# Properties

 Label 414.2.e.c Level $414$ Weight $2$ Character orbit 414.e Analytic conductor $3.306$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.30580664368$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.1481180578947.1 Defining polynomial: $$x^{10} - 2x^{9} + 6x^{8} - 11x^{7} + 22x^{6} - 45x^{5} + 66x^{4} - 99x^{3} + 162x^{2} - 162x + 243$$ x^10 - 2*x^9 + 6*x^8 - 11*x^7 + 22*x^6 - 45*x^5 + 66*x^4 - 99*x^3 + 162*x^2 - 162*x + 243 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 6x^{8} - 11x^{7} + 22x^{6} - 45x^{5} + 66x^{4} - 99x^{3} + 162x^{2} - 162x + 243$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{9} + 25\nu^{8} - 3\nu^{7} - 47\nu^{6} - 32\nu^{5} + 135\nu^{4} + 57\nu^{3} + 198\nu^{2} + 1188\nu + 648 ) / 1701$$ (v^9 + 25*v^8 - 3*v^7 - 47*v^6 - 32*v^5 + 135*v^4 + 57*v^3 + 198*v^2 + 1188*v + 648) / 1701 $$\beta_{3}$$ $$=$$ $$( 4\nu^{9} - 26\nu^{8} + 51\nu^{7} + \nu^{6} + 124\nu^{5} - 153\nu^{4} + 606\nu^{3} - 720\nu^{2} - 351\nu - 810 ) / 1701$$ (4*v^9 - 26*v^8 + 51*v^7 + v^6 + 124*v^5 - 153*v^4 + 606*v^3 - 720*v^2 - 351*v - 810) / 1701 $$\beta_{4}$$ $$=$$ $$( 2 \nu^{9} - 37 \nu^{8} + 42 \nu^{7} - 121 \nu^{6} + 218 \nu^{5} - 339 \nu^{4} + 528 \nu^{3} - 810 \nu^{2} + 891 \nu - 2997 ) / 1701$$ (2*v^9 - 37*v^8 + 42*v^7 - 121*v^6 + 218*v^5 - 339*v^4 + 528*v^3 - 810*v^2 + 891*v - 2997) / 1701 $$\beta_{5}$$ $$=$$ $$( - 10 \nu^{9} + 23 \nu^{8} - 75 \nu^{7} + 92 \nu^{6} - 37 \nu^{5} + 372 \nu^{4} - 318 \nu^{3} + 729 \nu^{2} - 162 \nu - 810 ) / 1701$$ (-10*v^9 + 23*v^8 - 75*v^7 + 92*v^6 - 37*v^5 + 372*v^4 - 318*v^3 + 729*v^2 - 162*v - 810) / 1701 $$\beta_{6}$$ $$=$$ $$( 11 \nu^{9} - 19 \nu^{8} + 51 \nu^{7} - 139 \nu^{6} + 425 \nu^{5} - 573 \nu^{4} + 1068 \nu^{3} - 1350 \nu^{2} + 3240 \nu - 2511 ) / 1701$$ (11*v^9 - 19*v^8 + 51*v^7 - 139*v^6 + 425*v^5 - 573*v^4 + 1068*v^3 - 1350*v^2 + 3240*v - 2511) / 1701 $$\beta_{7}$$ $$=$$ $$( - 14 \nu^{9} + 25 \nu^{8} - 15 \nu^{7} + 64 \nu^{6} - 5 \nu^{5} + 357 \nu^{4} - 321 \nu^{3} - 135 \nu^{2} - 162 \nu - 891 ) / 1701$$ (-14*v^9 + 25*v^8 - 15*v^7 + 64*v^6 - 5*v^5 + 357*v^4 - 321*v^3 - 135*v^2 - 162*v - 891) / 1701 $$\beta_{8}$$ $$=$$ $$( 11\nu^{9} - 10\nu^{8} + 33\nu^{7} - 58\nu^{6} + 83\nu^{5} - 132\nu^{4} + 204\nu^{3} - 189\nu^{2} + 891\nu + 162 ) / 567$$ (11*v^9 - 10*v^8 + 33*v^7 - 58*v^6 + 83*v^5 - 132*v^4 + 204*v^3 - 189*v^2 + 891*v + 162) / 567 $$\beta_{9}$$ $$=$$ $$( - 37 \nu^{9} + 68 \nu^{8} - 111 \nu^{7} + 281 \nu^{6} - 451 \nu^{5} + 1011 \nu^{4} - 1425 \nu^{3} + 2079 \nu^{2} - 3564 \nu + 3321 ) / 1701$$ (-37*v^9 + 68*v^8 - 111*v^7 + 281*v^6 - 451*v^5 + 1011*v^4 - 1425*v^3 + 2079*v^2 - 3564*v + 3321) / 1701
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} - 2$$ 2*b9 + b8 - 2*b7 + b6 + b4 - 2 $$\nu^{3}$$ $$=$$ $$-\beta_{8} - 2\beta_{7} + \beta_{5} + 3\beta_{3} + 3\beta_{2}$$ -b8 - 2*b7 + b5 + 3*b3 + 3*b2 $$\nu^{4}$$ $$=$$ $$-2\beta_{9} + \beta_{8} + 3\beta_{7} - 4\beta_{6} + 4\beta_{5} + 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + 5$$ -2*b9 + b8 + 3*b7 - 4*b6 + 4*b5 + 2*b4 + 3*b3 + 3*b2 + 5 $$\nu^{5}$$ $$=$$ $$4\beta_{9} + 5\beta_{8} + 2\beta_{7} + 5\beta_{6} + 3\beta_{5} + 2\beta_{4} - 3\beta_{3} - 6\beta_{2} - 6\beta _1 + 5$$ 4*b9 + 5*b8 + 2*b7 + 5*b6 + 3*b5 + 2*b4 - 3*b3 - 6*b2 - 6*b1 + 5 $$\nu^{6}$$ $$=$$ $$3\beta_{9} + \beta_{8} - \beta_{7} + 6\beta_{6} - \beta_{5} - 12\beta_{4} + 6\beta_{3} - 12\beta_{2} + 9\beta _1 - 12$$ 3*b9 + b8 - b7 + 6*b6 - b5 - 12*b4 + 6*b3 - 12*b2 + 9*b1 - 12 $$\nu^{7}$$ $$=$$ $$23\beta_{9} + 14\beta_{8} + 10\beta_{6} - 25\beta_{5} + 4\beta_{4} + 6\beta_{3} - 3\beta_{2} + 6\beta _1 - 35$$ 23*b9 + 14*b8 + 10*b6 - 25*b5 + 4*b4 + 6*b3 - 3*b2 + 6*b1 - 35 $$\nu^{8}$$ $$=$$ $$11 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 34 \beta_{6} - 27 \beta_{5} - 35 \beta_{4} - 15 \beta_{3} + 15 \beta_{2} - 36 \beta _1 - 56$$ 11*b9 - 2*b8 + 4*b7 + 34*b6 - 27*b5 - 35*b4 - 15*b3 + 15*b2 - 36*b1 - 56 $$\nu^{9}$$ $$=$$ $$- 63 \beta_{9} + 23 \beta_{8} + 22 \beta_{7} - 36 \beta_{6} + 52 \beta_{5} - 81 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 39 \beta _1 - 36$$ -63*b9 + 23*b8 + 22*b7 - 36*b6 + 52*b5 - 81*b4 + 3*b3 - 15*b2 - 39*b1 - 36

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$235$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 −1.24278 − 1.20644i 0.785454 − 1.54372i 1.64906 − 0.529718i −0.643944 + 1.60790i 0.452211 + 1.67198i −1.24278 + 1.20644i 0.785454 + 1.54372i 1.64906 + 0.529718i −0.643944 − 1.60790i 0.452211 − 1.67198i
−0.500000 0.866025i −1.66620 0.473061i −0.500000 + 0.866025i 0.274896 0.476134i 0.423416 + 1.67950i 0.708031 + 1.22635i 1.00000 2.55243 + 1.57642i −0.549792
139.2 −0.500000 0.866025i −0.944171 + 1.45208i −0.500000 + 0.866025i 1.85973 3.22115i 1.72963 + 0.0916356i 2.00591 + 3.47433i 1.00000 −1.21708 2.74203i −3.71947
139.3 −0.500000 0.866025i 0.365780 + 1.69299i −0.500000 + 0.866025i 0.231157 0.400376i 1.28328 1.16327i 0.165447 + 0.286563i 1.00000 −2.73241 + 1.23852i −0.462315
139.4 −0.500000 0.866025i 1.07051 1.36162i −0.500000 + 0.866025i −1.08471 + 1.87878i −1.71445 0.246277i 1.41120 + 2.44426i 1.00000 −0.708023 2.91525i 2.16943
139.5 −0.500000 0.866025i 1.67408 0.444362i −0.500000 + 0.866025i 1.21893 2.11124i −1.22187 1.22761i −1.79058 3.10138i 1.00000 2.60509 1.48779i −2.43785
277.1 −0.500000 + 0.866025i −1.66620 + 0.473061i −0.500000 0.866025i 0.274896 + 0.476134i 0.423416 1.67950i 0.708031 1.22635i 1.00000 2.55243 1.57642i −0.549792
277.2 −0.500000 + 0.866025i −0.944171 1.45208i −0.500000 0.866025i 1.85973 + 3.22115i 1.72963 0.0916356i 2.00591 3.47433i 1.00000 −1.21708 + 2.74203i −3.71947
277.3 −0.500000 + 0.866025i 0.365780 1.69299i −0.500000 0.866025i 0.231157 + 0.400376i 1.28328 + 1.16327i 0.165447 0.286563i 1.00000 −2.73241 1.23852i −0.462315
277.4 −0.500000 + 0.866025i 1.07051 + 1.36162i −0.500000 0.866025i −1.08471 1.87878i −1.71445 + 0.246277i 1.41120 2.44426i 1.00000 −0.708023 + 2.91525i 2.16943
277.5 −0.500000 + 0.866025i 1.67408 + 0.444362i −0.500000 0.866025i 1.21893 + 2.11124i −1.22187 + 1.22761i −1.79058 + 3.10138i 1.00000 2.60509 + 1.48779i −2.43785
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.e.c 10
3.b odd 2 1 1242.2.e.c 10
9.c even 3 1 inner 414.2.e.c 10
9.c even 3 1 3726.2.a.t 5
9.d odd 6 1 1242.2.e.c 10
9.d odd 6 1 3726.2.a.s 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.c 10 1.a even 1 1 trivial
414.2.e.c 10 9.c even 3 1 inner
1242.2.e.c 10 3.b odd 2 1
1242.2.e.c 10 9.d odd 6 1
3726.2.a.s 5 9.d odd 6 1
3726.2.a.t 5 9.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{10} - 5 T_{5}^{9} + 25 T_{5}^{8} - 46 T_{5}^{7} + 136 T_{5}^{6} - 205 T_{5}^{5} + 554 T_{5}^{4} - 483 T_{5}^{3} + 326 T_{5}^{2} - 105 T_{5} + 25$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{5}$$
$3$ $$T^{10} - T^{9} + 2 T^{7} - 2 T^{6} + \cdots + 243$$
$5$ $$T^{10} - 5 T^{9} + 25 T^{8} - 46 T^{7} + \cdots + 25$$
$7$ $$T^{10} - 5 T^{9} + 32 T^{8} - 89 T^{7} + \cdots + 361$$
$11$ $$T^{10} - 3 T^{9} + 46 T^{8} + \cdots + 710649$$
$13$ $$T^{10} - 8 T^{9} + 68 T^{8} - 122 T^{7} + \cdots + 361$$
$17$ $$(T^{5} + T^{4} - 45 T^{3} - 14 T^{2} + \cdots - 501)^{2}$$
$19$ $$(T^{5} + T^{4} - 93 T^{3} - 113 T^{2} + \cdots + 2043)^{2}$$
$23$ $$(T^{2} + T + 1)^{5}$$
$29$ $$T^{10} - 18 T^{9} + 241 T^{8} + \cdots + 463761$$
$31$ $$T^{10} - 8 T^{9} + 190 T^{8} + \cdots + 112550881$$
$37$ $$(T^{5} + 6 T^{4} - 55 T^{3} - 255 T^{2} + \cdots + 2043)^{2}$$
$41$ $$T^{10} - 24 T^{9} + \cdots + 157527601$$
$43$ $$T^{10} + 11 T^{9} + 179 T^{8} + \cdots + 1058841$$
$47$ $$T^{10} - 9 T^{9} + 91 T^{8} + \cdots + 305809$$
$53$ $$(T^{5} + 29 T^{4} + 323 T^{3} + 1717 T^{2} + \cdots + 3947)^{2}$$
$59$ $$T^{10} - 21 T^{9} + \cdots + 8393674689$$
$61$ $$T^{10} - 17 T^{9} + \cdots + 151560721$$
$67$ $$T^{10} - 3 T^{9} + 204 T^{8} + \cdots + 393824025$$
$71$ $$(T^{5} + 9 T^{4} - 202 T^{3} - 1875 T^{2} + \cdots + 17079)^{2}$$
$73$ $$(T^{5} + 7 T^{4} - 97 T^{3} - 769 T^{2} + \cdots - 89)^{2}$$
$79$ $$T^{10} - 15 T^{9} + 145 T^{8} + \cdots + 1089$$
$83$ $$T^{10} - 21 T^{9} + 323 T^{8} + \cdots + 2253001$$
$89$ $$(T^{5} + 9 T^{4} - 115 T^{3} - 273 T^{2} + \cdots - 849)^{2}$$
$97$ $$T^{10} + 32 T^{9} + \cdots + 1093955625$$