Properties

 Label 5054.2.a.bg Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5054 = 2 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5054.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$40.3563931816$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.19520000000.1 Defining polynomial: $$x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4$$ x^8 - 2*x^7 - 12*x^6 + 16*x^5 + 50*x^4 - 24*x^3 - 72*x^2 - 32*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 8\nu + 6 ) / 2$$ (v^4 - 2*v^3 - 6*v^2 + 8*v + 6) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 4\nu^{3} + 4\nu^{2} - 18\nu - 4 ) / 2$$ (-v^4 + 4*v^3 + 4*v^2 - 18*v - 4) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 10\nu^{2} + 16\nu - 2 ) / 2$$ (v^5 - 2*v^4 - 8*v^3 + 10*v^2 + 16*v - 2) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 8\nu^{2} + 18\nu + 4 ) / 2$$ (v^5 - 2*v^4 - 8*v^3 + 8*v^2 + 18*v + 4) / 2 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} - 12\nu^{5} + 17\nu^{4} + 48\nu^{3} - 32\nu^{2} - 64\nu - 16 ) / 2$$ (v^7 - 2*v^6 - 12*v^5 + 17*v^4 + 48*v^3 - 32*v^2 - 64*v - 16) / 2 $$\beta_{7}$$ $$=$$ $$( -2\nu^{7} + 5\nu^{6} + 22\nu^{5} - 43\nu^{4} - 84\nu^{3} + 86\nu^{2} + 116\nu + 24 ) / 2$$ (-2*v^7 + 5*v^6 + 22*v^5 - 43*v^4 - 84*v^3 + 86*v^2 + 116*v + 24) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta _1 + 3$$ -b5 + b4 + b1 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2$$ -b5 + b4 + b3 + b2 + 6*b1 + 2 $$\nu^{4}$$ $$=$$ $$-8\beta_{5} + 8\beta_{4} + 2\beta_{3} + 4\beta_{2} + 10\beta _1 + 16$$ -8*b5 + 8*b4 + 2*b3 + 4*b2 + 10*b1 + 16 $$\nu^{5}$$ $$=$$ $$-14\beta_{5} + 16\beta_{4} + 12\beta_{3} + 16\beta_{2} + 42\beta _1 + 20$$ -14*b5 + 16*b4 + 12*b3 + 16*b2 + 42*b1 + 20 $$\nu^{6}$$ $$=$$ $$2\beta_{7} + 4\beta_{6} - 66\beta_{5} + 70\beta_{4} + 30\beta_{3} + 56\beta_{2} + 92\beta _1 + 102$$ 2*b7 + 4*b6 - 66*b5 + 70*b4 + 30*b3 + 56*b2 + 92*b1 + 102 $$\nu^{7}$$ $$=$$ $$4\beta_{7} + 10\beta_{6} - 148\beta_{5} + 180\beta_{4} + 122\beta_{3} + 188\beta_{2} + 326\beta _1 + 188$$ 4*b7 + 10*b6 - 148*b5 + 180*b4 + 122*b3 + 188*b2 + 326*b1 + 188

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.35877 −0.761439 −2.15124 −1.91631 3.02989 −0.368905 −0.236286 2.04552
−1.00000 −2.35992 1.00000 −4.10066 2.35992 1.00000 −1.00000 2.56920 4.10066
1.2 −1.00000 −1.40760 1.00000 −2.26420 1.40760 1.00000 −1.00000 −1.01865 2.26420
1.3 −1.00000 −1.30521 1.00000 −0.772004 1.30521 1.00000 −1.00000 −1.29642 0.772004
1.4 −1.00000 0.282233 1.00000 2.81658 −0.282233 1.00000 −1.00000 −2.92034 −2.81658
1.5 −1.00000 1.02954 1.00000 −1.38232 −1.02954 1.00000 −1.00000 −1.94005 1.38232
1.6 −1.00000 1.57867 1.00000 0.329540 −1.57867 1.00000 −1.00000 −0.507804 −0.329540
1.7 −1.00000 3.04815 1.00000 3.90247 −3.04815 1.00000 −1.00000 6.29119 −3.90247
1.8 −1.00000 3.13415 1.00000 −0.529405 −3.13415 1.00000 −1.00000 6.82288 0.529405
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5054.2.a.bg 8
19.b odd 2 1 5054.2.a.bh yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.bg 8 1.a even 1 1 trivial
5054.2.a.bh yes 8 19.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5054))$$:

 $$T_{3}^{8} - 4T_{3}^{7} - 8T_{3}^{6} + 38T_{3}^{5} + 15T_{3}^{4} - 98T_{3}^{3} + 2T_{3}^{2} + 74T_{3} - 19$$ T3^8 - 4*T3^7 - 8*T3^6 + 38*T3^5 + 15*T3^4 - 98*T3^3 + 2*T3^2 + 74*T3 - 19 $$T_{5}^{8} + 2T_{5}^{7} - 22T_{5}^{6} - 46T_{5}^{5} + 90T_{5}^{4} + 254T_{5}^{3} + 138T_{5}^{2} - 18T_{5} - 19$$ T5^8 + 2*T5^7 - 22*T5^6 - 46*T5^5 + 90*T5^4 + 254*T5^3 + 138*T5^2 - 18*T5 - 19 $$T_{13}^{8} - 10T_{13}^{7} + 10T_{13}^{6} + 170T_{13}^{5} - 505T_{13}^{4} - 290T_{13}^{3} + 2600T_{13}^{2} - 2960T_{13} + 905$$ T13^8 - 10*T13^7 + 10*T13^6 + 170*T13^5 - 505*T13^4 - 290*T13^3 + 2600*T13^2 - 2960*T13 + 905

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{8}$$
$3$ $$T^{8} - 4 T^{7} - 8 T^{6} + 38 T^{5} + \cdots - 19$$
$5$ $$T^{8} + 2 T^{7} - 22 T^{6} - 46 T^{5} + \cdots - 19$$
$7$ $$(T - 1)^{8}$$
$11$ $$T^{8} + 12 T^{7} + 28 T^{6} + \cdots - 779$$
$13$ $$T^{8} - 10 T^{7} + 10 T^{6} + \cdots + 905$$
$17$ $$T^{8} + 6 T^{7} - 58 T^{6} + \cdots - 24719$$
$19$ $$T^{8}$$
$23$ $$T^{8} + 20 T^{7} + 120 T^{6} + \cdots - 5795$$
$29$ $$T^{8} - 8 T^{7} - 52 T^{6} + \cdots - 2179$$
$31$ $$T^{8} - 18 T^{7} + 78 T^{6} + \cdots + 4541$$
$37$ $$T^{8} - 36 T^{7} + 432 T^{6} + \cdots - 396379$$
$41$ $$T^{8} - 4 T^{7} - 148 T^{6} + \cdots - 9559$$
$43$ $$T^{8} + 16 T^{7} - 158 T^{6} + \cdots + 2944801$$
$47$ $$T^{8} + 10 T^{7} - 270 T^{6} + \cdots + 8597405$$
$53$ $$T^{8} - 32 T^{7} + 248 T^{6} + \cdots - 48299$$
$59$ $$T^{8} + 2 T^{7} - 122 T^{6} + \cdots + 29921$$
$61$ $$T^{8} - 2 T^{7} - 82 T^{6} + \cdots + 7421$$
$67$ $$T^{8} - 44 T^{7} + 662 T^{6} + \cdots - 51619$$
$71$ $$T^{8} - 8 T^{7} - 112 T^{6} + \cdots + 17936$$
$73$ $$T^{8} - 30 T^{7} - 10 T^{6} + \cdots + 27334525$$
$79$ $$T^{8} - 60 T^{7} + 1470 T^{6} + \cdots - 6438695$$
$83$ $$T^{8} + 28 T^{7} - 12 T^{6} + \cdots - 23760304$$
$89$ $$T^{8} + 22 T^{7} - 222 T^{6} + \cdots + 580621$$
$97$ $$T^{8} + 6 T^{7} - 378 T^{6} + \cdots - 1404719$$