# Properties

 Label 310.2.b.b 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: 8.0.2058981376.2 Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1$$ x^8 - 2*x^7 + 2*x^6 + 18*x^4 - 34*x^3 + 32*x^2 - 8*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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -97\nu^{7} + 102\nu^{6} - 30\nu^{5} - 432\nu^{4} - 1769\nu^{3} + 1637\nu^{2} - 408\nu - 4773 ) / 1631$$ (-97*v^7 + 102*v^6 - 30*v^5 - 432*v^4 - 1769*v^3 + 1637*v^2 - 408*v - 4773) / 1631 $$\beta_{2}$$ $$=$$ $$( 145\nu^{7} + 167\nu^{6} - 241\nu^{5} + 444\nu^{4} + 3132\nu^{3} + 2984\nu^{2} - 3930\nu + 4226 ) / 1631$$ (145*v^7 + 167*v^6 - 241*v^5 + 444*v^4 + 3132*v^3 + 2984*v^2 - 3930*v + 4226) / 1631 $$\beta_{3}$$ $$=$$ $$( 485\nu^{7} - 510\nu^{6} + 150\nu^{5} + 529\nu^{4} + 8845\nu^{3} - 8185\nu^{2} + 2040\nu + 4293 ) / 1631$$ (485*v^7 - 510*v^6 + 150*v^5 + 529*v^4 + 8845*v^3 - 8185*v^2 + 2040*v + 4293) / 1631 $$\beta_{4}$$ $$=$$ $$( -552\nu^{7} + 984\nu^{6} - 961\nu^{5} - 138\nu^{4} - 9966\nu^{3} + 16176\nu^{2} - 15353\nu + 2213 ) / 1631$$ (-552*v^7 + 984*v^6 - 961*v^5 - 138*v^4 - 9966*v^3 + 16176*v^2 - 15353*v + 2213) / 1631 $$\beta_{5}$$ $$=$$ $$( -137\nu^{7} + 305\nu^{6} - 309\nu^{5} + 24\nu^{4} - 2400\nu^{3} + 5281\nu^{2} - 4948\nu + 1236 ) / 233$$ (-137*v^7 + 305*v^6 - 309*v^5 + 24*v^4 - 2400*v^3 + 5281*v^2 - 4948*v + 1236) / 233 $$\beta_{6}$$ $$=$$ $$( -1084\nu^{7} + 2216\nu^{6} - 1899\nu^{5} - 271\nu^{4} - 19500\nu^{3} + 38219\nu^{2} - 30067\nu + 4334 ) / 1631$$ (-1084*v^7 + 2216*v^6 - 1899*v^5 - 271*v^4 - 19500*v^3 + 38219*v^2 - 30067*v + 4334) / 1631 $$\beta_{7}$$ $$=$$ $$( -1661\nu^{7} + 2890\nu^{6} - 2481\nu^{5} - 823\nu^{4} - 30006\nu^{3} + 49100\nu^{2} - 37656\nu + 1769 ) / 1631$$ (-1661*v^7 + 2890*v^6 - 2481*v^5 - 823*v^4 - 30006*v^3 + 49100*v^2 - 37656*v + 1769) / 1631
 $$\nu$$ $$=$$ $$( 2\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2$$ (2*b7 - b6 - b5 - b4 + b3 + b2 - b1 - 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2\beta_{4} - \beta_{2}$$ b5 - 2*b4 - b2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{6} + 5\beta_{5} - 3\beta_{4} - 5\beta_{3} + 3\beta_{2} + 3\beta _1 + 5 ) / 2$$ (-5*b6 + 5*b5 - 3*b4 - 5*b3 + 3*b2 + 3*b1 + 5) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{3} - 5\beta _1 - 12$$ -b3 - 5*b1 - 12 $$\nu^{5}$$ $$=$$ $$( -34\beta_{7} + 23\beta_{6} + 11\beta_{5} + 13\beta_{4} - 23\beta_{3} - 11\beta_{2} + 11\beta _1 + 21 ) / 2$$ (-34*b7 + 23*b6 + 11*b5 + 13*b4 - 23*b3 - 11*b2 + 11*b1 + 21) / 2 $$\nu^{6}$$ $$=$$ $$-\beta_{7} + 7\beta_{6} - 23\beta_{5} + 35\beta_{4} + 22\beta_{2}$$ -b7 + 7*b6 - 23*b5 + 35*b4 + 22*b2 $$\nu^{7}$$ $$=$$ $$( 103\beta_{6} - 105\beta_{5} + 61\beta_{4} + 103\beta_{3} - 43\beta_{2} - 43\beta _1 - 85 ) / 2$$ (103*b6 - 105*b5 + 61*b4 + 103*b3 - 43*b2 - 43*b1 - 85) / 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
 0.148421 + 0.148421i 1.52153 + 1.52153i −1.43917 − 1.43917i 0.769222 + 0.769222i 0.769222 − 0.769222i −1.43917 + 1.43917i 1.52153 − 1.52153i 0.148421 − 0.148421i
1.00000i 1.78165i −1.00000 −0.264435 2.22038i −1.78165 1.70316i 1.00000i −0.174289 −2.22038 + 0.264435i
249.2 1.00000i 0.287336i −1.00000 −0.437190 + 2.19291i −0.287336 1.04306i 1.00000i 2.91744 2.19291 + 0.437190i
249.3 1.00000i 1.55241i −1.00000 −2.23418 0.0917505i 1.55241 4.87834i 1.00000i 0.590025 −0.0917505 + 2.23418i
249.4 1.00000i 2.51658i −1.00000 1.93581 + 1.11922i 2.51658 0.461555i 1.00000i −3.33317 1.11922 1.93581i
249.5 1.00000i 2.51658i −1.00000 1.93581 1.11922i 2.51658 0.461555i 1.00000i −3.33317 1.11922 + 1.93581i
249.6 1.00000i 1.55241i −1.00000 −2.23418 + 0.0917505i 1.55241 4.87834i 1.00000i 0.590025 −0.0917505 2.23418i
249.7 1.00000i 0.287336i −1.00000 −0.437190 2.19291i −0.287336 1.04306i 1.00000i 2.91744 2.19291 0.437190i
249.8 1.00000i 1.78165i −1.00000 −0.264435 + 2.22038i −1.78165 1.70316i 1.00000i −0.174289 −2.22038 0.264435i
 $$n$$: 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.b 8
3.b odd 2 1 2790.2.d.l 8
4.b odd 2 1 2480.2.d.e 8
5.b even 2 1 inner 310.2.b.b 8
5.c odd 4 1 1550.2.a.n 4
5.c odd 4 1 1550.2.a.q 4
15.d odd 2 1 2790.2.d.l 8
20.d odd 2 1 2480.2.d.e 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 12T_{3}^{6} + 44T_{3}^{4} + 52T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(310, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$T^{8} + 12 T^{6} + 44 T^{4} + 52 T^{2} + \cdots + 4$$
$5$ $$T^{8} + 2 T^{7} + 4 T^{6} + 6 T^{5} + \cdots + 625$$
$7$ $$T^{8} + 28 T^{6} + 104 T^{4} + \cdots + 16$$
$11$ $$(T^{4} - 6 T^{3} + 6 T^{2} + 10 T - 2)^{2}$$
$13$ $$T^{8} + 40 T^{6} + 548 T^{4} + \cdots + 3844$$
$17$ $$T^{8} + 24 T^{6} + 152 T^{4} + \cdots + 144$$
$19$ $$(T^{4} + 2 T^{3} - 20 T^{2} - 64 T - 40)^{2}$$
$23$ $$T^{8} + 104 T^{6} + 2632 T^{4} + \cdots + 1296$$
$29$ $$(T^{4} - 4 T^{3} - 34 T^{2} + 198 T - 250)^{2}$$
$31$ $$(T - 1)^{8}$$
$37$ $$T^{8} + 92 T^{6} + 2748 T^{4} + \cdots + 88804$$
$41$ $$(T^{4} + 4 T^{3} - 56 T^{2} + 116 T - 36)^{2}$$
$43$ $$T^{8} + 168 T^{6} + 8148 T^{4} + \cdots + 252004$$
$47$ $$T^{8} + 188 T^{6} + 7272 T^{4} + \cdots + 138384$$
$53$ $$T^{8} + 296 T^{6} + 22116 T^{4} + \cdots + 11236$$
$59$ $$(T^{4} - 176 T^{2} - 608 T + 320)^{2}$$
$61$ $$(T^{4} - 138 T^{2} - 338 T + 1942)^{2}$$
$67$ $$T^{8} + 292 T^{6} + 17552 T^{4} + \cdots + 341056$$
$71$ $$(T^{4} - 12 T^{3} - 96 T^{2} + 640 T + 3504)^{2}$$
$73$ $$T^{8} + 392 T^{6} + \cdots + 28005264$$
$79$ $$(T^{4} - 12 T^{3} + 32 T^{2} + 32 T - 80)^{2}$$
$83$ $$T^{8} + 308 T^{6} + 28588 T^{4} + \cdots + 5008644$$
$89$ $$(T^{4} - 12 T^{3} - 240 T^{2} + 3392 T - 5760)^{2}$$
$97$ $$T^{8} + 788 T^{6} + \cdots + 1177038864$$