# Properties

 Label 143.4.a.c Level $143$ Weight $4$ Character orbit 143.a Self dual yes Analytic conductor $8.437$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988$$ x^9 - 59*x^7 - 12*x^6 + 1144*x^5 + 345*x^4 - 7888*x^3 - 2245*x^2 + 9710*x - 2988 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 5) q^{4} + (\beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{5} + (\beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_1 + 4) q^{6} + ( - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{7} + (\beta_{8} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} + 3 \beta_1 + 5) q^{8} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 10) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 + 1) * q^3 + (b4 - b3 - b2 + 5) * q^4 + (b8 + b4 - b3 + b2 - b1 + 4) * q^5 + (b8 + b5 - b4 + b3 + 3*b1 + 4) * q^6 + (-b8 - b6 - b5 - b4 - 2*b3 - b2 + 2) * q^7 + (b8 + 2*b6 - b5 + 3*b4 - b2 + 3*b1 + 5) * q^8 + (-2*b8 - b7 - b6 + b5 - b4 - b3 + 2*b2 + 10) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + 5) q^{4} + (\beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{5} + (\beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_1 + 4) q^{6} + ( - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{7} + (\beta_{8} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{2} + 3 \beta_1 + 5) q^{8} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 10) q^{9} + ( - \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_1 - 4) q^{10} - 11 q^{11} + ( - 3 \beta_{8} - \beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 20) q^{12} - 13 q^{13} + (4 \beta_{8} - 2 \beta_{7} + \beta_{6} + 4 \beta_{3} - \beta_{2} + 6 \beta_1 + 2) q^{14} + ( - 6 \beta_{8} + \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 6 \beta_{2} - \beta_1 + 37) q^{15} + ( - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + \cdots + 11) q^{16}+ \cdots + (22 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} + 11 \beta_{4} + \cdots - 110) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b3 + 1) * q^3 + (b4 - b3 - b2 + 5) * q^4 + (b8 + b4 - b3 + b2 - b1 + 4) * q^5 + (b8 + b5 - b4 + b3 + 3*b1 + 4) * q^6 + (-b8 - b6 - b5 - b4 - 2*b3 - b2 + 2) * q^7 + (b8 + 2*b6 - b5 + 3*b4 - b2 + 3*b1 + 5) * q^8 + (-2*b8 - b7 - b6 + b5 - b4 - b3 + 2*b2 + 10) * q^9 + (-b8 + 2*b7 - b4 + b3 - b2 + 5*b1 - 4) * q^10 - 11 * q^11 + (-3*b8 - b7 + b5 - 2*b4 - 2*b3 + b2 + b1 + 20) * q^12 - 13 * q^13 + (4*b8 - 2*b7 + b6 + 4*b3 - b2 + 6*b1 + 2) * q^14 + (-6*b8 + b7 - 2*b6 - 3*b4 + 6*b2 - b1 + 37) * q^15 + (-2*b8 + 2*b7 - b6 - 5*b5 + 4*b4 + 3*b3 - 5*b2 + 8*b1 + 11) * q^16 + (4*b8 - b7 + 4*b6 - 2*b5 - 3*b4 + 4*b3 + 5*b1 + 9) * q^17 + (b8 + b7 - 3*b6 + 6*b5 - 12*b4 + 11*b3 + 13*b1 + 1) * q^18 + (4*b8 + 3*b7 + 2*b6 - 2*b5 + b4 + b3 - 2*b2 - b1 + 8) * q^19 + (b8 - 2*b7 + 6*b6 + 2*b5 + 11*b4 + 3*b3 + 5*b2 - b1 + 36) * q^20 + (-4*b7 - b6 - 3*b5 - 6*b3 - 2*b2 - 11*b1 + 53) * q^21 - 11*b1 * q^22 + (5*b8 + b7 - 11*b6 + 3*b5 - 4*b4 + 4*b3 + 5*b2 - 5*b1 + 21) * q^23 + (-4*b8 - b7 - 5*b6 + b5 - 6*b4 + 4*b3 + b1 - 17) * q^24 + (5*b8 - 2*b7 - 6*b6 + 4*b5 - b4 + 7*b3 + 15*b2 - 11*b1 + 71) * q^25 - 13*b1 * q^26 + (6*b8 + 13*b6 - b5 + 18*b4 - 22*b3 - 8*b2 - 33*b1 + 39) * q^27 + (-12*b8 - b7 + b6 - 2*b5 - 12*b3 - 19*b2 - 5*b1 + 25) * q^28 + (3*b8 + 9*b7 + 14*b6 + 6*b4 - b3 - b2 - 2*b1 - 7) * q^29 + (16*b8 + 3*b7 - 2*b6 + 14*b5 - b4 + 34*b3 + 20*b2 + 25*b1 + 7) * q^30 + (-8*b8 + 4*b6 + 6*b5 - 4*b4 - 4*b3 - 10*b2 - 12*b1 + 68) * q^31 + (b8 - b7 - 4*b6 - 11*b5 + 31*b4 - 4*b3 - 5*b2 - 12*b1 + 72) * q^32 + (11*b3 - 11) * q^33 + (-16*b8 - 3*b7 - 16*b6 - 6*b5 - 11*b4 - 26*b3 - 26*b2 - 15*b1 + 33) * q^34 + (-15*b8 + 4*b7 - 4*b6 - 20*b5 - 11*b4 - 11*b3 + 5*b2 - 7*b1 - 46) * q^35 + (7*b8 - 4*b7 + 11*b6 + 2*b5 - 7*b4 - 38*b3 + 17*b2 - 33*b1 - 11) * q^36 + (-8*b8 - 3*b7 - 6*b6 + 3*b4 + 4*b3 - 41*b1 + 103) * q^37 + (3*b8 - b7 + 6*b6 - 7*b5 + 24*b4 - 17*b3 + 10*b2 - 4*b1 - 3) * q^38 + (13*b3 - 13) * q^39 + (-5*b8 - 10*b6 - 2*b5 + 13*b4 + 5*b3 - 17*b2 + 3*b1 + 62) * q^40 + (9*b8 - 7*b7 + 3*b6 + 19*b5 + 4*b4 + 3*b2 + 11*b1 - 17) * q^41 + (-10*b8 - 2*b7 - 11*b6 - 7*b5 - 36*b4 + 16*b3 - 32*b2 + 72*b1 - 148) * q^42 + (9*b8 - 6*b7 + 16*b6 + 2*b5 + 3*b4 + 7*b3 - 21*b2 + 13*b1 + 12) * q^43 + (-11*b4 + 11*b3 + 11*b2 - 55) * q^44 + (7*b8 + 7*b7 + 26*b6 + 2*b5 + 32*b4 - 73*b3 - 11*b2 - 82*b1 + 17) * q^45 + (-10*b8 + b7 + 19*b6 - 4*b5 - 11*b4 - 16*b3 + 35*b2 + 5*b1 - 111) * q^46 + (-8*b8 + 7*b7 - 20*b6 + 10*b5 - 51*b4 + 10*b3 + 12*b2 + 33*b1 - 43) * q^47 + (24*b8 + 2*b7 + 2*b6 - 6*b5 + 3*b4 + 20*b3 + 2*b2 - 32*b1 - 196) * q^48 + (19*b8 - 5*b7 - 10*b6 + 6*b5 - 4*b4 - 16*b3 + 17*b2 - 18*b1 + 35) * q^49 + (-25*b8 + 14*b7 - 6*b6 + 4*b5 - 43*b4 + 9*b3 + 13*b2 + 38*b1 - 168) * q^50 + (-10*b8 - 7*b7 - 28*b6 - 8*b5 - 35*b4 + 14*b3 - 24*b2 + 63*b1 - 85) * q^51 + (-13*b4 + 13*b3 + 13*b2 - 65) * q^52 + (10*b8 + b7 - 16*b6 + 18*b5 + 29*b4 + 41*b3 + 2*b2 + 33*b1 + 128) * q^53 + (10*b8 + 10*b7 - b6 + b5 - 10*b4 + 34*b3 - 8*b2 + 114*b1 - 278) * q^54 + (-11*b8 - 11*b4 + 11*b3 - 11*b2 + 11*b1 - 44) * q^55 + (-3*b7 + 4*b6 + b5 + 10*b4 - 3*b3 + 8*b2 + 48*b1 - 68) * q^56 + (-10*b8 + 2*b7 - 29*b6 + 7*b5 - 32*b4 + 77*b3 + 34*b2 + 61*b1 - 26) * q^57 + (31*b8 + 5*b7 + 6*b6 + 14*b5 + 62*b4 - 5*b3 + 51*b2 - 26*b1 + 75) * q^58 + (19*b8 - 3*b7 + 30*b6 - 6*b5 + 22*b4 - 5*b3 + 25*b2 - 70*b1 - 41) * q^59 + (-6*b8 + 11*b7 + 22*b6 + 39*b4 - 98*b3 - 6*b2 - 87*b1 - 83) * q^60 + (-13*b8 - 7*b7 - 6*b6 - 20*b5 - 2*b4 + 11*b3 + 3*b2 - 40*b1 + 43) * q^61 + (20*b8 - 14*b7 + 4*b6 + 22*b5 - 26*b4 + 18*b3 + 44*b2 + 104*b1 - 168) * q^62 + (-9*b8 - 18*b7 + 23*b6 - 11*b5 + 25*b4 - 63*b3 - 23*b2 - 72*b1 + 51) * q^63 + (14*b8 + 10*b7 + 38*b6 - 20*b5 + 68*b4 + 20*b3 - 28*b2 + 71*b1 - 124) * q^64 + (-13*b8 - 13*b4 + 13*b3 - 13*b2 + 13*b1 - 52) * q^65 + (-11*b8 - 11*b5 + 11*b4 - 11*b3 - 33*b1 - 44) * q^66 + (-5*b8 - 9*b7 - 8*b6 + 8*b5 + 32*b4 - 19*b3 - 71*b2 + 64*b1 - 45) * q^67 + (14*b8 - 29*b7 + 12*b5 - b4 + 16*b3 + 10*b2 + 101*b1 - 297) * q^68 + (2*b8 + 21*b7 + 7*b6 - 3*b5 - 35*b4 + 48*b3 + 22*b2 + 96*b1 - 100) * q^69 + (57*b8 - 6*b7 - 16*b6 + 2*b5 + 31*b4 + 97*b3 - 11*b2 - 85*b1 - 10) * q^70 + (-38*b8 - 11*b7 - 68*b6 - 24*b5 + b4 - 34*b3 - 36*b2 + 27*b1 + 211) * q^71 + (2*b7 - 32*b6 + 18*b5 - 63*b4 - 20*b3 - 2*b2 - 93*b1 - 280) * q^72 + (-7*b8 + 2*b7 + 44*b6 + 4*b5 + 59*b4 - 2*b3 + b2 - 37*b1 + 417) * q^73 + (3*b7 + 6*b6 - 8*b5 - 35*b4 + 72*b3 + 28*b2 + 121*b1 - 551) * q^74 + (-2*b8 + 27*b7 + 24*b6 - 18*b5 + 15*b4 - 105*b3 + 2*b2 - 23*b1 - 88) * q^75 + (-25*b8 + 10*b7 - 24*b6 + 7*b5 + 31*b4 + 52*b3 - 49*b2 + 46*b1 + 85) * q^76 + (11*b8 + 11*b6 + 11*b5 + 11*b4 + 22*b3 + 11*b2 - 22) * q^77 + (-13*b8 - 13*b5 + 13*b4 - 13*b3 - 39*b1 - 52) * q^78 + (-18*b8 + 6*b7 + 36*b6 - 22*b5 - 8*b4 - 80*b3 - 76*b2 - 20*b1 - 132) * q^79 + (-3*b8 + 12*b7 - 60*b5 - 5*b4 - 31*b3 - 49*b2 + 131*b1 - 276) * q^80 + (-68*b8 + b7 - 19*b6 - 29*b5 + 15*b4 + 2*b3 - 4*b2 - 84*b1 + 267) * q^81 + (-46*b8 + 7*b7 - 7*b6 + 24*b5 - 79*b4 - 56*b3 - 7*b2 + 25*b1 + 95) * q^82 + (20*b8 + 4*b7 + 51*b6 + 29*b5 - 17*b3 - 58*b2 - 139*b1 - 192) * q^83 + (-4*b8 - 36*b7 + 13*b6 - 5*b5 + 33*b4 - 93*b3 - 43*b2 - 89*b1 + 201) * q^84 + (18*b8 + b7 - 80*b6 - 22*b5 - 19*b4 + 66*b3 + 6*b2 - 57*b1 - 85) * q^85 + (-49*b8 - 18*b7 - 24*b6 - 26*b5 - 21*b4 - 97*b3 - 81*b2 + 37*b1 + 66) * q^86 + (-4*b8 + 8*b7 - 48*b6 + 58*b5 + 140*b3 + 126*b2 + 50*b1 + 202) * q^87 + (-11*b8 - 22*b6 + 11*b5 - 33*b4 + 11*b2 - 33*b1 - 55) * q^88 + (46*b8 + 26*b7 + 28*b6 - 12*b5 + 28*b4 + 52*b3 + 26*b2 + 62*b1 + 156) * q^89 + (87*b8 + 21*b7 + 14*b6 + 60*b5 - 36*b4 + 135*b3 + 81*b2 + 206*b1 - 601) * q^90 + (13*b8 + 13*b6 + 13*b5 + 13*b4 + 26*b3 + 13*b2 - 26) * q^91 + (16*b7 + 3*b6 + 60*b5 - 41*b4 + 82*b3 - 51*b2 - 150*b1 + 90) * q^92 + (-2*b8 - 16*b7 + 14*b6 + 20*b5 + 38*b4 - 118*b3 - 12*b2 - 6*b1 + 196) * q^93 + (34*b8 - 39*b7 + 8*b6 + 68*b5 - 65*b4 - 48*b3 + 142*b2 - 155*b1 + 229) * q^94 + (-4*b8 - 8*b7 - 70*b6 - 2*b5 - 72*b4 + 106*b3 + 76*b2 + 188*b1 + 202) * q^95 + (-28*b8 + 13*b7 + 37*b6 - 61*b5 + 43*b4 - 103*b3 - 5*b2 - 284*b1 - 368) * q^96 + (40*b8 + 4*b7 + 4*b6 + 30*b5 - 92*b4 + 54*b3 + 18*b2 + 64*b1 + 456) * q^97 + (-42*b8 + 13*b7 - 10*b6 + 15*b5 - 119*b4 - 18*b3 - 3*b2 + 38*b1 - 229) * q^98 + (22*b8 + 11*b7 + 11*b6 - 11*b5 + 11*b4 + 11*b3 - 22*b2 - 110) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + 8 q^{3} + 46 q^{4} + 30 q^{5} + 34 q^{6} + 25 q^{7} + 36 q^{8} + 91 q^{9}+O(q^{10})$$ 9 * q + 8 * q^3 + 46 * q^4 + 30 * q^5 + 34 * q^6 + 25 * q^7 + 36 * q^8 + 91 * q^9 $$9 q + 8 q^{3} + 46 q^{4} + 30 q^{5} + 34 q^{6} + 25 q^{7} + 36 q^{8} + 91 q^{9} - 22 q^{10} - 99 q^{11} + 181 q^{12} - 117 q^{13} + 351 q^{15} + 130 q^{16} + 53 q^{17} + 33 q^{18} + 69 q^{19} + 282 q^{20} + 463 q^{21} + 216 q^{23} - 121 q^{24} + 617 q^{25} + 275 q^{27} + 279 q^{28} - 91 q^{29} + 29 q^{30} + 636 q^{31} + 663 q^{32} - 88 q^{33} + 423 q^{34} - 358 q^{35} - 252 q^{36} + 967 q^{37} - 101 q^{38} - 104 q^{39} + 652 q^{40} - 226 q^{41} - 1186 q^{42} + 42 q^{43} - 506 q^{44} + 5 q^{45} - 1127 q^{46} - 269 q^{47} - 1820 q^{48} + 228 q^{49} - 1374 q^{50} - 589 q^{51} - 598 q^{52} + 1227 q^{53} - 2438 q^{54} - 330 q^{55} - 659 q^{56} - 71 q^{57} + 471 q^{58} - 613 q^{59} - 859 q^{60} + 427 q^{61} - 1714 q^{62} + 305 q^{63} - 1194 q^{64} - 390 q^{65} - 374 q^{66} - 271 q^{67} - 2835 q^{68} - 846 q^{69} - 102 q^{70} + 2279 q^{71} - 2400 q^{72} + 3602 q^{73} - 4955 q^{74} - 883 q^{75} + 1126 q^{76} - 275 q^{77} - 442 q^{78} - 1182 q^{79} - 2360 q^{80} + 2697 q^{81} + 1007 q^{82} - 1877 q^{83} + 1618 q^{84} - 441 q^{85} + 830 q^{86} + 1942 q^{87} - 396 q^{88} + 1258 q^{89} - 5669 q^{90} - 325 q^{91} + 1046 q^{92} + 1556 q^{93} + 1439 q^{94} + 2032 q^{95} - 3417 q^{96} + 4002 q^{97} - 1855 q^{98} - 1001 q^{99}+O(q^{100})$$ 9 * q + 8 * q^3 + 46 * q^4 + 30 * q^5 + 34 * q^6 + 25 * q^7 + 36 * q^8 + 91 * q^9 - 22 * q^10 - 99 * q^11 + 181 * q^12 - 117 * q^13 + 351 * q^15 + 130 * q^16 + 53 * q^17 + 33 * q^18 + 69 * q^19 + 282 * q^20 + 463 * q^21 + 216 * q^23 - 121 * q^24 + 617 * q^25 + 275 * q^27 + 279 * q^28 - 91 * q^29 + 29 * q^30 + 636 * q^31 + 663 * q^32 - 88 * q^33 + 423 * q^34 - 358 * q^35 - 252 * q^36 + 967 * q^37 - 101 * q^38 - 104 * q^39 + 652 * q^40 - 226 * q^41 - 1186 * q^42 + 42 * q^43 - 506 * q^44 + 5 * q^45 - 1127 * q^46 - 269 * q^47 - 1820 * q^48 + 228 * q^49 - 1374 * q^50 - 589 * q^51 - 598 * q^52 + 1227 * q^53 - 2438 * q^54 - 330 * q^55 - 659 * q^56 - 71 * q^57 + 471 * q^58 - 613 * q^59 - 859 * q^60 + 427 * q^61 - 1714 * q^62 + 305 * q^63 - 1194 * q^64 - 390 * q^65 - 374 * q^66 - 271 * q^67 - 2835 * q^68 - 846 * q^69 - 102 * q^70 + 2279 * q^71 - 2400 * q^72 + 3602 * q^73 - 4955 * q^74 - 883 * q^75 + 1126 * q^76 - 275 * q^77 - 442 * q^78 - 1182 * q^79 - 2360 * q^80 + 2697 * q^81 + 1007 * q^82 - 1877 * q^83 + 1618 * q^84 - 441 * q^85 + 830 * q^86 + 1942 * q^87 - 396 * q^88 + 1258 * q^89 - 5669 * q^90 - 325 * q^91 + 1046 * q^92 + 1556 * q^93 + 1439 * q^94 + 2032 * q^95 - 3417 * q^96 + 4002 * q^97 - 1855 * q^98 - 1001 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 59x^{7} - 12x^{6} + 1144x^{5} + 345x^{4} - 7888x^{3} - 2245x^{2} + 9710x - 2988$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - \nu^{8} + 357 \nu^{7} - 350 \nu^{6} - 14878 \nu^{5} + 9822 \nu^{4} + 154641 \nu^{3} - 82789 \nu^{2} - 171682 \nu + 322364 ) / 37760$$ (-v^8 + 357*v^7 - 350*v^6 - 14878*v^5 + 9822*v^4 + 154641*v^3 - 82789*v^2 - 171682*v + 322364) / 37760 $$\beta_{3}$$ $$=$$ $$( - 87 \nu^{8} - 621 \nu^{7} + 6990 \nu^{6} + 29774 \nu^{5} - 168206 \nu^{4} - 418873 \nu^{3} + 1290477 \nu^{2} + 1596466 \nu - 945052 ) / 75520$$ (-87*v^8 - 621*v^7 + 6990*v^6 + 29774*v^5 - 168206*v^4 - 418873*v^3 + 1290477*v^2 + 1596466*v - 945052) / 75520 $$\beta_{4}$$ $$=$$ $$( - 89 \nu^{8} + 93 \nu^{7} + 6290 \nu^{6} + 18 \nu^{5} - 148562 \nu^{4} - 109591 \nu^{3} + 1200419 \nu^{2} + 1253102 \nu - 1282084 ) / 75520$$ (-89*v^8 + 93*v^7 + 6290*v^6 + 18*v^5 - 148562*v^4 - 109591*v^3 + 1200419*v^2 + 1253102*v - 1282084) / 75520 $$\beta_{5}$$ $$=$$ $$( - 53 \nu^{8} - 119 \nu^{7} + 4810 \nu^{6} + 6346 \nu^{5} - 133194 \nu^{4} - 131707 \nu^{3} + 1212663 \nu^{2} + 1011254 \nu - 1429268 ) / 37760$$ (-53*v^8 - 119*v^7 + 4810*v^6 + 6346*v^5 - 133194*v^4 - 131707*v^3 + 1212663*v^2 + 1011254*v - 1429268) / 37760 $$\beta_{6}$$ $$=$$ $$( - 283 \nu^{8} + 551 \nu^{7} + 14550 \nu^{6} - 21674 \nu^{5} - 237974 \nu^{4} + 240843 \nu^{3} + 1287513 \nu^{2} - 549206 \nu - 612268 ) / 75520$$ (-283*v^8 + 551*v^7 + 14550*v^6 - 21674*v^5 - 237974*v^4 + 240843*v^3 + 1287513*v^2 - 549206*v - 612268) / 75520 $$\beta_{7}$$ $$=$$ $$( 311 \nu^{8} + 653 \nu^{7} - 16270 \nu^{6} - 35982 \nu^{5} + 263118 \nu^{4} + 525529 \nu^{3} - 1433101 \nu^{2} - 1848178 \nu + 1310876 ) / 37760$$ (311*v^8 + 653*v^7 - 16270*v^6 - 35982*v^5 + 263118*v^4 + 525529*v^3 - 1433101*v^2 - 1848178*v + 1310876) / 37760 $$\beta_{8}$$ $$=$$ $$( 145 \nu^{8} - 181 \nu^{7} - 7810 \nu^{6} + 5246 \nu^{5} + 134978 \nu^{4} - 6305 \nu^{3} - 783307 \nu^{2} - 483326 \nu + 495876 ) / 15104$$ (145*v^8 - 181*v^7 - 7810*v^6 + 5246*v^5 + 134978*v^4 - 6305*v^3 - 783307*v^2 - 483326*v + 495876) / 15104
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - \beta_{2} + 13$$ b4 - b3 - b2 + 13 $$\nu^{3}$$ $$=$$ $$\beta_{8} + 2\beta_{6} - \beta_{5} + 3\beta_{4} - \beta_{2} + 19\beta _1 + 5$$ b8 + 2*b6 - b5 + 3*b4 - b2 + 19*b1 + 5 $$\nu^{4}$$ $$=$$ $$-2\beta_{8} + 2\beta_{7} - \beta_{6} - 5\beta_{5} + 28\beta_{4} - 21\beta_{3} - 29\beta_{2} + 8\beta _1 + 259$$ -2*b8 + 2*b7 - b6 - 5*b5 + 28*b4 - 21*b3 - 29*b2 + 8*b1 + 259 $$\nu^{5}$$ $$=$$ $$33\beta_{8} - \beta_{7} + 60\beta_{6} - 43\beta_{5} + 127\beta_{4} - 4\beta_{3} - 37\beta_{2} + 404\beta _1 + 232$$ 33*b8 - b7 + 60*b6 - 43*b5 + 127*b4 - 4*b3 - 37*b2 + 404*b1 + 232 $$\nu^{6}$$ $$=$$ $$- 66 \beta_{8} + 90 \beta_{7} - 2 \beta_{6} - 220 \beta_{5} + 804 \beta_{4} - 436 \beta_{3} - 804 \beta_{2} + 391 \beta _1 + 5756$$ -66*b8 + 90*b7 - 2*b6 - 220*b5 + 804*b4 - 436*b3 - 804*b2 + 391*b1 + 5756 $$\nu^{7}$$ $$=$$ $$928\beta_{8} + 1662\beta_{6} - 1458\beta_{5} + 4309\beta_{4} - 275\beta_{3} - 1287\beta_{2} + 9292\beta _1 + 8519$$ 928*b8 + 1662*b6 - 1458*b5 + 4309*b4 - 275*b3 - 1287*b2 + 9292*b1 + 8519 $$\nu^{8}$$ $$=$$ $$- 1581 \beta_{8} + 3022 \beta_{7} + 814 \beta_{6} - 7503 \beta_{5} + 23557 \beta_{4} - 9536 \beta_{3} - 22023 \beta_{2} + 14755 \beta _1 + 138197$$ -1581*b8 + 3022*b7 + 814*b6 - 7503*b5 + 23557*b4 - 9536*b3 - 22023*b2 + 14755*b1 + 138197

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.87031 −4.11495 −3.76323 −1.62159 0.388321 0.765277 3.71870 4.08298 5.41479
−4.87031 −1.15688 15.7199 −10.2656 5.63435 15.3321 −37.5984 −25.6616 49.9968
1.2 −4.11495 9.51427 8.93279 14.5196 −39.1507 21.0410 −3.83839 63.5214 −59.7475
1.3 −3.76323 −4.70664 6.16188 20.6048 17.7122 −28.6315 6.91727 −4.84752 −77.5406
1.4 −1.62159 −2.83913 −5.37046 −8.40999 4.60389 −9.04976 21.6813 −18.9393 13.6375
1.5 0.388321 3.09988 −7.84921 16.8933 1.20375 26.1569 −6.15457 −17.3907 6.56001
1.6 0.765277 −9.54214 −7.41435 −17.1562 −7.30238 −4.60754 −11.7962 64.0525 −13.1292
1.7 3.71870 7.61710 5.82875 15.9808 28.3257 −21.9580 −8.07423 31.0202 59.4279
1.8 4.08298 7.19985 8.67073 −7.90460 29.3968 23.1330 2.73856 24.8378 −32.2743
1.9 5.41479 −1.18631 21.3199 5.73789 −6.42360 3.58372 72.1247 −25.5927 31.0694
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.a.c 9
3.b odd 2 1 1287.4.a.k 9
4.b odd 2 1 2288.4.a.r 9
11.b odd 2 1 1573.4.a.e 9
13.b even 2 1 1859.4.a.d 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.c 9 1.a even 1 1 trivial
1287.4.a.k 9 3.b odd 2 1
1573.4.a.e 9 11.b odd 2 1
1859.4.a.d 9 13.b even 2 1
2288.4.a.r 9 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{9} - 59T_{2}^{7} - 12T_{2}^{6} + 1144T_{2}^{5} + 345T_{2}^{4} - 7888T_{2}^{3} - 2245T_{2}^{2} + 9710T_{2} - 2988$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(143))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} - 59 T^{7} - 12 T^{6} + \cdots - 2988$$
$3$ $$T^{9} - 8 T^{8} - 135 T^{7} + \cdots + 283048$$
$5$ $$T^{9} - 30 T^{8} + \cdots - 5425892224$$
$7$ $$T^{9} - 25 T^{8} + \cdots - 18338418984$$
$11$ $$(T + 11)^{9}$$
$13$ $$(T + 13)^{9}$$
$17$ $$T^{9} - 53 T^{8} + \cdots + 63\!\cdots\!76$$
$19$ $$T^{9} - 69 T^{8} + \cdots + 14\!\cdots\!00$$
$23$ $$T^{9} - 216 T^{8} + \cdots + 22\!\cdots\!44$$
$29$ $$T^{9} + 91 T^{8} + \cdots + 27\!\cdots\!88$$
$31$ $$T^{9} - 636 T^{8} + \cdots + 56\!\cdots\!52$$
$37$ $$T^{9} - 967 T^{8} + \cdots - 72\!\cdots\!76$$
$41$ $$T^{9} + 226 T^{8} + \cdots - 58\!\cdots\!96$$
$43$ $$T^{9} - 42 T^{8} + \cdots + 51\!\cdots\!52$$
$47$ $$T^{9} + 269 T^{8} + \cdots - 51\!\cdots\!84$$
$53$ $$T^{9} - 1227 T^{8} + \cdots - 12\!\cdots\!32$$
$59$ $$T^{9} + 613 T^{8} + \cdots + 15\!\cdots\!52$$
$61$ $$T^{9} - 427 T^{8} + \cdots + 49\!\cdots\!64$$
$67$ $$T^{9} + 271 T^{8} + \cdots + 89\!\cdots\!72$$
$71$ $$T^{9} - 2279 T^{8} + \cdots + 15\!\cdots\!32$$
$73$ $$T^{9} - 3602 T^{8} + \cdots - 24\!\cdots\!24$$
$79$ $$T^{9} + 1182 T^{8} + \cdots + 19\!\cdots\!00$$
$83$ $$T^{9} + 1877 T^{8} + \cdots + 41\!\cdots\!88$$
$89$ $$T^{9} - 1258 T^{8} + \cdots - 94\!\cdots\!48$$
$97$ $$T^{9} - 4002 T^{8} + \cdots + 56\!\cdots\!64$$