Properties

 Label 495.2.n.f 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: 8.0.159390625.1 Defining polynomial: $$x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25$$ x^8 - x^7 + 6*x^6 - 11*x^5 + 21*x^4 - 5*x^3 + 10*x^2 + 25*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{2} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{4} + \beta_{6} q^{5} + (2 \beta_{6} - \beta_{5} + 3 \beta_{3} + 3 \beta_{2} - 2) q^{7} + (\beta_{7} + \beta_{4} + 2 \beta_{3}) q^{8}+O(q^{10})$$ q + (-b4 + b2 + b1) * q^2 + (b7 + 2*b6 + b5 + b3 + b2 - 2) * q^4 + b6 * q^5 + (2*b6 - b5 + 3*b3 + 3*b2 - 2) * q^7 + (b7 + b4 + 2*b3) * q^8 $$q + ( - \beta_{4} + \beta_{2} + \beta_1) q^{2} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{4} + \beta_{6} q^{5} + (2 \beta_{6} - \beta_{5} + 3 \beta_{3} + 3 \beta_{2} - 2) q^{7} + (\beta_{7} + \beta_{4} + 2 \beta_{3}) q^{8} + (\beta_{7} - \beta_1 + 1) q^{10} + (2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{11} + (2 \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{13} + (2 \beta_{7} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{14} + ( - \beta_{5} + 3 \beta_{3} + \beta_1 - 3) q^{16} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_1) q^{17} + ( - 4 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{20} + (\beta_{7} - 4 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{22} + ( - \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 3 \beta_{2} + \beta_1 + 5) q^{23} - \beta_{3} q^{25} + (\beta_{7} - 6 \beta_{6} - 3 \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 6) q^{26} + ( - \beta_{6} - \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{28} + (\beta_{7} + 3 \beta_{6} - 3) q^{29} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1) q^{31} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{32} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{2} - 1) q^{34} + (\beta_{6} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1) q^{35} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{3} - \beta_{2} + 3) q^{37} + (4 \beta_{6} + 2 \beta_{5} + \beta_{3} - 2 \beta_1 - 1) q^{38} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{40} + ( - 3 \beta_{7} - 3 \beta_{4} - 3 \beta_{3} + 4 \beta_1) q^{41} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + \beta_1 + 5) q^{43} + (2 \beta_{6} + 3 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 1) q^{44} + ( - 7 \beta_{7} - 8 \beta_{6} - 7 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 3 \beta_{2} + 9 \beta_1) q^{46} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{47} + ( - \beta_{6} + 5 \beta_{5} - \beta_{4} + 4 \beta_{3} - 5 \beta_1 - 4) q^{49} + (\beta_{6} + \beta_{5} - \beta_1) q^{50} + ( - 3 \beta_{7} - 3 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{52} + ( - \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{53} + ( - 2 \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + \beta_1) q^{55} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - \beta_{2} + \beta_1 - 2) q^{56} + (4 \beta_{7} + 4 \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta_1 + 1) q^{58} + ( - 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 5 \beta_{3} + 5 \beta_{2} - 2) q^{59} + ( - 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_1 + 1) q^{61} + (3 \beta_{7} - 8 \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + 8) q^{62} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + 7 \beta_{2} + 3 \beta_1) q^{64} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{2} + \beta_1 - 3) q^{65} + ( - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{2} + 3 \beta_1) q^{67} + ( - \beta_{7} - 5 \beta_{6} - \beta_{5} + \beta_{4} - 5 \beta_{3} - 6 \beta_{2}) q^{68} + ( - \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{70} + (2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 8 \beta_{3} - 3 \beta_1 - 8) q^{71} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - 4 \beta_{3} - 4 \beta_{2} - 2) q^{73} + ( - 2 \beta_{7} - 2 \beta_{4} - 6 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{74} + (4 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 5 \beta_{2} - 4 \beta_1 + 4) q^{76} + ( - 3 \beta_{7} - \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{77}+ \cdots + ( - 7 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 7 \beta_{2} + 13) q^{98}+O(q^{100})$$ q + (-b4 + b2 + b1) * q^2 + (b7 + 2*b6 + b5 + b3 + b2 - 2) * q^4 + b6 * q^5 + (2*b6 - b5 + 3*b3 + 3*b2 - 2) * q^7 + (b7 + b4 + 2*b3) * q^8 + (b7 - b1 + 1) * q^10 + (2*b6 + b4 + b3 + b2 - 2*b1) * q^11 + (2*b6 + b4 + 2*b3 - b2 - b1) * q^13 + (2*b7 + 2*b4 + b3 + 2*b2 - 2*b1 - 2) * q^14 + (-b5 + 3*b3 + b1 - 3) * q^16 + (b6 - b5 + b4 + b1) * q^17 + (-4*b3 - b2 + b1 + 1) * q^19 + (b7 - b6 + b5 - b4 - b3 + b2) * q^20 + (b7 - 4*b6 + b5 - b4 - 3*b3 - b2 - 2*b1 + 6) * q^22 + (-b7 - 3*b6 - 3*b5 + 3*b4 - 3*b2 + b1 + 5) * q^23 - b3 * q^25 + (b7 - 6*b6 - 3*b5 - 3*b3 - 3*b2 + 6) * q^26 + (-b6 - b5 + 4*b4 - 2*b3 + b1 + 2) * q^28 + (b7 + 3*b6 - 3) * q^29 + (2*b7 + 2*b6 + 2*b5 + b4 + 2*b3 - b2 - 3*b1) * q^31 + (2*b7 - 2*b6 - 2*b5 + 2*b4 - 2*b2 - 2*b1 + 1) * q^32 + (-b6 - 2*b5 + 2*b4 - b2 - 1) * q^34 + (b6 + b4 + b3 + 3*b2 - b1) * q^35 + (b7 - 3*b6 - 3*b5 - b3 - b2 + 3) * q^37 + (4*b6 + 2*b5 + b3 - 2*b1 - 1) * q^38 + (b7 + 2*b6 + b5 + 2*b3 + 2*b2 - 2) * q^40 + (-3*b7 - 3*b4 - 3*b3 + 4*b1) * q^41 + (-b7 - 2*b6 + 2*b5 - 2*b4 - 2*b2 + b1 + 5) * q^43 + (2*b6 + 3*b5 - 4*b4 - 4*b3 + 4*b2 + 3*b1 + 1) * q^44 + (-7*b7 - 8*b6 - 7*b5 - 2*b4 - 8*b3 - 3*b2 + 9*b1) * q^46 + (-b7 - b4 + b3 + b2 - 2*b1 - 1) * q^47 + (-b6 + 5*b5 - b4 + 4*b3 - 5*b1 - 4) * q^49 + (b6 + b5 - b1) * q^50 + (-3*b7 - 3*b4 - 6*b3 + 4*b2 + 4*b1 - 4) * q^52 + (-b7 - b5 - b4 - 2*b2 + 2*b1) * q^53 + (-2*b7 + b6 - b3 + b2 + b1) * q^55 + (-b7 - b6 - 3*b5 + 3*b4 - b2 + b1 - 2) * q^56 + (4*b7 + 4*b4 + b3 - b2 - 6*b1 + 1) * q^58 + (-2*b7 + 2*b6 + 3*b5 + 5*b3 + 5*b2 - 2) * q^59 + (-3*b6 - 2*b5 - 2*b4 - b3 + 2*b1 + 1) * q^61 + (3*b7 - 8*b6 - b5 - b3 - b2 + 8) * q^62 + (-2*b7 - 2*b6 - 2*b5 - b4 - 2*b3 + 7*b2 + 3*b1) * q^64 + (-b7 + 2*b6 + 2*b2 + b1 - 3) * q^65 + (-3*b7 - 2*b6 - 2*b2 + 3*b1) * q^67 + (-b7 - 5*b6 - b5 + b4 - 5*b3 - 6*b2) * q^68 + (-b6 + 2*b5 + b3 + b2 + 1) * q^70 + (2*b6 + 3*b5 + 2*b4 + 8*b3 - 3*b1 - 8) * q^71 + (-2*b7 + 2*b6 + b5 - 4*b3 - 4*b2 - 2) * q^73 + (-2*b7 - 2*b4 - 6*b3 + 7*b2 + 2*b1 - 7) * q^74 + (4*b7 - 5*b6 + 3*b5 - 3*b4 - 5*b2 - 4*b1 + 4) * q^76 + (-3*b7 - b6 + 3*b5 - 5*b4 + 7*b3 + 3*b2 + b1 - 1) * q^77 + (-3*b7 + 5*b6 - 3*b5 + 2*b4 + 5*b3 + 2*b2 + b1) * q^79 + (b7 + b4 + 3*b3 + 3*b2 - b1 - 3) * q^80 + (13*b6 + 2*b5 - 2*b4 + 5*b3 - 2*b1 - 5) * q^82 + (2*b6 + b5 - b4 - 3*b3 - b1 + 3) * q^83 + (b7 + b4 - b3) * q^85 + (-b7 + 3*b6 - b5 - 7*b4 + 3*b3 + 3*b2 + 8*b1) * q^86 + (4*b7 + 7*b6 + 5*b5 + 6*b3 + 6*b2 - 3*b1 - 3) * q^88 + (4*b7 - 6*b6 + 2*b5 - 2*b4 - 6*b2 - 4*b1 + 3) * q^89 + (b3 + 2*b2 - 2*b1 - 2) * q^91 + (-7*b7 + 8*b6 - 10*b3 - 10*b2 - 8) * q^92 + (3*b5 - 3*b4 - 4*b3 - 3*b1 + 4) * q^94 + (b7 - 3*b6 - 4*b3 - 4*b2 + 3) * q^95 + (-2*b7 + 2*b6 - 2*b5 - 2*b4 + 2*b3 + b2 + 4*b1) * q^97 + (-7*b6 + 2*b5 - 2*b4 - 7*b2 + 13) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{2} - 6 q^{4} + 2 q^{5} - 3 q^{7} + 2 q^{8}+O(q^{10})$$ 8 * q + 4 * q^2 - 6 * q^4 + 2 * q^5 - 3 * q^7 + 2 * q^8 $$8 q + 4 q^{2} - 6 q^{4} + 2 q^{5} - 3 q^{7} + 2 q^{8} + 6 q^{10} + 5 q^{11} + 4 q^{13} - 16 q^{14} - 20 q^{16} - q^{17} - q^{19} + q^{20} + 33 q^{22} + 18 q^{23} - 2 q^{25} + 14 q^{26} + 4 q^{28} - 19 q^{29} + 6 q^{31} - 12 q^{32} - 20 q^{34} + 8 q^{35} + 4 q^{37} + 6 q^{38} - 2 q^{40} + 4 q^{41} + 42 q^{43} + 28 q^{44} - 41 q^{46} - 4 q^{47} - 15 q^{49} + 4 q^{50} - 26 q^{52} - 3 q^{53} + 5 q^{55} - 30 q^{56} - 6 q^{58} + 19 q^{59} - 2 q^{61} + 38 q^{62} + 6 q^{64} - 14 q^{65} - 2 q^{67} - 35 q^{68} + 16 q^{70} - 40 q^{71} - 23 q^{73} - 48 q^{74} + 16 q^{76} + 28 q^{77} + 17 q^{79} - 15 q^{80} + 2 q^{82} + 25 q^{83} - 4 q^{85} + 31 q^{86} + 22 q^{88} - 12 q^{91} - 81 q^{92} + 33 q^{94} + q^{95} + 12 q^{97} + 84 q^{98}+O(q^{100})$$ 8 * q + 4 * q^2 - 6 * q^4 + 2 * q^5 - 3 * q^7 + 2 * q^8 + 6 * q^10 + 5 * q^11 + 4 * q^13 - 16 * q^14 - 20 * q^16 - q^17 - q^19 + q^20 + 33 * q^22 + 18 * q^23 - 2 * q^25 + 14 * q^26 + 4 * q^28 - 19 * q^29 + 6 * q^31 - 12 * q^32 - 20 * q^34 + 8 * q^35 + 4 * q^37 + 6 * q^38 - 2 * q^40 + 4 * q^41 + 42 * q^43 + 28 * q^44 - 41 * q^46 - 4 * q^47 - 15 * q^49 + 4 * q^50 - 26 * q^52 - 3 * q^53 + 5 * q^55 - 30 * q^56 - 6 * q^58 + 19 * q^59 - 2 * q^61 + 38 * q^62 + 6 * q^64 - 14 * q^65 - 2 * q^67 - 35 * q^68 + 16 * q^70 - 40 * q^71 - 23 * q^73 - 48 * q^74 + 16 * q^76 + 28 * q^77 + 17 * q^79 - 15 * q^80 + 2 * q^82 + 25 * q^83 - 4 * q^85 + 31 * q^86 + 22 * q^88 - 12 * q^91 - 81 * q^92 + 33 * q^94 + q^95 + 12 * q^97 + 84 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655$$ (555*v^7 - 2159*v^6 + 7489*v^5 - 18164*v^4 + 40069*v^3 - 84434*v^2 + 43855*v + 375) / 94655 $$\beta_{3}$$ $$=$$ $$( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655$$ (-970*v^7 - 1002*v^6 - 6608*v^5 + 9063*v^4 - 14943*v^3 + 27673*v^2 - 68120*v + 35160) / 94655 $$\beta_{4}$$ $$=$$ $$( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655$$ (-1604*v^7 + 4159*v^6 - 12059*v^5 + 28414*v^4 - 81659*v^3 + 38305*v^2 - 13500*v - 13875) / 94655 $$\beta_{5}$$ $$=$$ $$( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655$$ (-2052*v^7 + 2252*v^6 - 19912*v^5 + 21007*v^4 - 82042*v^3 + 35785*v^2 - 19395*v - 90925) / 94655 $$\beta_{6}$$ $$=$$ $$( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655$$ (-2667*v^7 + 6691*v^6 - 17466*v^5 + 50856*v^4 - 82441*v^3 + 72554*v^2 - 4035*v - 12035) / 94655 $$\beta_{7}$$ $$=$$ $$( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655$$ (4024*v^7 - 1464*v^6 + 21519*v^5 - 26434*v^4 + 59219*v^3 + 22635*v^2 + 54640*v + 66675) / 94655
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1$$ -b7 - b6 - b5 - b3 - 3*b2 + b1 $$\nu^{3}$$ $$=$$ $$2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1$$ 2*b6 + b5 - 4*b4 - b3 - b1 + 1 $$\nu^{4}$$ $$=$$ $$7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7$$ 7*b7 + 7*b6 + 2*b5 + 13*b3 + 13*b2 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12$$ -8*b7 - 11*b6 - 20*b5 + 20*b4 - 11*b2 + 8*b1 - 12 $$\nu^{6}$$ $$=$$ $$-19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36$$ -19*b7 - 19*b4 - 68*b3 - 36*b2 - 24*b1 + 36 $$\nu^{7}$$ $$=$$ $$111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1$$ 111*b7 + 81*b6 + 111*b5 - 55*b4 + 81*b3 + 148*b2 - 56*b1

Character values

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

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$-1 + \beta_{2} + \beta_{3} + \beta_{6}$$ $$1$$ $$1$$

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
 0.453245 + 1.39494i −0.762262 − 2.34600i 0.453245 − 1.39494i −0.762262 + 2.34600i −0.628998 − 0.456994i 1.43801 + 1.04478i −0.628998 + 0.456994i 1.43801 − 1.04478i
0.0756511 0.0549637i 0 −0.615332 + 1.89380i 0.809017 + 0.587785i 0 1.39815 4.30308i 0.115332 + 0.354955i 0 0.0935099
91.2 2.04238 1.48388i 0 1.35140 4.15918i 0.809017 + 0.587785i 0 0.646930 1.99105i −1.85140 5.69802i 0 2.52452
136.1 0.0756511 + 0.0549637i 0 −0.615332 1.89380i 0.809017 0.587785i 0 1.39815 + 4.30308i 0.115332 0.354955i 0 0.0935099
136.2 2.04238 + 1.48388i 0 1.35140 + 4.15918i 0.809017 0.587785i 0 0.646930 + 1.99105i −1.85140 + 5.69802i 0 2.52452
181.1 −0.697759 2.14748i 0 −2.50678 + 1.82128i −0.309017 + 0.951057i 0 −0.100294 + 0.0728678i 2.00678 + 1.45801i 0 2.25800
181.2 0.579725 + 1.78421i 0 −1.22929 + 0.893133i −0.309017 + 0.951057i 0 −3.44479 + 2.50279i 0.729292 + 0.529862i 0 −1.87603
361.1 −0.697759 + 2.14748i 0 −2.50678 1.82128i −0.309017 0.951057i 0 −0.100294 0.0728678i 2.00678 1.45801i 0 2.25800
361.2 0.579725 1.78421i 0 −1.22929 0.893133i −0.309017 0.951057i 0 −3.44479 2.50279i 0.729292 0.529862i 0 −1.87603
 $$n$$: 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.f 8
3.b odd 2 1 55.2.g.a 8
11.c even 5 1 inner 495.2.n.f 8
11.c even 5 1 5445.2.a.bg 4
11.d odd 10 1 5445.2.a.bu 4
12.b even 2 1 880.2.bo.e 8
15.d odd 2 1 275.2.h.b 8
15.e even 4 2 275.2.z.b 16
33.d even 2 1 605.2.g.n 8
33.f even 10 1 605.2.a.i 4
33.f even 10 2 605.2.g.g 8
33.f even 10 1 605.2.g.n 8
33.h odd 10 1 55.2.g.a 8
33.h odd 10 1 605.2.a.l 4
33.h odd 10 2 605.2.g.j 8
132.n odd 10 1 9680.2.a.cv 4
132.o even 10 1 880.2.bo.e 8
132.o even 10 1 9680.2.a.cs 4
165.o odd 10 1 275.2.h.b 8
165.o odd 10 1 3025.2.a.v 4
165.r even 10 1 3025.2.a.be 4
165.v even 20 2 275.2.z.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a 8 3.b odd 2 1
55.2.g.a 8 33.h odd 10 1
275.2.h.b 8 15.d odd 2 1
275.2.h.b 8 165.o odd 10 1
275.2.z.b 16 15.e even 4 2
275.2.z.b 16 165.v even 20 2
495.2.n.f 8 1.a even 1 1 trivial
495.2.n.f 8 11.c even 5 1 inner
605.2.a.i 4 33.f even 10 1
605.2.a.l 4 33.h odd 10 1
605.2.g.g 8 33.f even 10 2
605.2.g.j 8 33.h odd 10 2
605.2.g.n 8 33.d even 2 1
605.2.g.n 8 33.f even 10 1
880.2.bo.e 8 12.b even 2 1
880.2.bo.e 8 132.o even 10 1
3025.2.a.v 4 165.o odd 10 1
3025.2.a.be 4 165.r even 10 1
5445.2.a.bg 4 11.c even 5 1
5445.2.a.bu 4 11.d odd 10 1
9680.2.a.cs 4 132.o even 10 1
9680.2.a.cv 4 132.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 4T_{2}^{7} + 13T_{2}^{6} - 30T_{2}^{5} + 71T_{2}^{4} - 90T_{2}^{3} + 127T_{2}^{2} - 18T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 4 T^{7} + 13 T^{6} - 30 T^{5} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$7$ $$T^{8} + 3 T^{7} + 19 T^{6} + 87 T^{5} + \cdots + 25$$
$11$ $$T^{8} - 5 T^{7} + 26 T^{6} + \cdots + 14641$$
$13$ $$T^{8} - 4 T^{7} + 33 T^{6} - 20 T^{5} + \cdots + 121$$
$17$ $$T^{8} + T^{7} - 2 T^{6} - 5 T^{5} + \cdots + 121$$
$19$ $$T^{8} + T^{7} + 31 T^{6} + 151 T^{5} + \cdots + 625$$
$23$ $$(T^{4} - 9 T^{3} - 54 T^{2} + 706 T - 1669)^{2}$$
$29$ $$T^{8} + 19 T^{7} + 171 T^{6} + \cdots + 3025$$
$31$ $$T^{8} - 6 T^{7} + 57 T^{6} + \cdots + 10201$$
$37$ $$T^{8} - 4 T^{7} - 23 T^{6} + \cdots + 22801$$
$41$ $$T^{8} - 4 T^{7} - 3 T^{6} + \cdots + 249001$$
$43$ $$(T^{4} - 21 T^{3} + 121 T^{2} - 191 T - 59)^{2}$$
$47$ $$T^{8} + 4 T^{7} + 87 T^{6} + \cdots + 5041$$
$53$ $$T^{8} + 3 T^{7} + 37 T^{6} + 255 T^{5} + \cdots + 1$$
$59$ $$T^{8} - 19 T^{7} + 201 T^{6} + \cdots + 9150625$$
$61$ $$T^{8} + 2 T^{7} + 74 T^{6} + \cdots + 3025$$
$67$ $$(T^{4} + T^{3} - 82 T^{2} - 238 T - 101)^{2}$$
$71$ $$T^{8} + 40 T^{7} + 869 T^{6} + \cdots + 60824401$$
$73$ $$T^{8} + 23 T^{7} + 368 T^{6} + \cdots + 151321$$
$79$ $$T^{8} - 17 T^{7} + 154 T^{6} + \cdots + 4644025$$
$83$ $$T^{8} - 25 T^{7} + 271 T^{6} + \cdots + 841$$
$89$ $$(T^{4} - 150 T^{2} + 400 T + 725)^{2}$$
$97$ $$T^{8} - 12 T^{7} + 179 T^{6} + \cdots + 625$$