# Properties

 Label 171.3.p.e Level $171$ Weight $3$ Character orbit 171.p Analytic conductor $4.659$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 171.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.65941252056$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.6967728.1 Defining polynomial: $$x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1$$ x^6 - x^5 + 8*x^4 + 5*x^3 + 50*x^2 - 7*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 57) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 5) q^{7} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{8}+O(q^{10})$$ q - b3 * q^2 + (-b5 + b4 - 2*b3 + b2 - b1) * q^4 + (b5 + b3 - b2 + 1) * q^5 + (-b4 - b3 - 2*b2 + 5) * q^7 + (-2*b5 + b4 - 2*b3 + b2 - 4*b1 + 1) * q^8 $$q - \beta_{3} q^{2} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 5) q^{7} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{8} + (\beta_{5} + \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 5) q^{10} + ( - 3 \beta_{4} + 3 \beta_{3} + 2) q^{11} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_1 - 1) q^{13} + ( - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + \beta_{2} - 8 \beta_1 + 15) q^{14} + (2 \beta_{5} - 2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 6) q^{16} + ( - 4 \beta_{5} - 4 \beta_{2} + 2 \beta_1 + 4) q^{17} + ( - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 16 \beta_1 - 1) q^{19} + (\beta_{4} - 5 \beta_{3} - 4 \beta_{2} + 20) q^{20} + (3 \beta_{5} - 6 \beta_{4} + 13 \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 9) q^{22} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 10 \beta_1 + 11) q^{23} + (8 \beta_{3} - 4 \beta_{2} - 9 \beta_1 + 13) q^{25} + ( - 4 \beta_{4} + 9 \beta_{3} + 5 \beta_{2} - 7) q^{26} + ( - 8 \beta_{5} + 8 \beta_{4} - 20 \beta_{3} + 10 \beta_{2} + \beta_1 - 11) q^{28} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 6 \beta_1 - 12) q^{29} + (6 \beta_{5} - 3 \beta_{4} + 11 \beta_{3} - 8 \beta_{2} - 10 \beta_1 + 13) q^{31} + (2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} - 8 \beta_1 - 3) q^{32} + (6 \beta_{2} + 12 \beta_1 + 6) q^{34} + (9 \beta_{5} + 9 \beta_{3} - 9 \beta_{2} - 4 \beta_1 + 9) q^{35} + (18 \beta_{3} - 18 \beta_{2} + 6 \beta_1 + 15) q^{37} + ( - \beta_{5} - 8 \beta_{3} + 21 \beta_{2} + 10 \beta_1 - 33) q^{38} + ( - \beta_{5} + 2 \beta_{4} - 13 \beta_{3} + \beta_{2} - 14 \beta_1 + 27) q^{40} + (6 \beta_{5} - 12 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 2) q^{41} + (\beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{43} + (\beta_{5} - \beta_{4} + 26 \beta_{3} - 13 \beta_{2} + 46 \beta_1 - 33) q^{44} + (2 \beta_{5} - \beta_{4} - 13 \beta_{3} + 14 \beta_{2} + 28 \beta_1 - 28) q^{46} + ( - 4 \beta_{5} + 4 \beta_{4} - 32 \beta_{3} + 16 \beta_{2} + 6 \beta_1 - 22) q^{47} + ( - 6 \beta_{4} - 2 \beta_{3} - 8 \beta_{2} + 20) q^{49} + (8 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + \beta_{2} + 40 \beta_1 - 21) q^{50} + (\beta_{5} - 2 \beta_{4} + 21 \beta_{3} - \beta_{2} + 29 \beta_1 - 57) q^{52} + ( - 3 \beta_{5} - 3 \beta_{4} - 21 \beta_{2} - 26 \beta_1 - 5) q^{53} + (2 \beta_{5} - 10 \beta_{3} + 22 \beta_{2} - 48 \beta_1 - 22) q^{55} + ( - 12 \beta_{5} + 6 \beta_{4} - 29 \beta_{3} + 23 \beta_{2} + 12 \beta_1 - 29) q^{56} + ( - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 30) q^{58} + ( - 5 \beta_{5} + 10 \beta_{4} + 13 \beta_{3} + 5 \beta_{2} + 12 \beta_1 - 29) q^{59} + ( - 12 \beta_{5} + 12 \beta_{4} - 44 \beta_{3} + 22 \beta_{2} + 65 \beta_1 - 87) q^{61} + (11 \beta_{5} + 12 \beta_{3} - 13 \beta_{2} + 28 \beta_1 + 13) q^{62} + (\beta_{4} - 13 \beta_{3} - 12 \beta_{2} + 27) q^{64} + (2 \beta_{5} - \beta_{4} - 17 \beta_{3} + 18 \beta_{2} - 56 \beta_1 + 10) q^{65} + (\beta_{5} + \beta_{4} - 17 \beta_{2} - 13 \beta_1 + 4) q^{67} + (10 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} - 42) q^{68} + (9 \beta_{5} + 9 \beta_{4} - 23 \beta_{2} + 18 \beta_1 + 41) q^{70} + ( - 2 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 44 \beta_1 + 86) q^{71} + (20 \beta_{5} + 20 \beta_{2} + 41 \beta_1 - 20) q^{73} + (18 \beta_{5} + 21 \beta_{3} - 24 \beta_{2} + 90 \beta_1 + 24) q^{74} + (4 \beta_{5} - 18 \beta_{4} + 18 \beta_{3} + 4 \beta_{2} - 33 \beta_1 - 11) q^{76} + ( - 29 \beta_{4} + 25 \beta_{3} - 4 \beta_{2} + 22) q^{77} + ( - \beta_{5} + 2 \beta_{4} - 17 \beta_{3} + \beta_{2} + 19 \beta_1 - 39) q^{79} + ( - 9 \beta_{5} + 9 \beta_{4} - 18 \beta_{3} + 9 \beta_{2} + 8 \beta_1 - 17) q^{80} + (2 \beta_{5} - 2 \beta_{4} + 36 \beta_{3} - 18 \beta_{2} - 44 \beta_1 + 62) q^{82} + ( - 4 \beta_{3} - 4 \beta_{2} + 42) q^{83} + (14 \beta_{5} - 14 \beta_{4} + 12 \beta_{3} - 6 \beta_{2} - 40 \beta_1 + 46) q^{85} + (\beta_{5} + \beta_{4} - 6 \beta_{2} + 2 \beta_1 + 8) q^{86} + (2 \beta_{5} - \beta_{4} + 35 \beta_{3} - 34 \beta_{2} + 76 \beta_1 - 4) q^{88} + (\beta_{5} + \beta_{4} - \beta_{2} - 72 \beta_1 - 71) q^{89} + ( - 15 \beta_{5} - 15 \beta_{4} + 19 \beta_{2} + 21 \beta_1 + 2) q^{91} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 34 \beta_1 - 1) q^{92} + ( - 32 \beta_{5} + 16 \beta_{4} - 50 \beta_{3} + 34 \beta_{2} - 136 \beta_1 + 34) q^{94} + ( - 19 \beta_{5} + 12 \beta_{4} - 35 \beta_{3} + 11 \beta_{2} + 8 \beta_1 + 13) q^{95} + ( - 8 \beta_{5} + 16 \beta_{4} + 44 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 24) q^{97} + ( - 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 28 \beta_1 + 54) q^{98}+O(q^{100})$$ q - b3 * q^2 + (-b5 + b4 - 2*b3 + b2 - b1) * q^4 + (b5 + b3 - b2 + 1) * q^5 + (-b4 - b3 - 2*b2 + 5) * q^7 + (-2*b5 + b4 - 2*b3 + b2 - 4*b1 + 1) * q^8 + (b5 + b4 - 3*b2 + 2*b1 + 5) * q^10 + (-3*b4 + 3*b3 + 2) * q^11 + (-2*b5 - 2*b4 - b1 - 1) * q^13 + (-b5 + 2*b4 - 4*b3 + b2 - 8*b1 + 15) * q^14 + (2*b5 - 2*b3 + 6*b2 + 3*b1 - 6) * q^16 + (-4*b5 - 4*b2 + 2*b1 + 4) * q^17 + (-2*b5 + 3*b4 - b3 + 2*b2 - 16*b1 - 1) * q^19 + (b4 - 5*b3 - 4*b2 + 20) * q^20 + (3*b5 - 6*b4 + 13*b3 - 3*b2 + 6*b1 - 9) * q^22 + (-3*b5 + 3*b4 + 2*b3 - b2 - 10*b1 + 11) * q^23 + (8*b3 - 4*b2 - 9*b1 + 13) * q^25 + (-4*b4 + 9*b3 + 5*b2 - 7) * q^26 + (-8*b5 + 8*b4 - 20*b3 + 10*b2 + b1 - 11) * q^28 + (2*b5 + 2*b4 + 6*b2 - 6*b1 - 12) * q^29 + (6*b5 - 3*b4 + 11*b3 - 8*b2 - 10*b1 + 13) * q^31 + (2*b5 + 2*b4 - 5*b2 - 8*b1 - 3) * q^32 + (6*b2 + 12*b1 + 6) * q^34 + (9*b5 + 9*b3 - 9*b2 - 4*b1 + 9) * q^35 + (18*b3 - 18*b2 + 6*b1 + 15) * q^37 + (-b5 - 8*b3 + 21*b2 + 10*b1 - 33) * q^38 + (-b5 + 2*b4 - 13*b3 + b2 - 14*b1 + 27) * q^40 + (6*b5 - 12*b4 + 2*b3 - 6*b2 + 4*b1 - 2) * q^41 + (b5 + b3 - b2 + 3*b1 + 1) * q^43 + (b5 - b4 + 26*b3 - 13*b2 + 46*b1 - 33) * q^44 + (2*b5 - b4 - 13*b3 + 14*b2 + 28*b1 - 28) * q^46 + (-4*b5 + 4*b4 - 32*b3 + 16*b2 + 6*b1 - 22) * q^47 + (-6*b4 - 2*b3 - 8*b2 + 20) * q^49 + (8*b5 - 4*b4 + 3*b3 + b2 + 40*b1 - 21) * q^50 + (b5 - 2*b4 + 21*b3 - b2 + 29*b1 - 57) * q^52 + (-3*b5 - 3*b4 - 21*b2 - 26*b1 - 5) * q^53 + (2*b5 - 10*b3 + 22*b2 - 48*b1 - 22) * q^55 + (-12*b5 + 6*b4 - 29*b3 + 23*b2 + 12*b1 - 29) * q^56 + (-2*b4 + 4*b3 + 2*b2 - 30) * q^58 + (-5*b5 + 10*b4 + 13*b3 + 5*b2 + 12*b1 - 29) * q^59 + (-12*b5 + 12*b4 - 44*b3 + 22*b2 + 65*b1 - 87) * q^61 + (11*b5 + 12*b3 - 13*b2 + 28*b1 + 13) * q^62 + (b4 - 13*b3 - 12*b2 + 27) * q^64 + (2*b5 - b4 - 17*b3 + 18*b2 - 56*b1 + 10) * q^65 + (b5 + b4 - 17*b2 - 13*b1 + 4) * q^67 + (10*b4 - 6*b3 + 4*b2 - 42) * q^68 + (9*b5 + 9*b4 - 23*b2 + 18*b1 + 41) * q^70 + (-2*b5 + 4*b4 + 6*b3 + 2*b2 - 44*b1 + 86) * q^71 + (20*b5 + 20*b2 + 41*b1 - 20) * q^73 + (18*b5 + 21*b3 - 24*b2 + 90*b1 + 24) * q^74 + (4*b5 - 18*b4 + 18*b3 + 4*b2 - 33*b1 - 11) * q^76 + (-29*b4 + 25*b3 - 4*b2 + 22) * q^77 + (-b5 + 2*b4 - 17*b3 + b2 + 19*b1 - 39) * q^79 + (-9*b5 + 9*b4 - 18*b3 + 9*b2 + 8*b1 - 17) * q^80 + (2*b5 - 2*b4 + 36*b3 - 18*b2 - 44*b1 + 62) * q^82 + (-4*b3 - 4*b2 + 42) * q^83 + (14*b5 - 14*b4 + 12*b3 - 6*b2 - 40*b1 + 46) * q^85 + (b5 + b4 - 6*b2 + 2*b1 + 8) * q^86 + (2*b5 - b4 + 35*b3 - 34*b2 + 76*b1 - 4) * q^88 + (b5 + b4 - b2 - 72*b1 - 71) * q^89 + (-15*b5 - 15*b4 + 19*b2 + 21*b1 + 2) * q^91 + (-b5 - b3 + b2 - 34*b1 - 1) * q^92 + (-32*b5 + 16*b4 - 50*b3 + 34*b2 - 136*b1 + 34) * q^94 + (-19*b5 + 12*b4 - 35*b3 + 11*b2 + 8*b1 + 13) * q^95 + (-8*b5 + 16*b4 + 44*b3 + 8*b2 - 16*b1 + 24) * q^97 + (-2*b5 + 4*b4 - 6*b3 + 2*b2 - 28*b1 + 54) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} + 5 q^{4} + 2 q^{5} + 26 q^{7}+O(q^{10})$$ 6 * q + 3 * q^2 + 5 * q^4 + 2 * q^5 + 26 * q^7 $$6 q + 3 q^{2} + 5 q^{4} + 2 q^{5} + 26 q^{7} + 30 q^{10} - 15 q^{13} + 81 q^{14} + q^{16} + 10 q^{17} - 46 q^{19} + 124 q^{20} - 84 q^{22} + 24 q^{23} + 15 q^{25} - 58 q^{26} + 19 q^{28} - 66 q^{29} - 51 q^{32} + 90 q^{34} + 6 q^{35} - 83 q^{38} + 162 q^{40} - 24 q^{41} + 11 q^{43} - 176 q^{44} + 26 q^{47} + 96 q^{49} - 321 q^{52} - 180 q^{53} - 176 q^{55} - 188 q^{58} - 162 q^{59} - 141 q^{61} + 109 q^{62} + 166 q^{64} - 63 q^{67} - 212 q^{68} + 258 q^{70} + 372 q^{71} + 103 q^{73} + 315 q^{74} - 217 q^{76} + 16 q^{77} - 123 q^{79} - 6 q^{80} + 80 q^{82} + 252 q^{83} + 116 q^{85} + 39 q^{86} - 642 q^{89} + 87 q^{91} - 104 q^{92} + 214 q^{95} - 12 q^{97} + 264 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 + 5 * q^4 + 2 * q^5 + 26 * q^7 + 30 * q^10 - 15 * q^13 + 81 * q^14 + q^16 + 10 * q^17 - 46 * q^19 + 124 * q^20 - 84 * q^22 + 24 * q^23 + 15 * q^25 - 58 * q^26 + 19 * q^28 - 66 * q^29 - 51 * q^32 + 90 * q^34 + 6 * q^35 - 83 * q^38 + 162 * q^40 - 24 * q^41 + 11 * q^43 - 176 * q^44 + 26 * q^47 + 96 * q^49 - 321 * q^52 - 180 * q^53 - 176 * q^55 - 188 * q^58 - 162 * q^59 - 141 * q^61 + 109 * q^62 + 166 * q^64 - 63 * q^67 - 212 * q^68 + 258 * q^70 + 372 * q^71 + 103 * q^73 + 315 * q^74 - 217 * q^76 + 16 * q^77 - 123 * q^79 - 6 * q^80 + 80 * q^82 + 252 * q^83 + 116 * q^85 + 39 * q^86 - 642 * q^89 + 87 * q^91 - 104 * q^92 + 214 * q^95 - 12 * q^97 + 264 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 56\nu^{5} - 55\nu^{4} + 440\nu^{3} + 344\nu^{2} + 2750\nu + 8 ) / 393$$ (56*v^5 - 55*v^4 + 440*v^3 + 344*v^2 + 2750*v + 8) / 393 $$\beta_{2}$$ $$=$$ $$( 70\nu^{5} - 36\nu^{4} + 550\nu^{3} + 561\nu^{2} + 3634\nu + 927 ) / 393$$ (70*v^5 - 36*v^4 + 550*v^3 + 561*v^2 + 3634*v + 927) / 393 $$\beta_{3}$$ $$=$$ $$( -77\nu^{5} + 92\nu^{4} - 605\nu^{3} - 211\nu^{2} - 3683\nu + 1037 ) / 393$$ (-77*v^5 + 92*v^4 - 605*v^3 - 211*v^2 - 3683*v + 1037) / 393 $$\beta_{4}$$ $$=$$ $$( -79\nu^{5} + 108\nu^{4} - 733\nu^{3} - 111\nu^{2} - 3697\nu + 1149 ) / 393$$ (-79*v^5 + 108*v^4 - 733*v^3 - 111*v^2 - 3697*v + 1149) / 393 $$\beta_{5}$$ $$=$$ $$( 147\nu^{5} - 128\nu^{4} + 1155\nu^{3} + 772\nu^{2} + 8103\nu - 503 ) / 393$$ (147*v^5 - 128*v^4 + 1155*v^3 + 772*v^2 + 8103*v - 503) / 393
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{3} - \beta_{2} + 1 ) / 2$$ (b5 + b3 - b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$2\beta_{3} - \beta_{2} + 4\beta _1 - 3$$ 2*b3 - b2 + 4*b1 - 3 $$\nu^{3}$$ $$=$$ $$( -7\beta_{4} + 9\beta_{3} + 2\beta_{2} - 8 ) / 2$$ (-7*b4 + 9*b3 + 2*b2 - 8) / 2 $$\nu^{4}$$ $$=$$ $$-3\beta_{5} - 11\beta_{3} + 19\beta_{2} - 31\beta _1 - 19$$ -3*b5 - 11*b3 + 19*b2 - 31*b1 - 19 $$\nu^{5}$$ $$=$$ $$( -55\beta_{5} + 55\beta_{4} - 166\beta_{3} + 83\beta_{2} - 96\beta _1 + 13 ) / 2$$ (-55*b5 + 55*b4 - 166*b3 + 83*b2 - 96*b1 + 13) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 0.0702177 − 0.121621i −1.13654 + 1.96854i 1.56632 − 2.71294i 0.0702177 + 0.121621i −1.13654 − 1.96854i 1.56632 + 2.71294i
−1.99014 1.14901i 0 0.640435 + 1.10927i 0.140435 0.243241i 0 −5.24143 6.24860i 0 −0.558972 + 0.322723i
46.2 0.583430 + 0.336844i 0 −1.77307 3.07105i −2.27307 + 3.93708i 0 9.87987 5.08374i 0 −2.65236 + 1.53134i
46.3 2.90671 + 1.67819i 0 3.63264 + 6.29191i 3.13264 5.42589i 0 8.36156 10.9595i 0 18.2113 10.5143i
145.1 −1.99014 + 1.14901i 0 0.640435 1.10927i 0.140435 + 0.243241i 0 −5.24143 6.24860i 0 −0.558972 0.322723i
145.2 0.583430 0.336844i 0 −1.77307 + 3.07105i −2.27307 3.93708i 0 9.87987 5.08374i 0 −2.65236 1.53134i
145.3 2.90671 1.67819i 0 3.63264 6.29191i 3.13264 + 5.42589i 0 8.36156 10.9595i 0 18.2113 + 10.5143i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 145.3 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 171.3.p.e 6
3.b odd 2 1 57.3.g.a 6
12.b even 2 1 912.3.be.d 6
19.d odd 6 1 inner 171.3.p.e 6
57.f even 6 1 57.3.g.a 6
228.n odd 6 1 912.3.be.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.g.a 6 3.b odd 2 1
57.3.g.a 6 57.f even 6 1
171.3.p.e 6 1.a even 1 1 trivial
171.3.p.e 6 19.d odd 6 1 inner
912.3.be.d 6 12.b even 2 1
912.3.be.d 6 228.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(171, [\chi])$$:

 $$T_{2}^{6} - 3T_{2}^{5} - 4T_{2}^{4} + 21T_{2}^{3} + 40T_{2}^{2} - 63T_{2} + 27$$ T2^6 - 3*T2^5 - 4*T2^4 + 21*T2^3 + 40*T2^2 - 63*T2 + 27 $$T_{5}^{6} - 2T_{5}^{5} + 32T_{5}^{4} + 40T_{5}^{3} + 800T_{5}^{2} - 224T_{5} + 64$$ T5^6 - 2*T5^5 + 32*T5^4 + 40*T5^3 + 800*T5^2 - 224*T5 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} - 4 T^{4} + 21 T^{3} + \cdots + 27$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 2 T^{5} + 32 T^{4} + 40 T^{3} + \cdots + 64$$
$7$ $$(T^{3} - 13 T^{2} - 13 T + 433)^{2}$$
$11$ $$(T^{3} - 264 T + 304)^{2}$$
$13$ $$T^{6} + 15 T^{5} - 172 T^{4} + \cdots + 3518667$$
$17$ $$T^{6} - 10 T^{5} + 472 T^{4} + \cdots + 5184$$
$19$ $$T^{6} + 46 T^{5} + 1383 T^{4} + \cdots + 47045881$$
$23$ $$T^{6} - 24 T^{5} + 696 T^{4} + \cdots + 1517824$$
$29$ $$T^{6} + 66 T^{5} + 1520 T^{4} + \cdots + 19293888$$
$31$ $$T^{6} + 2033 T^{4} + \cdots + 124768803$$
$37$ $$T^{6} + 5913 T^{4} + \cdots + 1689765867$$
$41$ $$T^{6} + 24 T^{5} + \cdots + 1907539968$$
$43$ $$T^{6} - 11 T^{5} + 110 T^{4} + \cdots + 2209$$
$47$ $$T^{6} - 26 T^{5} + \cdots + 2266521664$$
$53$ $$T^{6} + 180 T^{5} + 11016 T^{4} + \cdots + 2495232$$
$59$ $$T^{6} + 162 T^{5} + \cdots + 48350430912$$
$61$ $$T^{6} + 141 T^{5} + \cdots + 558907255201$$
$67$ $$T^{6} + 63 T^{5} + \cdots + 2349816507$$
$71$ $$T^{6} - 372 T^{5} + \cdots + 100019515392$$
$73$ $$T^{6} - 103 T^{5} + \cdots + 214090364601$$
$79$ $$T^{6} + 123 T^{5} + \cdots + 2549342403$$
$83$ $$(T^{3} - 126 T^{2} + 4908 T - 57704)^{2}$$
$89$ $$T^{6} + 642 T^{5} + \cdots + 3516172210368$$
$97$ $$T^{6} + 12 T^{5} + \cdots + 3000768049152$$