# Properties

 Label 361.2.e.f Level $361$ Weight $2$ Character orbit 361.e Analytic conductor $2.883$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.e (of order $$9$$, degree $$6$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.88259951297$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i
−0.439693 2.49362i −0.500000 + 0.419550i −4.14543 + 1.50881i 1.26604 + 0.460802i 1.26604 + 1.06234i −0.766044 + 1.32683i 3.05303 + 5.28801i −0.446967 + 2.53487i 0.592396 3.35965i
54.1 0.673648 + 0.565258i −0.500000 + 0.181985i −0.213011 1.20805i −0.439693 + 2.49362i −0.439693 0.160035i 0.939693 + 1.62760i 1.41875 2.45734i −2.08125 + 1.74638i −1.70574 + 1.43128i
62.1 1.26604 0.460802i −0.500000 + 2.83564i −0.141559 + 0.118782i 0.673648 + 0.565258i 0.673648 + 3.82045i −0.173648 0.300767i −1.47178 + 2.54920i −4.97178 1.80958i 1.11334 + 0.405223i
99.1 1.26604 + 0.460802i −0.500000 2.83564i −0.141559 0.118782i 0.673648 0.565258i 0.673648 3.82045i −0.173648 + 0.300767i −1.47178 2.54920i −4.97178 + 1.80958i 1.11334 0.405223i
234.1 0.673648 0.565258i −0.500000 0.181985i −0.213011 + 1.20805i −0.439693 2.49362i −0.439693 + 0.160035i 0.939693 1.62760i 1.41875 + 2.45734i −2.08125 1.74638i −1.70574 1.43128i
245.1 −0.439693 + 2.49362i −0.500000 0.419550i −4.14543 1.50881i 1.26604 0.460802i 1.26604 1.06234i −0.766044 1.32683i 3.05303 5.28801i −0.446967 2.53487i 0.592396 + 3.35965i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 245.1 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 361.2.e.f 6
19.b odd 2 1 361.2.e.b 6
19.c even 3 1 19.2.e.a 6
19.c even 3 1 361.2.e.g 6
19.d odd 6 1 361.2.e.a 6
19.d odd 6 1 361.2.e.h 6
19.e even 9 1 19.2.e.a 6
19.e even 9 1 361.2.a.g 3
19.e even 9 2 361.2.c.i 6
19.e even 9 1 inner 361.2.e.f 6
19.e even 9 1 361.2.e.g 6
19.f odd 18 1 361.2.a.h 3
19.f odd 18 2 361.2.c.h 6
19.f odd 18 1 361.2.e.a 6
19.f odd 18 1 361.2.e.b 6
19.f odd 18 1 361.2.e.h 6
57.h odd 6 1 171.2.u.c 6
57.j even 18 1 3249.2.a.s 3
57.l odd 18 1 171.2.u.c 6
57.l odd 18 1 3249.2.a.z 3
76.g odd 6 1 304.2.u.b 6
76.k even 18 1 5776.2.a.bi 3
76.l odd 18 1 304.2.u.b 6
76.l odd 18 1 5776.2.a.br 3
95.i even 6 1 475.2.l.a 6
95.m odd 12 2 475.2.u.a 12
95.o odd 18 1 9025.2.a.x 3
95.p even 18 1 475.2.l.a 6
95.p even 18 1 9025.2.a.bd 3
95.q odd 36 2 475.2.u.a 12
133.g even 3 1 931.2.x.a 6
133.h even 3 1 931.2.v.b 6
133.k odd 6 1 931.2.x.b 6
133.m odd 6 1 931.2.w.a 6
133.t odd 6 1 931.2.v.a 6
133.u even 9 1 931.2.x.a 6
133.w even 9 1 931.2.v.b 6
133.x odd 18 1 931.2.x.b 6
133.y odd 18 1 931.2.w.a 6
133.z odd 18 1 931.2.v.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 19.c even 3 1
19.2.e.a 6 19.e even 9 1
171.2.u.c 6 57.h odd 6 1
171.2.u.c 6 57.l odd 18 1
304.2.u.b 6 76.g odd 6 1
304.2.u.b 6 76.l odd 18 1
361.2.a.g 3 19.e even 9 1
361.2.a.h 3 19.f odd 18 1
361.2.c.h 6 19.f odd 18 2
361.2.c.i 6 19.e even 9 2
361.2.e.a 6 19.d odd 6 1
361.2.e.a 6 19.f odd 18 1
361.2.e.b 6 19.b odd 2 1
361.2.e.b 6 19.f odd 18 1
361.2.e.f 6 1.a even 1 1 trivial
361.2.e.f 6 19.e even 9 1 inner
361.2.e.g 6 19.c even 3 1
361.2.e.g 6 19.e even 9 1
361.2.e.h 6 19.d odd 6 1
361.2.e.h 6 19.f odd 18 1
475.2.l.a 6 95.i even 6 1
475.2.l.a 6 95.p even 18 1
475.2.u.a 12 95.m odd 12 2
475.2.u.a 12 95.q odd 36 2
931.2.v.a 6 133.t odd 6 1
931.2.v.a 6 133.z odd 18 1
931.2.v.b 6 133.h even 3 1
931.2.v.b 6 133.w even 9 1
931.2.w.a 6 133.m odd 6 1
931.2.w.a 6 133.y odd 18 1
931.2.x.a 6 133.g even 3 1
931.2.x.a 6 133.u even 9 1
931.2.x.b 6 133.k odd 6 1
931.2.x.b 6 133.x odd 18 1
3249.2.a.s 3 57.j even 18 1
3249.2.a.z 3 57.l odd 18 1
5776.2.a.bi 3 76.k even 18 1
5776.2.a.br 3 76.l odd 18 1
9025.2.a.x 3 95.o odd 18 1
9025.2.a.bd 3 95.p even 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(361, [\chi])$$:

 $$T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9$$ T2^6 - 3*T2^5 + 9*T2^4 - 24*T2^3 + 36*T2^2 - 27*T2 + 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

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9$$
$3$ $$T^{6} + 3 T^{5} + 12 T^{4} + 19 T^{3} + \cdots + 1$$
$5$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9$$
$7$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$11$ $$T^{6} + 9 T^{4} - 18 T^{3} + 81 T^{2} + \cdots + 81$$
$13$ $$T^{6} - 15 T^{5} + 96 T^{4} + \cdots + 1369$$
$17$ $$T^{6} - 3 T^{5} + 18 T^{4} - 24 T^{3} + \cdots + 9$$
$19$ $$T^{6}$$
$23$ $$T^{6} - 6 T^{5} + 24 T^{3} + 144 T^{2} + \cdots + 576$$
$29$ $$T^{6} - 6 T^{5} - 84 T^{3} + \cdots + 12321$$
$31$ $$T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809$$
$37$ $$(T^{3} - 21 T - 17)^{2}$$
$41$ $$T^{6} + 6 T^{5} - 9 T^{4} + \cdots + 12321$$
$43$ $$T^{6} - 24 T^{5} + 249 T^{4} + \cdots + 26569$$
$47$ $$T^{6} - 15 T^{5} + 72 T^{4} - 84 T^{3} + \cdots + 9$$
$53$ $$T^{6} + 12 T^{5} + 99 T^{4} + \cdots + 2601$$
$59$ $$T^{6} + 6 T^{5} - 18 T^{4} + \cdots + 71289$$
$61$ $$T^{6} - 24 T^{5} + 276 T^{4} + \cdots + 32761$$
$67$ $$T^{6} - 6 T^{5} - 48 T^{4} + \cdots + 179776$$
$71$ $$T^{6} + 6 T^{5} + 144 T^{4} + \cdots + 788544$$
$73$ $$T^{6} - 24 T^{5} + 240 T^{4} + \cdots + 4096$$
$79$ $$T^{6} - 24 T^{5} + 366 T^{4} + \cdots + 654481$$
$83$ $$T^{6} + 189 T^{4} + 918 T^{3} + \cdots + 210681$$
$89$ $$T^{6} + 3 T^{5} + 99 T^{4} + \cdots + 3249$$
$97$ $$T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 16129$$
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