# Properties

 Label 83.2.a.b Level $83$ Weight $2$ Character orbit 83.a Self dual yes Analytic conductor $0.663$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$83$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 83.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.662758336777$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.9059636.1 Defining polynomial: $$x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1$$ x^6 - x^5 - 8*x^4 + 11*x^3 + 4*x^2 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{4} - 7\nu^{2} + 4\nu + 2$$ v^4 - 7*v^2 + 4*v + 2 $$\beta_{3}$$ $$=$$ $$\nu^{5} - \nu^{4} - 7\nu^{3} + 12\nu^{2} - 2\nu - 4$$ v^5 - v^4 - 7*v^3 + 12*v^2 - 2*v - 4 $$\beta_{4}$$ $$=$$ $$-\nu^{5} + \nu^{4} + 8\nu^{3} - 11\nu^{2} - 4\nu + 3$$ -v^5 + v^4 + 8*v^3 - 11*v^2 - 4*v + 3 $$\beta_{5}$$ $$=$$ $$-2\nu^{5} + 2\nu^{4} + 15\nu^{3} - 22\nu^{2} - \nu + 4$$ -2*v^5 + 2*v^4 + 15*v^3 - 22*v^2 - v + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 3$$ b5 - b4 + b3 - b1 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} + 7\beta _1 - 2$$ -b5 + 2*b4 + 7*b1 - 2 $$\nu^{4}$$ $$=$$ $$7\beta_{5} - 7\beta_{4} + 7\beta_{3} + \beta_{2} - 11\beta _1 + 19$$ 7*b5 - 7*b4 + 7*b3 + b2 - 11*b1 + 19 $$\nu^{5}$$ $$=$$ $$-12\beta_{5} + 19\beta_{4} - 4\beta_{3} + \beta_{2} + 52\beta _1 - 27$$ -12*b5 + 19*b4 - 4*b3 + b2 + 52*b1 - 27

## 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
 0.705771 2.14357 −0.236470 1.80570 −2.88130 −0.537266
−2.29018 1.96807 3.24494 −1.62860 −4.50724 3.12266 −2.85114 0.873293 3.72978
1.2 −1.66658 −2.04941 0.777479 2.79494 3.41549 3.61008 2.03743 1.20006 −4.65798
1.3 −0.429349 2.76735 −1.81566 1.71413 −1.18816 −3.46533 1.63825 4.65821 −0.735961
1.4 1.16417 1.13227 −0.644717 −1.45742 1.31815 3.35950 −3.07889 −1.71797 −1.69668
1.5 1.59835 −1.32239 0.554733 4.05060 −2.11364 −2.22837 −2.31005 −1.25129 6.47429
1.6 2.62359 −1.49589 4.88322 −3.47366 −3.92460 −1.39854 7.56440 −0.762314 −9.11346
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$83$$ $$-1$$

## 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 83.2.a.b 6
3.b odd 2 1 747.2.a.j 6
4.b odd 2 1 1328.2.a.l 6
5.b even 2 1 2075.2.a.g 6
7.b odd 2 1 4067.2.a.d 6
8.b even 2 1 5312.2.a.bn 6
8.d odd 2 1 5312.2.a.bo 6
83.b odd 2 1 6889.2.a.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.2.a.b 6 1.a even 1 1 trivial
747.2.a.j 6 3.b odd 2 1
1328.2.a.l 6 4.b odd 2 1
2075.2.a.g 6 5.b even 2 1
4067.2.a.d 6 7.b odd 2 1
5312.2.a.bn 6 8.b even 2 1
5312.2.a.bo 6 8.d odd 2 1
6889.2.a.e 6 83.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 7T_{2}^{3} + 20T_{2}^{2} - 12T_{2} - 8$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(83))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} - 9 T^{4} + 7 T^{3} + 20 T^{2} + \cdots - 8$$
$3$ $$T^{6} - T^{5} - 10 T^{4} + 5 T^{3} + \cdots - 25$$
$5$ $$T^{6} - 2 T^{5} - 20 T^{4} + 28 T^{3} + \cdots - 160$$
$7$ $$T^{6} - 3 T^{5} - 22 T^{4} + 55 T^{3} + \cdots - 409$$
$11$ $$T^{6} + 3 T^{5} - 26 T^{4} - 83 T^{3} + \cdots - 113$$
$13$ $$T^{6} - 14 T^{5} + 44 T^{4} + \cdots + 992$$
$17$ $$T^{6} + 5 T^{5} - 20 T^{4} - 77 T^{3} + \cdots - 275$$
$19$ $$T^{6} + 4 T^{5} - 68 T^{4} + \cdots + 6176$$
$23$ $$T^{6} + 5 T^{5} - 61 T^{4} + \cdots + 10912$$
$29$ $$T^{6} + T^{5} - 88 T^{4} - 181 T^{3} + \cdots - 55$$
$31$ $$T^{6} - 3 T^{5} - 66 T^{4} - 93 T^{3} + \cdots - 313$$
$37$ $$T^{6} - 39 T^{5} + 576 T^{4} + \cdots - 91499$$
$41$ $$T^{6} + T^{5} - 47 T^{4} - T^{3} + \cdots - 248$$
$43$ $$T^{6} + 8 T^{5} - 44 T^{4} + \cdots + 6400$$
$47$ $$T^{6} + 12 T^{5} - 96 T^{4} + \cdots + 25952$$
$53$ $$T^{6} - 14 T^{5} - 64 T^{4} + 1064 T^{3} + \cdots - 64$$
$59$ $$T^{6} + 17 T^{5} + 10 T^{4} + \cdots + 3527$$
$61$ $$T^{6} + 5 T^{5} - 208 T^{4} + \cdots - 47347$$
$67$ $$T^{6} - 16 T^{5} - 128 T^{4} + \cdots + 264256$$
$71$ $$T^{6} + 26 T^{5} + 168 T^{4} + \cdots + 7232$$
$73$ $$T^{6} + 6 T^{5} - 268 T^{4} + \cdots - 39136$$
$79$ $$T^{6} + 12 T^{5} - 12 T^{4} + \cdots - 160$$
$83$ $$(T - 1)^{6}$$
$89$ $$T^{6} + 22 T^{5} - 28 T^{4} + \cdots + 144896$$
$97$ $$T^{6} - 6 T^{5} - 300 T^{4} + \cdots - 101120$$