# Properties

 Label 5054.2.a.bf Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $1$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta_{7} - \beta_{5} - 1) q^{3} + q^{4} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{5} + ( - \beta_{7} + \beta_{5} + 1) q^{6} - q^{7} - q^{8} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10})$$ q - q^2 + (b7 - b5 - 1) * q^3 + q^4 + (-b7 + b6 + b1 + 1) * q^5 + (-b7 + b5 + 1) * q^6 - q^7 - q^8 + (-b7 + b6 + b5 + b4 + b3 + 2) * q^9 $$q - q^{2} + (\beta_{7} - \beta_{5} - 1) q^{3} + q^{4} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{5} + ( - \beta_{7} + \beta_{5} + 1) q^{6} - q^{7} - q^{8} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{9} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{10} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{11} + (\beta_{7} - \beta_{5} - 1) q^{12} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{13} + q^{14} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_1 - 2) q^{15} + q^{16} + (2 \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{17} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2) q^{18} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{20} + ( - \beta_{7} + \beta_{5} + 1) q^{21} + (\beta_{5} - \beta_{4} - \beta_1) q^{22} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{23} + ( - \beta_{7} + \beta_{5} + 1) q^{24} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_1 - 1) q^{25} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{26} + (\beta_{6} - 2 \beta_{4} - \beta_{3} + 2 \beta_1 - 3) q^{27} - q^{28} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{29} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_1 + 2) q^{30} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2) q^{31} - q^{32} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{33} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{34} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{35} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{36} + ( - 2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{37} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 3) q^{39} + (\beta_{7} - \beta_{6} - \beta_1 - 1) q^{40} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_1 - 5) q^{41} + (\beta_{7} - \beta_{5} - 1) q^{42} + ( - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{43} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{44} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 3 \beta_{2} + \beta_1 + 1) q^{45} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{46} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_1 + 3) q^{47} + (\beta_{7} - \beta_{5} - 1) q^{48} + q^{49} + (2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_1 + 1) q^{50} + ( - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{51} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{52} + ( - 3 \beta_{7} - 3 \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{53} + ( - \beta_{6} + 2 \beta_{4} + \beta_{3} - 2 \beta_1 + 3) q^{54} + (3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{55} + q^{56} + ( - \beta_{6} + 2 \beta_{4} + \beta_{2} + 2) q^{58} + (2 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{59} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_1 - 2) q^{60} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{61} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2) q^{62} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2) q^{63} + q^{64} + (5 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 5) q^{65} + (2 \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{66} + ( - 5 \beta_{7} + 4 \beta_{6} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{67} + (2 \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{68} + (7 \beta_{7} - 4 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{69} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{70} + ( - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{71} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2) q^{72} + (\beta_{7} - 4 \beta_{6} - \beta_{5} - 3 \beta_1 - 4) q^{73} + (2 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - \beta_1 - 3) q^{74} + (\beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{75} + (\beta_{5} - \beta_{4} - \beta_1) q^{77} + (3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 3) q^{78} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{79} + ( - \beta_{7} + \beta_{6} + \beta_1 + 1) q^{80} + ( - \beta_{7} - 10 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 6 \beta_1 - 1) q^{81} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_1 + 5) q^{82} + (7 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_1 + 6) q^{83} + ( - \beta_{7} + \beta_{5} + 1) q^{84} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} - 2) q^{85} + (3 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{86} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 5 \beta_{2} - 5 \beta_1 + 5) q^{87} + (\beta_{5} - \beta_{4} - \beta_1) q^{88} + ( - \beta_{7} + 7 \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{89} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - \beta_1 - 1) q^{90} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{91} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{92} + ( - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{93} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{5} - \beta_1 - 3) q^{94} + ( - \beta_{7} + \beta_{5} + 1) q^{96} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 6) q^{97} - q^{98} + (2 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q - q^2 + (b7 - b5 - 1) * q^3 + q^4 + (-b7 + b6 + b1 + 1) * q^5 + (-b7 + b5 + 1) * q^6 - q^7 - q^8 + (-b7 + b6 + b5 + b4 + b3 + 2) * q^9 + (b7 - b6 - b1 - 1) * q^10 + (-b5 + b4 + b1) * q^11 + (b7 - b5 - 1) * q^12 + (2*b7 - b6 - b5 - b3 - b2 - 2*b1 - 1) * q^13 + q^14 + (b7 - 2*b6 - b5 + b4 - b1 - 2) * q^15 + q^16 + (2*b7 - b6 - b4 - b3 - b2 - b1) * q^17 + (b7 - b6 - b5 - b4 - b3 - 2) * q^18 + (-b7 + b6 + b1 + 1) * q^20 + (-b7 + b5 + 1) * q^21 + (b5 - b4 - b1) * q^22 + (-2*b7 - b6 + 2*b5 + b3 + 2*b2 - b1 + 3) * q^23 + (-b7 + b5 + 1) * q^24 + (-2*b7 + b5 - b4 + b1 - 1) * q^25 + (-2*b7 + b6 + b5 + b3 + b2 + 2*b1 + 1) * q^26 + (b6 - 2*b4 - b3 + 2*b1 - 3) * q^27 - q^28 + (b6 - 2*b4 - b2 - 2) * q^29 + (-b7 + 2*b6 + b5 - b4 + b1 + 2) * q^30 + (b7 + b6 + b5 - b4 - 2) * q^31 - q^32 + (-2*b7 + b6 + b4 + b3 + 3*b2 + 2*b1 + 1) * q^33 + (-2*b7 + b6 + b4 + b3 + b2 + b1) * q^34 + (b7 - b6 - b1 - 1) * q^35 + (-b7 + b6 + b5 + b4 + b3 + 2) * q^36 + (-2*b7 + b6 + 3*b5 + b4 + 2*b3 + 3*b2 + b1 + 3) * q^37 + (-3*b7 + 3*b6 + 2*b5 - 2*b4 + 3) * q^39 + (b7 - b6 - b1 - 1) * q^40 + (b7 - 2*b6 + b5 - 2*b4 - 2*b1 - 5) * q^41 + (b7 - b5 - 1) * q^42 + (-3*b7 - b6 + 2*b5 - 2*b4 - b3 - 2*b2 + b1) * q^43 + (-b5 + b4 + b1) * q^44 + (2*b6 + 2*b5 - b4 + 3*b2 + b1 + 1) * q^45 + (2*b7 + b6 - 2*b5 - b3 - 2*b2 + b1 - 3) * q^46 + (-b7 + 3*b6 + 2*b5 + b1 + 3) * q^47 + (b7 - b5 - 1) * q^48 + q^49 + (2*b7 - b5 + b4 - b1 + 1) * q^50 + (-2*b7 + b5 - b4 + b3 - b2 - 2*b1 + 1) * q^51 + (2*b7 - b6 - b5 - b3 - b2 - 2*b1 - 1) * q^52 + (-3*b7 - 3*b6 + b3 - b2 + b1 - 4) * q^53 + (-b6 + 2*b4 + b3 - 2*b1 + 3) * q^54 + (3*b7 - 3*b6 - 2*b5 - b3 - b2) * q^55 + q^56 + (-b6 + 2*b4 + b2 + 2) * q^58 + (2*b7 - 5*b6 + 2*b5 - 2*b3 - b2 - 3*b1 - 2) * q^59 + (b7 - 2*b6 - b5 + b4 - b1 - 2) * q^60 + (-2*b6 - b5 - b4 - 2*b3 + 2*b2) * q^61 + (-b7 - b6 - b5 + b4 + 2) * q^62 + (b7 - b6 - b5 - b4 - b3 - 2) * q^63 + q^64 + (5*b7 - 2*b6 - 3*b5 + b4 - 2*b3 - 2*b2 - 3*b1 - 5) * q^65 + (2*b7 - b6 - b4 - b3 - 3*b2 - 2*b1 - 1) * q^66 + (-5*b7 + 4*b6 + 2*b4 + 3*b3 + b2 + b1 + 4) * q^67 + (2*b7 - b6 - b4 - b3 - b2 - b1) * q^68 + (7*b7 - 4*b5 - b4 - 3*b3 - b2 + 2*b1 - 7) * q^69 + (-b7 + b6 + b1 + 1) * q^70 + (-b7 - 2*b6 + 3*b5 - b4 - b3 + 4*b2 - 2*b1 + 1) * q^71 + (b7 - b6 - b5 - b4 - b3 - 2) * q^72 + (b7 - 4*b6 - b5 - 3*b1 - 4) * q^73 + (2*b7 - b6 - 3*b5 - b4 - 2*b3 - 3*b2 - b1 - 3) * q^74 + (b7 - 3*b6 + b5 + b4 - b3 - b2 - 2*b1 - 2) * q^75 + (b5 - b4 - b1) * q^77 + (3*b7 - 3*b6 - 2*b5 + 2*b4 - 3) * q^78 + (b7 + b6 - b5 - b4 + 2*b3 - b2 + 3*b1 - 2) * q^79 + (-b7 + b6 + b1 + 1) * q^80 + (-b7 - 10*b6 + b5 + b4 - b3 - 2*b2 - 6*b1 - 1) * q^81 + (-b7 + 2*b6 - b5 + 2*b4 + 2*b1 + 5) * q^82 + (7*b7 + b6 - b5 + b4 - b3 + b2 - 4*b1 + 6) * q^83 + (-b7 + b5 + 1) * q^84 + (b7 - b5 + b4 - b3 - 3*b2 - 2) * q^85 + (3*b7 + b6 - 2*b5 + 2*b4 + b3 + 2*b2 - b1) * q^86 + (2*b7 - 2*b6 + 2*b5 + b4 - 5*b2 - 5*b1 + 5) * q^87 + (b5 - b4 - b1) * q^88 + (-b7 + 7*b6 - b5 + b3 - 2*b2 + 2*b1) * q^89 + (-2*b6 - 2*b5 + b4 - 3*b2 - b1 - 1) * q^90 + (-2*b7 + b6 + b5 + b3 + b2 + 2*b1 + 1) * q^91 + (-2*b7 - b6 + 2*b5 + b3 + 2*b2 - b1 + 3) * q^92 + (-2*b7 - 3*b6 + 2*b5 + b3 - 3*b2 - 3*b1 + 2) * q^93 + (b7 - 3*b6 - 2*b5 - b1 - 3) * q^94 + (-b7 + b5 + 1) * q^96 + (2*b7 - 3*b6 - 2*b5 - 2*b3 - b2 + 3*b1 - 6) * q^97 - q^98 + (2*b7 - 3*b6 + b5 + b4 + 2*b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 $$8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100})$$ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * q^98 + 4 * 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
 −1.91631 2.04552 −0.368905 2.35877 −0.236286 −2.15124 3.02989 −0.761439
−1.00000 −3.35992 1.00000 −1.35877 3.35992 −1.00000 −1.00000 8.28904 1.35877
1.2 −1.00000 −2.40760 1.00000 1.76144 2.40760 −1.00000 −1.00000 2.79656 −1.76144
1.3 −1.00000 −2.30521 1.00000 3.15124 2.30521 −1.00000 −1.00000 2.31400 −3.15124
1.4 −1.00000 −0.717767 1.00000 2.91631 0.717767 −1.00000 −1.00000 −2.48481 −2.91631
1.5 −1.00000 0.0295377 1.00000 −2.02989 −0.0295377 −1.00000 −1.00000 −2.99913 2.02989
1.6 −1.00000 0.578669 1.00000 1.36891 −0.578669 −1.00000 −1.00000 −2.66514 −1.36891
1.7 −1.00000 2.04815 1.00000 1.23629 −2.04815 −1.00000 −1.00000 1.19490 −1.23629
1.8 −1.00000 2.13415 1.00000 −1.04552 −2.13415 −1.00000 −1.00000 1.55458 1.04552
 $$n$$: 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.bf 8
19.b odd 2 1 5054.2.a.bi yes 8

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

 $$T_{3}^{8} + 4T_{3}^{7} - 8T_{3}^{6} - 38T_{3}^{5} + 15T_{3}^{4} + 98T_{3}^{3} + 2T_{3}^{2} - 34T_{3} + 1$$ T3^8 + 4*T3^7 - 8*T3^6 - 38*T3^5 + 15*T3^4 + 98*T3^3 + 2*T3^2 - 34*T3 + 1 $$T_{5}^{8} - 6T_{5}^{7} + 2T_{5}^{6} + 42T_{5}^{5} - 50T_{5}^{4} - 82T_{5}^{3} + 122T_{5}^{2} + 46T_{5} - 79$$ T5^8 - 6*T5^7 + 2*T5^6 + 42*T5^5 - 50*T5^4 - 82*T5^3 + 122*T5^2 + 46*T5 - 79 $$T_{13}^{8} + 6T_{13}^{7} - 38T_{13}^{6} - 182T_{13}^{5} + 355T_{13}^{4} + 422T_{13}^{3} - 888T_{13}^{2} + 144T_{13} + 181$$ T13^8 + 6*T13^7 - 38*T13^6 - 182*T13^5 + 355*T13^4 + 422*T13^3 - 888*T13^2 + 144*T13 + 181

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{8}$$
$3$ $$T^{8} + 4 T^{7} - 8 T^{6} - 38 T^{5} + \cdots + 1$$
$5$ $$T^{8} - 6 T^{7} + 2 T^{6} + 42 T^{5} + \cdots - 79$$
$7$ $$(T + 1)^{8}$$
$11$ $$T^{8} - 4 T^{7} - 28 T^{6} + 68 T^{5} + \cdots - 419$$
$13$ $$T^{8} + 6 T^{7} - 38 T^{6} - 182 T^{5} + \cdots + 181$$
$17$ $$T^{8} - 2 T^{7} - 42 T^{6} + 86 T^{5} + \cdots - 779$$
$19$ $$T^{8}$$
$23$ $$T^{8} - 20 T^{7} + 80 T^{6} + \cdots + 1525$$
$29$ $$T^{8} + 16 T^{7} + 12 T^{6} + \cdots - 7619$$
$31$ $$T^{8} + 22 T^{7} + 158 T^{6} + \cdots - 919$$
$37$ $$T^{8} - 12 T^{7} - 112 T^{6} + \cdots - 410699$$
$41$ $$T^{8} + 36 T^{7} + 452 T^{6} + \cdots + 515341$$
$43$ $$T^{8} - 190 T^{6} - 80 T^{5} + \cdots + 3305$$
$47$ $$T^{8} - 6 T^{7} - 78 T^{6} + 2 T^{5} + \cdots + 121$$
$53$ $$T^{8} + 16 T^{7} - 48 T^{6} + \cdots - 24979$$
$59$ $$T^{8} + 14 T^{7} - 218 T^{6} + \cdots - 3744859$$
$61$ $$T^{8} - 10 T^{7} - 170 T^{6} + \cdots - 411095$$
$67$ $$T^{8} - 20 T^{7} - 130 T^{6} + \cdots - 4692995$$
$71$ $$T^{8} - 240 T^{6} - 320 T^{5} + \cdots + 14480$$
$73$ $$T^{8} + 18 T^{7} - 2 T^{6} + \cdots - 58519$$
$79$ $$T^{8} + 4 T^{7} - 338 T^{6} + \cdots + 1204961$$
$83$ $$T^{8} - 36 T^{7} + 172 T^{6} + \cdots + 24520336$$
$89$ $$T^{8} + 18 T^{7} - 102 T^{6} + \cdots - 122399$$
$97$ $$T^{8} + 26 T^{7} - 58 T^{6} + \cdots - 7416859$$