# Properties

 Label 722.2.c.m Level $722$ Weight $2$ Character orbit 722.c Analytic conductor $5.765$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.76519902594$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.324000000.2 Defining polynomial: $$x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25$$ x^8 + 5*x^6 + 20*x^4 + 25*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{6} - 15 ) / 20$$ (v^6 - 15) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - 35\nu ) / 20$$ (v^7 - 35*v) / 20 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 4\nu^{4} + 20\nu^{2} + 5 ) / 20$$ (v^6 + 4*v^4 + 20*v^2 + 5) / 20 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 4\nu^{5} + 20\nu^{3} + 5\nu ) / 20$$ (v^7 + 4*v^5 + 20*v^3 + 5*v) / 20 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 8\nu^{4} + 20\nu^{2} + 25 ) / 20$$ (v^6 + 8*v^4 + 20*v^2 + 25) / 20 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 6\nu^{5} + 20\nu^{3} + 25\nu ) / 10$$ (v^7 + 6*v^5 + 20*v^3 + 25*v) / 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{6} + 2\beta_{4} - \beta_{2}$$ -b6 + 2*b4 - b2 $$\nu^{3}$$ $$=$$ $$-\beta_{7} + 3\beta_{5} - \beta_{3}$$ -b7 + 3*b5 - b3 $$\nu^{4}$$ $$=$$ $$5\beta_{6} - 5\beta_{4} - 5$$ 5*b6 - 5*b4 - 5 $$\nu^{5}$$ $$=$$ $$5\beta_{7} - 10\beta_{5} - 10\beta_1$$ 5*b7 - 10*b5 - 10*b1 $$\nu^{6}$$ $$=$$ $$20\beta_{2} + 15$$ 20*b2 + 15 $$\nu^{7}$$ $$=$$ $$20\beta_{3} + 35\beta_1$$ 20*b3 + 35*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/722\mathbb{Z}\right)^\times$$.

 $$n$$ $$363$$ $$\chi(n)$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
429.1
 −0.951057 + 1.64728i 0.587785 − 1.01807i 0.951057 − 1.64728i −0.587785 + 1.01807i −0.951057 − 1.64728i 0.587785 + 1.01807i 0.951057 + 1.64728i −0.587785 − 1.01807i
−0.500000 + 0.866025i −1.39680 + 2.41933i −0.500000 0.866025i 1.17229 2.03046i −1.39680 2.41933i −1.28408 1.00000 −2.40211 4.16058i 1.17229 + 2.03046i
429.2 −0.500000 + 0.866025i −0.642040 + 1.11205i −0.500000 0.866025i −1.84786 + 3.20059i −0.642040 1.11205i −0.442463 1.00000 0.675571 + 1.17012i −1.84786 3.20059i
429.3 −0.500000 + 0.866025i −0.221232 + 0.383185i −0.500000 0.866025i 0.445746 0.772054i −0.221232 0.383185i 2.52015 1.00000 1.40211 + 2.42853i 0.445746 + 0.772054i
429.4 −0.500000 + 0.866025i 1.26007 2.18251i −0.500000 0.866025i 1.22982 2.13012i 1.26007 + 2.18251i −2.79360 1.00000 −1.67557 2.90217i 1.22982 + 2.13012i
653.1 −0.500000 0.866025i −1.39680 2.41933i −0.500000 + 0.866025i 1.17229 + 2.03046i −1.39680 + 2.41933i −1.28408 1.00000 −2.40211 + 4.16058i 1.17229 2.03046i
653.2 −0.500000 0.866025i −0.642040 1.11205i −0.500000 + 0.866025i −1.84786 3.20059i −0.642040 + 1.11205i −0.442463 1.00000 0.675571 1.17012i −1.84786 + 3.20059i
653.3 −0.500000 0.866025i −0.221232 0.383185i −0.500000 + 0.866025i 0.445746 + 0.772054i −0.221232 + 0.383185i 2.52015 1.00000 1.40211 2.42853i 0.445746 0.772054i
653.4 −0.500000 0.866025i 1.26007 + 2.18251i −0.500000 + 0.866025i 1.22982 + 2.13012i 1.26007 2.18251i −2.79360 1.00000 −1.67557 + 2.90217i 1.22982 2.13012i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 653.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.c.m 8
19.b odd 2 1 722.2.c.n 8
19.c even 3 1 722.2.a.n yes 4
19.c even 3 1 inner 722.2.c.m 8
19.d odd 6 1 722.2.a.m 4
19.d odd 6 1 722.2.c.n 8
19.e even 9 6 722.2.e.r 24
19.f odd 18 6 722.2.e.s 24
57.f even 6 1 6498.2.a.ca 4
57.h odd 6 1 6498.2.a.bx 4
76.f even 6 1 5776.2.a.bv 4
76.g odd 6 1 5776.2.a.bt 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m 4 19.d odd 6 1
722.2.a.n yes 4 19.c even 3 1
722.2.c.m 8 1.a even 1 1 trivial
722.2.c.m 8 19.c even 3 1 inner
722.2.c.n 8 19.b odd 2 1
722.2.c.n 8 19.d odd 6 1
722.2.e.r 24 19.e even 9 6
722.2.e.s 24 19.f odd 18 6
5776.2.a.bt 4 76.g odd 6 1
5776.2.a.bv 4 76.f even 6 1
6498.2.a.bx 4 57.h odd 6 1
6498.2.a.ca 4 57.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(722, [\chi])$$:

 $$T_{3}^{8} + 2T_{3}^{7} + 10T_{3}^{6} + 12T_{3}^{5} + 64T_{3}^{4} + 88T_{3}^{3} + 120T_{3}^{2} + 48T_{3} + 16$$ T3^8 + 2*T3^7 + 10*T3^6 + 12*T3^5 + 64*T3^4 + 88*T3^3 + 120*T3^2 + 48*T3 + 16 $$T_{5}^{8} - 2T_{5}^{7} + 15T_{5}^{6} - 42T_{5}^{5} + 204T_{5}^{4} - 428T_{5}^{3} + 815T_{5}^{2} - 608T_{5} + 361$$ T5^8 - 2*T5^7 + 15*T5^6 - 42*T5^5 + 204*T5^4 - 428*T5^3 + 815*T5^2 - 608*T5 + 361 $$T_{7}^{4} + 2T_{7}^{3} - 6T_{7}^{2} - 12T_{7} - 4$$ T7^4 + 2*T7^3 - 6*T7^2 - 12*T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{4}$$
$3$ $$T^{8} + 2 T^{7} + 10 T^{6} + 12 T^{5} + \cdots + 16$$
$5$ $$T^{8} - 2 T^{7} + 15 T^{6} - 42 T^{5} + \cdots + 361$$
$7$ $$(T^{4} + 2 T^{3} - 6 T^{2} - 12 T - 4)^{2}$$
$11$ $$(T^{4} - 2 T^{3} - 26 T^{2} + 12 T + 76)^{2}$$
$13$ $$T^{8} + 18 T^{7} + 215 T^{6} + \cdots + 3721$$
$17$ $$T^{8} + 6 T^{7} + 45 T^{6} + 94 T^{5} + \cdots + 361$$
$19$ $$T^{8}$$
$23$ $$T^{8} - 10 T^{7} + 150 T^{6} + \cdots + 1488400$$
$29$ $$T^{8} - 2 T^{7} + 35 T^{6} - 122 T^{5} + \cdots + 361$$
$31$ $$(T^{4} - 26 T^{3} + 246 T^{2} - 996 T + 1436)^{2}$$
$37$ $$(T^{4} - 4 T^{3} - 19 T^{2} + 46 T - 19)^{2}$$
$41$ $$T^{8} - 12 T^{7} + 125 T^{6} + \cdots + 128881$$
$43$ $$T^{8} - 10 T^{7} + 90 T^{6} + \cdots + 10000$$
$47$ $$T^{8} - 12 T^{7} + 140 T^{6} + \cdots + 5776$$
$53$ $$T^{8} + 8 T^{7} + 95 T^{6} + \cdots + 32761$$
$59$ $$T^{8} - 8 T^{7} + 60 T^{6} + \cdots + 26896$$
$61$ $$T^{8} + 115 T^{6} + 300 T^{5} + \cdots + 1050625$$
$67$ $$T^{8} + 10 T^{7} + 130 T^{6} + \cdots + 400$$
$71$ $$T^{8} + 100 T^{6} - 720 T^{5} + \cdots + 400$$
$73$ $$T^{8} - 14 T^{7} + 145 T^{6} + \cdots + 19321$$
$79$ $$T^{8} + 22 T^{7} + 390 T^{6} + \cdots + 22316176$$
$83$ $$(T^{4} + 12 T^{3} - 196 T^{2} - 2832 T - 6884)^{2}$$
$89$ $$T^{8} - 16 T^{7} + 285 T^{6} + \cdots + 1343281$$
$97$ $$T^{8} + 28 T^{7} + 685 T^{6} + \cdots + 63664441$$