$T^{8} + 18 T^{7} + 327 T^{6} + \cdots + 2396304$$
$89$ $$(T^{4} + 6 T^{3} - 132 T^{2} - 48 T + 2064)^{2}$$
$97$ $$T^{8} + 26 T^{7} + 520 T^{6} + \cdots + 1510441$$