Properties

 Label 310.2.b.a Level $310$ Weight $2$ Character orbit 310.b Analytic conductor $2.475$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$310 = 2 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 310.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.47536246266$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.619810816.2 Defining polynomial: $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319$$ (64*v^7 + 16*v^6 + 4*v^5 - 127*v^4 + 944*v^3 - 276*v^2 + 378*v + 63) / 319 $$\beta_{2}$$ $$=$$ $$( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319$$ (-63*v^7 + 64*v^6 + 16*v^5 + 130*v^4 - 1009*v^3 + 1448*v^2 - 402*v - 67) / 319 $$\beta_{3}$$ $$=$$ $$( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319$$ (-67*v^7 + 63*v^6 - 64*v^5 + 118*v^4 - 1068*v^3 + 1545*v^2 - 1263*v + 268) / 319 $$\beta_{4}$$ $$=$$ $$( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319$$ (83*v^7 - 59*v^6 + 65*v^5 - 70*v^4 + 1304*v^3 - 1614*v^2 + 1198*v + 306) / 319 $$\beta_{5}$$ $$=$$ $$( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319$$ (-172*v^7 - 43*v^6 + 69*v^5 + 441*v^4 - 2218*v^3 + 662*v^2 + 619*v - 269) / 319 $$\beta_{6}$$ $$=$$ $$( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319$$ (-196*v^7 - 49*v^6 - 92*v^5 + 369*v^4 - 2572*v^3 + 1244*v^2 - 1038*v - 173) / 319 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2$$ v^7 - 2*v^4 + 14*v^3 - 8*v^2 + v + 2
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2$$ (-b7 - b5 + b4 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2}$$ b6 - b5 + b4 - b3 + 2*b2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2$$ (3*b7 + 5*b5 - 5*b4 - 2*b2 + 3*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7$$ -b7 - b5 + 5*b4 + 4*b3 - 7 $$\nu^{5}$$ $$=$$ $$( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2$$ (11*b7 + 2*b6 + 9*b5 - 11*b4 - 12*b3 + 12*b2 - 11*b1 + 12) / 2 $$\nu^{6}$$ $$=$$ $$-15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1$$ -15*b6 + 16*b5 - 16*b4 + 16*b3 - 28*b2 + 7*b1 $$\nu^{7}$$ $$=$$ $$( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2$$ (-43*b7 + 16*b6 - 89*b5 + 105*b4 + 60*b2 - 43*b1 - 60) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/310\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$251$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −1.49094 − 1.49094i 1.18254 + 1.18254i 0.561103 + 0.561103i −0.252709 − 0.252709i −0.252709 + 0.252709i 0.561103 − 0.561103i 1.18254 − 1.18254i −1.49094 + 1.49094i
1.00000i 3.44579i −1.00000 −1.82630 + 1.29021i −3.44579 2.40146i 1.00000i −8.87345 1.29021 + 1.82630i
249.2 1.00000i 1.79682i −1.00000 1.60536 + 1.55654i −1.79682 3.47817i 1.00000i −0.228545 1.55654 1.60536i
249.3 1.00000i 0.370326i −1.00000 1.45220 1.70032i 0.370326 4.27844i 1.00000i 2.86286 −1.70032 1.45220i
249.4 1.00000i 0.872276i −1.00000 −2.23127 0.146426i 0.872276 2.79827i 1.00000i 2.23913 −0.146426 + 2.23127i
249.5 1.00000i 0.872276i −1.00000 −2.23127 + 0.146426i 0.872276 2.79827i 1.00000i 2.23913 −0.146426 2.23127i
249.6 1.00000i 0.370326i −1.00000 1.45220 + 1.70032i 0.370326 4.27844i 1.00000i 2.86286 −1.70032 + 1.45220i
249.7 1.00000i 1.79682i −1.00000 1.60536 1.55654i −1.79682 3.47817i 1.00000i −0.228545 1.55654 + 1.60536i
249.8 1.00000i 3.44579i −1.00000 −1.82630 1.29021i −3.44579 2.40146i 1.00000i −8.87345 1.29021 1.82630i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 249.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 310.2.b.a 8
3.b odd 2 1 2790.2.d.m 8
4.b odd 2 1 2480.2.d.d 8
5.b even 2 1 inner 310.2.b.a 8
5.c odd 4 1 1550.2.a.o 4
5.c odd 4 1 1550.2.a.p 4
15.d odd 2 1 2790.2.d.m 8
20.d odd 2 1 2480.2.d.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
310.2.b.a 8 1.a even 1 1 trivial
310.2.b.a 8 5.b even 2 1 inner
1550.2.a.o 4 5.c odd 4 1
1550.2.a.p 4 5.c odd 4 1
2480.2.d.d 8 4.b odd 2 1
2480.2.d.d 8 20.d odd 2 1
2790.2.d.m 8 3.b odd 2 1
2790.2.d.m 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 16T_{3}^{6} + 52T_{3}^{4} + 36T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(310, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$T^{8} + 16 T^{6} + 52 T^{4} + 36 T^{2} + \cdots + 4$$
$5$ $$T^{8} + 2 T^{7} - 4 T^{6} + 6 T^{5} + \cdots + 625$$
$7$ $$T^{8} + 44 T^{6} + 680 T^{4} + \cdots + 10000$$
$11$ $$(T^{4} + 8 T^{3} + 2 T^{2} - 62 T - 82)^{2}$$
$13$ $$T^{8} + 84 T^{6} + 2044 T^{4} + \cdots + 36100$$
$17$ $$T^{8} + 128 T^{6} + 5256 T^{4} + \cdots + 2704$$
$19$ $$(T^{4} + 6 T^{3} + 4 T^{2} - 16 T - 8)^{2}$$
$23$ $$T^{8} + 32 T^{6} + 184 T^{4} + \cdots + 16$$
$29$ $$(T^{4} - 6 T^{3} - 38 T^{2} + 202 T - 2)^{2}$$
$31$ $$(T + 1)^{8}$$
$37$ $$T^{8} + 152 T^{6} + 4804 T^{4} + \cdots + 37636$$
$41$ $$(T^{4} - 104 T^{2} - 44 T + 1468)^{2}$$
$43$ $$T^{8} + 172 T^{6} + 8044 T^{4} + \cdots + 6724$$
$47$ $$T^{8} + 172 T^{6} + 8584 T^{4} + \cdots + 795664$$
$53$ $$T^{8} + 148 T^{6} + 6364 T^{4} + \cdots + 3364$$
$59$ $$(T^{4} - 4 T^{3} - 64 T^{2} - 128 T - 64)^{2}$$
$61$ $$(T^{4} - 2 T^{3} - 62 T^{2} + 10 T + 574)^{2}$$
$67$ $$T^{8} + 308 T^{6} + \cdots + 16321600$$
$71$ $$(T^{4} + 12 T^{3} + 32 T^{2} - 16)^{2}$$
$73$ $$T^{8} + 400 T^{6} + 42712 T^{4} + \cdots + 2999824$$
$79$ $$(T^{4} - 16 T^{3} - 80 T^{2} + 1328 T + 2800)^{2}$$
$83$ $$T^{8} + 168 T^{6} + 8308 T^{4} + \cdots + 36100$$
$89$ $$(T^{4} + 12 T^{3} - 176 T^{2} + \cdots + 10048)^{2}$$
$97$ $$T^{8} + 436 T^{6} + 53528 T^{4} + \cdots + 4080400$$