# Properties

 Label 273.2.cd.c Level $273$ Weight $2$ Character orbit 273.cd Analytic conductor $2.180$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.cd (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{24}^{6}$$ $$-\zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
44.1
 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 + 0.258819i −0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i
−0.965926 + 0.258819i 1.62484 0.599900i −0.866025 + 0.500000i 0.565826 0.151613i −1.41421 + 1.00000i −2.38014 1.15539i 2.12132 2.12132i 2.28024 1.94949i −0.507306 + 0.292893i
44.2 0.965926 0.258819i 1.10721 + 1.33195i −0.866025 + 0.500000i −3.29788 + 0.883663i 1.41421 + 1.00000i 2.38014 + 1.15539i −2.12132 + 2.12132i −0.548188 + 2.94949i −2.95680 + 1.70711i
86.1 −0.258819 0.965926i 0.599900 + 1.62484i 0.866025 0.500000i 0.883663 + 3.29788i 1.41421 1.00000i 1.15539 2.38014i −2.12132 2.12132i −2.28024 + 1.94949i 2.95680 1.70711i
86.2 0.258819 + 0.965926i −1.33195 + 1.10721i 0.866025 0.500000i −0.151613 0.565826i −1.41421 1.00000i −1.15539 + 2.38014i 2.12132 + 2.12132i 0.548188 2.94949i 0.507306 0.292893i
200.1 −0.258819 + 0.965926i 0.599900 1.62484i 0.866025 + 0.500000i 0.883663 3.29788i 1.41421 + 1.00000i 1.15539 + 2.38014i −2.12132 + 2.12132i −2.28024 1.94949i 2.95680 + 1.70711i
200.2 0.258819 0.965926i −1.33195 1.10721i 0.866025 + 0.500000i −0.151613 + 0.565826i −1.41421 + 1.00000i −1.15539 2.38014i 2.12132 2.12132i 0.548188 + 2.94949i 0.507306 + 0.292893i
242.1 −0.965926 0.258819i 1.62484 + 0.599900i −0.866025 0.500000i 0.565826 + 0.151613i −1.41421 1.00000i −2.38014 + 1.15539i 2.12132 + 2.12132i 2.28024 + 1.94949i −0.507306 0.292893i
242.2 0.965926 + 0.258819i 1.10721 1.33195i −0.866025 0.500000i −3.29788 0.883663i 1.41421 1.00000i 2.38014 1.15539i −2.12132 2.12132i −0.548188 2.94949i −2.95680 1.70711i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 242.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
39.f even 4 1 inner
273.cd even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cd.c 8
3.b odd 2 1 273.2.cd.d yes 8
7.c even 3 1 inner 273.2.cd.c 8
13.d odd 4 1 273.2.cd.d yes 8
21.h odd 6 1 273.2.cd.d yes 8
39.f even 4 1 inner 273.2.cd.c 8
91.z odd 12 1 273.2.cd.d yes 8
273.cd even 12 1 inner 273.2.cd.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.c 8 1.a even 1 1 trivial
273.2.cd.c 8 7.c even 3 1 inner
273.2.cd.c 8 39.f even 4 1 inner
273.2.cd.c 8 273.cd even 12 1 inner
273.2.cd.d yes 8 3.b odd 2 1
273.2.cd.d yes 8 13.d odd 4 1
273.2.cd.d yes 8 21.h odd 6 1
273.2.cd.d yes 8 91.z odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{8} - T_{2}^{4} + 1$$ T2^8 - T2^4 + 1 $$T_{5}^{8} + 4T_{5}^{7} + 8T_{5}^{6} + 48T_{5}^{5} + 92T_{5}^{4} - 96T_{5}^{3} + 32T_{5}^{2} - 32T_{5} + 16$$ T5^8 + 4*T5^7 + 8*T5^6 + 48*T5^5 + 92*T5^4 - 96*T5^3 + 32*T5^2 - 32*T5 + 16 $$T_{19}^{8} + 16 T_{19}^{7} + 128 T_{19}^{6} + 1056 T_{19}^{5} + 7487 T_{19}^{4} + 32736 T_{19}^{3} + 123008 T_{19}^{2} + 476656 T_{19} + 923521$$ T19^8 + 16*T19^7 + 128*T19^6 + 1056*T19^5 + 7487*T19^4 + 32736*T19^3 + 123008*T19^2 + 476656*T19 + 923521

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{4} + 1$$
$3$ $$T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81$$
$5$ $$T^{8} + 4 T^{7} + 8 T^{6} + 48 T^{5} + \cdots + 16$$
$7$ $$T^{8} + 23T^{4} + 2401$$
$11$ $$T^{8} - 8 T^{7} + 32 T^{6} - 272 T^{5} + \cdots + 1$$
$13$ $$(T^{2} - 4 T + 13)^{4}$$
$17$ $$(T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49)^{2}$$
$19$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 923521$$
$23$ $$(T^{4} - 16 T^{3} + 194 T^{2} - 992 T + 3844)^{2}$$
$29$ $$(T^{2} + 1)^{4}$$
$31$ $$T^{8} + 8 T^{7} + 32 T^{6} + 192 T^{5} + \cdots + 256$$
$37$ $$T^{8} - 256 T^{4} + 65536$$
$41$ $$(T^{2} - 8 T + 32)^{4}$$
$43$ $$(T^{2} + 36)^{4}$$
$47$ $$T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 2401$$
$53$ $$T^{8} - 82 T^{6} + 6195 T^{4} + \cdots + 279841$$
$59$ $$T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 25411681$$
$61$ $$(T^{2} - T + 1)^{4}$$
$67$ $$T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 4879681$$
$71$ $$(T^{4} - 16 T^{3} + 128 T^{2} - 112 T + 49)^{2}$$
$73$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 71639296$$
$79$ $$(T^{4} + 2 T^{2} + 4)^{2}$$
$83$ $$(T^{4} + 16 T^{3} + 128 T^{2} - 1088 T + 4624)^{2}$$
$89$ $$T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 9834496$$
$97$ $$(T^{2} + 14 T + 98)^{4}$$