Properties

 Label 495.2.n.a Level $495$ Weight $2$ Character orbit 495.n Analytic conductor $3.953$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.n (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 Defining polynomial: $$x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1$$ x^8 - 3*x^7 + 5*x^6 - 3*x^5 + 4*x^4 + 3*x^3 + 5*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{2} + ( - \beta_{6} - \beta_{5} + 2 \beta_1) q^{4} - \beta_{4} q^{5} + ( - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q + (-b7 + b6 + b5 - b4 + b3 - b2 - b1 - 1) * q^2 + (-b6 - b5 + 2*b1) * q^4 - b4 * q^5 + (-b6 - b5 + 2*b3 + b1) * q^7 + (-b7 + 2*b6 - b4 + b3 - b2 - 2*b1) * q^8 $$q + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{2} + ( - \beta_{6} - \beta_{5} + 2 \beta_1) q^{4} - \beta_{4} q^{5} + ( - \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{8} + (\beta_{5} - 1) q^{10} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{11} + (2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{13} + ( - 2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{14} + ( - 3 \beta_{7} + \beta_{6} - 2 \beta_{4} - 3) q^{16} + (2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{17} + ( - 2 \beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{6} - 2 \beta_{5} + \beta_{2} + 2 \beta_1) q^{20} + ( - 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 3 \beta_1 - 1) q^{22} + ( - \beta_{7} - 3 \beta_{5} + \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{23} + \beta_{7} q^{25} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 1) q^{26} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - 2) q^{28} + ( - \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 6 \beta_1 - 3) q^{29} + ( - \beta_{7} + 5 \beta_{6} + 4 \beta_{5} - \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 4) q^{31} + (\beta_{7} + 2 \beta_{5} - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{32} + (2 \beta_{7} - 3 \beta_{5} - 2 \beta_{3} + 1) q^{34} + ( - 2 \beta_{7} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_1 - 2) q^{35} + ( - 6 \beta_{6} - 6 \beta_{5} + \beta_{4} + 4 \beta_{3} + 3 \beta_1 + 1) q^{37} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{2} + 1) q^{38} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1 - 1) q^{40} + (4 \beta_{7} - \beta_{6} - 4 \beta_{4} + 4 \beta_{3} + \beta_1) q^{41} + ( - 3 \beta_{7} - 5 \beta_{5} + 3 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 4) q^{43} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - \beta_1) q^{44} + (2 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 1) q^{46} + (\beta_{6} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - \beta_1) q^{47} + ( - \beta_{7} - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{2} - 1) q^{49} + ( - \beta_{6} + \beta_{4}) q^{50} + (\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{52} + (7 \beta_{7} + 3 \beta_{5} + 7 \beta_{4} - \beta_{3} - 3 \beta_1 + 1) q^{53} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{55} + ( - \beta_{7} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{56} + (11 \beta_{7} + 3 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{58} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 5 \beta_1 - 2) q^{59} + (5 \beta_{7} - 7 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} + 5) q^{61} + (\beta_{6} + \beta_{5} - 6 \beta_{4} - \beta_1 - 6) q^{62} + ( - 3 \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{64} + ( - \beta_{7} + \beta_{5} + \beta_{3} + 1) q^{65} + (7 \beta_{7} + 3 \beta_{5} - 7 \beta_{3} + 3) q^{67} + (3 \beta_{6} + 4 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 2) q^{68} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{70} + (5 \beta_{7} - 3 \beta_{6} - 5 \beta_{5} + 9 \beta_{4} + 5 \beta_{2} + 5) q^{71} + ( - 4 \beta_{4} + \beta_{3} + 5 \beta_1 - 4) q^{73} + ( - 8 \beta_{7} + 4 \beta_{6} - 7 \beta_{4} + 7 \beta_{3} + \beta_{2} - 4 \beta_1) q^{74} + ( - 3 \beta_{7} + 7 \beta_{5} + 3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 5) q^{76} + (\beta_{7} + 7 \beta_{4} - 4 \beta_{3} + \beta_{2} - 4 \beta_1 + 5) q^{77} + (4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{79} + (2 \beta_{7} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2}) q^{80} + ( - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} - 3 \beta_{2} - 3) q^{82} + (5 \beta_{7} - \beta_{6} - 3 \beta_{5} + 5 \beta_{4} + 3 \beta_{2} + 5) q^{83} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{2} + 2 \beta_1) q^{85} + (\beta_{7} + 4 \beta_{6} + 6 \beta_{5} + \beta_{4} + \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 1) q^{86} + ( - 3 \beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + \cdots + 4) q^{88}+ \cdots + ( - 2 \beta_{7} + 7 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{98}+O(q^{100})$$ q + (-b7 + b6 + b5 - b4 + b3 - b2 - b1 - 1) * q^2 + (-b6 - b5 + 2*b1) * q^4 - b4 * q^5 + (-b6 - b5 + 2*b3 + b1) * q^7 + (-b7 + 2*b6 - b4 + b3 - b2 - 2*b1) * q^8 + (b5 - 1) * q^10 + (-b7 - b6 - 3*b5 + b4 + b3 + b1) * q^11 + (2*b7 + b6 + b5 + 2*b4 - b3 - b2 - b1 + 1) * q^13 + (-2*b7 + b6 - b4 + b3 - b2 - b1) * q^14 + (-3*b7 + b6 - 2*b4 - 3) * q^16 + (2*b6 + 2*b5 + b4 - 2*b2) * q^17 + (-2*b7 - b6 - b2 + b1) * q^19 + (-b6 - 2*b5 + b2 + 2*b1) * q^20 + (-3*b7 + 2*b6 + b5 - 4*b4 + 3*b3 - 3*b1 - 1) * q^22 + (-b7 - 3*b5 + b3 + 4*b2 + 4*b1 - 2) * q^23 + b7 * q^25 + (-2*b6 - 2*b5 + b4 + 1) * q^26 + (-2*b7 + b6 - b5 + b4 + b2 - 2) * q^28 + (-b6 - b5 - 3*b4 - 3*b3 + 6*b1 - 3) * q^29 + (-b7 + 5*b6 + 4*b5 - b4 - 4*b3 - 5*b2 - 4*b1 + 4) * q^31 + (b7 + 2*b5 - b3 - 2*b2 - 2*b1) * q^32 + (2*b7 - 3*b5 - 2*b3 + 1) * q^34 + (-2*b7 - b5 - 2*b4 + 2*b3 + b1 - 2) * q^35 + (-6*b6 - 6*b5 + b4 + 4*b3 + 3*b1 + 1) * q^37 + (b7 + 3*b6 - 2*b5 - 3*b4 + 2*b2 + 1) * q^38 + (2*b6 + 2*b5 - b4 - b1 - 1) * q^40 + (4*b7 - b6 - 4*b4 + 4*b3 + b1) * q^41 + (-3*b7 - 5*b5 + 3*b3 + 5*b2 + 5*b1 - 4) * q^43 + (-2*b7 - b6 + 2*b5 + 3*b4 - 4*b3 - 2*b2 - b1) * q^44 + (2*b7 + 3*b6 + 5*b5 + 2*b4 + b3 - 3*b2 - 5*b1 - 1) * q^46 + (b6 - 2*b4 + 2*b3 + 4*b2 - b1) * q^47 + (-b7 - 3*b5 - 2*b4 + 3*b2 - 1) * q^49 + (-b6 + b4) * q^50 + (b7 + b6 - b4 + b3 - b2 - b1) * q^52 + (7*b7 + 3*b5 + 7*b4 - b3 - 3*b1 + 1) * q^53 + (-2*b7 + 2*b6 - b5 - b4 + b1 - 1) * q^55 + (-b7 + b3 - 2*b2 - 2*b1 + 3) * q^56 + (11*b7 + 3*b6 + 2*b4 - 2*b3 + 7*b2 - 3*b1) * q^58 + (-2*b6 - 2*b5 - 2*b4 - 4*b3 + 5*b1 - 2) * q^59 + (5*b7 - 7*b6 - b5 + 2*b4 + b2 + 5) * q^61 + (b6 + b5 - 6*b4 - b1 - 6) * q^62 + (-3*b7 - b6 - 2*b5 - 3*b4 - 4*b3 + b2 + 2*b1 + 4) * q^64 + (-b7 + b5 + b3 + 1) * q^65 + (7*b7 + 3*b5 - 7*b3 + 3) * q^67 + (3*b6 + 4*b5 + 2*b3 - 3*b2 - 4*b1 - 2) * q^68 + (b6 + b5 - b4 - b3 - 1) * q^70 + (5*b7 - 3*b6 - 5*b5 + 9*b4 + 5*b2 + 5) * q^71 + (-4*b4 + b3 + 5*b1 - 4) * q^73 + (-8*b7 + 4*b6 - 7*b4 + 7*b3 + b2 - 4*b1) * q^74 + (-3*b7 + 7*b5 + 3*b3 - 4*b2 - 4*b1 - 5) * q^76 + (b7 + 7*b4 - 4*b3 + b2 - 4*b1 + 5) * q^77 + (4*b7 - 2*b6 + 2*b5 + 4*b4 + b3 + 2*b2 - 2*b1 - 1) * q^79 + (2*b7 + 3*b4 - 3*b3 + b2) * q^80 + (-3*b7 - 3*b6 + 3*b5 + 4*b4 - 3*b2 - 3) * q^82 + (5*b7 - b6 - 3*b5 + 5*b4 + 3*b2 + 5) * q^83 + (-b7 - 2*b6 + 2*b2 + 2*b1) * q^85 + (b7 + 4*b6 + 6*b5 + b4 + b3 - 4*b2 - 6*b1 - 1) * q^86 + (-3*b7 - b6 - 4*b5 - 4*b4 + 4*b3 + 3*b2 + 2*b1 + 4) * q^88 + (3*b7 + 8*b5 - 3*b3 - 4*b2 - 4*b1 + 5) * q^89 + (4*b7 - b6 + b4 - b3 + b1) * q^91 + (7*b4 - 10*b3 + 2*b1 + 7) * q^92 + (-3*b7 - b6 + 7*b5 + 4*b4 - 7*b2 - 3) * q^94 + (-b6 - b5 - 2*b3 + 2*b1) * q^95 + (5*b7 - 4*b6 - 5*b5 + 5*b4 - 7*b3 + 4*b2 + 5*b1 + 7) * q^97 + (-2*b7 + 7*b5 + 2*b3 - 2*b2 - 2*b1 - 4) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 2 q^{4} + 2 q^{5} + 3 q^{7} + q^{8}+O(q^{10})$$ 8 * q - 4 * q^2 + 2 * q^4 + 2 * q^5 + 3 * q^7 + q^8 $$8 q - 4 q^{2} + 2 q^{4} + 2 q^{5} + 3 q^{7} + q^{8} - 6 q^{10} - 3 q^{11} - 4 q^{13} + 4 q^{14} - 12 q^{16} + 2 q^{19} + 3 q^{20} + 9 q^{22} + 6 q^{23} - 2 q^{25} - 2 q^{26} - 11 q^{28} - 10 q^{29} + 19 q^{31} - 12 q^{32} - 6 q^{34} - 3 q^{35} - q^{37} + 20 q^{38} - q^{40} + 9 q^{41} - 17 q^{44} - 22 q^{46} + 19 q^{47} + q^{49} - 4 q^{50} - 2 q^{52} - 25 q^{53} + 3 q^{55} + 16 q^{56} - 12 q^{58} - 13 q^{59} + 13 q^{61} - 35 q^{62} + 39 q^{64} + 14 q^{65} + 2 q^{67} - 19 q^{68} - 4 q^{70} + 11 q^{71} - 7 q^{73} + 43 q^{74} - 38 q^{76} + 7 q^{77} - 22 q^{79} - 13 q^{80} - 35 q^{82} + 21 q^{83} + 10 q^{85} - 20 q^{86} + 59 q^{88} + 20 q^{89} - 11 q^{91} + 28 q^{92} - 35 q^{94} - 2 q^{95} + 31 q^{97} - 22 q^{98}+O(q^{100})$$ 8 * q - 4 * q^2 + 2 * q^4 + 2 * q^5 + 3 * q^7 + q^8 - 6 * q^10 - 3 * q^11 - 4 * q^13 + 4 * q^14 - 12 * q^16 + 2 * q^19 + 3 * q^20 + 9 * q^22 + 6 * q^23 - 2 * q^25 - 2 * q^26 - 11 * q^28 - 10 * q^29 + 19 * q^31 - 12 * q^32 - 6 * q^34 - 3 * q^35 - q^37 + 20 * q^38 - q^40 + 9 * q^41 - 17 * q^44 - 22 * q^46 + 19 * q^47 + q^49 - 4 * q^50 - 2 * q^52 - 25 * q^53 + 3 * q^55 + 16 * q^56 - 12 * q^58 - 13 * q^59 + 13 * q^61 - 35 * q^62 + 39 * q^64 + 14 * q^65 + 2 * q^67 - 19 * q^68 - 4 * q^70 + 11 * q^71 - 7 * q^73 + 43 * q^74 - 38 * q^76 + 7 * q^77 - 22 * q^79 - 13 * q^80 - 35 * q^82 + 21 * q^83 + 10 * q^85 - 20 * q^86 + 59 * q^88 + 20 * q^89 - 11 * q^91 + 28 * q^92 - 35 * q^94 - 2 * q^95 + 31 * q^97 - 22 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8$$ (-v^7 + 2*v^6 - 3*v^5 - 4*v^3 - 7*v^2 - 12*v - 7) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8$$ (v^7 - 7*v^5 + 20*v^4 - 16*v^3 + 19*v^2 + 6*v + 9) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8$$ (-v^7 + 4*v^6 - 9*v^5 + 12*v^4 - 16*v^3 + 13*v^2 - 10*v - 1) / 8 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8$$ (-3*v^7 + 10*v^6 - 17*v^5 + 8*v^4 - 4*v^3 - 13*v^2 - 8*v - 5) / 8 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8$$ (3*v^7 - 12*v^6 + 23*v^5 - 20*v^4 + 16*v^3 + v^2 + 6*v - 1) / 8 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8$$ (-5*v^7 + 18*v^6 - 35*v^5 + 32*v^4 - 28*v^3 - 11*v^2 - 12*v - 7) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ b6 + b5 + b4 - b2 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1$$ b7 + 3*b6 + 2*b5 + b4 - 3*b2 - 2*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1$$ 3*b7 + 4*b6 + b4 - b3 - 5*b2 - 4*b1 $$\nu^{5}$$ $$=$$ $$4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1$$ 4*b7 - 6*b5 - 4*b3 - 6*b2 - 6*b1 - 1 $$\nu^{6}$$ $$=$$ $$-16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6$$ -16*b6 - 16*b5 - 6*b4 - 6*b3 - 7*b1 - 6 $$\nu^{7}$$ $$=$$ $$-16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16$$ -16*b7 - 51*b6 - 29*b5 - 23*b4 + 29*b2 - 16

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1
 0.418926 − 1.28932i −0.227943 + 0.701538i 0.418926 + 1.28932i −0.227943 − 0.701538i 1.69513 − 1.23158i −0.386111 + 0.280526i 1.69513 + 1.23158i −0.386111 − 0.280526i
−1.90578 + 1.38463i 0 1.09676 3.37549i 0.809017 + 0.587785i 0 0.0598032 0.184055i 1.12773 + 3.47080i 0 −2.35567
91.2 −0.212253 + 0.154211i 0 −0.596764 + 1.83665i 0.809017 + 0.587785i 0 −0.986854 + 3.03722i −0.318714 0.980901i 0 −0.262360
136.1 −1.90578 1.38463i 0 1.09676 + 3.37549i 0.809017 0.587785i 0 0.0598032 + 0.184055i 1.12773 3.47080i 0 −2.35567
136.2 −0.212253 0.154211i 0 −0.596764 1.83665i 0.809017 0.587785i 0 −0.986854 3.03722i −0.318714 + 0.980901i 0 −0.262360
181.1 −0.338464 1.04169i 0 0.647481 0.470423i −0.309017 + 0.951057i 0 0.570387 0.414410i −2.48141 1.80285i 0 1.09529
181.2 0.456498 + 1.40496i 0 −0.147481 + 0.107152i −0.309017 + 0.951057i 0 1.85666 1.34895i 2.17239 + 1.57833i 0 −1.47726
361.1 −0.338464 + 1.04169i 0 0.647481 + 0.470423i −0.309017 0.951057i 0 0.570387 + 0.414410i −2.48141 + 1.80285i 0 1.09529
361.2 0.456498 1.40496i 0 −0.147481 0.107152i −0.309017 0.951057i 0 1.85666 + 1.34895i 2.17239 1.57833i 0 −1.47726
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.n.a 8
3.b odd 2 1 165.2.m.d 8
11.c even 5 1 inner 495.2.n.a 8
11.c even 5 1 5445.2.a.bt 4
11.d odd 10 1 5445.2.a.bf 4
15.d odd 2 1 825.2.n.g 8
15.e even 4 2 825.2.bx.f 16
33.f even 10 1 1815.2.a.w 4
33.h odd 10 1 165.2.m.d 8
33.h odd 10 1 1815.2.a.p 4
165.o odd 10 1 825.2.n.g 8
165.o odd 10 1 9075.2.a.di 4
165.r even 10 1 9075.2.a.cm 4
165.v even 20 2 825.2.bx.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.d 8 3.b odd 2 1
165.2.m.d 8 33.h odd 10 1
495.2.n.a 8 1.a even 1 1 trivial
495.2.n.a 8 11.c even 5 1 inner
825.2.n.g 8 15.d odd 2 1
825.2.n.g 8 165.o odd 10 1
825.2.bx.f 16 15.e even 4 2
825.2.bx.f 16 165.v even 20 2
1815.2.a.p 4 33.h odd 10 1
1815.2.a.w 4 33.f even 10 1
5445.2.a.bf 4 11.d odd 10 1
5445.2.a.bt 4 11.c even 5 1
9075.2.a.cm 4 165.r even 10 1
9075.2.a.di 4 165.o odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 4T_{2}^{7} + 9T_{2}^{6} + 13T_{2}^{5} + 24T_{2}^{4} + 21T_{2}^{3} + 21T_{2}^{2} + 7T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{7} + 9 T^{6} + 13 T^{5} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$7$ $$T^{8} - 3 T^{7} + 11 T^{6} - 39 T^{5} + \cdots + 1$$
$11$ $$T^{8} + 3 T^{7} - 22 T^{6} + \cdots + 14641$$
$13$ $$T^{8} + 4 T^{7} + 29 T^{6} + 33 T^{5} + \cdots + 121$$
$17$ $$T^{8} + 47 T^{6} + 250 T^{5} + 559 T^{4} + \cdots + 1$$
$19$ $$T^{8} - 2 T^{7} + 25 T^{6} - 137 T^{5} + \cdots + 1$$
$23$ $$(T^{4} - 3 T^{3} - 55 T^{2} + 129 T + 449)^{2}$$
$29$ $$T^{8} + 10 T^{7} + 113 T^{6} + \cdots + 249001$$
$31$ $$T^{8} - 19 T^{7} + 235 T^{6} + \cdots + 1560001$$
$37$ $$T^{8} + T^{7} + 37 T^{6} + \cdots + 12538681$$
$41$ $$T^{8} - 9 T^{7} + 205 T^{6} + \cdots + 1681$$
$43$ $$(T^{4} - 110 T^{2} + 375 T + 275)^{2}$$
$47$ $$T^{8} - 19 T^{7} + 255 T^{6} + \cdots + 961$$
$53$ $$T^{8} + 25 T^{7} + 432 T^{6} + \cdots + 410881$$
$59$ $$T^{8} + 13 T^{7} + 165 T^{6} + \cdots + 346921$$
$61$ $$T^{8} - 13 T^{7} + 210 T^{6} + \cdots + 1324801$$
$67$ $$(T^{4} - T^{3} - 100 T^{2} + 112 T + 619)^{2}$$
$71$ $$T^{8} - 11 T^{7} + 225 T^{6} + \cdots + 78961$$
$73$ $$T^{8} + 7 T^{7} + 48 T^{6} + \cdots + 866761$$
$79$ $$T^{8} + 22 T^{7} + 283 T^{6} + \cdots + 28561$$
$83$ $$T^{8} - 21 T^{7} + 275 T^{6} + \cdots + 1681$$
$89$ $$(T^{4} - 10 T^{3} - 89 T^{2} + 310 T + 209)^{2}$$
$97$ $$T^{8} - 31 T^{7} + 430 T^{6} + \cdots + 1343281$$