# Properties

 Label 63.3.n.a Level $63$ Weight $3$ Character orbit 63.n 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.n (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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + 1) q^{2} + 3 \beta_{2} q^{3} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{4} + (\beta_{5} + \beta_{4} + 4 \beta_{2} + 3) q^{5} + ( - 3 \beta_{5} - 3 \beta_{4} - 3) q^{6} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{7} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{8} + ( - 9 \beta_{2} - 9) q^{9}+O(q^{10})$$ q + (b5 + 1) * q^2 + 3*b2 * q^3 + (-b3 - 4*b2 + b1) * q^4 + (b5 + b4 + 4*b2 + 3) * q^5 + (-3*b5 - 3*b4 - 3) * q^6 + (-b5 - 2*b4 - b1 - 2) * q^7 + (3*b5 + 3*b4 - 2*b3 - 2*b2 + b1 + 2) * q^8 + (-9*b2 - 9) * q^9 $$q + (\beta_{5} + 1) q^{2} + 3 \beta_{2} q^{3} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{4} + (\beta_{5} + \beta_{4} + 4 \beta_{2} + 3) q^{5} + ( - 3 \beta_{5} - 3 \beta_{4} - 3) q^{6} + ( - \beta_{5} - 2 \beta_{4} - \beta_1 - 2) q^{7} + (3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{8} + ( - 9 \beta_{2} - 9) q^{9} + ( - 2 \beta_{5} - 4 \beta_{4} - \beta_{3} - 8 \beta_{2} - 10) q^{10} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 - 4) q^{11} + (3 \beta_{3} + 12 \beta_{2} + 12) q^{12} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2} + 6) q^{13} + ( - 4 \beta_{5} + 2 \beta_{3} + 9 \beta_{2} - \beta_1 + 11) q^{14} + ( - 3 \beta_{4} - 6 \beta_{2} - 12) q^{15} + (4 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 11 \beta_{2} - 7) q^{16} + (6 \beta_{5} + 2 \beta_{3} + 5 \beta_{2} - 4 \beta_1 + 1) q^{17} + 9 \beta_{4} q^{18} + ( - 2 \beta_{3} + 7 \beta_{2} + 2 \beta_1) 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} + 3 \beta_1 - 3) q^{21} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 19 \beta_{2} + 18) q^{22} + (3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{23} + ( - 9 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 6) q^{24} + (4 \beta_{5} - 4 \beta_{4} - \beta_1 + 9) q^{25} + ( - 2 \beta_{4} - 2 \beta_{3} - 9 \beta_{2} - 2 \beta_1 - 18) q^{26} + 27 q^{27} + ( - \beta_{5} - 11 \beta_{4} + 3 \beta_{3} + 31 \beta_{2} + 2) q^{28} + ( - \beta_{4} - 6 \beta_{2} - 12) q^{29} + ( - 6 \beta_{5} + 6 \beta_{4} + 3 \beta_1 + 18) q^{30} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 12 \beta_{2} + 4 \beta_1 + 4) q^{31} + (5 \beta_{4} - 2 \beta_{3} - 30 \beta_{2} - 2 \beta_1 - 60) q^{32} + (6 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 12) q^{33} + ( - 22 \beta_{5} - 11 \beta_{4} - 42 \beta_{2} - 22) q^{34} + ( - 10 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 10 \beta_{2} + 3 \beta_1 - 3) q^{35} + ( - 9 \beta_1 - 36) q^{36} + (14 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 14) q^{37} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 3) q^{38} + (3 \beta_{5} - 3 \beta_{4} + 3 \beta_1 - 12) q^{39} + (2 \beta_{5} - 2 \beta_{4} - 19) q^{40} + (10 \beta_{5} + 4 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 5) q^{41} + (12 \beta_{5} + 12 \beta_{4} - 3 \beta_{3} + 18 \beta_{2} - 3 \beta_1 - 15) q^{42} + (8 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} + 38 \beta_{2} + 3 \beta_1 + 8) q^{43} + ( - 26 \beta_{4} + 2 \beta_{3} + 21 \beta_{2} + 2 \beta_1 + 42) q^{44} + ( - 9 \beta_{5} - 18 \beta_{2} + 9) q^{45} + ( - 2 \beta_{5} - 4 \beta_{4} - 21 \beta_{2} - 23) q^{46} + ( - 18 \beta_{5} - 2 \beta_{3} + 12 \beta_{2} + 4 \beta_1 - 30) q^{47} + (12 \beta_{5} - 12 \beta_{4} + 6 \beta_1 + 45) q^{48} + (12 \beta_{5} + 12 \beta_{4} - 3 \beta_{3} + 18 \beta_{2} - 3 \beta_1 + 34) q^{49} + (2 \beta_{5} - 3 \beta_{3} - 31 \beta_{2} + 6 \beta_1 + 33) q^{50} + ( - 18 \beta_{5} - 18 \beta_{4} - 12 \beta_{3} - 30 \beta_{2} + 6 \beta_1 - 33) q^{51} + ( - 11 \beta_{5} + 11 \beta_{4} - 21) q^{52} + ( - 15 \beta_{5} - 3 \beta_{3} + \beta_{2} + 6 \beta_1 - 16) q^{53} + (27 \beta_{5} + 27) q^{54} + ( - 3 \beta_{5} + 3 \beta_{4} - \beta_1 + 28) q^{55} + ( - 12 \beta_{5} - 24 \beta_{4} + 35 \beta_{2} + 9 \beta_1 + 46) q^{56} + (6 \beta_{3} - 21 \beta_{2} - 21) q^{57} + ( - 6 \beta_{5} + 6 \beta_{4} + \beta_1 + 2) q^{58} + (6 \beta_{4} + 3 \beta_{3} + 17 \beta_{2} + 3 \beta_1 + 34) q^{59} + (21 \beta_{5} + 3 \beta_{3} + 21 \beta_{2} - 6 \beta_1) q^{60} + ( - 6 \beta_{5} - 12 \beta_{4} + 8 \beta_{3} - 16 \beta_{2} - 22) q^{61} + (24 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} - 40 \beta_{2} + 6 \beta_1 + 4) q^{62} + (18 \beta_{5} + 9 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} + 27) q^{63} + ( - 20 \beta_{5} + 20 \beta_{4} - 3 \beta_1 - 22) q^{64} + (8 \beta_{5} + 19 \beta_{2} - 11) q^{65} + ( - 3 \beta_{5} + 3 \beta_{4} - 15 \beta_1 - 60) q^{66} + ( - 12 \beta_{5} - 6 \beta_{4} + 7 \beta_{3} - 41 \beta_{2} - 7 \beta_1 - 12) q^{67} + (18 \beta_{5} + 18 \beta_{4} + 6 \beta_{3} + 136 \beta_{2} - 3 \beta_1 + 86) q^{68} + ( - 9 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{69} + (26 \beta_{5} + 19 \beta_{4} + 4 \beta_{3} + 74 \beta_{2} - 3 \beta_1 + 55) q^{70} + ( - 6 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + 32 \beta_{2} - 2 \beta_1 + 10) q^{71} + ( - 27 \beta_{5} + 9 \beta_{3} + 9 \beta_{2} - 18 \beta_1 - 36) q^{72} + (18 \beta_{5} + 36 \beta_{4} + 7 \beta_{3} - 18 \beta_{2}) q^{73} + ( - 18 \beta_{5} - 18 \beta_{4} - 4 \beta_{3} - 102 \beta_{2} + 2 \beta_1 - 69) q^{74} + ( - 24 \beta_{5} - 12 \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 3 \beta_1 - 24) q^{75} + (8 \beta_{5} + 16 \beta_{4} + 3 \beta_{3} - 26 \beta_{2} - 18) q^{76} + (17 \beta_{5} + 20 \beta_{4} + 7 \beta_{3} - 35 \beta_{2} - 11 \beta_1 - 29) q^{77} + ( - 6 \beta_{5} - 6 \beta_{3} - 27 \beta_{2} + 12 \beta_1 + 21) q^{78} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 29 \beta_{2} - 27) q^{79} + (7 \beta_{5} + 2 \beta_{3} + 12 \beta_{2} - 4 \beta_1 - 5) q^{80} + 81 \beta_{2} q^{81} + ( - 34 \beta_{5} - 17 \beta_{4} + 2 \beta_{3} - 68 \beta_{2} - 2 \beta_1 - 34) q^{82} + (9 \beta_{3} + 8 \beta_{2} + 9 \beta_1 + 16) q^{83} + ( - 30 \beta_{5} + 3 \beta_{4} - 84 \beta_{2} - 9 \beta_1 - 123) q^{84} + ( - 23 \beta_{5} - 46 \beta_{4} - 12 \beta_{3} - 72 \beta_{2} - 95) q^{85} + ( - 29 \beta_{5} - 29 \beta_{4} - 14 \beta_{3} - 70 \beta_{2} + 7 \beta_1 - 64) q^{86} + ( - 3 \beta_{5} - 18 \beta_{2} + 15) q^{87} + (23 \beta_{5} - 23 \beta_{4} + 12 \beta_1 + 161) q^{88} + (8 \beta_{4} + 5 \beta_{3} + 28 \beta_{2} + 5 \beta_1 + 56) q^{89} + (36 \beta_{5} + 18 \beta_{4} + 9 \beta_{3} + 72 \beta_{2} - 9 \beta_1 + 36) q^{90} + ( - 12 \beta_{5} - 18 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 13) q^{91} + (9 \beta_{4} - 2 \beta_{3} + 20 \beta_{2} - 2 \beta_1 + 40) q^{92} + ( - 6 \beta_{5} - 12 \beta_{4} + 12 \beta_{3} + 36 \beta_{2} + 30) q^{93} + ( - 12 \beta_{5} - 6 \beta_{4} + 12 \beta_{3} + 138 \beta_{2} - 12 \beta_1 - 12) q^{94} + ( - \beta_{4} + 2 \beta_{3} - 16 \beta_{2} + 2 \beta_1 - 32) q^{95} + (15 \beta_{5} - 6 \beta_{3} - 90 \beta_{2} + 12 \beta_1 + 105) q^{96} + ( - 24 \beta_{5} - 12 \beta_{4} - \beta_{3} + 42 \beta_{2} + \beta_1 - 24) q^{97} + ( - 5 \beta_{5} - 9 \beta_{4} - 12 \beta_{3} - 96 \beta_{2} - 9 \beta_1 - 110) q^{98} + (18 \beta_{5} - 9 \beta_{3} + 18 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100})$$ q + (b5 + 1) * q^2 + 3*b2 * q^3 + (-b3 - 4*b2 + b1) * q^4 + (b5 + b4 + 4*b2 + 3) * q^5 + (-3*b5 - 3*b4 - 3) * q^6 + (-b5 - 2*b4 - b1 - 2) * q^7 + (3*b5 + 3*b4 - 2*b3 - 2*b2 + b1 + 2) * q^8 + (-9*b2 - 9) * q^9 + (-2*b5 - 4*b4 - b3 - 8*b2 - 10) * q^10 + (-2*b5 - 2*b4 + 2*b3 - 4*b2 - b1 - 4) * q^11 + (3*b3 + 12*b2 + 12) * q^12 + (b5 + 2*b4 - b3 + 5*b2 + 6) * q^13 + (-4*b5 + 2*b3 + 9*b2 - b1 + 11) * q^14 + (-3*b4 - 6*b2 - 12) * q^15 + (4*b5 + 8*b4 - 2*b3 - 11*b2 - 7) * q^16 + (6*b5 + 2*b3 + 5*b2 - 4*b1 + 1) * q^17 + 9*b4 * q^18 + (-2*b3 + 7*b2 + 2*b1) * q^19 + (7*b4 + b3 + 7*b2 + b1 + 14) * q^20 + (-3*b5 + 3*b4 - 3*b3 - 3*b2 + 3*b1 - 3) * q^21 + (-b5 - 2*b4 + 5*b3 + 19*b2 + 18) * q^22 + (3*b5 + 3*b4 + 2*b3 - 2*b2 - b1 + 2) * q^23 + (-9*b4 + 3*b3 + 3*b2 + 3*b1 + 6) * q^24 + (4*b5 - 4*b4 - b1 + 9) * q^25 + (-2*b4 - 2*b3 - 9*b2 - 2*b1 - 18) * q^26 + 27 * q^27 + (-b5 - 11*b4 + 3*b3 + 31*b2 + 2) * q^28 + (-b4 - 6*b2 - 12) * q^29 + (-6*b5 + 6*b4 + 3*b1 + 18) * q^30 + (4*b5 + 2*b4 - 4*b3 - 12*b2 + 4*b1 + 4) * q^31 + (5*b4 - 2*b3 - 30*b2 - 2*b1 - 60) * q^32 + (6*b4 - 3*b3 + 6*b2 - 3*b1 + 12) * q^33 + (-22*b5 - 11*b4 - 42*b2 - 22) * q^34 + (-10*b5 - 4*b4 - 3*b3 - 10*b2 + 3*b1 - 3) * q^35 + (-9*b1 - 36) * q^36 + (14*b5 + 7*b4 + 5*b3 + 3*b2 - 5*b1 + 14) * q^37 + (-b5 - b4 - 4*b3 - 4*b2 + 2*b1 - 3) * q^38 + (3*b5 - 3*b4 + 3*b1 - 12) * q^39 + (2*b5 - 2*b4 - 19) * q^40 + (10*b5 + 4*b3 + 5*b2 - 8*b1 + 5) * q^41 + (12*b5 + 12*b4 - 3*b3 + 18*b2 - 3*b1 - 15) * q^42 + (8*b5 + 4*b4 - 3*b3 + 38*b2 + 3*b1 + 8) * q^43 + (-26*b4 + 2*b3 + 21*b2 + 2*b1 + 42) * q^44 + (-9*b5 - 18*b2 + 9) * q^45 + (-2*b5 - 4*b4 - 21*b2 - 23) * q^46 + (-18*b5 - 2*b3 + 12*b2 + 4*b1 - 30) * q^47 + (12*b5 - 12*b4 + 6*b1 + 45) * q^48 + (12*b5 + 12*b4 - 3*b3 + 18*b2 - 3*b1 + 34) * q^49 + (2*b5 - 3*b3 - 31*b2 + 6*b1 + 33) * q^50 + (-18*b5 - 18*b4 - 12*b3 - 30*b2 + 6*b1 - 33) * q^51 + (-11*b5 + 11*b4 - 21) * q^52 + (-15*b5 - 3*b3 + b2 + 6*b1 - 16) * q^53 + (27*b5 + 27) * q^54 + (-3*b5 + 3*b4 - b1 + 28) * q^55 + (-12*b5 - 24*b4 + 35*b2 + 9*b1 + 46) * q^56 + (6*b3 - 21*b2 - 21) * q^57 + (-6*b5 + 6*b4 + b1 + 2) * q^58 + (6*b4 + 3*b3 + 17*b2 + 3*b1 + 34) * q^59 + (21*b5 + 3*b3 + 21*b2 - 6*b1) * q^60 + (-6*b5 - 12*b4 + 8*b3 - 16*b2 - 22) * q^61 + (24*b5 + 24*b4 - 12*b3 - 40*b2 + 6*b1 + 4) * q^62 + (18*b5 + 9*b4 + 9*b3 + 9*b2 + 27) * q^63 + (-20*b5 + 20*b4 - 3*b1 - 22) * q^64 + (8*b5 + 19*b2 - 11) * q^65 + (-3*b5 + 3*b4 - 15*b1 - 60) * q^66 + (-12*b5 - 6*b4 + 7*b3 - 41*b2 - 7*b1 - 12) * q^67 + (18*b5 + 18*b4 + 6*b3 + 136*b2 - 3*b1 + 86) * q^68 + (-9*b4 - 3*b3 + 3*b2 - 3*b1 + 6) * q^69 + (26*b5 + 19*b4 + 4*b3 + 74*b2 - 3*b1 + 55) * q^70 + (-6*b5 - 6*b4 + 4*b3 + 32*b2 - 2*b1 + 10) * q^71 + (-27*b5 + 9*b3 + 9*b2 - 18*b1 - 36) * q^72 + (18*b5 + 36*b4 + 7*b3 - 18*b2) * q^73 + (-18*b5 - 18*b4 - 4*b3 - 102*b2 + 2*b1 - 69) * q^74 + (-24*b5 - 12*b4 - 3*b3 + 15*b2 + 3*b1 - 24) * q^75 + (8*b5 + 16*b4 + 3*b3 - 26*b2 - 18) * q^76 + (17*b5 + 20*b4 + 7*b3 - 35*b2 - 11*b1 - 29) * q^77 + (-6*b5 - 6*b3 - 27*b2 + 12*b1 + 21) * q^78 + (2*b5 + 4*b4 + 3*b3 - 29*b2 - 27) * q^79 + (7*b5 + 2*b3 + 12*b2 - 4*b1 - 5) * q^80 + 81*b2 * q^81 + (-34*b5 - 17*b4 + 2*b3 - 68*b2 - 2*b1 - 34) * q^82 + (9*b3 + 8*b2 + 9*b1 + 16) * q^83 + (-30*b5 + 3*b4 - 84*b2 - 9*b1 - 123) * q^84 + (-23*b5 - 46*b4 - 12*b3 - 72*b2 - 95) * q^85 + (-29*b5 - 29*b4 - 14*b3 - 70*b2 + 7*b1 - 64) * q^86 + (-3*b5 - 18*b2 + 15) * q^87 + (23*b5 - 23*b4 + 12*b1 + 161) * q^88 + (8*b4 + 5*b3 + 28*b2 + 5*b1 + 56) * q^89 + (36*b5 + 18*b4 + 9*b3 + 72*b2 - 9*b1 + 36) * q^90 + (-12*b5 - 18*b4 - 9*b3 - 2*b2 + 6*b1 + 13) * q^91 + (9*b4 - 2*b3 + 20*b2 - 2*b1 + 40) * q^92 + (-6*b5 - 12*b4 + 12*b3 + 36*b2 + 30) * q^93 + (-12*b5 - 6*b4 + 12*b3 + 138*b2 - 12*b1 - 12) * q^94 + (-b4 + 2*b3 - 16*b2 + 2*b1 - 32) * q^95 + (15*b5 - 6*b3 - 90*b2 + 12*b1 + 105) * q^96 + (-24*b5 - 12*b4 - b3 + 42*b2 + b1 - 24) * q^97 + (-5*b5 - 9*b4 - 12*b3 - 96*b2 - 9*b1 - 110) * q^98 + (18*b5 - 9*b3 + 18*b2 + 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 9 q^{3} + 13 q^{4} - 5 q^{7} - 27 q^{9}+O(q^{10})$$ 6 * q + 3 * q^2 - 9 * q^3 + 13 * q^4 - 5 * q^7 - 27 * q^9 $$6 q + 3 q^{2} - 9 q^{3} + 13 q^{4} - 5 q^{7} - 27 q^{9} - 19 q^{10} + 39 q^{12} + 11 q^{13} + 51 q^{14} - 45 q^{15} - 47 q^{16} - 33 q^{17} - 27 q^{18} - 19 q^{19} + 45 q^{20} - 6 q^{21} + 65 q^{22} + 63 q^{24} + 52 q^{25} - 81 q^{26} + 162 q^{27} - 42 q^{28} - 51 q^{29} + 114 q^{30} + 46 q^{31} - 291 q^{32} + 27 q^{33} + 93 q^{34} + 57 q^{35} - 234 q^{36} + 7 q^{37} - 66 q^{39} - 114 q^{40} - 27 q^{41} - 225 q^{42} - 99 q^{43} + 273 q^{44} + 135 q^{45} - 57 q^{46} - 156 q^{47} + 282 q^{48} + 69 q^{49} + 294 q^{50} - 126 q^{52} - 45 q^{53} + 81 q^{54} + 166 q^{55} + 297 q^{56} - 57 q^{57} + 14 q^{58} + 144 q^{59} - 135 q^{60} - 22 q^{61} + 63 q^{63} - 138 q^{64} - 147 q^{65} - 390 q^{66} + 98 q^{67} + 45 q^{69} - 29 q^{70} - 189 q^{72} - 101 q^{73} - 78 q^{75} - 99 q^{76} - 195 q^{77} + 243 q^{78} - 90 q^{79} - 93 q^{80} - 243 q^{81} + 151 q^{82} + 99 q^{83} - 423 q^{84} - 159 q^{85} + 153 q^{87} + 990 q^{88} + 243 q^{89} - 171 q^{90} + 177 q^{91} + 147 q^{92} + 138 q^{93} - 444 q^{94} - 135 q^{95} + 873 q^{96} - 161 q^{97} - 360 q^{98} - 81 q^{99}+O(q^{100})$$ 6 * q + 3 * q^2 - 9 * q^3 + 13 * q^4 - 5 * q^7 - 27 * q^9 - 19 * q^10 + 39 * q^12 + 11 * q^13 + 51 * q^14 - 45 * q^15 - 47 * q^16 - 33 * q^17 - 27 * q^18 - 19 * q^19 + 45 * q^20 - 6 * q^21 + 65 * q^22 + 63 * q^24 + 52 * q^25 - 81 * q^26 + 162 * q^27 - 42 * q^28 - 51 * q^29 + 114 * q^30 + 46 * q^31 - 291 * q^32 + 27 * q^33 + 93 * q^34 + 57 * q^35 - 234 * q^36 + 7 * q^37 - 66 * q^39 - 114 * q^40 - 27 * q^41 - 225 * q^42 - 99 * q^43 + 273 * q^44 + 135 * q^45 - 57 * q^46 - 156 * q^47 + 282 * q^48 + 69 * q^49 + 294 * q^50 - 126 * q^52 - 45 * q^53 + 81 * q^54 + 166 * q^55 + 297 * q^56 - 57 * q^57 + 14 * q^58 + 144 * q^59 - 135 * q^60 - 22 * q^61 + 63 * q^63 - 138 * q^64 - 147 * q^65 - 390 * q^66 + 98 * q^67 + 45 * q^69 - 29 * q^70 - 189 * q^72 - 101 * q^73 - 78 * q^75 - 99 * q^76 - 195 * q^77 + 243 * q^78 - 90 * q^79 - 93 * q^80 - 243 * q^81 + 151 * q^82 + 99 * q^83 - 423 * q^84 - 159 * q^85 + 153 * q^87 + 990 * q^88 + 243 * q^89 - 171 * q^90 + 177 * q^91 + 147 * q^92 + 138 * q^93 - 444 * q^94 - 135 * q^95 + 873 * q^96 - 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 $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −0.140998 + 0.244215i 1.98253 − 3.43384i −1.34153 + 2.32360i −0.140998 − 0.244215i 1.98253 + 3.43384i −1.34153 − 2.32360i
−2.67824 + 1.54629i −1.50000 + 2.59808i 2.78200 4.81856i 6.55667i 9.27771i −5.24223 4.63886i 4.83675i −4.50000 7.79423i −10.1385 17.5604i
2.2 0.895777 0.517177i −1.50000 + 2.59808i −1.46506 + 2.53755i 2.42975i 3.10306i 6.82589 + 1.55153i 7.16819i −4.50000 7.79423i 1.25661 + 2.17651i
2.3 3.28247 1.89513i −1.50000 + 2.59808i 5.18306 8.97732i 0.326165i 11.3708i −4.08365 + 5.68540i 24.1293i −4.50000 7.79423i −0.618126 1.07063i
32.1 −2.67824 1.54629i −1.50000 2.59808i 2.78200 + 4.81856i 6.55667i 9.27771i −5.24223 + 4.63886i 4.83675i −4.50000 + 7.79423i −10.1385 + 17.5604i
32.2 0.895777 + 0.517177i −1.50000 2.59808i −1.46506 2.53755i 2.42975i 3.10306i 6.82589 1.55153i 7.16819i −4.50000 + 7.79423i 1.25661 2.17651i
32.3 3.28247 + 1.89513i −1.50000 2.59808i 5.18306 + 8.97732i 0.326165i 11.3708i −4.08365 5.68540i 24.1293i −4.50000 + 7.79423i −0.618126 + 1.07063i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.3 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.n 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.n.a yes 6
3.b odd 2 1 189.3.n.a 6
7.b odd 2 1 441.3.n.c 6
7.c even 3 1 63.3.j.a 6
7.c even 3 1 441.3.r.b 6
7.d odd 6 1 441.3.j.c 6
7.d odd 6 1 441.3.r.c 6
9.c even 3 1 189.3.j.a 6
9.d odd 6 1 63.3.j.a 6
21.h odd 6 1 189.3.j.a 6
63.g even 3 1 189.3.n.a 6
63.i even 6 1 441.3.r.c 6
63.j odd 6 1 441.3.r.b 6
63.n odd 6 1 inner 63.3.n.a yes 6
63.o even 6 1 441.3.j.c 6
63.s even 6 1 441.3.n.c 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3T_{2}^{5} - 8T_{2}^{4} + 33T_{2}^{3} + 100T_{2}^{2} - 231T_{2} + 147$$ acting on $$S_{3}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} - 8 T^{4} + 33 T^{3} + \cdots + 147$$
$3$ $$(T^{2} + 3 T + 9)^{3}$$
$5$ $$T^{6} + 49 T^{4} + 259 T^{2} + \cdots + 27$$
$7$ $$T^{6} + 5 T^{5} - 22 T^{4} + \cdots + 117649$$
$11$ $$T^{6} + 457 T^{4} + 60859 T^{2} + \cdots + 1982907$$
$13$ $$T^{6} - 11 T^{5} + 182 T^{4} + \cdots + 499849$$
$17$ $$T^{6} + 33 T^{5} - 252 T^{4} + \cdots + 31434507$$
$19$ $$T^{6} + 19 T^{5} + 422 T^{4} + \cdots + 214369$$
$23$ $$T^{6} + 393 T^{4} + 31491 T^{2} + \cdots + 128547$$
$29$ $$T^{6} + 51 T^{5} + 1144 T^{4} + \cdots + 694083$$
$31$ $$T^{6} - 46 T^{5} + 2440 T^{4} + \cdots + 21455424$$
$37$ $$T^{6} - 7 T^{5} + 2230 T^{4} + \cdots + 564110001$$
$41$ $$T^{6} + 27 T^{5} + \cdots + 1387137027$$
$43$ $$T^{6} + 99 T^{5} + \cdots + 432265681$$
$47$ $$T^{6} + 156 T^{5} + \cdots + 29274835968$$
$53$ $$T^{6} + 45 T^{5} + \cdots + 5141134827$$
$59$ $$T^{6} - 144 T^{5} + \cdots + 813189888$$
$61$ $$T^{6} + 22 T^{5} + \cdots + 25823204416$$
$67$ $$T^{6} - 98 T^{5} + \cdots + 44264793664$$
$71$ $$T^{6} + 5568 T^{4} + \cdots + 109734912$$
$73$ $$T^{6} + 101 T^{5} + \cdots + 653628591729$$
$79$ $$T^{6} + 90 T^{5} + \cdots + 279290944$$
$83$ $$T^{6} - 99 T^{5} + \cdots + 165842832483$$
$89$ $$T^{6} - 243 T^{5} + \cdots + 15857178627$$
$97$ $$T^{6} + 161 T^{5} + \cdots + 36908941689$$