# Properties

 Label 57.3.g.b Level $57$ Weight $3$ Character orbit 57.g Analytic conductor $1.553$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 57.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.55313750685$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.92607408.1 Defining polynomial: $$x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43$$ x^6 - 3*x^5 + 20*x^4 - 35*x^3 + 94*x^2 - 77*x + 43 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_{3} - 2) q^{3} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{4} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1) q^{6} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{7} + (2 \beta_{5} + 10 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{8} + (3 \beta_{3} + 3) q^{9}+O(q^{10})$$ q - b4 * q^2 + (-b3 - 2) * q^3 + (b5 - b4 + 2*b3 + b1 + 1) * q^4 + (b5 - b4 - 2*b3 + b2 - 2*b1) * q^5 + (b4 - b3 - b1) * q^6 + (2*b4 + b3 + b2 + b1 - 3) * q^7 + (2*b5 + 10*b3 + b2 + 2*b1 + 4) * q^8 + (3*b3 + 3) * q^9 $$q - \beta_{4} q^{2} + ( - \beta_{3} - 2) q^{3} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{4} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + (\beta_{4} - \beta_{3} - \beta_1) q^{6} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{7} + (2 \beta_{5} + 10 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{8} + (3 \beta_{3} + 3) q^{9} + ( - \beta_{5} - \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - \beta_1 + 6) q^{10} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 8) q^{11} + ( - 2 \beta_{5} - 5 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{12} + ( - 2 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 1) q^{13} + ( - \beta_{5} + 8 \beta_{4} - 4 \beta_{3} + \beta_{2} - 8) q^{14} + ( - \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{15} + ( - 2 \beta_{5} + 4 \beta_{4} + 11 \beta_{3} - 2 \beta_{2} + 8 \beta_1) q^{16} + (2 \beta_{5} + 4 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{17} + (3 \beta_{3} + 3 \beta_1) q^{18} + ( - 2 \beta_{5} - 2 \beta_{4} - 7 \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 5) q^{19} + ( - 10 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 2) q^{20} + (\beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 6) q^{21} + (\beta_{5} + 11 \beta_{4} + 8 \beta_{3} - \beta_{2} + 16) q^{22} + ( - 3 \beta_{5} - 3 \beta_{4} + 17 \beta_{3} + 3 \beta_1 + 14) q^{23} + ( - 3 \beta_{5} - 2 \beta_{4} - 16 \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{24} + ( - 2 \beta_{4} - 7 \beta_{3} + 2 \beta_1 - 9) q^{25} + ( - 14 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 14) q^{26} + ( - 6 \beta_{3} - 3) q^{27} + ( - 4 \beta_{5} + 10 \beta_{4} - 35 \beta_{3} - 10 \beta_1 - 25) q^{28} + (4 \beta_{5} + 2 \beta_{4} + 14 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 12) q^{29} + (2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 - 18) q^{30} + (2 \beta_{5} + 31 \beta_{3} + \beta_{2} + 5 \beta_1 + 13) q^{31} + (3 \beta_{4} + 9 \beta_{3} + 3 \beta_1 - 6) q^{32} + (3 \beta_{5} + 3 \beta_{4} + 8 \beta_{3} - 3 \beta_{2} + 16) q^{33} + (4 \beta_{5} + 18 \beta_{3} + 8 \beta_{2} - 18) q^{34} + (3 \beta_{5} + 7 \beta_{4} + 16 \beta_{3} + 3 \beta_{2} + 14 \beta_1) q^{35} + (3 \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 6 \beta_1) q^{36} + (8 \beta_{5} - 14 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 1) q^{37} + ( - 3 \beta_{5} + 2 \beta_{4} - 22 \beta_{3} - 5 \beta_{2} - 14 \beta_1 + 18) q^{38} + (8 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 3) q^{39} + (\beta_{5} - 23 \beta_{4} - 4 \beta_{3} - \beta_{2} - 8) q^{40} + (2 \beta_{5} + 2 \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + 24) q^{41} + (3 \beta_{5} - 8 \beta_{4} + 20 \beta_{3} + 8 \beta_1 + 12) q^{42} + ( - \beta_{5} - 3 \beta_{4} - 39 \beta_{3} - \beta_{2} - 6 \beta_1) q^{43} + (\beta_{5} + 5 \beta_{4} - 31 \beta_{3} - 5 \beta_1 - 26) q^{44} + ( - 6 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{45} + (6 \beta_{5} + 53 \beta_{3} + 3 \beta_{2} + 17 \beta_1 + 18) q^{46} + ( - 4 \beta_{5} - 18 \beta_{3} - 18) q^{47} + (2 \beta_{5} - 12 \beta_{4} - 15 \beta_{3} + 4 \beta_{2} - 12 \beta_1 + 3) q^{48} + ( - 24 \beta_{4} - 12 \beta_{3} - 6 \beta_{2} - 12 \beta_1 - 18) q^{49} + (4 \beta_{5} + 17 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 10) q^{50} + ( - 2 \beta_{5} - 12 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 12 \beta_1 - 16) q^{51} + ( - \beta_{5} - 27 \beta_{4} + 35 \beta_{3} + \beta_{2} + 70) q^{52} + ( - 9 \beta_{5} + 9 \beta_{4} + 13 \beta_{3} - 18 \beta_{2} + 9 \beta_1 - 4) q^{53} + ( - 3 \beta_{4} - 6 \beta_{3} - 6 \beta_1) q^{54} + ( - 2 \beta_{5} + 16 \beta_{4} - 34 \beta_{3} - 2 \beta_{2} + 32 \beta_1) q^{55} + ( - 12 \beta_{5} - 91 \beta_{3} - 6 \beta_{2} - 31 \beta_1 - 30) q^{56} + (7 \beta_{5} + 7 \beta_{4} + 15 \beta_{3} - \beta_{2} + 8 \beta_1 + 8) q^{57} + (44 \beta_{4} + 22 \beta_{3} + 2 \beta_{2} + 22 \beta_1 + 2) q^{58} + ( - \beta_{5} - 11 \beta_{4} - 4 \beta_{3} + \beta_{2} - 8) q^{59} + ( - 5 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} + 5 \beta_{2} - 4) q^{60} + (10 \beta_{4} + 9 \beta_{3} - 10 \beta_1 + 19) q^{61} + (5 \beta_{5} + 20 \beta_{4} + 62 \beta_{3} + 5 \beta_{2} + 40 \beta_1) q^{62} + ( - 3 \beta_{5} + 3 \beta_{4} - 12 \beta_{3} - 3 \beta_1 - 9) q^{63} + ( - 14 \beta_{4} - 7 \beta_{3} + 11 \beta_{2} - 7 \beta_1 - 3) q^{64} + ( - 22 \beta_{5} - 3 \beta_{3} - 11 \beta_{2} - 19 \beta_1 + 8) q^{65} + ( - 3 \beta_{5} - 11 \beta_{4} - 13 \beta_{3} + 11 \beta_1 - 24) q^{66} + ( - 9 \beta_{5} + 13 \beta_{4} + 10 \beta_{3} - 18 \beta_{2} + 13 \beta_1 + 3) q^{67} + (20 \beta_{4} + 10 \beta_{3} - 8 \beta_{2} + 10 \beta_1 - 52) q^{68} + (6 \beta_{5} - 37 \beta_{3} + 3 \beta_{2} - 9 \beta_1 - 14) q^{69} + (7 \beta_{5} + 29 \beta_{4} + 61 \beta_{3} + 14 \beta_{2} + 29 \beta_1 - 32) q^{70} + ( - 4 \beta_{5} + 30 \beta_{4} + 18 \beta_{3} + 4 \beta_{2} + 36) q^{71} + (3 \beta_{5} + 6 \beta_{4} + 18 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 12) q^{72} + ( - 8 \beta_{5} - 20 \beta_{4} + 3 \beta_{3} - 8 \beta_{2} - 40 \beta_1) q^{73} + ( - 12 \beta_{5} - 5 \beta_{4} - 82 \beta_{3} - 12 \beta_{2} - 10 \beta_1) q^{74} + (12 \beta_{3} - 6 \beta_1 + 9) q^{75} + (4 \beta_{5} - 28 \beta_{4} - 91 \beta_{3} - 6 \beta_{2} - 32 \beta_1 - 5) q^{76} + ( - 6 \beta_{4} - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 42) q^{77} + ( - 4 \beta_{5} + 21 \beta_{4} - 14 \beta_{3} + 4 \beta_{2} - 28) q^{78} + ( - 11 \beta_{5} - \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 6) q^{79} + (3 \beta_{5} - \beta_{4} + 105 \beta_{3} + \beta_1 + 104) q^{80} + 9 \beta_{3} q^{81} + ( - 2 \beta_{5} - 14 \beta_{4} - 2 \beta_{3} + 14 \beta_1 - 16) q^{82} + ( - 48 \beta_{4} - 24 \beta_{3} - 6 \beta_{2} - 24 \beta_1 - 80) q^{83} + (8 \beta_{5} + 80 \beta_{3} + 4 \beta_{2} + 30 \beta_1 + 25) q^{84} + (34 \beta_{5} + 6 \beta_{4} + 34 \beta_{3} - 6 \beta_1 + 40) q^{85} + ( - 3 \beta_{5} - 44 \beta_{4} - 58 \beta_{3} - 6 \beta_{2} - 44 \beta_1 + 14) q^{86} + ( - 4 \beta_{4} - 2 \beta_{3} - 12 \beta_{2} - 2 \beta_1 + 36) q^{87} + ( - 18 \beta_{5} - 111 \beta_{3} - 9 \beta_{2} + 5 \beta_1 - 58) q^{88} + (15 \beta_{5} + 9 \beta_{4} + 19 \beta_{3} + 30 \beta_{2} + 9 \beta_1 - 10) q^{89} + (3 \beta_{5} - 3 \beta_{4} + 18 \beta_{3} - 3 \beta_{2} + 36) q^{90} + (3 \beta_{5} + 35 \beta_{4} + 68 \beta_{3} + 6 \beta_{2} + 35 \beta_1 - 33) q^{91} + (29 \beta_{5} + 29 \beta_{4} + 78 \beta_{3} + 29 \beta_{2} + 58 \beta_1) q^{92} + ( - 3 \beta_{5} - 5 \beta_{4} - 49 \beta_{3} - 3 \beta_{2} - 10 \beta_1) q^{93} + ( - 18 \beta_{3} - 26 \beta_1 + 4) q^{94} + ( - \beta_{5} + 31 \beta_{4} - 60 \beta_{3} + 11 \beta_{2} + 30 \beta_1 - 16) q^{95} + ( - 6 \beta_{4} - 3 \beta_{3} - 3 \beta_1 + 18) q^{96} + ( - 14 \beta_{4} - 30 \beta_{3} - 60) q^{97} + (12 \beta_{5} - 30 \beta_{4} + 54 \beta_{3} - 12 \beta_{2} + 108) q^{98} + ( - 9 \beta_{5} - 3 \beta_{4} - 21 \beta_{3} + 3 \beta_1 - 24) q^{99}+O(q^{100})$$ q - b4 * q^2 + (-b3 - 2) * q^3 + (b5 - b4 + 2*b3 + b1 + 1) * q^4 + (b5 - b4 - 2*b3 + b2 - 2*b1) * q^5 + (b4 - b3 - b1) * q^6 + (2*b4 + b3 + b2 + b1 - 3) * q^7 + (2*b5 + 10*b3 + b2 + 2*b1 + 4) * q^8 + (3*b3 + 3) * q^9 + (-b5 - b4 - 7*b3 - 2*b2 - b1 + 6) * q^10 + (-2*b4 - b3 + 3*b2 - b1 - 8) * q^11 + (-2*b5 - 5*b3 - b2 - 3*b1 - 1) * q^12 + (-2*b5 - 4*b4 - 3*b3 - 4*b2 - 4*b1 - 1) * q^13 + (-b5 + 8*b4 - 4*b3 + b2 - 8) * q^14 + (-b5 + 3*b4 + 3*b3 - 2*b2 + 3*b1) * q^15 + (-2*b5 + 4*b4 + 11*b3 - 2*b2 + 8*b1) * q^16 + (2*b5 + 4*b4 - 8*b3 + 2*b2 + 8*b1) * q^17 + (3*b3 + 3*b1) * q^18 + (-2*b5 - 2*b4 - 7*b3 + 3*b2 - 5*b1 - 5) * q^19 + (-10*b4 - 5*b3 - 5*b2 - 5*b1 + 2) * q^20 + (b5 - 3*b4 + 3*b3 - b2 + 6) * q^21 + (b5 + 11*b4 + 8*b3 - b2 + 16) * q^22 + (-3*b5 - 3*b4 + 17*b3 + 3*b1 + 14) * q^23 + (-3*b5 - 2*b4 - 16*b3 - 3*b2 - 4*b1) * q^24 + (-2*b4 - 7*b3 + 2*b1 - 9) * q^25 + (-14*b4 - 7*b3 - 4*b2 - 7*b1 + 14) * q^26 + (-6*b3 - 3) * q^27 + (-4*b5 + 10*b4 - 35*b3 - 10*b1 - 25) * q^28 + (4*b5 + 2*b4 + 14*b3 + 8*b2 + 2*b1 - 12) * q^29 + (2*b4 + b3 + 3*b2 + b1 - 18) * q^30 + (2*b5 + 31*b3 + b2 + 5*b1 + 13) * q^31 + (3*b4 + 9*b3 + 3*b1 - 6) * q^32 + (3*b5 + 3*b4 + 8*b3 - 3*b2 + 16) * q^33 + (4*b5 + 18*b3 + 8*b2 - 18) * q^34 + (3*b5 + 7*b4 + 16*b3 + 3*b2 + 14*b1) * q^35 + (3*b5 + 3*b4 + 9*b3 + 3*b2 + 6*b1) * q^36 + (8*b5 - 14*b3 + 4*b2 - 12*b1 - 1) * q^37 + (-3*b5 + 2*b4 - 22*b3 - 5*b2 - 14*b1 + 18) * q^38 + (8*b4 + 4*b3 + 6*b2 + 4*b1 + 3) * q^39 + (b5 - 23*b4 - 4*b3 - b2 - 8) * q^40 + (2*b5 + 2*b4 + 12*b3 - 2*b2 + 24) * q^41 + (3*b5 - 8*b4 + 20*b3 + 8*b1 + 12) * q^42 + (-b5 - 3*b4 - 39*b3 - b2 - 6*b1) * q^43 + (b5 + 5*b4 - 31*b3 - 5*b1 - 26) * q^44 + (-6*b4 - 3*b3 + 3*b2 - 3*b1) * q^45 + (6*b5 + 53*b3 + 3*b2 + 17*b1 + 18) * q^46 + (-4*b5 - 18*b3 - 18) * q^47 + (2*b5 - 12*b4 - 15*b3 + 4*b2 - 12*b1 + 3) * q^48 + (-24*b4 - 12*b3 - 6*b2 - 12*b1 - 18) * q^49 + (4*b5 + 17*b3 + 2*b2 - 3*b1 + 10) * q^50 + (-2*b5 - 12*b4 + 4*b3 - 4*b2 - 12*b1 - 16) * q^51 + (-b5 - 27*b4 + 35*b3 + b2 + 70) * q^52 + (-9*b5 + 9*b4 + 13*b3 - 18*b2 + 9*b1 - 4) * q^53 + (-3*b4 - 6*b3 - 6*b1) * q^54 + (-2*b5 + 16*b4 - 34*b3 - 2*b2 + 32*b1) * q^55 + (-12*b5 - 91*b3 - 6*b2 - 31*b1 - 30) * q^56 + (7*b5 + 7*b4 + 15*b3 - b2 + 8*b1 + 8) * q^57 + (44*b4 + 22*b3 + 2*b2 + 22*b1 + 2) * q^58 + (-b5 - 11*b4 - 4*b3 + b2 - 8) * q^59 + (-5*b5 + 15*b4 - 2*b3 + 5*b2 - 4) * q^60 + (10*b4 + 9*b3 - 10*b1 + 19) * q^61 + (5*b5 + 20*b4 + 62*b3 + 5*b2 + 40*b1) * q^62 + (-3*b5 + 3*b4 - 12*b3 - 3*b1 - 9) * q^63 + (-14*b4 - 7*b3 + 11*b2 - 7*b1 - 3) * q^64 + (-22*b5 - 3*b3 - 11*b2 - 19*b1 + 8) * q^65 + (-3*b5 - 11*b4 - 13*b3 + 11*b1 - 24) * q^66 + (-9*b5 + 13*b4 + 10*b3 - 18*b2 + 13*b1 + 3) * q^67 + (20*b4 + 10*b3 - 8*b2 + 10*b1 - 52) * q^68 + (6*b5 - 37*b3 + 3*b2 - 9*b1 - 14) * q^69 + (7*b5 + 29*b4 + 61*b3 + 14*b2 + 29*b1 - 32) * q^70 + (-4*b5 + 30*b4 + 18*b3 + 4*b2 + 36) * q^71 + (3*b5 + 6*b4 + 18*b3 + 6*b2 + 6*b1 - 12) * q^72 + (-8*b5 - 20*b4 + 3*b3 - 8*b2 - 40*b1) * q^73 + (-12*b5 - 5*b4 - 82*b3 - 12*b2 - 10*b1) * q^74 + (12*b3 - 6*b1 + 9) * q^75 + (4*b5 - 28*b4 - 91*b3 - 6*b2 - 32*b1 - 5) * q^76 + (-6*b4 - 3*b3 - 5*b2 - 3*b1 + 42) * q^77 + (-4*b5 + 21*b4 - 14*b3 + 4*b2 - 28) * q^78 + (-11*b5 - b4 - 3*b3 + 11*b2 - 6) * q^79 + (3*b5 - b4 + 105*b3 + b1 + 104) * q^80 + 9*b3 * q^81 + (-2*b5 - 14*b4 - 2*b3 + 14*b1 - 16) * q^82 + (-48*b4 - 24*b3 - 6*b2 - 24*b1 - 80) * q^83 + (8*b5 + 80*b3 + 4*b2 + 30*b1 + 25) * q^84 + (34*b5 + 6*b4 + 34*b3 - 6*b1 + 40) * q^85 + (-3*b5 - 44*b4 - 58*b3 - 6*b2 - 44*b1 + 14) * q^86 + (-4*b4 - 2*b3 - 12*b2 - 2*b1 + 36) * q^87 + (-18*b5 - 111*b3 - 9*b2 + 5*b1 - 58) * q^88 + (15*b5 + 9*b4 + 19*b3 + 30*b2 + 9*b1 - 10) * q^89 + (3*b5 - 3*b4 + 18*b3 - 3*b2 + 36) * q^90 + (3*b5 + 35*b4 + 68*b3 + 6*b2 + 35*b1 - 33) * q^91 + (29*b5 + 29*b4 + 78*b3 + 29*b2 + 58*b1) * q^92 + (-3*b5 - 5*b4 - 49*b3 - 3*b2 - 10*b1) * q^93 + (-18*b3 - 26*b1 + 4) * q^94 + (-b5 + 31*b4 - 60*b3 + 11*b2 + 30*b1 - 16) * q^95 + (-6*b4 - 3*b3 - 3*b1 + 18) * q^96 + (-14*b4 - 30*b3 - 60) * q^97 + (12*b5 - 30*b4 + 54*b3 - 12*b2 + 108) * q^98 + (-9*b5 - 3*b4 - 21*b3 + 3*b1 - 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 9 q^{3} + 5 q^{4} + 4 q^{5} - 3 q^{6} - 22 q^{7} + 9 q^{9}+O(q^{10})$$ 6 * q + 3 * q^2 - 9 * q^3 + 5 * q^4 + 4 * q^5 - 3 * q^6 - 22 * q^7 + 9 * q^9 $$6 q + 3 q^{2} - 9 q^{3} + 5 q^{4} + 4 q^{5} - 3 q^{6} - 22 q^{7} + 9 q^{9} + 54 q^{10} - 36 q^{11} - 3 q^{13} - 57 q^{14} - 12 q^{15} - 23 q^{16} + 38 q^{17} - 10 q^{19} + 32 q^{20} + 33 q^{21} + 36 q^{22} + 54 q^{23} + 39 q^{24} - 21 q^{25} + 118 q^{26} - 101 q^{28} - 102 q^{29} - 108 q^{30} - 63 q^{32} + 54 q^{33} - 150 q^{34} - 24 q^{35} - 15 q^{36} + 119 q^{38} + 6 q^{39} + 30 q^{40} + 96 q^{41} + 57 q^{42} + 107 q^{43} - 94 q^{44} + 24 q^{45} - 50 q^{47} + 69 q^{48} - 48 q^{49} - 114 q^{51} + 399 q^{52} - 90 q^{53} + 9 q^{54} + 148 q^{55} - 3 q^{57} - 116 q^{58} - 48 q^{60} + 27 q^{61} - 121 q^{62} - 33 q^{63} + 46 q^{64} - 36 q^{66} - 39 q^{67} - 388 q^{68} - 354 q^{70} + 84 q^{71} - 117 q^{72} - 77 q^{73} + 219 q^{74} + 215 q^{76} + 260 q^{77} - 177 q^{78} + 9 q^{79} + 312 q^{80} - 27 q^{81} - 4 q^{82} - 348 q^{83} + 68 q^{85} + 249 q^{86} + 204 q^{87} - 72 q^{89} + 162 q^{90} - 393 q^{91} - 118 q^{92} + 129 q^{93} + 104 q^{95} + 126 q^{96} - 228 q^{97} + 540 q^{98} - 54 q^{99}+O(q^{100})$$ 6 * q + 3 * q^2 - 9 * q^3 + 5 * q^4 + 4 * q^5 - 3 * q^6 - 22 * q^7 + 9 * q^9 + 54 * q^10 - 36 * q^11 - 3 * q^13 - 57 * q^14 - 12 * q^15 - 23 * q^16 + 38 * q^17 - 10 * q^19 + 32 * q^20 + 33 * q^21 + 36 * q^22 + 54 * q^23 + 39 * q^24 - 21 * q^25 + 118 * q^26 - 101 * q^28 - 102 * q^29 - 108 * q^30 - 63 * q^32 + 54 * q^33 - 150 * q^34 - 24 * q^35 - 15 * q^36 + 119 * q^38 + 6 * q^39 + 30 * q^40 + 96 * q^41 + 57 * q^42 + 107 * q^43 - 94 * q^44 + 24 * q^45 - 50 * q^47 + 69 * q^48 - 48 * q^49 - 114 * q^51 + 399 * q^52 - 90 * q^53 + 9 * q^54 + 148 * q^55 - 3 * q^57 - 116 * q^58 - 48 * q^60 + 27 * q^61 - 121 * q^62 - 33 * q^63 + 46 * q^64 - 36 * q^66 - 39 * q^67 - 388 * q^68 - 354 * q^70 + 84 * q^71 - 117 * q^72 - 77 * q^73 + 219 * q^74 + 215 * q^76 + 260 * q^77 - 177 * q^78 + 9 * q^79 + 312 * q^80 - 27 * q^81 - 4 * q^82 - 348 * q^83 + 68 * q^85 + 249 * q^86 + 204 * q^87 - 72 * q^89 + 162 * q^90 - 393 * q^91 - 118 * q^92 + 129 * q^93 + 104 * q^95 + 126 * q^96 - 228 * q^97 + 540 * q^98 - 54 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 94x^{2} - 77x + 43$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu + 6$$ v^2 - v + 6 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} - 56\nu + 11 ) / 23$$ (-2*v^5 + 5*v^4 - 26*v^3 + 34*v^2 - 56*v + 11) / 23 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 9\nu^{4} - 10\nu^{3} + 98\nu^{2} - 87\nu + 52 ) / 23$$ (v^5 + 9*v^4 - 10*v^3 + 98*v^2 - 87*v + 52) / 23 $$\beta_{5}$$ $$=$$ $$\nu^{3} - 2\nu^{2} + 9\nu - 7$$ v^3 - 2*v^2 + 9*v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 - 6$$ b2 + b1 - 6 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{2} - 7\beta _1 - 5$$ b5 + 2*b2 - 7*b1 - 5 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 2\beta_{4} + \beta_{3} - 6\beta_{2} - 14\beta _1 + 45$$ 2*b5 + 2*b4 + b3 - 6*b2 - 14*b1 + 45 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 5\beta_{4} - 9\beta_{3} - 24\beta_{2} + 45\beta _1 + 81$$ -8*b5 + 5*b4 - 9*b3 - 24*b2 + 45*b1 + 81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.5 + 2.93068i 0.5 + 0.630453i 0.5 − 2.69511i 0.5 − 2.93068i 0.5 − 0.630453i 0.5 + 2.69511i
−1.78805 + 1.03233i −1.50000 + 0.866025i 0.131406 0.227602i −3.20750 5.55555i 1.78805 3.09699i −2.26281 7.71601i 1.50000 2.59808i 11.4703 + 6.62239i
31.2 0.204011 0.117786i −1.50000 + 0.866025i −1.97225 + 3.41604i 2.88028 + 4.98878i −0.204011 + 0.353358i 1.94451 1.87150i 1.50000 2.59808i 1.17522 + 0.678513i
31.3 3.08403 1.78057i −1.50000 + 0.866025i 4.34085 7.51857i 2.32722 + 4.03087i −3.08403 + 5.34170i −10.6817 16.6722i 1.50000 2.59808i 14.3545 + 8.28756i
46.1 −1.78805 1.03233i −1.50000 0.866025i 0.131406 + 0.227602i −3.20750 + 5.55555i 1.78805 + 3.09699i −2.26281 7.71601i 1.50000 + 2.59808i 11.4703 6.62239i
46.2 0.204011 + 0.117786i −1.50000 0.866025i −1.97225 3.41604i 2.88028 4.98878i −0.204011 0.353358i 1.94451 1.87150i 1.50000 + 2.59808i 1.17522 0.678513i
46.3 3.08403 + 1.78057i −1.50000 0.866025i 4.34085 + 7.51857i 2.32722 4.03087i −3.08403 5.34170i −10.6817 16.6722i 1.50000 + 2.59808i 14.3545 8.28756i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.g.b 6
3.b odd 2 1 171.3.p.c 6
4.b odd 2 1 912.3.be.f 6
19.d odd 6 1 inner 57.3.g.b 6
57.f even 6 1 171.3.p.c 6
76.f even 6 1 912.3.be.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.b 6 1.a even 1 1 trivial
57.3.g.b 6 19.d odd 6 1 inner
171.3.p.c 6 3.b odd 2 1
171.3.p.c 6 57.f even 6 1
912.3.be.f 6 4.b odd 2 1
912.3.be.f 6 76.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3T_{2}^{5} - 4T_{2}^{4} + 21T_{2}^{3} + 46T_{2}^{2} - 21T_{2} + 3$$ acting on $$S_{3}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} - 4 T^{4} + 21 T^{3} + \cdots + 3$$
$3$ $$(T^{2} + 3 T + 3)^{3}$$
$5$ $$T^{6} - 4 T^{5} + 56 T^{4} + \cdots + 29584$$
$7$ $$(T^{3} + 11 T^{2} - T - 47)^{2}$$
$11$ $$(T^{3} + 18 T^{2} - 96 T - 1084)^{2}$$
$13$ $$T^{6} + 3 T^{5} - 340 T^{4} + \cdots + 2883$$
$17$ $$T^{6} - 38 T^{5} + 1408 T^{4} + \cdots + 52186176$$
$19$ $$T^{6} + 10 T^{5} - 249 T^{4} + \cdots + 47045881$$
$23$ $$T^{6} - 54 T^{5} + 2352 T^{4} + \cdots + 21049744$$
$29$ $$T^{6} + 102 T^{5} + \cdots + 487228608$$
$31$ $$T^{6} + 2345 T^{4} + \cdots + 112326483$$
$37$ $$T^{6} + 4713 T^{4} + \cdots + 3652564347$$
$41$ $$T^{6} - 96 T^{5} + 3824 T^{4} + \cdots + 1354752$$
$43$ $$T^{6} - 107 T^{5} + \cdots + 1477402969$$
$47$ $$T^{6} + 50 T^{5} + 1976 T^{4} + \cdots + 5798464$$
$53$ $$T^{6} + 90 T^{5} + \cdots + 6327041328$$
$59$ $$T^{6} - 1048 T^{4} + \cdots + 25509168$$
$61$ $$T^{6} - 27 T^{5} + \cdots + 1215986641$$
$67$ $$T^{6} + 39 T^{5} + \cdots + 11787475467$$
$71$ $$T^{6} - 84 T^{5} + \cdots + 149905029888$$
$73$ $$T^{6} + 77 T^{5} + \cdots + 434693631969$$
$79$ $$T^{6} - 9 T^{5} + \cdots + 32898206883$$
$83$ $$(T^{3} + 174 T^{2} - 4140 T - 1174072)^{2}$$
$89$ $$T^{6} + 72 T^{5} + \cdots + 591631797168$$
$97$ $$T^{6} + 228 T^{5} + \cdots + 68659968$$