# Properties

 Label 143.2.e.c Level $143$ Weight $2$ Character orbit 143.e Analytic conductor $1.142$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 143.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14186074890$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 9x^{10} - 2x^{9} + 59x^{8} - 13x^{7} + 175x^{6} - 50x^{5} + 380x^{4} - 64x^{3} + 280x^{2} + 48x + 144$$ x^12 + 9*x^10 - 2*x^9 + 59*x^8 - 13*x^7 + 175*x^6 - 50*x^5 + 380*x^4 - 64*x^3 + 280*x^2 + 48*x + 144 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{6} q^{2} + (\beta_{8} + \beta_{6}) q^{3} + ( - \beta_{10} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{4} + ( - \beta_{7} - 1) q^{5} + ( - \beta_{11} - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{3} + \beta_{2} + 1) q^{8} + (\beta_{11} + \beta_{10} - \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_1) q^{9}+O(q^{10})$$ q - b6 * q^2 + (b8 + b6) * q^3 + (-b10 + b6 + 2*b5 + b4 - b2) * q^4 + (-b7 - 1) * q^5 + (-b11 - b6 - 3*b5 - b4 + b2) * q^6 + (-b7 - b5 - b4 - b1) * q^7 + (-b11 - b3 + b2 + 1) * q^8 + (b11 + b10 - b7 + 2*b5 + b4 - b1) * q^9 $$q - \beta_{6} q^{2} + (\beta_{8} + \beta_{6}) q^{3} + ( - \beta_{10} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{4} + ( - \beta_{7} - 1) q^{5} + ( - \beta_{11} - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{3} + \beta_{2} + 1) q^{8} + (\beta_{11} + \beta_{10} - \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_1) q^{9} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{10} + (\beta_{5} + 1) q^{11} + ( - \beta_{10} - \beta_{8} - 2 \beta_{7} - 2 \beta_{2} - 3) q^{12} + (\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5}) q^{13} + ( - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{3} - \beta_{2} + 2) q^{14} + ( - \beta_{9} - 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{15} + (\beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + 2 \beta_1 - 2) q^{16} + ( - \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{2} + \beta_1) q^{17} + (3 \beta_{10} - \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{2} + 2) q^{18} + (\beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{11} + 3 \beta_{10} + \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{20} + (2 \beta_{11} - 2 \beta_{9} + \beta_{7} + 2 \beta_{3} - 2) q^{21} + ( - \beta_{6} + \beta_{2}) q^{22} + (\beta_{6} - \beta_{3} + 2 \beta_1) q^{23} + (\beta_{8} + 3 \beta_{6} + 3 \beta_{5} - \beta_{3} - 2 \beta_1 + 3) q^{24} + (2 \beta_{7} + \beta_{2} - 1) q^{25} + ( - \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{3} + \beta_{2} + \beta_1) q^{26} + ( - \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{2} + 1) q^{27} + ( - \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_1 + 2) q^{28} + (2 \beta_{9} - 3 \beta_{8} + 2 \beta_{4} - 2) q^{29} + (2 \beta_{11} - 2 \beta_{10} - 2 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} + \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{30} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{3} - 2) q^{31} + (\beta_{11} + \beta_{10} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1) q^{32} + ( - \beta_{10} + \beta_{6} - \beta_{2}) q^{33} + ( - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \beta_{2}) q^{34} + ( - \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_1) q^{35} + ( - \beta_{9} - \beta_{8} - 3 \beta_{6} - 6 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_1 - 5) q^{36} + (\beta_{9} - \beta_{8} + 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 3) q^{37} + ( - \beta_{9} + \beta_{7} - 2 \beta_{2} - 4) q^{38} + (2 \beta_{9} - 3 \beta_{7} + 4 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{39} + (2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{3} - 4 \beta_{2} - 2) q^{40} + ( - \beta_{9} + \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} + 5) q^{41} + ( - 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 1) q^{42}+ \cdots + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{3} - 1) q^{99}+O(q^{100})$$ q - b6 * q^2 + (b8 + b6) * q^3 + (-b10 + b6 + 2*b5 + b4 - b2) * q^4 + (-b7 - 1) * q^5 + (-b11 - b6 - 3*b5 - b4 + b2) * q^6 + (-b7 - b5 - b4 - b1) * q^7 + (-b11 - b3 + b2 + 1) * q^8 + (b11 + b10 - b7 + 2*b5 + b4 - b1) * q^9 + (b6 + b5 - b3 - b1 + 1) * q^10 + (b5 + 1) * q^11 + (-b10 - b8 - 2*b7 - 2*b2 - 3) * q^12 + (b11 + b8 + b7 + b5) * q^13 + (-b11 + b9 + b7 - b3 - b2 + 2) * q^14 + (-b9 - 2*b8 - 2*b6 - 2*b5 - b4 + b3 - 1) * q^15 + (b9 - b8 - b5 + b4 + 2*b1 - 2) * q^16 + (-b11 - b10 + b7 + b6 - b2 + b1) * q^17 + (3*b10 - b9 + 3*b8 + 3*b7 + 2*b2 + 2) * q^18 + (b11 - b10 + b7 - b6 - b5 + b2 + b1) * q^19 + (-b11 + 3*b10 + b7 - 2*b6 - 4*b5 - b4 + 2*b2 + b1) * q^20 + (2*b11 - 2*b9 + b7 + 2*b3 - 2) * q^21 + (-b6 + b2) * q^22 + (b6 - b3 + 2*b1) * q^23 + (b8 + 3*b6 + 3*b5 - b3 - 2*b1 + 3) * q^24 + (2*b7 + b2 - 1) * q^25 + (-b11 + b10 + 2*b8 + 2*b7 + b3 + b2 + b1) * q^26 + (-b10 + b9 - b8 - 2*b7 - 2*b2 + 1) * q^27 + (-b8 - 2*b6 + 2*b5 + b3 + b1 + 2) * q^28 + (2*b9 - 3*b8 + 2*b4 - 2) * q^29 + (2*b11 - 2*b10 - 2*b7 + 4*b6 + 5*b5 + b4 - 4*b2 - 2*b1) * q^30 + (b11 - b10 + b9 - b8 + b3 - 2) * q^31 + (b11 + b10 - 2*b7 - b6 + b5 + b4 + b2 - 2*b1) * q^32 + (-b10 + b6 - b2) * q^33 + (-2*b10 + b9 - 2*b8 - 3*b7 + b2) * q^34 + (-b10 + b7 - b6 - 2*b5 + b2 + b1) * q^35 + (-b9 - b8 - 3*b6 - 6*b5 - b4 + 4*b3 + b1 - 5) * q^36 + (b9 - b8 + 2*b6 + 4*b5 + b4 + b3 - b1 + 3) * q^37 + (-b9 + b7 - 2*b2 - 4) * q^38 + (2*b9 - 3*b7 + 4*b5 + b4 - b3 - b2 - b1) * q^39 + (2*b11 - b10 - b9 - b8 - b7 + 2*b3 - 4*b2 - 2) * q^40 + (-b9 + b6 + 4*b5 - b4 - b3 + 5) * q^41 + (-2*b9 + 4*b8 + 4*b6 - 3*b5 - 2*b4 + b3 - 3*b1 - 1) * q^42 + (b10 + 2*b7 - 2*b5 - b4 + 2*b1) * q^43 + (-b10 - b9 - b8 - b2 - 1) * q^44 + (-b11 + 2*b7 - 3*b6 - 5*b5 + 3*b2 + 2*b1) * q^45 + (2*b11 + 3*b10 - b6 - 3*b5 - b4 + b2) * q^46 + (-b11 + b10 + b8 - 2*b7 - b3 + b2 - 1) * q^47 + (-3*b11 + 2*b10 - 2*b6 - 8*b5 - 3*b4 + 2*b2) * q^48 + (-2*b9 + 2*b8 + b6 - b5 - 2*b4 + b3 - b1 + 1) * q^49 + (-b9 - b8 - 6*b5 - b4 + 2*b3 + 2*b1 - 5) * q^50 + (2*b10 - b9 + 2*b8 + b7 + 2*b2 - 2) * q^51 + (-3*b10 - b9 - 3*b7 - b6 - b4 + b3 + b1 - 3) * q^52 + (-b10 - b9 - b8 + 4*b7 - 4*b2 + 2) * q^53 + (3*b9 + b8 + 11*b5 + 3*b4 - 3*b3 - 2*b1 + 8) * q^54 + (-b5 + b1 - 1) * q^55 + (-3*b10 - b7 + 2*b6 + 3*b5 - 2*b2 - b1) * q^56 + (-2*b11 + 3*b10 + 3*b9 + 3*b8 - 3*b7 - 2*b3 + 3*b2) * q^57 + (3*b11 + 3*b10 + b5 + 2*b4) * q^58 + (-b11 - 2*b10 + 2*b7 + 2*b6 + 2*b5 + 3*b4 - 2*b2 + 2*b1) * q^59 + (-2*b11 + 2*b10 + b9 + 2*b8 + 6*b7 - 2*b3 + 5*b2 + 11) * q^60 + (2*b7 + b6 - 3*b5 + b4 - b2 + 2*b1) * q^61 + (b9 + b8 + b6 + 2*b5 + b4 - b3 - 2*b1 + 1) * q^62 + (2*b9 - 3*b8 - 5*b6 + 3*b5 + 2*b4 - 3*b3 + 3*b1 + 1) * q^63 + (-b11 - b3 - 4) * q^64 + (-b11 + b10 - b9 - 2*b8 - b7 - 4*b6 - 3*b5 - 2*b4 + 2*b2 - b1 - 3) * q^65 + (-b11 + b9 - b3 + b2 + 2) * q^66 + (-b9 - b6 + 4*b5 - b4 - 2*b3 + 5) * q^67 + (-b8 - b5 - 3*b3 - b1 - 1) * q^68 + (-b11 + 2*b10 - 2*b7 - b5 - 2*b4 - 2*b1) * q^69 + (-2*b10 - b9 - 2*b8 - b7 - 3*b2 - 5) * q^70 + (-b11 - 3*b10 - 4*b7 + b6 - b4 - b2 - 4*b1) * q^71 + (2*b11 - 4*b10 - 3*b7 + 5*b6 + 12*b5 + 4*b4 - 5*b2 - 3*b1) * q^72 + (b10 + b8 + b7 - 2) * q^73 + (b10 - b7 - 6*b6 - 7*b5 - b4 + 6*b2 - b1) * q^74 + (3*b9 + b8 + 2*b6 + 7*b5 + 3*b4 - 3*b3 + 4) * q^75 + (b9 + 4*b8 + 5*b6 + 3*b5 + b4 - b3 + 3*b1 + 2) * q^76 + (b9 - b7) * q^77 + (-b11 + 2*b10 + 2*b9 + b8 + 3*b7 - b6 + 9*b5 + 3*b4 - 3*b3 + 4*b2 + 7) * q^78 + (b11 - 2*b10 - 5*b9 - 2*b8 - b7 + b3 + 1) * q^79 + (b9 + b8 + 3*b6 + 8*b5 + b4 - 3*b1 + 7) * q^80 + (-2*b9 - 2*b8 - b6 - 8*b5 - 2*b4 + 2*b3 - 3*b1 - 6) * q^81 + (3*b10 + 2*b7 - 5*b6 - 7*b5 - 2*b4 + 5*b2 + 2*b1) * q^82 + (-b11 - b10 - 2*b9 - b8 + b7 - b3 + 2*b2) * q^83 + (-3*b11 - 2*b10 + 3*b7 - b6 - 10*b5 - 2*b4 + b2 + 3*b1) * q^84 + (b10 + 2*b6 + 2*b5 - 2*b2) * q^85 + (3*b11 + b10 + b9 + b8 - 2*b7 + 3*b3 - 2*b2) * q^86 + (-2*b11 - 3*b10 + b7 + 3*b6 - 10*b5 - 2*b4 - 3*b2 + b1) * q^87 + (b6 + b5 - b3 + 1) * q^88 + (-3*b9 + 3*b8 - 4*b6 - 3*b5 - 3*b4 + b3 - b1) * q^89 + (2*b11 - 5*b10 - 3*b9 - 5*b8 - 4*b7 + 2*b3 - 8*b2 - 12) * q^90 + (b11 + b10 - 3*b8 + 2*b7 - 2*b6 + 3*b5 + 2*b4 - 2*b2 - 1) * q^91 + (b11 + 6*b10 + 6*b8 + b3 - 2*b2 + 3) * q^92 + (-b6 - 4*b5 + b3 + 2*b1 - 4) * q^93 + (-b9 - 2*b8 - 2*b5 - b4 - b3 - 1) * q^94 + (3*b10 - 5*b7 - 2*b6 + 3*b5 - 2*b4 + 2*b2 - 5*b1) * q^95 + (-4*b10 + b9 - 4*b8 - 2*b7 - 4*b2 + 2) * q^96 + (-3*b11 + b7 - 2*b6 - 3*b5 + 2*b4 + 2*b2 + b1) * q^97 + (-3*b11 - 3*b10 - b7 - 6*b5 - 3*b4 - b1) * q^98 + (b11 + b10 - b9 + b8 - b7 + b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} + 3 q^{7} + 6 q^{8} - 7 q^{9}+O(q^{10})$$ 12 * q - q^3 - 8 * q^4 - 12 * q^5 + 12 * q^6 + 3 * q^7 + 6 * q^8 - 7 * q^9 $$12 q - q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} + 3 q^{7} + 6 q^{8} - 7 q^{9} + 3 q^{10} + 6 q^{11} - 34 q^{12} - 4 q^{13} + 24 q^{14} - 4 q^{15} - 8 q^{16} - 2 q^{17} + 12 q^{18} + 10 q^{19} + 15 q^{20} - 24 q^{21} - 3 q^{23} + 14 q^{24} - 12 q^{25} - 3 q^{26} + 20 q^{27} + 16 q^{28} - 3 q^{29} - 19 q^{30} - 10 q^{31} - q^{32} + q^{33} + 10 q^{34} + 13 q^{35} - 20 q^{36} + 25 q^{37} - 54 q^{38} - 12 q^{39} - 16 q^{40} + 24 q^{41} - 13 q^{42} + 8 q^{43} - 16 q^{44} + 27 q^{45} + 18 q^{46} - 20 q^{47} + 28 q^{48} + q^{49} - 26 q^{50} - 34 q^{51} - 39 q^{52} + 20 q^{53} + 47 q^{54} - 6 q^{55} - 15 q^{56} + 6 q^{58} - 4 q^{59} + 122 q^{60} + 21 q^{61} + 5 q^{62} + 6 q^{63} - 54 q^{64} - 32 q^{65} + 24 q^{66} + 21 q^{67} - 14 q^{68} - 5 q^{69} - 62 q^{70} - 3 q^{71} - 50 q^{72} - 26 q^{73} + 38 q^{74} + 23 q^{75} + 8 q^{76} + 6 q^{77} + 36 q^{78} - 8 q^{79} + 44 q^{80} - 34 q^{81} + 33 q^{82} - 16 q^{83} + 47 q^{84} - 13 q^{85} + 22 q^{86} + 51 q^{87} + 3 q^{88} - 9 q^{89} - 140 q^{90} - 19 q^{91} + 30 q^{92} - 21 q^{93} - 10 q^{94} - 27 q^{95} + 38 q^{96} + 15 q^{97} + 21 q^{98} - 14 q^{99}+O(q^{100})$$ 12 * q - q^3 - 8 * q^4 - 12 * q^5 + 12 * q^6 + 3 * q^7 + 6 * q^8 - 7 * q^9 + 3 * q^10 + 6 * q^11 - 34 * q^12 - 4 * q^13 + 24 * q^14 - 4 * q^15 - 8 * q^16 - 2 * q^17 + 12 * q^18 + 10 * q^19 + 15 * q^20 - 24 * q^21 - 3 * q^23 + 14 * q^24 - 12 * q^25 - 3 * q^26 + 20 * q^27 + 16 * q^28 - 3 * q^29 - 19 * q^30 - 10 * q^31 - q^32 + q^33 + 10 * q^34 + 13 * q^35 - 20 * q^36 + 25 * q^37 - 54 * q^38 - 12 * q^39 - 16 * q^40 + 24 * q^41 - 13 * q^42 + 8 * q^43 - 16 * q^44 + 27 * q^45 + 18 * q^46 - 20 * q^47 + 28 * q^48 + q^49 - 26 * q^50 - 34 * q^51 - 39 * q^52 + 20 * q^53 + 47 * q^54 - 6 * q^55 - 15 * q^56 + 6 * q^58 - 4 * q^59 + 122 * q^60 + 21 * q^61 + 5 * q^62 + 6 * q^63 - 54 * q^64 - 32 * q^65 + 24 * q^66 + 21 * q^67 - 14 * q^68 - 5 * q^69 - 62 * q^70 - 3 * q^71 - 50 * q^72 - 26 * q^73 + 38 * q^74 + 23 * q^75 + 8 * q^76 + 6 * q^77 + 36 * q^78 - 8 * q^79 + 44 * q^80 - 34 * q^81 + 33 * q^82 - 16 * q^83 + 47 * q^84 - 13 * q^85 + 22 * q^86 + 51 * q^87 + 3 * q^88 - 9 * q^89 - 140 * q^90 - 19 * q^91 + 30 * q^92 - 21 * q^93 - 10 * q^94 - 27 * q^95 + 38 * q^96 + 15 * q^97 + 21 * q^98 - 14 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 9x^{10} - 2x^{9} + 59x^{8} - 13x^{7} + 175x^{6} - 50x^{5} + 380x^{4} - 64x^{3} + 280x^{2} + 48x + 144$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 433477 \nu^{11} + 1209797 \nu^{10} - 3549502 \nu^{9} + 8605189 \nu^{8} - 25150864 \nu^{7} + 55816400 \nu^{6} - 68042765 \nu^{5} + \cdots - 736079934 ) / 321804478$$ (-433477*v^11 + 1209797*v^10 - 3549502*v^9 + 8605189*v^8 - 25150864*v^7 + 55816400*v^6 - 68042765*v^5 + 59215674*v^4 - 157945465*v^3 + 26061228*v^2 - 19608912*v - 736079934) / 321804478 $$\beta_{3}$$ $$=$$ $$( 6513113 \nu^{11} + 23332617 \nu^{10} + 26020485 \nu^{9} + 178582811 \nu^{8} + 134283553 \nu^{7} + 1222140490 \nu^{6} - 896911294 \nu^{5} + \cdots + 5403214872 ) / 3861653736$$ (6513113*v^11 + 23332617*v^10 + 26020485*v^9 + 178582811*v^8 + 134283553*v^7 + 1222140490*v^6 - 896911294*v^5 + 3156182165*v^4 - 3200910050*v^3 + 8285100052*v^2 - 14013942892*v + 5403214872) / 3861653736 $$\beta_{4}$$ $$=$$ $$( - 7065514 \nu^{11} + 107212017 \nu^{10} + 28665066 \nu^{9} + 964117505 \nu^{8} + 21141184 \nu^{7} + 5746960873 \nu^{6} + 1240962365 \nu^{5} + \cdots + 6598596768 ) / 3861653736$$ (-7065514*v^11 + 107212017*v^10 + 28665066*v^9 + 964117505*v^8 + 21141184*v^7 + 5746960873*v^6 + 1240962365*v^5 + 15007459715*v^4 - 801817694*v^3 + 22370540164*v^2 + 3755120972*v + 6598596768) / 3861653736 $$\beta_{5}$$ $$=$$ $$( - 8004451 \nu^{11} + 19111125 \nu^{10} - 66838335 \nu^{9} + 173491463 \nu^{8} - 467890835 \nu^{7} + 1128351970 \nu^{6} - 1347413182 \nu^{5} + \cdots + 792512880 ) / 3861653736$$ (-8004451*v^11 + 19111125*v^10 - 66838335*v^9 + 173491463*v^8 - 467890835*v^7 + 1128351970*v^6 - 1347413182*v^5 + 3074872625*v^4 - 3180734450*v^3 + 7063924276*v^2 - 1569012700*v + 792512880) / 3861653736 $$\beta_{6}$$ $$=$$ $$( 6270543 \nu^{11} - 14271937 \nu^{10} + 52640327 \nu^{9} - 139070707 \nu^{8} + 367287379 \nu^{7} - 905086370 \nu^{6} + 1075242122 \nu^{5} + \cdots - 3736832616 ) / 1287217912$$ (6270543*v^11 - 14271937*v^10 + 52640327*v^9 - 139070707*v^8 + 367287379*v^7 - 905086370*v^6 + 1075242122*v^5 - 2838009929*v^4 + 2548952590*v^3 - 5672461452*v^2 + 1490577052*v - 3736832616) / 1287217912 $$\beta_{7}$$ $$=$$ $$( 6370375 \nu^{11} + 1733908 \nu^{10} + 52494187 \nu^{9} + 1457258 \nu^{8} + 341431369 \nu^{7} + 17788581 \nu^{6} + 891550025 \nu^{5} + \cdots + 384213648 ) / 1287217912$$ (6370375*v^11 + 1733908*v^10 + 52494187*v^9 + 1457258*v^8 + 341431369*v^7 + 17788581*v^6 + 891550025*v^5 - 46347690*v^4 + 2183879804*v^3 + 224077860*v^2 + 392242176*v + 384213648) / 1287217912 $$\beta_{8}$$ $$=$$ $$( 26508257 \nu^{11} + 24694089 \nu^{10} + 156505095 \nu^{9} + 156404459 \nu^{8} + 773028499 \nu^{7} + 1155926710 \nu^{6} - 94460776 \nu^{5} + \cdots + 5595887088 ) / 3861653736$$ (26508257*v^11 + 24694089*v^10 + 156505095*v^9 + 156404459*v^8 + 773028499*v^7 + 1155926710*v^6 - 94460776*v^5 + 3277228427*v^4 - 2700205100*v^3 + 8685938968*v^2 - 11400311692*v + 5595887088) / 3861653736 $$\beta_{9}$$ $$=$$ $$( - 15688761 \nu^{11} - 26880018 \nu^{10} - 140572891 \nu^{9} - 151808712 \nu^{8} - 808207929 \nu^{7} - 854883141 \nu^{6} + \cdots + 3212162936 ) / 1287217912$$ (-15688761*v^11 - 26880018*v^10 - 140572891*v^9 - 151808712*v^8 - 808207929*v^7 - 854883141*v^6 - 2070071595*v^5 - 840492498*v^4 - 2892144582*v^3 - 1255268076*v^2 - 1086587424*v + 3212162936) / 1287217912 $$\beta_{10}$$ $$=$$ $$( 52748602 \nu^{11} - 7877727 \nu^{10} + 529245270 \nu^{9} - 169452311 \nu^{8} + 3481751012 \nu^{7} - 1189783075 \nu^{6} + 11213080081 \nu^{5} + \cdots + 2349022944 ) / 3861653736$$ (52748602*v^11 - 7877727*v^10 + 529245270*v^9 - 169452311*v^8 + 3481751012*v^7 - 1189783075*v^6 + 11213080081*v^5 - 4054671365*v^4 + 22493758034*v^3 - 6046044724*v^2 + 16255017388*v + 2349022944) / 3861653736 $$\beta_{11}$$ $$=$$ $$( 69931387 \nu^{11} - 2525721 \nu^{10} + 603909759 \nu^{9} - 161095715 \nu^{8} + 3962892875 \nu^{7} - 1008677518 \nu^{6} + 11595511594 \nu^{5} + \cdots + 3069002640 ) / 3861653736$$ (69931387*v^11 - 2525721*v^10 + 603909759*v^9 - 161095715*v^8 + 3962892875*v^7 - 1008677518*v^6 + 11595511594*v^5 - 3712354445*v^4 + 25545813962*v^3 - 5596165732*v^2 + 18720849004*v + 3069002640) / 3861653736
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 3\beta_{5} - \beta_{2}$$ b6 + 3*b5 - b2 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + 4\beta_{7} - \beta_{3} + 1$$ -b11 + 4*b7 - b3 + 1 $$\nu^{4}$$ $$=$$ $$-\beta_{9} - \beta_{8} - 7\beta_{6} - 13\beta_{5} - \beta_{4} - 12$$ -b9 - b8 - 7*b6 - 13*b5 - b4 - 12 $$\nu^{5}$$ $$=$$ $$7\beta_{11} - 2\beta_{10} - 18\beta_{7} + \beta_{6} + 8\beta_{5} - \beta_{2} - 18\beta_1$$ 7*b11 - 2*b10 - 18*b7 + b6 + 8*b5 - b2 - 18*b1 $$\nu^{6}$$ $$=$$ $$-\beta_{11} + 9\beta_{10} + 9\beta_{9} + 9\beta_{8} - \beta_{3} + 41\beta_{2} + 55$$ -b11 + 9*b10 + 9*b9 + 9*b8 - b3 + 41*b2 + 55 $$\nu^{7}$$ $$=$$ $$-\beta_{9} - 19\beta_{8} - 12\beta_{6} - 51\beta_{5} - \beta_{4} + 41\beta_{3} + 86\beta _1 - 50$$ -b9 - 19*b8 - 12*b6 - 51*b5 - b4 + 41*b3 + 86*b1 - 50 $$\nu^{8}$$ $$=$$ $$12\beta_{11} - 61\beta_{10} - 2\beta_{7} + 228\beta_{6} + 330\beta_{5} + 59\beta_{4} - 228\beta_{2} - 2\beta_1$$ 12*b11 - 61*b10 - 2*b7 + 228*b6 + 330*b5 + 59*b4 - 228*b2 - 2*b1 $$\nu^{9}$$ $$=$$ $$-228\beta_{11} + 132\beta_{10} + 14\beta_{9} + 132\beta_{8} + 426\beta_{7} - 228\beta_{3} + 99\beta_{2} + 293$$ -228*b11 + 132*b10 + 14*b9 + 132*b8 + 426*b7 - 228*b3 + 99*b2 + 293 $$\nu^{10}$$ $$=$$ $$-346\beta_{9} - 374\beta_{8} - 1242\beta_{6} - 1737\beta_{5} - 346\beta_{4} + 99\beta_{3} + 32\beta _1 - 1391$$ -346*b9 - 374*b8 - 1242*b6 - 1737*b5 - 346*b4 + 99*b3 + 32*b1 - 1391 $$\nu^{11}$$ $$=$$ $$1242 \beta_{11} - 819 \beta_{10} - 2160 \beta_{7} + 703 \beta_{6} + 1811 \beta_{5} + 127 \beta_{4} - 703 \beta_{2} - 2160 \beta_1$$ 1242*b11 - 819*b10 - 2160*b7 + 703*b6 + 1811*b5 + 127*b4 - 703*b2 - 2160*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/143\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 −1.17629 + 2.03740i 1.11519 − 1.93157i −0.903935 + 1.56566i 0.790735 − 1.36959i 0.542744 − 0.940060i −0.368446 + 0.638166i −1.17629 − 2.03740i 1.11519 + 1.93157i −0.903935 − 1.56566i 0.790735 + 1.36959i 0.542744 + 0.940060i −0.368446 − 0.638166i
−1.26732 + 2.19506i 1.49512 2.58962i −2.21221 3.83165i −3.35258 3.78959 + 6.56377i −0.959205 1.66139i 6.14501 −2.97076 5.14551i 4.24880 7.35913i
100.2 −0.987312 + 1.71007i 0.285746 0.494927i −0.949569 1.64470i 1.23039 0.564241 + 0.977294i 1.27902 + 2.21532i −0.199164 1.33670 + 2.31523i −1.21478 + 2.10405i
100.3 −0.134198 + 0.232438i −1.17558 + 2.03617i 0.963982 + 1.66967i −2.80787 −0.315522 0.546500i −1.19233 2.06518i −1.05425 −1.26400 2.18931i 0.376811 0.652655i
100.4 0.249477 0.432106i 0.120298 0.208363i 0.875523 + 1.51645i 0.581470 −0.0600233 0.103963i −0.705086 1.22125i 1.87160 1.47106 + 2.54794i 0.145063 0.251257i
100.5 0.910859 1.57765i −1.55764 + 2.69791i −0.659327 1.14199i 0.0854874 2.83757 + 4.91482i 2.25971 + 3.91393i 1.24122 −3.35247 5.80665i 0.0778669 0.134869i
100.6 1.22850 2.12782i 0.332058 0.575141i −2.01840 3.49598i −1.73689 −0.815863 1.41312i 0.817900 + 1.41664i −5.00442 1.27948 + 2.21612i −2.13376 + 3.69579i
133.1 −1.26732 2.19506i 1.49512 + 2.58962i −2.21221 + 3.83165i −3.35258 3.78959 6.56377i −0.959205 + 1.66139i 6.14501 −2.97076 + 5.14551i 4.24880 + 7.35913i
133.2 −0.987312 1.71007i 0.285746 + 0.494927i −0.949569 + 1.64470i 1.23039 0.564241 0.977294i 1.27902 2.21532i −0.199164 1.33670 2.31523i −1.21478 2.10405i
133.3 −0.134198 0.232438i −1.17558 2.03617i 0.963982 1.66967i −2.80787 −0.315522 + 0.546500i −1.19233 + 2.06518i −1.05425 −1.26400 + 2.18931i 0.376811 + 0.652655i
133.4 0.249477 + 0.432106i 0.120298 + 0.208363i 0.875523 1.51645i 0.581470 −0.0600233 + 0.103963i −0.705086 + 1.22125i 1.87160 1.47106 2.54794i 0.145063 + 0.251257i
133.5 0.910859 + 1.57765i −1.55764 2.69791i −0.659327 + 1.14199i 0.0854874 2.83757 4.91482i 2.25971 3.91393i 1.24122 −3.35247 + 5.80665i 0.0778669 + 0.134869i
133.6 1.22850 + 2.12782i 0.332058 + 0.575141i −2.01840 + 3.49598i −1.73689 −0.815863 + 1.41312i 0.817900 1.41664i −5.00442 1.27948 2.21612i −2.13376 3.69579i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.2.e.c 12
13.c even 3 1 inner 143.2.e.c 12
13.c even 3 1 1859.2.a.k 6
13.e even 6 1 1859.2.a.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.c 12 1.a even 1 1 trivial
143.2.e.c 12 13.c even 3 1 inner
1859.2.a.k 6 13.c even 3 1
1859.2.a.l 6 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 10 T_{2}^{10} - 2 T_{2}^{9} + 76 T_{2}^{8} - 15 T_{2}^{7} + 235 T_{2}^{6} - 76 T_{2}^{5} + 551 T_{2}^{4} - 114 T_{2}^{3} + 97 T_{2}^{2} + 15 T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(143, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 10 T^{10} - 2 T^{9} + 76 T^{8} + \cdots + 9$$
$3$ $$T^{12} + T^{11} + 13 T^{10} + 2 T^{9} + \cdots + 4$$
$5$ $$(T^{6} + 6 T^{5} + 6 T^{4} - 15 T^{3} - 14 T^{2} + \cdots - 1)^{2}$$
$7$ $$T^{12} - 3 T^{11} + 25 T^{10} + \cdots + 14884$$
$11$ $$(T^{2} - T + 1)^{6}$$
$13$ $$T^{12} + 4 T^{11} + 17 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} + 2 T^{11} + 38 T^{10} + \cdots + 19321$$
$19$ $$T^{12} - 10 T^{11} + 109 T^{10} + \cdots + 91204$$
$23$ $$T^{12} + 3 T^{11} + 65 T^{10} + \cdots + 350464$$
$29$ $$T^{12} + 3 T^{11} + 135 T^{10} + \cdots + 395651881$$
$31$ $$(T^{6} + 5 T^{5} - 31 T^{4} - 57 T^{3} + \cdots + 120)^{2}$$
$37$ $$T^{12} - 25 T^{11} + 449 T^{10} + \cdots + 26739241$$
$41$ $$T^{12} - 24 T^{11} + 394 T^{10} + \cdots + 41306329$$
$43$ $$T^{12} - 8 T^{11} + 107 T^{10} + \cdots + 2096704$$
$47$ $$(T^{6} + 10 T^{5} - 15 T^{4} - 183 T^{3} + \cdots - 2)^{2}$$
$53$ $$(T^{6} - 10 T^{5} - 156 T^{4} + 1997 T^{3} + \cdots + 121)^{2}$$
$59$ $$T^{12} + 4 T^{11} + 165 T^{10} + \cdots + 338486404$$
$61$ $$T^{12} - 21 T^{11} + 297 T^{10} + \cdots + 429025$$
$67$ $$T^{12} - 21 T^{11} + 347 T^{10} + \cdots + 8868484$$
$71$ $$T^{12} + 3 T^{11} + \cdots + 10252777536$$
$73$ $$(T^{6} + 13 T^{5} + 49 T^{4} + 5 T^{3} + \cdots + 296)^{2}$$
$79$ $$(T^{6} + 4 T^{5} - 408 T^{4} + \cdots - 1572016)^{2}$$
$83$ $$(T^{6} + 8 T^{5} - 177 T^{4} - 397 T^{3} + \cdots - 12114)^{2}$$
$89$ $$T^{12} + 9 T^{11} + \cdots + 420375876496$$
$97$ $$T^{12} - 15 T^{11} + 492 T^{10} + \cdots + 58186384$$