Properties

 Label 114.2.i.c Level $114$ Weight $2$ Character orbit 114.i Analytic conductor $0.910$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.i (of order $$9$$, degree $$6$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.910294583043$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i
0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i −0.613341 3.47843i 0.939693 0.342020i −1.85844 + 3.21891i −0.500000 0.866025i 0.766044 + 0.642788i −2.70574 2.27038i
43.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i −0.0923963 0.0775297i −0.173648 0.984808i 2.14543 + 3.71599i −0.500000 + 0.866025i −0.939693 0.342020i 0.113341 + 0.0412527i
55.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i 2.20574 0.802823i −0.766044 + 0.642788i −1.78699 3.09516i −0.500000 + 0.866025i 0.173648 + 0.984808i −0.407604 2.31164i
61.1 −0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i −0.0923963 + 0.0775297i −0.173648 + 0.984808i 2.14543 3.71599i −0.500000 0.866025i −0.939693 + 0.342020i 0.113341 0.0412527i
73.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i −0.613341 + 3.47843i 0.939693 + 0.342020i −1.85844 3.21891i −0.500000 + 0.866025i 0.766044 0.642788i −2.70574 + 2.27038i
85.1 0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 2.20574 + 0.802823i −0.766044 0.642788i −1.78699 + 3.09516i −0.500000 0.866025i 0.173648 0.984808i −0.407604 + 2.31164i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 85.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.i.c 6
3.b odd 2 1 342.2.u.b 6
4.b odd 2 1 912.2.bo.d 6
19.e even 9 1 inner 114.2.i.c 6
19.e even 9 1 2166.2.a.r 3
19.f odd 18 1 2166.2.a.p 3
57.j even 18 1 6498.2.a.bu 3
57.l odd 18 1 342.2.u.b 6
57.l odd 18 1 6498.2.a.bp 3
76.l odd 18 1 912.2.bo.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 1.a even 1 1 trivial
114.2.i.c 6 19.e even 9 1 inner
342.2.u.b 6 3.b odd 2 1
342.2.u.b 6 57.l odd 18 1
912.2.bo.d 6 4.b odd 2 1
912.2.bo.d 6 76.l odd 18 1
2166.2.a.p 3 19.f odd 18 1
2166.2.a.r 3 19.e even 9 1
6498.2.a.bp 3 57.l odd 18 1
6498.2.a.bu 3 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 3T_{5}^{5} + 12T_{5}^{4} - 46T_{5}^{3} + 60T_{5}^{2} + 12T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6} - T^{3} + 1$$
$5$ $$T^{6} - 3 T^{5} + 12 T^{4} - 46 T^{3} + \cdots + 1$$
$7$ $$T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249$$
$11$ $$T^{6} + 21 T^{4} + 74 T^{3} + \cdots + 1369$$
$13$ $$T^{6} - 9 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 1$$
$17$ $$T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9$$
$19$ $$T^{6} + 107T^{3} + 6859$$
$23$ $$T^{6} - 27 T^{5} + 378 T^{4} + \cdots + 157609$$
$29$ $$T^{6} + 3 T^{5} + 12 T^{4} + 46 T^{3} + \cdots + 1$$
$31$ $$T^{6} + 15 T^{5} + 153 T^{4} + \cdots + 11881$$
$37$ $$(T^{3} + 3 T^{2} - 45 T + 17)^{2}$$
$41$ $$T^{6} + 15 T^{5} + 87 T^{4} + \cdots + 2809$$
$43$ $$T^{6} - 3 T^{5} - 3 T^{4} + \cdots + 63001$$
$47$ $$T^{6} - 15 T^{5} + 51 T^{4} + \cdots + 5041$$
$53$ $$T^{6} - 6 T^{5} - 30 T^{4} + \cdots + 395641$$
$59$ $$T^{6} + 27 T^{5} + 198 T^{4} + \cdots + 81$$
$61$ $$T^{6} + 15 T^{5} + 84 T^{4} + \cdots + 289$$
$67$ $$T^{6} + 3 T^{5} + 72 T^{4} + \cdots + 7017201$$
$71$ $$T^{6} - 3 T^{5} - 87 T^{4} + \cdots + 26569$$
$73$ $$T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249$$
$79$ $$T^{6} - 27 T^{5} + 351 T^{4} + \cdots + 32761$$
$83$ $$T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 2601$$
$89$ $$T^{6} - 42 T^{5} + 756 T^{4} + \cdots + 239121$$
$97$ $$T^{6} - 18 T^{5} + 171 T^{4} + \cdots + 218089$$
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