# Properties

 Label 418.2.b.c Level $418$ Weight $2$ Character orbit 418.b Analytic conductor $3.338$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.14584320320.1 Defining polynomial: $$x^{8} - x^{7} + 4x^{6} + 11x^{5} - 11x^{4} + 32x^{3} + 44x^{2} - 18x + 46$$ x^8 - x^7 + 4*x^6 + 11*x^5 - 11*x^4 + 32*x^3 + 44*x^2 - 18*x + 46 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_{3} q^{3} + q^{4} + \beta_{2} q^{5} + \beta_{3} q^{6} + (\beta_{5} + \beta_{3}) q^{7} - q^{8} + (\beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{9}+O(q^{10})$$ q - q^2 - b3 * q^3 + q^4 + b2 * q^5 + b3 * q^6 + (b5 + b3) * q^7 - q^8 + (b6 - b4 - b2 - 2*b1 - 2) * q^9 $$q - q^{2} - \beta_{3} q^{3} + q^{4} + \beta_{2} q^{5} + \beta_{3} q^{6} + (\beta_{5} + \beta_{3}) q^{7} - q^{8} + (\beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{9} - \beta_{2} q^{10} + (\beta_{5} + \beta_{4} - \beta_1 - 1) q^{11} - \beta_{3} q^{12} + ( - \beta_{2} - 2 \beta_1 + 1) q^{13} + ( - \beta_{5} - \beta_{3}) q^{14} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4}) q^{15} + q^{16} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{17} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{18} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{19} + \beta_{2} q^{20} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 + 3) q^{21} + ( - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{22} + ( - \beta_{2} + \beta_1 + 2) q^{23} + \beta_{3} q^{24} + (\beta_{6} - \beta_{4} - 2 \beta_{2} - \beta_1) q^{25} + (\beta_{2} + 2 \beta_1 - 1) q^{26} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 3 \beta_{3}) q^{27} + (\beta_{5} + \beta_{3}) q^{28} + ( - \beta_1 - 2) q^{29} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{30} + (3 \beta_{7} - \beta_{6} - \beta_{4}) q^{31} - q^{32} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{33} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{34} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4}) q^{35} + (\beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{36} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{37} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{38} + ( - 3 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}) q^{39} - \beta_{2} q^{40} + (2 \beta_{2} + 3 \beta_1 + 3) q^{41} + (\beta_{6} - \beta_{4} - 2 \beta_1 - 3) q^{42} + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4}) q^{43} + (\beta_{5} + \beta_{4} - \beta_1 - 1) q^{44} + (\beta_{6} - \beta_{4} - \beta_1 - 1) q^{45} + (\beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{47} - \beta_{3} q^{48} + ( - \beta_{6} + \beta_{4} - \beta_1 + 1) q^{49} + ( - \beta_{6} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{50} + (\beta_{6} - \beta_{4} - \beta_{2} - 5 \beta_1 - 4) q^{51} + ( - \beta_{2} - 2 \beta_1 + 1) q^{52} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{53} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 3 \beta_{3}) q^{54} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{55} + ( - \beta_{5} - \beta_{3}) q^{56} + (\beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{57} + (\beta_1 + 2) q^{58} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3}) q^{59} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4}) q^{60} + ( - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3}) q^{61} + ( - 3 \beta_{7} + \beta_{6} + \beta_{4}) q^{62} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 4 \beta_{3}) q^{63} + q^{64} + ( - \beta_{6} + \beta_{4} + 3 \beta_{2} - \beta_1 - 3) q^{65} + (3 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{66} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{67} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{68} + (3 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{69} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4}) q^{70} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{71} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{72} + (3 \beta_{7} + \beta_{5} + \beta_{3}) q^{73} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{74} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3}) q^{75} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{76} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{77} + (3 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{78} + (3 \beta_{6} - 3 \beta_{4} - 2 \beta_1 - 2) q^{79} + \beta_{2} q^{80} + ( - 2 \beta_{6} + 2 \beta_{4} + 5 \beta_{2} + 11 \beta_1 + 6) q^{81} + ( - 2 \beta_{2} - 3 \beta_1 - 3) q^{82} + ( - 5 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{83} + ( - \beta_{6} + \beta_{4} + 2 \beta_1 + 3) q^{84} + (\beta_{6} + \beta_{4}) q^{85} + (3 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4}) q^{86} + ( - 2 \beta_{7} + \beta_{5} + 3 \beta_{3}) q^{87} + ( - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{88} + (6 \beta_{7} - 2 \beta_{6} - 6 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{89} + ( - \beta_{6} + \beta_{4} + \beta_1 + 1) q^{90} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{91} + ( - \beta_{2} + \beta_1 + 2) q^{92} + ( - \beta_{6} + \beta_{4} - \beta_{2} - 5 \beta_1 - 5) q^{93} + (2 \beta_{2} - 2 \beta_1 - 4) q^{94} + ( - \beta_{7} - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{95} + \beta_{3} q^{96} + ( - \beta_{6} - \beta_{4}) q^{97} + (\beta_{6} - \beta_{4} + \beta_1 - 1) q^{98} + ( - \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 4) q^{99}+O(q^{100})$$ q - q^2 - b3 * q^3 + q^4 + b2 * q^5 + b3 * q^6 + (b5 + b3) * q^7 - q^8 + (b6 - b4 - b2 - 2*b1 - 2) * q^9 - b2 * q^10 + (b5 + b4 - b1 - 1) * q^11 - b3 * q^12 + (-b2 - 2*b1 + 1) * q^13 + (-b5 - b3) * q^14 + (-b7 + b6 + 2*b5 + b4) * q^15 + q^16 + (b7 - b5 - b3) * q^17 + (-b6 + b4 + b2 + 2*b1 + 2) * q^18 + (-2*b7 + 2*b6 + b5 - b2 - b1) * q^19 + b2 * q^20 + (-b6 + b4 + 2*b1 + 3) * q^21 + (-b5 - b4 + b1 + 1) * q^22 + (-b2 + b1 + 2) * q^23 + b3 * q^24 + (b6 - b4 - 2*b2 - b1) * q^25 + (b2 + 2*b1 - 1) * q^26 + (-b7 - 2*b6 - 2*b4 + 3*b3) * q^27 + (b5 + b3) * q^28 + (-b1 - 2) * q^29 + (b7 - b6 - 2*b5 - b4) * q^30 + (3*b7 - b6 - b4) * q^31 - q^32 + (-3*b7 + b6 + b5 + b3 - 2*b2 - 2*b1 - 1) * q^33 + (-b7 + b5 + b3) * q^34 + (2*b7 - 2*b6 - b5 - 2*b4) * q^35 + (b6 - b4 - b2 - 2*b1 - 2) * q^36 + (-2*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3) * q^37 + (2*b7 - 2*b6 - b5 + b2 + b1) * q^38 + (-3*b7 - b6 - b4 + b3) * q^39 - b2 * q^40 + (2*b2 + 3*b1 + 3) * q^41 + (b6 - b4 - 2*b1 - 3) * q^42 + (-3*b7 + b6 + 3*b5 + b4) * q^43 + (b5 + b4 - b1 - 1) * q^44 + (b6 - b4 - b1 - 1) * q^45 + (b2 - b1 - 2) * q^46 + (-2*b2 + 2*b1 + 4) * q^47 - b3 * q^48 + (-b6 + b4 - b1 + 1) * q^49 + (-b6 + b4 + 2*b2 + b1) * q^50 + (b6 - b4 - b2 - 5*b1 - 4) * q^51 + (-b2 - 2*b1 + 1) * q^52 + (b7 + b6 + 3*b5 + b4 + 3*b3) * q^53 + (b7 + 2*b6 + 2*b4 - 3*b3) * q^54 + (2*b7 - 3*b6 - 2*b5 - b4 - b3 - b2 - b1) * q^55 + (-b5 - b3) * q^56 + (b7 - b6 - b5 - 3*b4 + 3*b3 - b2 + 2*b1 + 2) * q^57 + (b1 + 2) * q^58 + (-3*b7 + 3*b6 + b5 + 3*b4 - b3) * q^59 + (-b7 + b6 + 2*b5 + b4) * q^60 + (-2*b7 + 3*b6 + 2*b5 + 3*b4 - 2*b3) * q^61 + (-3*b7 + b6 + b4) * q^62 + (2*b7 + b6 + b5 + b4 - 4*b3) * q^63 + q^64 + (-b6 + b4 + 3*b2 - b1 - 3) * q^65 + (3*b7 - b6 - b5 - b3 + 2*b2 + 2*b1 + 1) * q^66 + (2*b7 - 3*b6 - 3*b5 - 3*b4 + b3) * q^67 + (b7 - b5 - b3) * q^68 + (3*b7 - b6 - 3*b5 - b4 - 3*b3) * q^69 + (-2*b7 + 2*b6 + b5 + 2*b4) * q^70 + (b7 + b6 - b5 + b4 + 2*b3) * q^71 + (-b6 + b4 + b2 + 2*b1 + 2) * q^72 + (3*b7 + b5 + b3) * q^73 + (2*b7 - 2*b6 - 2*b5 - 2*b4 + 2*b3) * q^74 + (2*b7 - 3*b6 - 3*b5 - 3*b4 + 3*b3) * q^75 + (-2*b7 + 2*b6 + b5 - b2 - b1) * q^76 + (-b6 - 2*b5 + 2*b4 - 2*b3 + 2*b2 + 2*b1 - 2) * q^77 + (3*b7 + b6 + b4 - b3) * q^78 + (3*b6 - 3*b4 - 2*b1 - 2) * q^79 + b2 * q^80 + (-2*b6 + 2*b4 + 5*b2 + 11*b1 + 6) * q^81 + (-2*b2 - 3*b1 - 3) * q^82 + (-5*b7 + 2*b6 + b5 + 2*b4 - 2*b3) * q^83 + (-b6 + b4 + 2*b1 + 3) * q^84 + (b6 + b4) * q^85 + (3*b7 - b6 - 3*b5 - b4) * q^86 + (-2*b7 + b5 + 3*b3) * q^87 + (-b5 - b4 + b1 + 1) * q^88 + (6*b7 - 2*b6 - 6*b5 - 2*b4 - 2*b3) * q^89 + (-b6 + b4 + b1 + 1) * q^90 + (2*b6 + 2*b5 + 2*b4 - b3) * q^91 + (-b2 + b1 + 2) * q^92 + (-b6 + b4 - b2 - 5*b1 - 5) * q^93 + (2*b2 - 2*b1 - 4) * q^94 + (-b7 - 3*b5 - 2*b4 - 2*b3 + 2*b2 - 2) * q^95 + b3 * q^96 + (-b6 - b4) * q^97 + (b6 - b4 + b1 - 1) * q^98 + (-b7 - 3*b6 + b5 + b4 + 4*b3 + 2*b2 + 6*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{8} - 14 q^{9}+O(q^{10})$$ 8 * q - 8 * q^2 + 8 * q^4 + 2 * q^5 - 8 * q^8 - 14 * q^9 $$8 q - 8 q^{2} + 8 q^{4} + 2 q^{5} - 8 q^{8} - 14 q^{9} - 2 q^{10} - 6 q^{11} + 10 q^{13} + 8 q^{16} + 14 q^{18} + 2 q^{20} + 20 q^{21} + 6 q^{22} + 12 q^{23} - 2 q^{25} - 10 q^{26} - 14 q^{29} - 8 q^{32} - 8 q^{33} - 14 q^{36} - 2 q^{40} + 22 q^{41} - 20 q^{42} - 6 q^{44} - 6 q^{45} - 12 q^{46} + 24 q^{47} + 10 q^{49} + 2 q^{50} - 24 q^{51} + 10 q^{52} + 10 q^{57} + 14 q^{58} + 8 q^{64} - 16 q^{65} + 8 q^{66} + 14 q^{72} - 16 q^{77} - 12 q^{79} + 2 q^{80} + 36 q^{81} - 22 q^{82} + 20 q^{84} + 6 q^{88} + 6 q^{90} + 12 q^{92} - 32 q^{93} - 24 q^{94} - 12 q^{95} - 10 q^{98} + 24 q^{99}+O(q^{100})$$ 8 * q - 8 * q^2 + 8 * q^4 + 2 * q^5 - 8 * q^8 - 14 * q^9 - 2 * q^10 - 6 * q^11 + 10 * q^13 + 8 * q^16 + 14 * q^18 + 2 * q^20 + 20 * q^21 + 6 * q^22 + 12 * q^23 - 2 * q^25 - 10 * q^26 - 14 * q^29 - 8 * q^32 - 8 * q^33 - 14 * q^36 - 2 * q^40 + 22 * q^41 - 20 * q^42 - 6 * q^44 - 6 * q^45 - 12 * q^46 + 24 * q^47 + 10 * q^49 + 2 * q^50 - 24 * q^51 + 10 * q^52 + 10 * q^57 + 14 * q^58 + 8 * q^64 - 16 * q^65 + 8 * q^66 + 14 * q^72 - 16 * q^77 - 12 * q^79 + 2 * q^80 + 36 * q^81 - 22 * q^82 + 20 * q^84 + 6 * q^88 + 6 * q^90 + 12 * q^92 - 32 * q^93 - 24 * q^94 - 12 * q^95 - 10 * q^98 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 4x^{6} + 11x^{5} - 11x^{4} + 32x^{3} + 44x^{2} - 18x + 46$$ :

 $$\beta_{1}$$ $$=$$ $$( 53\nu^{7} + 1084\nu^{6} - 556\nu^{5} + 2891\nu^{4} + 14726\nu^{3} - 4474\nu^{2} + 10896\nu + 54404 ) / 23578$$ (53*v^7 + 1084*v^6 - 556*v^5 + 2891*v^4 + 14726*v^3 - 4474*v^2 + 10896*v + 54404) / 23578 $$\beta_{2}$$ $$=$$ $$( -355\nu^{7} + 302\nu^{6} - 2504\nu^{5} - 3349\nu^{4} + 1014\nu^{3} - 26086\nu^{2} + 12432\nu - 4506 ) / 23578$$ (-355*v^7 + 302*v^6 - 2504*v^5 - 3349*v^4 + 1014*v^3 - 26086*v^2 + 12432*v - 4506) / 23578 $$\beta_{3}$$ $$=$$ $$( -955\nu^{7} - 848\nu^{6} + 1566\nu^{5} - 23621\nu^{4} + 11362\nu^{3} + 4544\nu^{2} - 107360\nu + 42008 ) / 23578$$ (-955*v^7 - 848*v^6 + 1566*v^5 - 23621*v^4 + 11362*v^3 + 4544*v^2 - 107360*v + 42008) / 23578 $$\beta_{4}$$ $$=$$ $$( 1247\nu^{7} - 2522\nu^{6} + 11386\nu^{5} + 1735\nu^{4} + 2150\nu^{3} + 67788\nu^{2} + 5014\nu + 53088 ) / 23578$$ (1247*v^7 - 2522*v^6 + 11386*v^5 + 1735*v^4 + 2150*v^3 + 67788*v^2 + 5014*v + 53088) / 23578 $$\beta_{5}$$ $$=$$ $$( 686\nu^{7} - 650\nu^{6} + 2813\nu^{5} + 4944\nu^{4} - 4915\nu^{3} + 8599\nu^{2} + 7126\nu - 1388 ) / 11789$$ (686*v^7 - 650*v^6 + 2813*v^5 + 4944*v^4 - 4915*v^3 + 8599*v^2 + 7126*v - 1388) / 11789 $$\beta_{6}$$ $$=$$ $$( 1501\nu^{7} - 4000\nu^{6} + 10056\nu^{5} - 1315\nu^{4} - 19364\nu^{3} + 58358\nu^{2} - 1490\nu + 5968 ) / 23578$$ (1501*v^7 - 4000*v^6 + 10056*v^5 - 1315*v^4 - 19364*v^3 + 58358*v^2 - 1490*v + 5968) / 23578 $$\beta_{7}$$ $$=$$ $$( 1727\nu^{7} - 1602\nu^{6} + 8130\nu^{5} + 13237\nu^{4} - 10844\nu^{3} + 43284\nu^{2} + 48976\nu + 1730 ) / 23578$$ (1727*v^7 - 1602*v^6 + 8130*v^5 + 13237*v^4 - 10844*v^3 + 43284*v^2 + 48976*v + 1730) / 23578
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{5} + \beta_{2} ) / 2$$ (b7 - b5 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 - 1 ) / 2$$ (b7 + b6 - 2*b5 - b4 - 2*b2 + b1 - 1) / 2 $$\nu^{3}$$ $$=$$ $$-3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta _1 - 5$$ -3*b7 + b6 + b5 + b4 - b3 + 2*b1 - 5 $$\nu^{4}$$ $$=$$ $$( -15\beta_{7} - 2\beta_{6} + 9\beta_{5} + 8\beta_{4} - 6\beta_{3} - 3\beta_{2} - 4\beta _1 + 4 ) / 2$$ (-15*b7 - 2*b6 + 9*b5 + 8*b4 - 6*b3 - 3*b2 - 4*b1 + 4) / 2 $$\nu^{5}$$ $$=$$ $$( 11\beta_{7} - 21\beta_{6} + 6\beta_{5} + 13\beta_{4} + 6\beta_{3} + 14\beta_{2} - 23\beta _1 + 21 ) / 2$$ (11*b7 - 21*b6 + 6*b5 + 13*b4 + 6*b3 + 14*b2 - 23*b1 + 21) / 2 $$\nu^{6}$$ $$=$$ $$58\beta_{7} - 16\beta_{6} - 22\beta_{5} - 20\beta_{4} + 22\beta_{3} - \beta_{2} - 7\beta _1 + 19$$ 58*b7 - 16*b6 - 22*b5 - 20*b4 + 22*b3 - b2 - 7*b1 + 19 $$\nu^{7}$$ $$=$$ $$( 107\beta_{7} + 72\beta_{6} - 47\beta_{5} - 122\beta_{4} + 46\beta_{3} - 23\beta_{2} + 126\beta _1 - 134 ) / 2$$ (107*b7 + 72*b6 - 47*b5 - 122*b4 + 46*b3 - 23*b2 + 126*b1 - 134) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 0.274776 − 0.839339i 0.682410 + 2.29682i −1.66113 + 0.0964267i 1.20394 + 1.50360i 1.20394 − 1.50360i −1.66113 − 0.0964267i 0.682410 − 2.29682i 0.274776 + 0.839339i
−1.00000 3.37171i 1.00000 0.549551 3.37171i 2.61555i −1.00000 −8.36845 −0.549551
417.2 −1.00000 2.00783i 1.00000 1.36482 2.00783i 0.331974i −1.00000 −1.03137 −1.36482
417.3 −1.00000 1.58627i 1.00000 −3.32225 1.58627i 2.41983i −1.00000 0.483751 3.32225
417.4 −1.00000 1.04112i 1.00000 2.40788 1.04112i 3.19266i −1.00000 1.91607 −2.40788
417.5 −1.00000 1.04112i 1.00000 2.40788 1.04112i 3.19266i −1.00000 1.91607 −2.40788
417.6 −1.00000 1.58627i 1.00000 −3.32225 1.58627i 2.41983i −1.00000 0.483751 3.32225
417.7 −1.00000 2.00783i 1.00000 1.36482 2.00783i 0.331974i −1.00000 −1.03137 −1.36482
417.8 −1.00000 3.37171i 1.00000 0.549551 3.37171i 2.61555i −1.00000 −8.36845 −0.549551
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 417.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.b.c 8
3.b odd 2 1 3762.2.g.h 8
11.b odd 2 1 418.2.b.d yes 8
19.b odd 2 1 418.2.b.d yes 8
33.d even 2 1 3762.2.g.g 8
57.d even 2 1 3762.2.g.g 8
209.d even 2 1 inner 418.2.b.c 8
627.b odd 2 1 3762.2.g.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.b.c 8 1.a even 1 1 trivial
418.2.b.c 8 209.d even 2 1 inner
418.2.b.d yes 8 11.b odd 2 1
418.2.b.d yes 8 19.b odd 2 1
3762.2.g.g 8 33.d even 2 1
3762.2.g.g 8 57.d even 2 1
3762.2.g.h 8 3.b odd 2 1
3762.2.g.h 8 627.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$:

 $$T_{3}^{8} + 19T_{3}^{6} + 104T_{3}^{4} + 207T_{3}^{2} + 125$$ T3^8 + 19*T3^6 + 104*T3^4 + 207*T3^2 + 125 $$T_{13}^{4} - 5T_{13}^{3} - 16T_{13}^{2} + 79T_{13} - 1$$ T13^4 - 5*T13^3 - 16*T13^2 + 79*T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{8}$$
$3$ $$T^{8} + 19 T^{6} + 104 T^{4} + \cdots + 125$$
$5$ $$(T^{4} - T^{3} - 9 T^{2} + 16 T - 6)^{2}$$
$7$ $$T^{8} + 23 T^{6} + 172 T^{4} + \cdots + 45$$
$11$ $$T^{8} + 6 T^{7} + 10 T^{6} + \cdots + 14641$$
$13$ $$(T^{4} - 5 T^{3} - 16 T^{2} + 79 T - 1)^{2}$$
$17$ $$T^{8} + 38 T^{6} + 365 T^{4} + \cdots + 1280$$
$19$ $$T^{8} + 64 T^{5} + 302 T^{4} + \cdots + 130321$$
$23$ $$(T^{4} - 6 T^{3} - 7 T^{2} + 44 T + 48)^{2}$$
$29$ $$(T^{4} + 7 T^{3} + 12 T^{2} - T - 9)^{2}$$
$31$ $$T^{8} + 121 T^{6} + 4229 T^{4} + \cdots + 95220$$
$37$ $$T^{8} + 152 T^{6} + 7120 T^{4} + \cdots + 184320$$
$41$ $$(T^{4} - 11 T^{3} - 21 T^{2} + 212 T + 450)^{2}$$
$43$ $$T^{8} + 145 T^{6} + 4673 T^{4} + \cdots + 162000$$
$47$ $$(T^{4} - 12 T^{3} - 28 T^{2} + 352 T + 768)^{2}$$
$53$ $$T^{8} + 266 T^{6} + 19229 T^{4} + \cdots + 2366720$$
$59$ $$T^{8} + 290 T^{6} + 23949 T^{4} + \cdots + 1458000$$
$61$ $$T^{8} + 288 T^{6} + 25652 T^{4} + \cdots + 3732480$$
$67$ $$T^{8} + 259 T^{6} + 16856 T^{4} + \cdots + 595125$$
$71$ $$T^{8} + 313 T^{6} + 31417 T^{4} + \cdots + 4762880$$
$73$ $$T^{8} + 182 T^{6} + 11341 T^{4} + \cdots + 2592000$$
$79$ $$(T^{4} + 6 T^{3} - 162 T^{2} - 1504 T - 3200)^{2}$$
$83$ $$T^{8} + 265 T^{6} + 16109 T^{4} + \cdots + 36980$$
$89$ $$T^{8} + 568 T^{6} + \cdots + 184832000$$
$97$ $$T^{8} + 64 T^{6} + 1460 T^{4} + \cdots + 46080$$