# Properties

 Label 63.3.j.a Level $63$ Weight $3$ Character orbit 63.j Analytic conductor $1.717$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 63.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.71662566547$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.63369648.1 Defining polynomial: $$x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9$$ x^6 - x^5 + 12*x^4 + 17*x^3 + 118*x^2 + 33*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{4} - 1) q^{2} + 3 q^{3} + ( - \beta_1 - 4) q^{4} + ( - \beta_{4} - 2 \beta_{2} - 4) q^{5} + ( - 3 \beta_{5} - 3 \beta_{4} - 3) q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b5 - b4 - 1) * q^2 + 3 * q^3 + (-b1 - 4) * q^4 + (-b4 - 2*b2 - 4) * q^5 + (-3*b5 - 3*b4 - 3) * q^6 + (-b5 + b4 + b3 + b2) * q^7 + (3*b5 + 3*b4 - 2*b3 - 2*b2 + b1 + 2) * q^8 + 9 * q^9 $$q + ( - \beta_{5} - \beta_{4} - 1) q^{2} + 3 q^{3} + ( - \beta_1 - 4) q^{4} + ( - \beta_{4} - 2 \beta_{2} - 4) q^{5} + ( - 3 \beta_{5} - 3 \beta_{4} - 3) q^{6} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{7} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{8} + 9 q^{9} + (4 \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} - \beta_1 + 4) q^{10} + (2 \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{11} + ( - 3 \beta_1 - 12) q^{12} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2} + 6) q^{13} + ( - 4 \beta_{5} - \beta_{3} - 15 \beta_{2} - \beta_1 - 13) q^{14} + ( - 3 \beta_{4} - 6 \beta_{2} - 12) q^{15} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_1 + 15) q^{16} + (6 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 10) q^{17} + ( - 9 \beta_{5} - 9 \beta_{4} - 9) q^{18} + (2 \beta_{3} - 7 \beta_{2} - 7) q^{19} + (7 \beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_1 + 14) q^{20} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2}) q^{21} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 19 \beta_{2} + 18) q^{22} + ( - 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{23} + (9 \beta_{5} + 9 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 6) q^{24} + (4 \beta_{5} + 8 \beta_{4} + \beta_{3} - 5 \beta_{2} - 1) q^{25} + ( - 2 \beta_{5} - 2 \beta_{3} - 9 \beta_{2} + 4 \beta_1 + 7) q^{26} + 27 q^{27} + (11 \beta_{5} + \beta_{4} - 3 \beta_{3} - 31 \beta_{2} + 3 \beta_1 - 17) q^{28} + ( - \beta_{4} - 6 \beta_{2} - 12) q^{29} + (12 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 24 \beta_{2} - 3 \beta_1 + 12) q^{30} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_1 - 14) q^{31} + ( - 5 \beta_{5} - 5 \beta_{4} + 4 \beta_{3} + 60 \beta_{2} - 2 \beta_1 + 25) q^{32} + (6 \beta_{5} - 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{33} + ( - 22 \beta_{5} - 11 \beta_{4} - 42 \beta_{2} - 22) q^{34} + ( - 4 \beta_{5} - 10 \beta_{4} - 3 \beta_{3} - 10 \beta_{2} - 21) q^{35} + ( - 9 \beta_1 - 36) q^{36} + ( - 7 \beta_{5} - 14 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 10) q^{37} + (\beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 1) q^{38} + (3 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 18) q^{39} + ( - 4 \beta_{5} - 2 \beta_{4} - 21 \beta_{2} - 4) q^{40} + (10 \beta_{5} + 4 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 5) q^{41} + ( - 12 \beta_{5} - 3 \beta_{3} - 45 \beta_{2} - 3 \beta_1 - 39) q^{42} + (8 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 38 \beta_{2} + 3 \beta_1 + 8) q^{43} + ( - 26 \beta_{5} + 2 \beta_{3} + 21 \beta_{2} - 4 \beta_1 - 47) q^{44} + ( - 9 \beta_{4} - 18 \beta_{2} - 36) q^{45} + (4 \beta_{5} + 2 \beta_{4} + 21 \beta_{2} + 4) q^{46} + (18 \beta_{5} + 18 \beta_{4} + 4 \beta_{3} - 24 \beta_{2} - 2 \beta_1 + 6) q^{47} + (12 \beta_{5} - 12 \beta_{4} + 6 \beta_1 + 45) q^{48} + (12 \beta_{4} - 6 \beta_{3} + 22 \beta_{2} + 3 \beta_1 + 18) q^{49} + (2 \beta_{5} - 3 \beta_{3} - 31 \beta_{2} + 6 \beta_1 + 33) q^{50} + (18 \beta_{4} + 6 \beta_{3} + 15 \beta_{2} + 6 \beta_1 + 30) q^{51} + ( - 11 \beta_{5} - 22 \beta_{4} + 10 \beta_{2} - 1) q^{52} + ( - 15 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{53} + ( - 27 \beta_{5} - 27 \beta_{4} - 27) q^{54} + ( - 3 \beta_{5} + 3 \beta_{4} - \beta_1 + 28) q^{55} + (12 \beta_{5} - 12 \beta_{4} + 9 \beta_{3} + 23 \beta_{2} + 70) q^{56} + (6 \beta_{3} - 21 \beta_{2} - 21) q^{57} + (12 \beta_{5} + 6 \beta_{4} + \beta_{3} + 8 \beta_{2} - \beta_1 + 12) q^{58} + ( - 6 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 34 \beta_{2} + 3 \beta_1 - 23) q^{59} + (21 \beta_{4} + 3 \beta_{3} + 21 \beta_{2} + 3 \beta_1 + 42) q^{60} + ( - 6 \beta_{5} + 6 \beta_{4} - 8 \beta_1 + 10) q^{61} + (24 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} - 40 \beta_{2} + 6 \beta_1 + 4) q^{62} + ( - 9 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} + 9 \beta_{2}) q^{63} + ( - 20 \beta_{5} + 20 \beta_{4} - 3 \beta_1 - 22) q^{64} + ( - 8 \beta_{5} - 8 \beta_{4} - 38 \beta_{2} - 27) q^{65} + ( - 3 \beta_{5} - 6 \beta_{4} + 15 \beta_{3} + 57 \beta_{2} + 54) q^{66} + (6 \beta_{5} - 6 \beta_{4} + 7 \beta_1 - 35) q^{67} + ( - 18 \beta_{4} - 3 \beta_{3} - 68 \beta_{2} - 3 \beta_1 - 136) q^{68} + ( - 9 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{69} + (26 \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 45 \beta_{2} - 4 \beta_1 - 3) q^{70} + ( - 6 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + 32 \beta_{2} - 2 \beta_1 + 10) q^{71} + (27 \beta_{5} + 27 \beta_{4} - 18 \beta_{3} - 18 \beta_{2} + 9 \beta_1 + 18) q^{72} + ( - 36 \beta_{5} - 18 \beta_{4} - 7 \beta_{3} + 18 \beta_{2} + 7 \beta_1 - 36) q^{73} + (18 \beta_{5} + 2 \beta_{3} + 51 \beta_{2} - 4 \beta_1 - 33) q^{74} + (12 \beta_{5} + 24 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 3) q^{75} + (8 \beta_{5} + 16 \beta_{4} + 3 \beta_{3} - 26 \beta_{2} - 18) q^{76} + ( - 3 \beta_{5} - 20 \beta_{4} + 4 \beta_{3} + 46 \beta_{2} - 11 \beta_1 + 32) q^{77} + ( - 6 \beta_{5} - 6 \beta_{3} - 27 \beta_{2} + 12 \beta_1 + 21) q^{78} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_1 + 31) q^{79} + (7 \beta_{4} + 2 \beta_{3} + 12 \beta_{2} + 2 \beta_1 + 24) q^{80} + 81 q^{81} + (17 \beta_{5} + 34 \beta_{4} - 2 \beta_{3} + 68 \beta_{2} + 85) q^{82} + (9 \beta_{3} + 8 \beta_{2} + 9 \beta_1 + 16) q^{83} + (33 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 93 \beta_{2} + 9 \beta_1 - 51) q^{84} + ( - 23 \beta_{5} - 46 \beta_{4} - 12 \beta_{3} - 72 \beta_{2} - 95) q^{85} + (29 \beta_{4} + 7 \beta_{3} + 35 \beta_{2} + 7 \beta_1 + 70) q^{86} + ( - 3 \beta_{4} - 18 \beta_{2} - 36) q^{87} + (23 \beta_{5} + 46 \beta_{4} - 12 \beta_{3} - 138 \beta_{2} - 115) q^{88} + (8 \beta_{5} + 5 \beta_{3} + 28 \beta_{2} - 10 \beta_1 - 20) q^{89} + (36 \beta_{5} + 18 \beta_{4} + 9 \beta_{3} + 72 \beta_{2} - 9 \beta_1 + 36) q^{90} + (18 \beta_{5} + 12 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 45) q^{91} + (9 \beta_{4} - 2 \beta_{3} + 20 \beta_{2} - 2 \beta_1 + 40) q^{92} + ( - 6 \beta_{5} + 6 \beta_{4} - 12 \beta_1 - 42) q^{93} + (6 \beta_{5} - 6 \beta_{4} + 12 \beta_1 + 144) q^{94} + (\beta_{5} + \beta_{4} - 4 \beta_{3} + 32 \beta_{2} + 2 \beta_1 + 17) q^{95} + ( - 15 \beta_{5} - 15 \beta_{4} + 12 \beta_{3} + 180 \beta_{2} - 6 \beta_1 + 75) q^{96} + ( - 24 \beta_{5} - 12 \beta_{4} - \beta_{3} + 42 \beta_{2} + \beta_1 - 24) q^{97} + ( - 9 \beta_{5} - 5 \beta_{4} - 12 \beta_{3} - 96 \beta_{2} + 21 \beta_1) q^{98} + (18 \beta_{5} - 9 \beta_{3} + 18 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100})$$ q + (-b5 - b4 - 1) * q^2 + 3 * q^3 + (-b1 - 4) * q^4 + (-b4 - 2*b2 - 4) * q^5 + (-3*b5 - 3*b4 - 3) * q^6 + (-b5 + b4 + b3 + b2) * q^7 + (3*b5 + 3*b4 - 2*b3 - 2*b2 + b1 + 2) * q^8 + 9 * q^9 + (4*b5 + 2*b4 + b3 + 8*b2 - b1 + 4) * q^10 + (2*b5 - b3 + 2*b2 + 2*b1) * q^11 + (-3*b1 - 12) * q^12 + (b5 + 2*b4 - b3 + 5*b2 + 6) * q^13 + (-4*b5 - b3 - 15*b2 - b1 - 13) * q^14 + (-3*b4 - 6*b2 - 12) * q^15 + (4*b5 - 4*b4 + 2*b1 + 15) * q^16 + (6*b4 + 2*b3 + 5*b2 + 2*b1 + 10) * q^17 + (-9*b5 - 9*b4 - 9) * q^18 + (2*b3 - 7*b2 - 7) * q^19 + (7*b4 + b3 + 7*b2 + b1 + 14) * q^20 + (-3*b5 + 3*b4 + 3*b3 + 3*b2) * q^21 + (-b5 - 2*b4 + 5*b3 + 19*b2 + 18) * q^22 + (-3*b4 - b3 + b2 - b1 + 2) * q^23 + (9*b5 + 9*b4 - 6*b3 - 6*b2 + 3*b1 + 6) * q^24 + (4*b5 + 8*b4 + b3 - 5*b2 - 1) * q^25 + (-2*b5 - 2*b3 - 9*b2 + 4*b1 + 7) * q^26 + 27 * q^27 + (11*b5 + b4 - 3*b3 - 31*b2 + 3*b1 - 17) * q^28 + (-b4 - 6*b2 - 12) * q^29 + (12*b5 + 6*b4 + 3*b3 + 24*b2 - 3*b1 + 12) * q^30 + (-2*b5 + 2*b4 - 4*b1 - 14) * q^31 + (-5*b5 - 5*b4 + 4*b3 + 60*b2 - 2*b1 + 25) * q^32 + (6*b5 - 3*b3 + 6*b2 + 6*b1) * q^33 + (-22*b5 - 11*b4 - 42*b2 - 22) * q^34 + (-4*b5 - 10*b4 - 3*b3 - 10*b2 - 21) * q^35 + (-9*b1 - 36) * q^36 + (-7*b5 - 14*b4 - 5*b3 - 3*b2 - 10) * q^37 + (b5 + 2*b3 + 2*b2 - 4*b1 - 1) * q^38 + (3*b5 + 6*b4 - 3*b3 + 15*b2 + 18) * q^39 + (-4*b5 - 2*b4 - 21*b2 - 4) * q^40 + (10*b5 + 4*b3 + 5*b2 - 8*b1 + 5) * q^41 + (-12*b5 - 3*b3 - 45*b2 - 3*b1 - 39) * q^42 + (8*b5 + 4*b4 - 3*b3 + 38*b2 + 3*b1 + 8) * q^43 + (-26*b5 + 2*b3 + 21*b2 - 4*b1 - 47) * q^44 + (-9*b4 - 18*b2 - 36) * q^45 + (4*b5 + 2*b4 + 21*b2 + 4) * q^46 + (18*b5 + 18*b4 + 4*b3 - 24*b2 - 2*b1 + 6) * q^47 + (12*b5 - 12*b4 + 6*b1 + 45) * q^48 + (12*b4 - 6*b3 + 22*b2 + 3*b1 + 18) * q^49 + (2*b5 - 3*b3 - 31*b2 + 6*b1 + 33) * q^50 + (18*b4 + 6*b3 + 15*b2 + 6*b1 + 30) * q^51 + (-11*b5 - 22*b4 + 10*b2 - 1) * q^52 + (-15*b4 - 3*b3 + b2 - 3*b1 + 2) * q^53 + (-27*b5 - 27*b4 - 27) * q^54 + (-3*b5 + 3*b4 - b1 + 28) * q^55 + (12*b5 - 12*b4 + 9*b3 + 23*b2 + 70) * q^56 + (6*b3 - 21*b2 - 21) * q^57 + (12*b5 + 6*b4 + b3 + 8*b2 - b1 + 12) * q^58 + (-6*b5 - 6*b4 - 6*b3 - 34*b2 + 3*b1 - 23) * q^59 + (21*b4 + 3*b3 + 21*b2 + 3*b1 + 42) * q^60 + (-6*b5 + 6*b4 - 8*b1 + 10) * q^61 + (24*b5 + 24*b4 - 12*b3 - 40*b2 + 6*b1 + 4) * q^62 + (-9*b5 + 9*b4 + 9*b3 + 9*b2) * q^63 + (-20*b5 + 20*b4 - 3*b1 - 22) * q^64 + (-8*b5 - 8*b4 - 38*b2 - 27) * q^65 + (-3*b5 - 6*b4 + 15*b3 + 57*b2 + 54) * q^66 + (6*b5 - 6*b4 + 7*b1 - 35) * q^67 + (-18*b4 - 3*b3 - 68*b2 - 3*b1 - 136) * q^68 + (-9*b4 - 3*b3 + 3*b2 - 3*b1 + 6) * q^69 + (26*b5 + 7*b4 + 3*b3 + 45*b2 - 4*b1 - 3) * q^70 + (-6*b5 - 6*b4 + 4*b3 + 32*b2 - 2*b1 + 10) * q^71 + (27*b5 + 27*b4 - 18*b3 - 18*b2 + 9*b1 + 18) * q^72 + (-36*b5 - 18*b4 - 7*b3 + 18*b2 + 7*b1 - 36) * q^73 + (18*b5 + 2*b3 + 51*b2 - 4*b1 - 33) * q^74 + (12*b5 + 24*b4 + 3*b3 - 15*b2 - 3) * q^75 + (8*b5 + 16*b4 + 3*b3 - 26*b2 - 18) * q^76 + (-3*b5 - 20*b4 + 4*b3 + 46*b2 - 11*b1 + 32) * q^77 + (-6*b5 - 6*b3 - 27*b2 + 12*b1 + 21) * q^78 + (2*b5 - 2*b4 - 3*b1 + 31) * q^79 + (7*b4 + 2*b3 + 12*b2 + 2*b1 + 24) * q^80 + 81 * q^81 + (17*b5 + 34*b4 - 2*b3 + 68*b2 + 85) * q^82 + (9*b3 + 8*b2 + 9*b1 + 16) * q^83 + (33*b5 + 3*b4 - 9*b3 - 93*b2 + 9*b1 - 51) * q^84 + (-23*b5 - 46*b4 - 12*b3 - 72*b2 - 95) * q^85 + (29*b4 + 7*b3 + 35*b2 + 7*b1 + 70) * q^86 + (-3*b4 - 18*b2 - 36) * q^87 + (23*b5 + 46*b4 - 12*b3 - 138*b2 - 115) * q^88 + (8*b5 + 5*b3 + 28*b2 - 10*b1 - 20) * q^89 + (36*b5 + 18*b4 + 9*b3 + 72*b2 - 9*b1 + 36) * q^90 + (18*b5 + 12*b4 + 9*b3 + 2*b2 - 3*b1 + 45) * q^91 + (9*b4 - 2*b3 + 20*b2 - 2*b1 + 40) * q^92 + (-6*b5 + 6*b4 - 12*b1 - 42) * q^93 + (6*b5 - 6*b4 + 12*b1 + 144) * q^94 + (b5 + b4 - 4*b3 + 32*b2 + 2*b1 + 17) * q^95 + (-15*b5 - 15*b4 + 12*b3 + 180*b2 - 6*b1 + 75) * q^96 + (-24*b5 - 12*b4 - b3 + 42*b2 + b1 - 24) * q^97 + (-9*b5 - 5*b4 - 12*b3 - 96*b2 + 21*b1) * q^98 + (18*b5 - 9*b3 + 18*b2 + 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{7} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 - 26 * q^4 - 15 * q^5 - 2 * q^7 + 54 * q^9 $$6 q + 18 q^{3} - 26 q^{4} - 15 q^{5} - 2 q^{7} + 54 q^{9} - 19 q^{10} - 9 q^{11} - 78 q^{12} + 11 q^{13} - 24 q^{14} - 45 q^{15} + 94 q^{16} + 33 q^{17} - 19 q^{19} + 45 q^{20} - 6 q^{21} + 65 q^{22} + 15 q^{23} - 26 q^{25} + 81 q^{26} + 162 q^{27} - 42 q^{28} - 51 q^{29} - 57 q^{30} - 92 q^{31} - 27 q^{33} + 93 q^{34} - 57 q^{35} - 234 q^{36} + 7 q^{37} - 21 q^{38} + 33 q^{39} + 57 q^{40} - 27 q^{41} - 72 q^{42} - 99 q^{43} - 273 q^{44} - 135 q^{45} - 57 q^{46} + 282 q^{48} + 6 q^{49} + 294 q^{50} + 99 q^{51} + 63 q^{52} + 45 q^{53} + 166 q^{55} + 360 q^{56} - 57 q^{57} - 7 q^{58} + 135 q^{60} + 44 q^{61} - 18 q^{63} - 138 q^{64} + 195 q^{66} - 196 q^{67} - 567 q^{68} + 45 q^{69} - 257 q^{70} - 101 q^{73} - 411 q^{74} - 78 q^{75} - 99 q^{76} + 105 q^{77} + 243 q^{78} + 180 q^{79} + 93 q^{80} + 486 q^{81} + 151 q^{82} + 99 q^{83} - 126 q^{84} - 159 q^{85} + 249 q^{86} - 153 q^{87} - 495 q^{88} - 243 q^{89} - 171 q^{90} + 177 q^{91} + 147 q^{92} - 276 q^{93} + 888 q^{94} - 161 q^{97} + 360 q^{98} - 81 q^{99}+O(q^{100})$$ 6 * q + 18 * q^3 - 26 * q^4 - 15 * q^5 - 2 * q^7 + 54 * q^9 - 19 * q^10 - 9 * q^11 - 78 * q^12 + 11 * q^13 - 24 * q^14 - 45 * q^15 + 94 * q^16 + 33 * q^17 - 19 * q^19 + 45 * q^20 - 6 * q^21 + 65 * q^22 + 15 * q^23 - 26 * q^25 + 81 * q^26 + 162 * q^27 - 42 * q^28 - 51 * q^29 - 57 * q^30 - 92 * q^31 - 27 * q^33 + 93 * q^34 - 57 * q^35 - 234 * q^36 + 7 * q^37 - 21 * q^38 + 33 * q^39 + 57 * q^40 - 27 * q^41 - 72 * q^42 - 99 * q^43 - 273 * q^44 - 135 * q^45 - 57 * q^46 + 282 * q^48 + 6 * q^49 + 294 * q^50 + 99 * q^51 + 63 * q^52 + 45 * q^53 + 166 * q^55 + 360 * q^56 - 57 * q^57 - 7 * q^58 + 135 * q^60 + 44 * q^61 - 18 * q^63 - 138 * q^64 + 195 * q^66 - 196 * q^67 - 567 * q^68 + 45 * q^69 - 257 * q^70 - 101 * q^73 - 411 * q^74 - 78 * q^75 - 99 * q^76 + 105 * q^77 + 243 * q^78 + 180 * q^79 + 93 * q^80 + 486 * q^81 + 151 * q^82 + 99 * q^83 - 126 * q^84 - 159 * q^85 + 249 * q^86 - 153 * q^87 - 495 * q^88 - 243 * q^89 - 171 * q^90 + 177 * q^91 + 147 * q^92 - 276 * q^93 + 888 * q^94 - 161 * q^97 + 360 * q^98 - 81 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 12x^{4} + 17x^{3} + 118x^{2} + 33x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{5} - 24\nu^{4} + 288\nu^{3} - 236\nu^{2} - 66\nu + 2241 ) / 1449$$ (2*v^5 - 24*v^4 + 288*v^3 - 236*v^2 - 66*v + 2241) / 1449 $$\beta_{2}$$ $$=$$ $$( 44\nu^{5} - 45\nu^{4} + 540\nu^{3} + 604\nu^{2} + 5310\nu + 36 ) / 1449$$ (44*v^5 - 45*v^4 + 540*v^3 + 604*v^2 + 5310*v + 36) / 1449 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 3\nu^{4} + 36\nu^{3} + 16\nu^{2} + 354\nu + 99 ) / 63$$ (2*v^5 - 3*v^4 + 36*v^3 + 16*v^2 + 354*v + 99) / 63 $$\beta_{4}$$ $$=$$ $$( 83\nu^{5} - 30\nu^{4} + 843\nu^{3} + 2281\nu^{2} + 9819\nu + 5337 ) / 1449$$ (83*v^5 - 30*v^4 + 843*v^3 + 2281*v^2 + 9819*v + 5337) / 1449 $$\beta_{5}$$ $$=$$ $$( 32\nu^{5} - 62\nu^{4} + 422\nu^{3} + 249\nu^{2} + 3130\nu - 1335 ) / 483$$ (32*v^5 - 62*v^4 + 422*v^3 + 249*v^2 + 3130*v - 1335) / 483
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} - \beta_1 ) / 2$$ (b3 - b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + \beta_{3} - 7\beta_{2} - 6$$ b5 + 2*b4 + b3 - 7*b2 - 6 $$\nu^{3}$$ $$=$$ $$( -2\beta_{5} + 2\beta_{4} + 13\beta _1 - 33 ) / 2$$ (-2*b5 + 2*b4 + 13*b1 - 33) / 2 $$\nu^{4}$$ $$=$$ $$-24\beta_{5} - 12\beta_{4} - 19\beta_{3} + 94\beta_{2} + 19\beta _1 - 24$$ -24*b5 - 12*b4 - 19*b3 + 94*b2 + 19*b1 - 24 $$\nu^{5}$$ $$=$$ $$( -52\beta_{5} - 104\beta_{4} - 187\beta_{3} + 571\beta_{2} + 519 ) / 2$$ (-52*b5 - 104*b4 - 187*b3 + 571*b2 + 519) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.140998 + 0.244215i 1.98253 − 3.43384i −1.34153 + 2.32360i −1.34153 − 2.32360i 1.98253 + 3.43384i −0.140998 − 0.244215i
3.09257i 3.00000 −5.56399 −5.67824 3.27834i 9.27771i 6.63848 + 2.22048i 4.83675i 9.00000 −10.1385 + 17.5604i
11.2 1.03435i 3.00000 2.93011 −2.10422 1.21487i 3.10306i −4.75661 5.13563i 7.16819i 9.00000 1.25661 2.17651i
11.3 3.79027i 3.00000 −10.3661 0.282467 + 0.163082i 11.3708i −2.88187 + 6.37925i 24.1293i 9.00000 −0.618126 + 1.07063i
23.1 3.79027i 3.00000 −10.3661 0.282467 0.163082i 11.3708i −2.88187 6.37925i 24.1293i 9.00000 −0.618126 1.07063i
23.2 1.03435i 3.00000 2.93011 −2.10422 + 1.21487i 3.10306i −4.75661 + 5.13563i 7.16819i 9.00000 1.25661 + 2.17651i
23.3 3.09257i 3.00000 −5.56399 −5.67824 + 3.27834i 9.27771i 6.63848 2.22048i 4.83675i 9.00000 −10.1385 17.5604i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.3.j.a 6
3.b odd 2 1 189.3.j.a 6
7.b odd 2 1 441.3.j.c 6
7.c even 3 1 63.3.n.a yes 6
7.c even 3 1 441.3.r.b 6
7.d odd 6 1 441.3.n.c 6
7.d odd 6 1 441.3.r.c 6
9.c even 3 1 189.3.n.a 6
9.d odd 6 1 63.3.n.a yes 6
21.h odd 6 1 189.3.n.a 6
63.h even 3 1 189.3.j.a 6
63.i even 6 1 441.3.j.c 6
63.j odd 6 1 inner 63.3.j.a 6
63.n odd 6 1 441.3.r.b 6
63.o even 6 1 441.3.n.c 6
63.s even 6 1 441.3.r.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.j.a 6 1.a even 1 1 trivial
63.3.j.a 6 63.j odd 6 1 inner
63.3.n.a yes 6 7.c even 3 1
63.3.n.a yes 6 9.d odd 6 1
189.3.j.a 6 3.b odd 2 1
189.3.j.a 6 63.h even 3 1
189.3.n.a 6 9.c even 3 1
189.3.n.a 6 21.h odd 6 1
441.3.j.c 6 7.b odd 2 1
441.3.j.c 6 63.i even 6 1
441.3.n.c 6 7.d odd 6 1
441.3.n.c 6 63.o even 6 1
441.3.r.b 6 7.c even 3 1
441.3.r.b 6 63.n odd 6 1
441.3.r.c 6 7.d odd 6 1
441.3.r.c 6 63.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 25T_{2}^{4} + 163T_{2}^{2} + 147$$ acting on $$S_{3}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 25 T^{4} + 163 T^{2} + \cdots + 147$$
$3$ $$(T - 3)^{6}$$
$5$ $$T^{6} + 15 T^{5} + 88 T^{4} + 195 T^{3} + \cdots + 27$$
$7$ $$T^{6} + 2 T^{5} - T^{4} - 532 T^{3} + \cdots + 117649$$
$11$ $$T^{6} + 9 T^{5} - 188 T^{4} + \cdots + 1982907$$
$13$ $$T^{6} - 11 T^{5} + 182 T^{4} + \cdots + 499849$$
$17$ $$T^{6} - 33 T^{5} - 252 T^{4} + \cdots + 31434507$$
$19$ $$T^{6} + 19 T^{5} + 422 T^{4} + \cdots + 214369$$
$23$ $$T^{6} - 15 T^{5} - 84 T^{4} + \cdots + 128547$$
$29$ $$T^{6} + 51 T^{5} + 1144 T^{4} + \cdots + 694083$$
$31$ $$(T^{3} + 46 T^{2} - 324 T - 4632)^{2}$$
$37$ $$T^{6} - 7 T^{5} + 2230 T^{4} + \cdots + 564110001$$
$41$ $$T^{6} + 27 T^{5} + \cdots + 1387137027$$
$43$ $$T^{6} + 99 T^{5} + \cdots + 432265681$$
$47$ $$T^{6} + 10128 T^{4} + \cdots + 29274835968$$
$53$ $$T^{6} - 45 T^{5} + \cdots + 5141134827$$
$59$ $$T^{6} + 4896 T^{4} + \cdots + 813189888$$
$61$ $$(T^{3} - 22 T^{2} - 4996 T + 160696)^{2}$$
$67$ $$(T^{3} + 98 T^{2} - 1156 T - 210392)^{2}$$
$71$ $$T^{6} + 5568 T^{4} + \cdots + 109734912$$
$73$ $$T^{6} + 101 T^{5} + \cdots + 653628591729$$
$79$ $$(T^{3} - 90 T^{2} + 2268 T - 16712)^{2}$$
$83$ $$T^{6} - 99 T^{5} + \cdots + 165842832483$$
$89$ $$T^{6} + 243 T^{5} + \cdots + 15857178627$$
$97$ $$T^{6} + 161 T^{5} + \cdots + 36908941689$$