# Properties

 Label 418.2.f.a Level $418$ Weight $2$ Character orbit 418.f Analytic conductor $3.338$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
115.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i −0.309017 − 0.951057i 0.809017 + 0.587785i
0.309017 + 0.951057i −1.00000 0.726543i −0.809017 + 0.587785i −0.881966 + 2.71441i 0.381966 1.17557i −2.30902 + 1.67760i −0.809017 0.587785i −0.454915 1.40008i −2.85410
191.1 −0.809017 0.587785i −1.00000 + 3.07768i 0.309017 + 0.951057i −3.11803 + 2.26538i 2.61803 1.90211i −1.19098 3.66547i 0.309017 0.951057i −6.04508 4.39201i 3.85410
229.1 0.309017 0.951057i −1.00000 + 0.726543i −0.809017 0.587785i −0.881966 2.71441i 0.381966 + 1.17557i −2.30902 1.67760i −0.809017 + 0.587785i −0.454915 + 1.40008i −2.85410
267.1 −0.809017 + 0.587785i −1.00000 3.07768i 0.309017 0.951057i −3.11803 2.26538i 2.61803 + 1.90211i −1.19098 + 3.66547i 0.309017 + 0.951057i −6.04508 + 4.39201i 3.85410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.f.a 4
11.c even 5 1 inner 418.2.f.a 4
11.c even 5 1 4598.2.a.be 2
11.d odd 10 1 4598.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.a 4 1.a even 1 1 trivial
418.2.f.a 4 11.c even 5 1 inner
4598.2.a.v 2 11.d odd 10 1
4598.2.a.be 2 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 4T_{3}^{3} + 16T_{3}^{2} + 24T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16$$
$5$ $$T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121$$
$7$ $$T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121$$
$11$ $$T^{4} - T^{3} + 21 T^{2} - 11 T + 121$$
$13$ $$T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256$$
$17$ $$T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1$$
$19$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$23$ $$(T^{2} + 5 T - 5)^{2}$$
$29$ $$T^{4} + 16 T^{3} + 136 T^{2} + \cdots + 1936$$
$31$ $$T^{4} + 40 T^{2} - 200 T + 400$$
$37$ $$T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16$$
$41$ $$T^{4} - 14 T^{3} + 76 T^{2} - 24 T + 16$$
$43$ $$(T^{2} + 13 T + 41)^{2}$$
$47$ $$T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25$$
$53$ $$T^{4} + 14 T^{3} + 96 T^{2} + \cdots + 5776$$
$59$ $$T^{4} - 22 T^{3} + 244 T^{2} + \cdots + 5776$$
$61$ $$T^{4} + 3 T^{3} + 54 T^{2} - 108 T + 81$$
$67$ $$(T^{2} + 14 T + 44)^{2}$$
$71$ $$T^{4} + 40 T^{2} - 200 T + 400$$
$73$ $$T^{4} + 18 T^{3} + 244 T^{2} + \cdots + 5776$$
$79$ $$T^{4} - 12 T^{3} + 64 T^{2} + \cdots + 1936$$
$83$ $$T^{4} - 2 T^{3} + 64 T^{2} + 247 T + 361$$
$89$ $$(T^{2} + 6 T - 116)^{2}$$
$97$ $$T^{4} + 28 T^{3} + 304 T^{2} + \cdots + 256$$