# Properties

 Label 931.2.x.a Level $931$ Weight $2$ Character orbit 931.x Analytic conductor $7.434$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$931 = 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 931.x (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.43407242818$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/931\mathbb{Z}\right)^\times$$.

 $$n$$ $$248$$ $$344$$ $$\chi(n)$$ $$-\zeta_{18}^{3}$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
226.1
 −0.766044 − 0.642788i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 − 0.342020i
−1.03209 0.866025i −0.500000 + 2.83564i −0.0320889 0.181985i 0.152704 0.866025i 2.97178 2.49362i 0 −1.47178 + 2.54920i −4.97178 1.80958i −0.907604 + 0.761570i
557.1 2.37939 + 0.866025i −0.500000 + 0.419550i 3.37939 + 2.83564i −1.03209 + 0.866025i −1.55303 + 0.565258i 0 3.05303 + 5.28801i −0.446967 + 2.53487i −3.20574 + 1.16679i
655.1 −1.03209 + 0.866025i −0.500000 2.83564i −0.0320889 + 0.181985i 0.152704 + 0.866025i 2.97178 + 2.49362i 0 −1.47178 2.54920i −4.97178 + 1.80958i −0.907604 0.761570i
765.1 0.152704 0.866025i −0.500000 + 0.181985i 1.15270 + 0.419550i 2.37939 0.866025i 0.0812519 + 0.460802i 0 1.41875 2.45734i −2.08125 + 1.74638i −0.386659 2.19285i
802.1 0.152704 + 0.866025i −0.500000 0.181985i 1.15270 0.419550i 2.37939 + 0.866025i 0.0812519 0.460802i 0 1.41875 + 2.45734i −2.08125 1.74638i −0.386659 + 2.19285i
814.1 2.37939 0.866025i −0.500000 0.419550i 3.37939 2.83564i −1.03209 0.866025i −1.55303 0.565258i 0 3.05303 5.28801i −0.446967 2.53487i −3.20574 1.16679i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 814.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.w even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.x.a 6
7.b odd 2 1 931.2.x.b 6
7.c even 3 1 19.2.e.a 6
7.c even 3 1 931.2.v.b 6
7.d odd 6 1 931.2.v.a 6
7.d odd 6 1 931.2.w.a 6
19.e even 9 1 931.2.v.b 6
21.h odd 6 1 171.2.u.c 6
28.g odd 6 1 304.2.u.b 6
35.j even 6 1 475.2.l.a 6
35.l odd 12 2 475.2.u.a 12
133.g even 3 1 361.2.e.f 6
133.h even 3 1 361.2.e.g 6
133.j odd 6 1 361.2.e.a 6
133.n odd 6 1 361.2.e.b 6
133.r odd 6 1 361.2.e.h 6
133.u even 9 1 19.2.e.a 6
133.u even 9 1 361.2.e.f 6
133.u even 9 1 361.2.e.g 6
133.w even 9 1 361.2.a.g 3
133.w even 9 2 361.2.c.i 6
133.w even 9 1 inner 931.2.x.a 6
133.x odd 18 1 931.2.w.a 6
133.y odd 18 1 931.2.v.a 6
133.z odd 18 1 931.2.x.b 6
133.bd odd 18 1 361.2.e.a 6
133.bd odd 18 1 361.2.e.b 6
133.bd odd 18 1 361.2.e.h 6
133.be odd 18 1 361.2.a.h 3
133.be odd 18 2 361.2.c.h 6
399.br even 18 1 3249.2.a.s 3
399.ca odd 18 1 3249.2.a.z 3
399.ch odd 18 1 171.2.u.c 6
532.bs even 18 1 5776.2.a.bi 3
532.bt odd 18 1 5776.2.a.br 3
532.cf odd 18 1 304.2.u.b 6
665.ct odd 18 1 9025.2.a.x 3
665.db even 18 1 475.2.l.a 6
665.dc even 18 1 9025.2.a.bd 3
665.dj odd 36 2 475.2.u.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 7.c even 3 1
19.2.e.a 6 133.u even 9 1
171.2.u.c 6 21.h odd 6 1
171.2.u.c 6 399.ch odd 18 1
304.2.u.b 6 28.g odd 6 1
304.2.u.b 6 532.cf odd 18 1
361.2.a.g 3 133.w even 9 1
361.2.a.h 3 133.be odd 18 1
361.2.c.h 6 133.be odd 18 2
361.2.c.i 6 133.w even 9 2
361.2.e.a 6 133.j odd 6 1
361.2.e.a 6 133.bd odd 18 1
361.2.e.b 6 133.n odd 6 1
361.2.e.b 6 133.bd odd 18 1
361.2.e.f 6 133.g even 3 1
361.2.e.f 6 133.u even 9 1
361.2.e.g 6 133.h even 3 1
361.2.e.g 6 133.u even 9 1
361.2.e.h 6 133.r odd 6 1
361.2.e.h 6 133.bd odd 18 1
475.2.l.a 6 35.j even 6 1
475.2.l.a 6 665.db even 18 1
475.2.u.a 12 35.l odd 12 2
475.2.u.a 12 665.dj odd 36 2
931.2.v.a 6 7.d odd 6 1
931.2.v.a 6 133.y odd 18 1
931.2.v.b 6 7.c even 3 1
931.2.v.b 6 19.e even 9 1
931.2.w.a 6 7.d odd 6 1
931.2.w.a 6 133.x odd 18 1
931.2.x.a 6 1.a even 1 1 trivial
931.2.x.a 6 133.w even 9 1 inner
931.2.x.b 6 7.b odd 2 1
931.2.x.b 6 133.z odd 18 1
3249.2.a.s 3 399.br even 18 1
3249.2.a.z 3 399.ca odd 18 1
5776.2.a.bi 3 532.bs even 18 1
5776.2.a.br 3 532.bt odd 18 1
9025.2.a.x 3 665.ct odd 18 1
9025.2.a.bd 3 665.dc even 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(931, [\chi])$$:

 $$T_{2}^{6} - 3T_{2}^{5} + 3T_{2}^{3} + 9T_{2}^{2} + 9$$ T2^6 - 3*T2^5 + 3*T2^3 + 9*T2^2 + 9 $$T_{3}^{6} + 3T_{3}^{5} + 12T_{3}^{4} + 19T_{3}^{3} + 15T_{3}^{2} + 6T_{3} + 1$$ T3^6 + 3*T3^5 + 12*T3^4 + 19*T3^3 + 15*T3^2 + 6*T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9$$
$3$ $$T^{6} + 3 T^{5} + 12 T^{4} + 19 T^{3} + \cdots + 1$$
$5$ $$T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9$$
$7$ $$T^{6}$$
$11$ $$(T^{3} - 9 T - 9)^{2}$$
$13$ $$T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1369$$
$17$ $$T^{6} - 3 T^{5} + 18 T^{4} - 24 T^{3} + \cdots + 9$$
$19$ $$T^{6} - 6 T^{5} + 42 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 6 T^{5} + 24 T^{3} + 144 T^{2} + \cdots + 576$$
$29$ $$T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 12321$$
$31$ $$T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809$$
$37$ $$T^{6} + 21 T^{4} - 34 T^{3} + \cdots + 289$$
$41$ $$T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 12321$$
$43$ $$T^{6} + 3 T^{5} + 60 T^{4} + \cdots + 26569$$
$47$ $$T^{6} - 15 T^{5} + 72 T^{4} - 84 T^{3} + \cdots + 9$$
$53$ $$T^{6} - 15 T^{5} + 90 T^{4} + \cdots + 2601$$
$59$ $$T^{6} + 6 T^{5} - 18 T^{4} + \cdots + 71289$$
$61$ $$T^{6} - 24 T^{5} + 276 T^{4} + \cdots + 32761$$
$67$ $$T^{6} - 24 T^{5} + 456 T^{4} + \cdots + 179776$$
$71$ $$T^{6} + 6 T^{5} - 36 T^{4} + \cdots + 788544$$
$73$ $$T^{6} - 24 T^{5} + 240 T^{4} + \cdots + 4096$$
$79$ $$T^{6} - 15 T^{5} + 87 T^{4} + \cdots + 654481$$
$83$ $$T^{6} + 189 T^{4} + 918 T^{3} + \cdots + 210681$$
$89$ $$T^{6} + 3 T^{5} + 99 T^{4} + \cdots + 3249$$
$97$ $$T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 16129$$