# Properties

 Label 418.2.h.a Level $418$ Weight $2$ Character orbit 418.h Analytic conductor $3.338$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 41 x^{18} + 707 x^{16} + 6667 x^{14} + 37400 x^{12} + 126976 x^{10} + 253280 x^{8} + 273276 x^{6} + 137476 x^{4} + 30336 x^{2} + 2304$$ x^20 + 41*x^18 + 707*x^16 + 6667*x^14 + 37400*x^12 + 126976*x^10 + 253280*x^8 + 273276*x^6 + 137476*x^4 + 30336*x^2 + 2304 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{10} q^{2} + \beta_{3} q^{3} + (\beta_{10} - 1) q^{4} + \beta_{8} q^{5} - \beta_{4} q^{6} + ( - \beta_{14} - \beta_{13}) q^{7} + q^{8} + ( - \beta_{10} + \beta_{9} - \beta_{4} + \beta_{3} + 1) q^{9}+O(q^{10})$$ q - b10 * q^2 + b3 * q^3 + (b10 - 1) * q^4 + b8 * q^5 - b4 * q^6 + (-b14 - b13) * q^7 + q^8 + (-b10 + b9 - b4 + b3 + 1) * q^9 $$q - \beta_{10} q^{2} + \beta_{3} q^{3} + (\beta_{10} - 1) q^{4} + \beta_{8} q^{5} - \beta_{4} q^{6} + ( - \beta_{14} - \beta_{13}) q^{7} + q^{8} + ( - \beta_{10} + \beta_{9} - \beta_{4} + \beta_{3} + 1) q^{9} + ( - \beta_{8} - \beta_{2}) q^{10} - \beta_{5} q^{11} + (\beta_{4} - \beta_{3}) q^{12} + (\beta_{19} + \beta_{18} + \beta_{17} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{13} + \beta_{14} q^{14} + ( - \beta_{19} + \beta_{18} + \beta_{15} - 2 \beta_{13} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \cdots + 1) q^{15}+ \cdots + ( - 2 \beta_{18} - 2 \beta_{17} - 2 \beta_{14} + \beta_{13} - 3 \beta_{12} + \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{99}+O(q^{100})$$ q - b10 * q^2 + b3 * q^3 + (b10 - 1) * q^4 + b8 * q^5 - b4 * q^6 + (-b14 - b13) * q^7 + q^8 + (-b10 + b9 - b4 + b3 + 1) * q^9 + (-b8 - b2) * q^10 - b5 * q^11 + (b4 - b3) * q^12 + (b19 + b18 + b17 + b12 - b11 - b9 - b7 + b6 + b4 - b3 - b1) * q^13 + b14 * q^14 + (-b19 + b18 + b15 - 2*b13 + b12 + b10 - b9 + b7 - b6 - b3 - b1 + 1) * q^15 - b10 * q^16 + (-b16 + b15 - b9 - b1) * q^17 + (b1 - 1) * q^18 + (b16 - b15 + b12 + b1 + 1) * q^19 + b2 * q^20 + (-b14 - 2*b13 + 2*b10 - b9 + b8 - b4 - b1) * q^21 + b19 * q^22 + (b17 + b11 + b8 + b2) * q^23 + b3 * q^24 + (b17 - 2*b14 - b13 + b12 - b11 + 2*b10 - 2*b9 + b8 - b7 + b6 + b5 + b4 + b2 - b1 - 2) * q^25 + b7 * q^26 + (b12 - b11 - 2*b10 + 1) * q^27 + b13 * q^28 + (b17 + b12 - b11 - b7 + b6 + b5 - b1) * q^29 + (-b16 + 2*b14 + 2*b13 - b12 + b11 - 2*b10 + b9 + b6 - b5 - b4 + b3 + 1) * q^30 + (2*b17 + b16 - b14 - b13 + b12 - b11 + b9 - b7 + b6 + b5) * q^31 + (b10 - 1) * q^32 + (-b18 + b17 + b15 - 2*b14 - b13 - b11 + b10 - b9 - b7 + b5 + b4 + b2 - b1 - 1) * q^33 - b15 * q^34 + (b19 - b18 + b16 - b15 - b14 - b6 + b4 - b3 + 2*b1) * q^35 + (b10 - b9 + b4 - b3 - b1) * q^36 + (2*b19 + 2*b18 - b16 + 2*b14 + 2*b13 - b9 - 2*b8 + 2*b6 - 2*b5 - b2 - b1) * q^37 + (b17 + b15 - b10 - b3 - b1) * q^38 + (-2*b17 - b16 + b14 + b13 - b12 + b11 + b9 + b7 - b6 - b5 - b4 + b3 + b1) * q^39 + b8 * q^40 + (b19 + b18 + 2*b17 + b14 + 2*b13 - b11 - b10 - b9 - b8 - b7 + 2*b6 + b5 + b4 - b3 - 2*b1) * q^41 + (2*b14 + b13 - 2*b10 + b9 - b8 - b3 - b2 + 2) * q^42 + (-b16 + b15 - 2*b14 - b11 - b9 + b8 - b7 + b6 + b5 + 2*b2 - 2*b1) * q^43 + (-b19 + b5) * q^44 + (b16 - 2*b15 + b14 - b13 + b12 + b11 + b9 - 2*b2 + 2*b1 + 1) * q^45 + (-b12 - b11 - b7 + b6 + b5 - b2 - b1) * q^46 + (-b17 + 2*b14 + b13 - b12 + b11 - b8 + b7 - b6 - b5 - b2 + b1) * q^47 - b4 * q^48 + (-b16 + 2*b15 - b12 - b11 - b9 - b7 - b4 - b3 - 2*b1 + 1) * q^49 + (b14 - b13 + b7 - b6 - b5 - b4 - b3 - b2 - b1 + 2) * q^50 + (b19 + b18 - b17 + 2*b14 + b13 - b12 + b11 - b10 + b9 - 2*b8 + b7 - b6 - 2*b5 - b3 - 2*b2 + b1 + 1) * q^51 + (-b19 - b18 - b17 - b12 + b11 + b9 - b6 - b4 + b3 + b1) * q^52 + (-b15 + b13 - b12 - b10 + b9 - b7 + b6 + b5 - b4 + b3 + b1 - 1) * q^53 + (b11 + b10 + b7 - b6 - b5 + b1 - 2) * q^54 + (-2*b18 - b17 - b15 + b13 - b10 + b9 + b8 - b4 + 3*b3 + b2 + 2*b1 - 1) * q^55 + (-b14 - b13) * q^56 + (-2*b19 + b14 + b11 - 3*b10 + 2*b9 + b8 + b7 - b4 + 2*b3 + 2*b1) * q^57 + (b7 - b6 - b5 + b1) * q^58 + (-2*b19 - b17 - b16 + b15 - b14 - b12 + 2*b10 + b7 + b6 + b5 - b4 + b3 - b1 - 4) * q^59 + (b19 - b18 + b16 - b15 - 2*b14 - b11 + b10 - b7 + b5 + b4 + b1 - 2) * q^60 + (-b17 - b15 + 2*b12 + b11 - b10 + 2*b7 - 2*b6 - 2*b5 + b4 + 2*b1 - 1) * q^61 + (-b17 - b16 + b15 + b14 - b12 - b9 + b7 - b6 - b5) * q^62 + (-b19 + b18 + b15 - 2*b13 - b9 + b8 + b5 + b4 - b3 - b2 - 2*b1) * q^63 + q^64 + (b14 - b13 + b12 + b11 + 2*b7 - 2*b6 - 2*b5 - b4 - b3 + b2 + 1) * q^65 + (b18 - b17 - b16 + b14 - b13 - b9 + b8 + b7 - b6 - b5 - b4 - b1 + 1) * q^66 + (-b17 - b15 + b13 - b12 + b11 + b10 + b9 - b7 + b6 + b5 + b3 + b1 + 1) * q^67 + (b16 + b9 + b1) * q^68 + (b16 - b14 - b13 - b12 + b11 + 2*b10 - b9 - 2*b8 + 2*b4 - 2*b3 - b2 - 1) * q^69 + (-b19 + b18 + b15 - b13 - b9 + b5 + b4 - b3 - 2*b1) * q^70 + (b19 - b18 - b17 + 2*b14 - b12 - b10 + b9 + b7 - 2*b6 - b5 - b4 + 2) * q^71 + (-b10 + b9 - b4 + b3 + 1) * q^72 + (b19 - b18 + b17 + b16 - b15 - 2*b14 + b12 + b9 + b8 - b7 + b5 + 2*b3 + 2*b2) * q^73 + (-2*b18 + b16 - b15 - 2*b14 + b9 + b8 + 2*b5 + 2*b2 + b1) * q^74 + (-2*b17 + b16 - b14 - b13 - 2*b12 + 2*b11 - 2*b10 + 3*b9 + b7 - b6 - b5 + b4 - b3 + 3*b1 + 1) * q^75 + (-b17 - b16 - b12 + b10 + b3 - 1) * q^76 + (b15 + 2*b14 + b13 - b12 - 3*b10 + b9 - b8 - b7 - 2*b4 - 2*b2 - b1 + 1) * q^77 + (b17 + b16 - b15 - b14 + b12 - b7 + b6 + b5 + b4 + b1) * q^78 + (-3*b19 - b18 + b16 + b15 - 2*b14 - 4*b13 + 4*b10 + b8 - 3*b6 + 2*b5 + b4 - b3) * q^79 + (-b8 - b2) * q^80 + (-b17 + b14 + 2*b13 + b12 + b10 - b9 + b7 - b6 - b5 - b4) * q^81 + (-b19 - b18 - 2*b17 - 2*b14 - b13 - b12 + b10 + b9 + b8 + b7 - b6 - b4 + b3 + b2 + b1 - 1) * q^82 + (-b14 - b13 + b12 - b11 + 4*b10 - 2*b9 - 2*b8 + 2*b4 - 2*b3 - b2 - b1 - 2) * q^83 + (-b14 + b13 + b4 + b3 + b2 + b1 - 2) * q^84 + (-b19 + b18 + b17 + 3*b15 - 2*b13 + b12 - b11 + b10 - b9 - b8 + b7 - b6 + 2*b4 - b3 + b2 - b1 + 1) * q^85 + (-b15 - 2*b13 + b12 + b8 + b7 - b6 - b5 - b2 + b1) * q^86 + (-2*b17 + 4*b14 + 4*b13 - b12 + b11 - 4*b10 + 4*b9 - 2*b8 + b7 - 2*b5 - 3*b4 + 3*b3 - b2 + 3*b1 + 2) * q^87 - b5 * q^88 + (-2*b15 + 2*b13 - b12 + 2*b9 - b8 - b7 + b6 + b5 + 2*b4 + 2*b3 + b2 + 3*b1) * q^89 + (2*b17 + b16 + b15 + b14 + 2*b13 - b11 - b10 - 2*b8 - b7 + b6 + b5 + b4 - b3 - b1) * q^90 + (-b19 + b18 + 2*b13 + b5 + b4) * q^91 + (-b17 + b12 - b8 + b7 - b6 - b5 + b1) * q^92 + (-2*b18 - b17 + b14 + 2*b13 - b12 + b11 - b10 + 2*b9 - 2*b5 - 4*b4 + 3*b3 + 2*b1) * q^93 + (-b14 + b13 - b7 + b6 + b5 + b2 - b1) * q^94 + (-2*b19 - b17 - b16 + b14 - b12 + b11 + b10 - b9 + b8 + b7 - b4 - 2*b3 - 3*b1 + 1) * q^95 + (b4 - b3) * q^96 + (2*b17 + 2*b16 - 2*b15 - 2*b14 + 2*b12 - b10 - 2*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 + 2*b1 + 2) * q^97 + (b19 + b18 - b17 - b16 - b15 + b12 - b10 - b9 + b7 - b5 + 2*b4 - b3) * q^98 + (-2*b18 - 2*b17 - 2*b14 + b13 - 3*b12 + b9 + b8 - b7 - b6 + b5 + b3 + b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 10 q^{2} + 3 q^{3} - 10 q^{4} + 2 q^{5} - 3 q^{6} + 20 q^{8} + 11 q^{9}+O(q^{10})$$ 20 * q - 10 * q^2 + 3 * q^3 - 10 * q^4 + 2 * q^5 - 3 * q^6 + 20 * q^8 + 11 * q^9 $$20 q - 10 q^{2} + 3 q^{3} - 10 q^{4} + 2 q^{5} - 3 q^{6} + 20 q^{8} + 11 q^{9} + 2 q^{10} + q^{11} - 5 q^{13} - 6 q^{14} + 12 q^{15} - 10 q^{16} - 6 q^{17} - 22 q^{18} + 18 q^{19} - 4 q^{20} + 14 q^{21} + q^{22} - 4 q^{23} + 3 q^{24} - 20 q^{25} + 10 q^{26} + 6 q^{28} - 5 q^{29} - 10 q^{32} - 8 q^{33} + 6 q^{34} + 12 q^{35} + 11 q^{36} - 12 q^{38} + 2 q^{40} + q^{41} + 14 q^{42} - 3 q^{43} - 2 q^{44} + 12 q^{45} + 8 q^{46} + q^{47} - 3 q^{48} + 8 q^{49} + 40 q^{50} + 12 q^{51} - 5 q^{52} - 24 q^{53} - 27 q^{54} - 20 q^{55} - 32 q^{57} + 10 q^{58} - 51 q^{59} - 12 q^{60} - 27 q^{61} - 12 q^{63} + 20 q^{64} + 8 q^{65} + 13 q^{66} + 27 q^{67} - 12 q^{70} + 33 q^{71} + 11 q^{72} + 9 q^{73} + 12 q^{74} - 6 q^{76} - 22 q^{77} + 24 q^{79} + 2 q^{80} + 12 q^{81} + q^{82} - 28 q^{84} + 12 q^{85} + 3 q^{86} + q^{88} + 21 q^{89} - 6 q^{90} + 12 q^{91} - 4 q^{92} - 10 q^{93} - 2 q^{94} + 24 q^{95} + 24 q^{97} - 4 q^{98} + 21 q^{99}+O(q^{100})$$ 20 * q - 10 * q^2 + 3 * q^3 - 10 * q^4 + 2 * q^5 - 3 * q^6 + 20 * q^8 + 11 * q^9 + 2 * q^10 + q^11 - 5 * q^13 - 6 * q^14 + 12 * q^15 - 10 * q^16 - 6 * q^17 - 22 * q^18 + 18 * q^19 - 4 * q^20 + 14 * q^21 + q^22 - 4 * q^23 + 3 * q^24 - 20 * q^25 + 10 * q^26 + 6 * q^28 - 5 * q^29 - 10 * q^32 - 8 * q^33 + 6 * q^34 + 12 * q^35 + 11 * q^36 - 12 * q^38 + 2 * q^40 + q^41 + 14 * q^42 - 3 * q^43 - 2 * q^44 + 12 * q^45 + 8 * q^46 + q^47 - 3 * q^48 + 8 * q^49 + 40 * q^50 + 12 * q^51 - 5 * q^52 - 24 * q^53 - 27 * q^54 - 20 * q^55 - 32 * q^57 + 10 * q^58 - 51 * q^59 - 12 * q^60 - 27 * q^61 - 12 * q^63 + 20 * q^64 + 8 * q^65 + 13 * q^66 + 27 * q^67 - 12 * q^70 + 33 * q^71 + 11 * q^72 + 9 * q^73 + 12 * q^74 - 6 * q^76 - 22 * q^77 + 24 * q^79 + 2 * q^80 + 12 * q^81 + q^82 - 28 * q^84 + 12 * q^85 + 3 * q^86 + q^88 + 21 * q^89 - 6 * q^90 + 12 * q^91 - 4 * q^92 - 10 * q^93 - 2 * q^94 + 24 * q^95 + 24 * q^97 - 4 * q^98 + 21 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 41 x^{18} + 707 x^{16} + 6667 x^{14} + 37400 x^{12} + 126976 x^{10} + 253280 x^{8} + 273276 x^{6} + 137476 x^{4} + 30336 x^{2} + 2304$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{2}$$ $$=$$ $$( 53 \nu^{18} + 2845 \nu^{16} + 61804 \nu^{14} + 710321 \nu^{12} + 4698655 \nu^{10} + 18093662 \nu^{8} + 38709820 \nu^{6} + 40363248 \nu^{4} + 15032156 \nu^{2} + \cdots + 1650300 ) / 47556$$ (53*v^18 + 2845*v^16 + 61804*v^14 + 710321*v^12 + 4698655*v^10 + 18093662*v^8 + 38709820*v^6 + 40363248*v^4 + 15032156*v^2 + 1650300) / 47556 $$\beta_{3}$$ $$=$$ $$( 2603 \nu^{18} + 96159 \nu^{16} + 1450417 \nu^{14} + 11480565 \nu^{12} + 50950576 \nu^{10} + 125250072 \nu^{8} + 157669240 \nu^{6} + 87843316 \nu^{4} + \cdots + 1787040 ) / 126816$$ (2603*v^18 + 96159*v^16 + 1450417*v^14 + 11480565*v^12 + 50950576*v^10 + 125250072*v^8 + 157669240*v^6 + 87843316*v^4 + 21371196*v^2 + 63408*v + 1787040) / 126816 $$\beta_{4}$$ $$=$$ $$( 2603 \nu^{18} + 96159 \nu^{16} + 1450417 \nu^{14} + 11480565 \nu^{12} + 50950576 \nu^{10} + 125250072 \nu^{8} + 157669240 \nu^{6} + 87843316 \nu^{4} + \cdots + 1787040 ) / 126816$$ (2603*v^18 + 96159*v^16 + 1450417*v^14 + 11480565*v^12 + 50950576*v^10 + 125250072*v^8 + 157669240*v^6 + 87843316*v^4 + 21371196*v^2 - 63408*v + 1787040) / 126816 $$\beta_{5}$$ $$=$$ $$( 107 \nu^{19} - 97800 \nu^{18} + 73339 \nu^{17} - 3695736 \nu^{16} + 2599705 \nu^{15} - 57543960 \nu^{14} + 38281121 \nu^{13} - 476930472 \nu^{12} + \cdots - 185170176 ) / 4565376$$ (107*v^19 - 97800*v^18 + 73339*v^17 - 3695736*v^16 + 2599705*v^15 - 57543960*v^14 + 38281121*v^13 - 476930472*v^12 + 295384336*v^11 - 2268888960*v^10 + 1264414016*v^9 - 6227561472*v^8 + 2919941344*v^7 - 9417456192*v^6 + 3218346132*v^5 - 7051263840*v^4 + 1249989452*v^3 - 2107154400*v^2 + 163079520*v - 185170176) / 4565376 $$\beta_{6}$$ $$=$$ $$( - 107 \nu^{19} - 97800 \nu^{18} - 73339 \nu^{17} - 3695736 \nu^{16} - 2599705 \nu^{15} - 57543960 \nu^{14} - 38281121 \nu^{13} - 476930472 \nu^{12} + \cdots - 185170176 ) / 4565376$$ (-107*v^19 - 97800*v^18 - 73339*v^17 - 3695736*v^16 - 2599705*v^15 - 57543960*v^14 - 38281121*v^13 - 476930472*v^12 - 295384336*v^11 - 2268888960*v^10 - 1264414016*v^9 - 6227561472*v^8 - 2919941344*v^7 - 9417456192*v^6 - 3218346132*v^5 - 7051263840*v^4 - 1249989452*v^3 - 2107154400*v^2 - 163079520*v - 185170176) / 4565376 $$\beta_{7}$$ $$=$$ $$( 1461 \nu^{18} + 50809 \nu^{16} + 698211 \nu^{14} + 4718987 \nu^{12} + 15238980 \nu^{10} + 13209104 \nu^{8} - 42132048 \nu^{6} - 90775684 \nu^{4} - 37037932 \nu^{2} + \cdots - 3576288 ) / 63408$$ (1461*v^18 + 50809*v^16 + 698211*v^14 + 4718987*v^12 + 15238980*v^10 + 13209104*v^8 - 42132048*v^6 - 90775684*v^4 - 37037932*v^2 - 3576288) / 63408 $$\beta_{8}$$ $$=$$ $$( - 9151 \nu^{19} - 1696 \nu^{18} - 384143 \nu^{17} - 91040 \nu^{16} - 6770837 \nu^{15} - 1977728 \nu^{14} - 64982941 \nu^{13} - 22730272 \nu^{12} + \cdots - 52809600 ) / 3043584$$ (-9151*v^19 - 1696*v^18 - 384143*v^17 - 91040*v^16 - 6770837*v^15 - 1977728*v^14 - 64982941*v^13 - 22730272*v^12 - 367640240*v^11 - 150356960*v^10 - 1235475904*v^9 - 578997184*v^8 - 2342158112*v^7 - 1238714240*v^6 - 2165660292*v^5 - 1291623936*v^4 - 656550556*v^3 - 481028992*v^2 - 41087712*v - 52809600) / 3043584 $$\beta_{9}$$ $$=$$ $$( - 999 \nu^{19} - 41263 \nu^{17} - 716829 \nu^{15} - 6804269 \nu^{13} - 38321328 \nu^{11} - 129781328 \nu^{9} - 254263344 \nu^{7} - 258637028 \nu^{5} + \cdots - 507264 ) / 253632$$ (-999*v^19 - 41263*v^17 - 716829*v^15 - 6804269*v^13 - 38321328*v^11 - 129781328*v^9 - 254263344*v^7 - 258637028*v^5 - 107549660*v^3 - 126816*v^2 - 13944864*v - 507264) / 253632 $$\beta_{10}$$ $$=$$ $$( - 6205 \nu^{19} - 233581 \nu^{17} - 3617663 \nu^{15} - 29765399 \nu^{13} - 140222480 \nu^{11} - 380281472 \nu^{9} - 569601824 \nu^{7} - 434323660 \nu^{5} + \cdots + 507264 ) / 1014528$$ (-6205*v^19 - 233581*v^17 - 3617663*v^15 - 29765399*v^13 - 140222480*v^11 - 380281472*v^9 - 569601824*v^7 - 434323660*v^5 - 150292052*v^3 - 17265312*v + 507264) / 1014528 $$\beta_{11}$$ $$=$$ $$( 18615 \nu^{19} - 79232 \nu^{18} + 700743 \nu^{17} - 2877280 \nu^{16} + 10852989 \nu^{15} - 42346624 \nu^{14} + 89296197 \nu^{13} - 322856288 \nu^{12} + \cdots + 30557568 ) / 3043584$$ (18615*v^19 - 79232*v^18 + 700743*v^17 - 2877280*v^16 + 10852989*v^15 - 42346624*v^14 + 89296197*v^13 - 322856288*v^12 + 420667440*v^11 - 1346216320*v^10 + 1140844416*v^9 - 2940288320*v^8 + 1708805472*v^7 - 2793758080*v^6 + 1302970980*v^5 - 470506752*v^4 + 452397948*v^3 + 190428544*v^2 + 60926688*v + 30557568) / 3043584 $$\beta_{12}$$ $$=$$ $$( - 18615 \nu^{19} - 79232 \nu^{18} - 700743 \nu^{17} - 2877280 \nu^{16} - 10852989 \nu^{15} - 42346624 \nu^{14} - 89296197 \nu^{13} - 322856288 \nu^{12} + \cdots + 30557568 ) / 3043584$$ (-18615*v^19 - 79232*v^18 - 700743*v^17 - 2877280*v^16 - 10852989*v^15 - 42346624*v^14 - 89296197*v^13 - 322856288*v^12 - 420667440*v^11 - 1346216320*v^10 - 1140844416*v^9 - 2940288320*v^8 - 1708805472*v^7 - 2793758080*v^6 - 1302970980*v^5 - 470506752*v^4 - 452397948*v^3 + 190428544*v^2 - 60926688*v + 30557568) / 3043584 $$\beta_{13}$$ $$=$$ $$( - 75833 \nu^{19} + 124320 \nu^{18} - 2792329 \nu^{17} + 4505568 \nu^{16} - 41911795 \nu^{15} + 66110784 \nu^{14} - 329104619 \nu^{13} + 501541536 \nu^{12} + \cdots - 121507200 ) / 9130752$$ (-75833*v^19 + 124320*v^18 - 2792329*v^17 + 4505568*v^16 - 41911795*v^15 + 66110784*v^14 - 329104619*v^13 + 501541536*v^12 - 1439821168*v^11 + 2072105952*v^10 - 3437352704*v^9 + 4432691712*v^8 - 4019293792*v^7 + 3932054592*v^6 - 1722956316*v^5 + 164995200*v^4 - 60631748*v^3 - 621225600*v^2 + 64682976*v - 121507200) / 9130752 $$\beta_{14}$$ $$=$$ $$( - 75833 \nu^{19} - 124320 \nu^{18} - 2792329 \nu^{17} - 4505568 \nu^{16} - 41911795 \nu^{15} - 66110784 \nu^{14} - 329104619 \nu^{13} - 501541536 \nu^{12} + \cdots + 121507200 ) / 9130752$$ (-75833*v^19 - 124320*v^18 - 2792329*v^17 - 4505568*v^16 - 41911795*v^15 - 66110784*v^14 - 329104619*v^13 - 501541536*v^12 - 1439821168*v^11 - 2072105952*v^10 - 3437352704*v^9 - 4432691712*v^8 - 4019293792*v^7 - 3932054592*v^6 - 1722956316*v^5 - 164995200*v^4 - 60631748*v^3 + 621225600*v^2 + 64682976*v + 121507200) / 9130752 $$\beta_{15}$$ $$=$$ $$( - 35533 \nu^{19} - 225984 \nu^{18} - 1369613 \nu^{17} - 8419488 \nu^{16} - 21888863 \nu^{15} - 128542848 \nu^{14} - 187878583 \nu^{13} - 1035922080 \nu^{12} + \cdots - 272207232 ) / 4565376$$ (-35533*v^19 - 225984*v^18 - 1369613*v^17 - 8419488*v^16 - 21888863*v^15 - 128542848*v^14 - 187878583*v^13 - 1035922080*v^12 - 937451648*v^11 - 4728523488*v^10 - 2746801600*v^9 - 12182984256*v^8 - 4531487552*v^7 - 16689242976*v^6 - 3750420108*v^5 - 10847241792*v^4 - 1170060436*v^3 - 3011218752*v^2 - 98823072*v - 272207232) / 4565376 $$\beta_{16}$$ $$=$$ $$( - 13271 \nu^{19} - 499123 \nu^{17} - 7718701 \nu^{15} - 63320081 \nu^{13} - 296279848 \nu^{11} - 789384824 \nu^{9} - 1121558728 \nu^{7} + \cdots - 2282688 ) / 1141344$$ (-13271*v^19 - 499123*v^17 - 7718701*v^15 - 63320081*v^13 - 296279848*v^11 - 789384824*v^9 - 1121558728*v^7 - 711343428*v^5 - 101056748*v^3 - 570672*v^2 + 13340352*v - 2282688) / 1141344 $$\beta_{17}$$ $$=$$ $$( 20051 \nu^{19} + 50132 \nu^{18} + 703651 \nu^{17} + 1841620 \nu^{16} + 9815041 \nu^{15} + 27559852 \nu^{14} + 68225753 \nu^{13} + 215604668 \nu^{12} + \cdots + 21851520 ) / 1521792$$ (20051*v^19 + 50132*v^18 + 703651*v^17 + 1841620*v^16 + 9815041*v^15 + 27559852*v^14 + 68225753*v^13 + 215604668*v^12 + 235787200*v^11 + 939164080*v^10 + 292951904*v^9 + 2234363072*v^8 - 336148352*v^7 + 2633567488*v^6 - 1026664236*v^5 + 1261113072*v^4 - 483639316*v^3 + 258690512*v^2 - 51638880*v + 21851520) / 1521792 $$\beta_{18}$$ $$=$$ $$( - 145213 \nu^{19} + 346440 \nu^{18} - 5609741 \nu^{17} + 12897096 \nu^{16} - 89878367 \nu^{15} + 196613592 \nu^{14} - 773611639 \nu^{13} + \cdots + 275428224 ) / 9130752$$ (-145213*v^19 + 346440*v^18 - 5609741*v^17 + 12897096*v^16 - 89878367*v^15 + 196613592*v^14 - 773611639*v^13 + 1580261592*v^12 - 3872186096*v^11 + 7177801344*v^10 - 11387849152*v^9 + 18319904832*v^8 - 18888164768*v^7 + 24606711168*v^6 - 15829337676*v^5 + 15267180384*v^4 - 5188465492*v^3 + 3790348320*v^2 - 491515680*v + 275428224) / 9130752 $$\beta_{19}$$ $$=$$ $$( - 145213 \nu^{19} - 346440 \nu^{18} - 5609741 \nu^{17} - 12897096 \nu^{16} - 89878367 \nu^{15} - 196613592 \nu^{14} - 773611639 \nu^{13} + \cdots - 275428224 ) / 9130752$$ (-145213*v^19 - 346440*v^18 - 5609741*v^17 - 12897096*v^16 - 89878367*v^15 - 196613592*v^14 - 773611639*v^13 - 1580261592*v^12 - 3872186096*v^11 - 7177801344*v^10 - 11387849152*v^9 - 18319904832*v^8 - 18888164768*v^7 - 24606711168*v^6 - 15829337676*v^5 - 15267180384*v^4 - 5188465492*v^3 - 3790348320*v^2 - 491515680*v - 275428224) / 9130752
 $$\nu$$ $$=$$ $$-\beta_{4} + \beta_{3}$$ -b4 + b3 $$\nu^{2}$$ $$=$$ $$\beta _1 - 4$$ b1 - 4 $$\nu^{3}$$ $$=$$ $$-\beta_{12} + \beta_{11} + 2\beta_{10} + 6\beta_{4} - 6\beta_{3} - 1$$ -b12 + b11 + 2*b10 + 6*b4 - 6*b3 - 1 $$\nu^{4}$$ $$=$$ $$\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 9\beta _1 + 26$$ b14 - b13 - b12 - b11 - b7 + b6 + b5 + b4 + b3 - 9*b1 + 26 $$\nu^{5}$$ $$=$$ $$2 \beta_{14} + 2 \beta_{13} + 10 \beta_{12} - 10 \beta_{11} - 28 \beta_{10} + 4 \beta_{9} - \beta_{6} + \beta_{5} - 44 \beta_{4} + 44 \beta_{3} + 2 \beta _1 + 14$$ 2*b14 + 2*b13 + 10*b12 - 10*b11 - 28*b10 + 4*b9 - b6 + b5 - 44*b4 + 44*b3 + 2*b1 + 14 $$\nu^{6}$$ $$=$$ $$\beta_{16} - 2 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + 13 \beta_{12} + 13 \beta_{11} + \beta_{9} + 14 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} - 16 \beta_{4} - 16 \beta_{3} + \beta_{2} + 78 \beta _1 - 191$$ b16 - 2*b15 - 13*b14 + 13*b13 + 13*b12 + 13*b11 + b9 + 14*b7 - 13*b6 - 13*b5 - 16*b4 - 16*b3 + b2 + 78*b1 - 191 $$\nu^{7}$$ $$=$$ $$- 2 \beta_{19} - 2 \beta_{18} + 4 \beta_{17} + 2 \beta_{16} - 32 \beta_{14} - 32 \beta_{13} - 86 \beta_{12} + 86 \beta_{11} + 312 \beta_{10} - 62 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 13 \beta_{6} - 9 \beta_{5} + 352 \beta_{4} - 352 \beta_{3} + \cdots - 156$$ -2*b19 - 2*b18 + 4*b17 + 2*b16 - 32*b14 - 32*b13 - 86*b12 + 86*b11 + 312*b10 - 62*b9 + 4*b8 - 2*b7 + 13*b6 - 9*b5 + 352*b4 - 352*b3 + 2*b2 - 32*b1 - 156 $$\nu^{8}$$ $$=$$ $$2 \beta_{19} - 2 \beta_{18} - 15 \beta_{16} + 30 \beta_{15} + 129 \beta_{14} - 129 \beta_{13} - 131 \beta_{12} - 131 \beta_{11} - 15 \beta_{9} - 152 \beta_{7} + 137 \beta_{6} + 137 \beta_{5} + 188 \beta_{4} + 188 \beta_{3} - 15 \beta_{2} + \cdots + 1497$$ 2*b19 - 2*b18 - 15*b16 + 30*b15 + 129*b14 - 129*b13 - 131*b12 - 131*b11 - 15*b9 - 152*b7 + 137*b6 + 137*b5 + 188*b4 + 188*b3 - 15*b2 - 672*b1 + 1497 $$\nu^{9}$$ $$=$$ $$32 \beta_{19} + 32 \beta_{18} - 76 \beta_{17} - 32 \beta_{16} + 376 \beta_{14} + 376 \beta_{13} + 714 \beta_{12} - 714 \beta_{11} - 3148 \beta_{10} + 720 \beta_{9} - 68 \beta_{8} + 38 \beta_{7} - 131 \beta_{6} + 55 \beta_{5} - 2950 \beta_{4} + \cdots + 1574$$ 32*b19 + 32*b18 - 76*b17 - 32*b16 + 376*b14 + 376*b13 + 714*b12 - 714*b11 - 3148*b10 + 720*b9 - 68*b8 + 38*b7 - 131*b6 + 55*b5 - 2950*b4 + 2950*b3 - 34*b2 + 382*b1 + 1574 $$\nu^{10}$$ $$=$$ $$- 40 \beta_{19} + 40 \beta_{18} + 159 \beta_{16} - 318 \beta_{15} - 1175 \beta_{14} + 1175 \beta_{13} + 1219 \beta_{12} + 1219 \beta_{11} + 159 \beta_{9} + 1512 \beta_{7} - 1351 \beta_{6} - 1351 \beta_{5} - 1954 \beta_{4} + \cdots - 12219$$ -40*b19 + 40*b18 + 159*b16 - 318*b15 - 1175*b14 + 1175*b13 + 1219*b12 + 1219*b11 + 159*b9 + 1512*b7 - 1351*b6 - 1351*b5 - 1954*b4 - 1954*b3 + 169*b2 + 5792*b1 - 12219 $$\nu^{11}$$ $$=$$ $$- 358 \beta_{19} - 358 \beta_{18} + 984 \beta_{17} + 370 \beta_{16} - 3930 \beta_{14} - 3930 \beta_{13} - 5900 \beta_{12} + 5900 \beta_{11} + 30160 \beta_{10} - 7474 \beta_{9} + 812 \beta_{8} - 492 \beta_{7} + \cdots - 15080$$ -358*b19 - 358*b18 + 984*b17 + 370*b16 - 3930*b14 - 3930*b13 - 5900*b12 + 5900*b11 + 30160*b10 - 7474*b9 + 812*b8 - 492*b7 + 1211*b6 - 227*b5 + 25342*b4 - 25342*b3 + 406*b2 - 4044*b1 - 15080 $$\nu^{12}$$ $$=$$ $$530 \beta_{19} - 530 \beta_{18} - 1483 \beta_{16} + 2966 \beta_{15} + 10389 \beta_{14} - 10389 \beta_{13} - 10985 \beta_{12} - 10985 \beta_{11} - 1483 \beta_{9} - 14396 \beta_{7} + 12863 \beta_{6} + \cdots + 102383$$ 530*b19 - 530*b18 - 1483*b16 + 2966*b15 + 10389*b14 - 10389*b13 - 10985*b12 - 10985*b11 - 1483*b9 - 14396*b7 + 12863*b6 + 12863*b5 + 19068*b4 + 19068*b3 - 1759*b2 - 50072*b1 + 102383 $$\nu^{13}$$ $$=$$ $$3496 \beta_{19} + 3496 \beta_{18} - 10848 \beta_{17} - 3822 \beta_{16} + 38784 \beta_{14} + 38784 \beta_{13} + 49030 \beta_{12} - 49030 \beta_{11} - 280608 \beta_{10} + 73214 \beta_{9} - 8496 \beta_{8} + \cdots + 140304$$ 3496*b19 + 3496*b18 - 10848*b17 - 3822*b16 + 38784*b14 + 38784*b13 + 49030*b12 - 49030*b11 - 280608*b10 + 73214*b9 - 8496*b8 + 5424*b7 - 10759*b6 - 89*b5 - 220540*b4 + 220540*b3 - 4248*b2 + 40120*b1 + 140304 $$\nu^{14}$$ $$=$$ $$- 5848 \beta_{19} + 5848 \beta_{18} + 13079 \beta_{16} - 26158 \beta_{15} - 91123 \beta_{14} + 91123 \beta_{13} + 97555 \beta_{12} + 97555 \beta_{11} + 13079 \beta_{9} + 133516 \beta_{7} + \cdots - 872755$$ -5848*b19 + 5848*b18 + 13079*b16 - 26158*b15 - 91123*b14 + 91123*b13 + 97555*b12 + 97555*b11 + 13079*b9 + 133516*b7 - 119687*b6 - 119687*b5 - 179406*b4 - 179406*b3 + 17829*b2 + 434500*b1 - 872755 $$\nu^{15}$$ $$=$$ $$- 32006 \beta_{19} - 32006 \beta_{18} + 109776 \beta_{17} + 37506 \beta_{16} - 370610 \beta_{14} - 370610 \beta_{13} - 411048 \beta_{12} + 411048 \beta_{11} + 2565248 \beta_{10} + \cdots - 1282624$$ -32006*b19 - 32006*b18 + 109776*b17 + 37506*b16 - 370610*b14 - 370610*b13 - 411048*b12 + 411048*b11 + 2565248*b10 - 693250*b9 + 83716*b8 - 54888*b7 + 93555*b6 + 16221*b5 + 1932290*b4 - 1932290*b3 + 41858*b2 - 382760*b1 - 1282624 $$\nu^{16}$$ $$=$$ $$58066 \beta_{19} - 58066 \beta_{18} - 112531 \beta_{16} + 225062 \beta_{15} + 799765 \beta_{14} - 799765 \beta_{13} - 860129 \beta_{12} - 860129 \beta_{11} - 112531 \beta_{9} - 1217252 \beta_{7} + \cdots + 7525551$$ 58066*b19 - 58066*b18 - 112531*b16 + 225062*b15 + 799765*b14 - 799765*b13 - 860129*b12 - 860129*b11 - 112531*b9 - 1217252*b7 + 1096043*b6 + 1096043*b5 + 1650300*b4 + 1650300*b3 - 178279*b2 - 3783988*b1 + 7525551 $$\nu^{17}$$ $$=$$ $$283128 \beta_{19} + 283128 \beta_{18} - 1055440 \beta_{17} - 357554 \beta_{16} + 3472380 \beta_{14} + 3472380 \beta_{13} + 3477142 \beta_{12} - 3477142 \beta_{11} - 23193696 \beta_{10} + \cdots + 11596848$$ 283128*b19 + 283128*b18 - 1055440*b17 - 357554*b16 + 3472380*b14 + 3472380*b13 + 3477142*b12 - 3477142*b11 - 23193696*b10 + 6426402*b9 - 801224*b8 + 527720*b7 - 803059*b6 - 252381*b5 - 16990108*b4 + 16990108*b3 - 400612*b2 + 3562144*b1 + 11596848 $$\nu^{18}$$ $$=$$ $$- 537344 \beta_{19} + 537344 \beta_{18} + 959079 \beta_{16} - 1918158 \beta_{15} - 7045071 \beta_{14} + 7045071 \beta_{13} + 7554695 \beta_{12} + 7554695 \beta_{11} + 959079 \beta_{9} + \cdots - 65395747$$ -537344*b19 + 537344*b18 + 959079*b16 - 1918158*b15 - 7045071*b14 + 7045071*b13 + 7554695*b12 + 7554695*b11 + 959079*b9 + 10968608*b7 - 9925479*b6 - 9925479*b5 - 14961106*b4 - 14961106*b3 + 1762733*b2 + 33056664*b1 - 65395747 $$\nu^{19}$$ $$=$$ $$- 2455502 \beta_{19} - 2455502 \beta_{18} + 9820672 \beta_{17} + 3343206 \beta_{16} - 32121518 \beta_{14} - 32121518 \beta_{13} - 29649348 \beta_{12} + 29649348 \beta_{11} + \cdots - 104107396$$ -2455502*b19 - 2455502*b18 + 9820672*b17 + 3343206*b16 - 32121518*b14 - 32121518*b13 - 29649348*b12 + 29649348*b11 + 208214792*b10 - 58756854*b9 + 7558636*b8 - 4910336*b7 + 6835091*b6 + 2985581*b5 + 149668274*b4 - 149668274*b3 + 3779318*b2 - 32617160*b1 - 104107396

## Character values

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

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$1 - \beta_{10}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 − 2.51965i − 2.37734i − 2.02516i − 0.634765i − 0.579050i 0.401261i 1.50467i 2.01980i 2.96323i 2.97905i 2.51965i 2.37734i 2.02516i 0.634765i 0.579050i − 0.401261i − 1.50467i − 2.01980i − 2.96323i − 2.97905i
−0.500000 + 0.866025i −2.18208 1.25983i −0.500000 0.866025i −2.02560 + 3.50844i 2.18208 1.25983i 1.09641i 1.00000 1.67433 + 2.90002i −2.02560 3.50844i
65.2 −0.500000 + 0.866025i −2.05883 1.18867i −0.500000 0.866025i 2.05852 3.56546i 2.05883 1.18867i 2.89251i 1.00000 1.32587 + 2.29647i 2.05852 + 3.56546i
65.3 −0.500000 + 0.866025i −1.75384 1.01258i −0.500000 0.866025i −0.202207 + 0.350232i 1.75384 1.01258i 1.38521i 1.00000 0.550630 + 0.953720i −0.202207 0.350232i
65.4 −0.500000 + 0.866025i −0.549723 0.317383i −0.500000 0.866025i −0.464143 + 0.803919i 0.549723 0.317383i 0.602460i 1.00000 −1.29854 2.24913i −0.464143 0.803919i
65.5 −0.500000 + 0.866025i −0.501472 0.289525i −0.500000 0.866025i 1.06779 1.84947i 0.501472 0.289525i 0.637717i 1.00000 −1.33235 2.30770i 1.06779 + 1.84947i
65.6 −0.500000 + 0.866025i 0.347503 + 0.200631i −0.500000 0.866025i −1.32940 + 2.30258i −0.347503 + 0.200631i 3.90204i 1.00000 −1.41949 2.45864i −1.32940 2.30258i
65.7 −0.500000 + 0.866025i 1.30309 + 0.752337i −0.500000 0.866025i 1.28873 2.23214i −1.30309 + 0.752337i 1.03498i 1.00000 −0.367978 0.637356i 1.28873 + 2.23214i
65.8 −0.500000 + 0.866025i 1.74919 + 1.00990i −0.500000 0.866025i 0.0648685 0.112356i −1.74919 + 1.00990i 4.47991i 1.00000 0.539789 + 0.934941i 0.0648685 + 0.112356i
65.9 −0.500000 + 0.866025i 2.56624 + 1.48162i −0.500000 0.866025i −1.17566 + 2.03631i −2.56624 + 1.48162i 0.571027i 1.00000 2.89038 + 5.00628i −1.17566 2.03631i
65.10 −0.500000 + 0.866025i 2.57993 + 1.48952i −0.500000 0.866025i 1.71710 2.97410i −2.57993 + 1.48952i 4.12917i 1.00000 2.93737 + 5.08767i 1.71710 + 2.97410i
373.1 −0.500000 0.866025i −2.18208 + 1.25983i −0.500000 + 0.866025i −2.02560 3.50844i 2.18208 + 1.25983i 1.09641i 1.00000 1.67433 2.90002i −2.02560 + 3.50844i
373.2 −0.500000 0.866025i −2.05883 + 1.18867i −0.500000 + 0.866025i 2.05852 + 3.56546i 2.05883 + 1.18867i 2.89251i 1.00000 1.32587 2.29647i 2.05852 3.56546i
373.3 −0.500000 0.866025i −1.75384 + 1.01258i −0.500000 + 0.866025i −0.202207 0.350232i 1.75384 + 1.01258i 1.38521i 1.00000 0.550630 0.953720i −0.202207 + 0.350232i
373.4 −0.500000 0.866025i −0.549723 + 0.317383i −0.500000 + 0.866025i −0.464143 0.803919i 0.549723 + 0.317383i 0.602460i 1.00000 −1.29854 + 2.24913i −0.464143 + 0.803919i
373.5 −0.500000 0.866025i −0.501472 + 0.289525i −0.500000 + 0.866025i 1.06779 + 1.84947i 0.501472 + 0.289525i 0.637717i 1.00000 −1.33235 + 2.30770i 1.06779 1.84947i
373.6 −0.500000 0.866025i 0.347503 0.200631i −0.500000 + 0.866025i −1.32940 2.30258i −0.347503 0.200631i 3.90204i 1.00000 −1.41949 + 2.45864i −1.32940 + 2.30258i
373.7 −0.500000 0.866025i 1.30309 0.752337i −0.500000 + 0.866025i 1.28873 + 2.23214i −1.30309 0.752337i 1.03498i 1.00000 −0.367978 + 0.637356i 1.28873 2.23214i
373.8 −0.500000 0.866025i 1.74919 1.00990i −0.500000 + 0.866025i 0.0648685 + 0.112356i −1.74919 1.00990i 4.47991i 1.00000 0.539789 0.934941i 0.0648685 0.112356i
373.9 −0.500000 0.866025i 2.56624 1.48162i −0.500000 + 0.866025i −1.17566 2.03631i −2.56624 1.48162i 0.571027i 1.00000 2.89038 5.00628i −1.17566 + 2.03631i
373.10 −0.500000 0.866025i 2.57993 1.48952i −0.500000 + 0.866025i 1.71710 + 2.97410i −2.57993 1.48952i 4.12917i 1.00000 2.93737 5.08767i 1.71710 2.97410i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.h.a 20
11.b odd 2 1 418.2.h.b yes 20
19.d odd 6 1 418.2.h.b yes 20
209.g even 6 1 inner 418.2.h.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.h.a 20 1.a even 1 1 trivial
418.2.h.a 20 209.g even 6 1 inner
418.2.h.b yes 20 11.b odd 2 1
418.2.h.b yes 20 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{20} + 5 T_{13}^{19} + 77 T_{13}^{18} + 352 T_{13}^{17} + 3849 T_{13}^{16} + 15391 T_{13}^{15} + 94191 T_{13}^{14} + 209168 T_{13}^{13} + 965371 T_{13}^{12} + 1069185 T_{13}^{11} + 7201739 T_{13}^{10} + 2382406 T_{13}^{9} + \cdots + 2985984$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{10}$$
$3$ $$T^{20} - 3 T^{19} - 16 T^{18} + \cdots + 2304$$
$5$ $$T^{20} - 2 T^{19} + 37 T^{18} + \cdots + 9216$$
$7$ $$T^{20} + 66 T^{18} + 1677 T^{16} + \cdots + 5184$$
$11$ $$T^{20} - T^{19} + 7 T^{18} + \cdots + 25937424601$$
$13$ $$T^{20} + 5 T^{19} + 77 T^{18} + \cdots + 2985984$$
$17$ $$T^{20} + 6 T^{19} - 66 T^{18} + \cdots + 73547776$$
$19$ $$T^{20} - 18 T^{19} + \cdots + 6131066257801$$
$23$ $$T^{20} + 4 T^{19} + \cdots + 22265110462464$$
$29$ $$T^{20} + 5 T^{19} + \cdots + 29561988096$$
$31$ $$T^{20} + 344 T^{18} + \cdots + 949829566464$$
$37$ $$T^{20} + 514 T^{18} + \cdots + 135789302016$$
$41$ $$T^{20} - T^{19} + \cdots + 42\!\cdots\!44$$
$43$ $$T^{20} + 3 T^{19} + \cdots + 497465017344$$
$47$ $$T^{20} - T^{19} + 143 T^{18} + \cdots + 112896$$
$53$ $$T^{20} + 24 T^{19} + \cdots + 2828537856$$
$59$ $$T^{20} + 51 T^{19} + \cdots + 20\!\cdots\!84$$
$61$ $$T^{20} + 27 T^{19} + \cdots + 3718911688704$$
$67$ $$T^{20} - 27 T^{19} + \cdots + 36790308864$$
$71$ $$T^{20} - 33 T^{19} + \cdots + 13\!\cdots\!96$$
$73$ $$T^{20} - 9 T^{19} + \cdots + 5707473897024$$
$79$ $$T^{20} - 24 T^{19} + \cdots + 16\!\cdots\!56$$
$83$ $$T^{20} + 550 T^{18} + \cdots + 4811139551761$$
$89$ $$T^{20} - 21 T^{19} + \cdots + 13\!\cdots\!84$$
$97$ $$T^{20} - 24 T^{19} + \cdots + 13\!\cdots\!89$$