# Properties

 Label 1805.2.a.v Level $1805$ Weight $2$ Character orbit 1805.a Self dual yes Analytic conductor $14.413$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1$$ x^9 - 3*x^8 - 6*x^7 + 16*x^6 + 12*x^5 - 27*x^4 - 8*x^3 + 15*x^2 - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 3x^{8} - 6x^{7} + 16x^{6} + 12x^{5} - 27x^{4} - 8x^{3} + 15x^{2} - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{6} - 3\nu^{5} - 4\nu^{4} + 10\nu^{3} + 4\nu^{2} - 7\nu$$ v^6 - 3*v^5 - 4*v^4 + 10*v^3 + 4*v^2 - 7*v $$\beta_{3}$$ $$=$$ $$\nu^{7} - 3\nu^{6} - 4\nu^{5} + 10\nu^{4} + 4\nu^{3} - 7\nu^{2}$$ v^7 - 3*v^6 - 4*v^5 + 10*v^4 + 4*v^3 - 7*v^2 $$\beta_{4}$$ $$=$$ $$\nu^{8} - 3\nu^{7} - 5\nu^{6} + 13\nu^{5} + 7\nu^{4} - 14\nu^{3} - \nu^{2} + \nu - 1$$ v^8 - 3*v^7 - 5*v^6 + 13*v^5 + 7*v^4 - 14*v^3 - v^2 + v - 1 $$\beta_{5}$$ $$=$$ $$\nu^{7} - 3\nu^{6} - 5\nu^{5} + 13\nu^{4} + 7\nu^{3} - 14\nu^{2} - 2\nu + 2$$ v^7 - 3*v^6 - 5*v^5 + 13*v^4 + 7*v^3 - 14*v^2 - 2*v + 2 $$\beta_{6}$$ $$=$$ $$-\nu^{8} + 3\nu^{7} + 5\nu^{6} - 13\nu^{5} - 7\nu^{4} + 14\nu^{3} + 2\nu^{2} - 3\nu$$ -v^8 + 3*v^7 + 5*v^6 - 13*v^5 - 7*v^4 + 14*v^3 + 2*v^2 - 3*v $$\beta_{7}$$ $$=$$ $$\nu^{8} - 3\nu^{7} - 5\nu^{6} + 13\nu^{5} + 8\nu^{4} - 17\nu^{3} - 4\nu^{2} + 8\nu$$ v^8 - 3*v^7 - 5*v^6 + 13*v^5 + 8*v^4 - 17*v^3 - 4*v^2 + 8*v $$\beta_{8}$$ $$=$$ $$2\nu^{8} - 7\nu^{7} - 7\nu^{6} + 30\nu^{5} + 5\nu^{4} - 34\nu^{3} - \nu^{2} + 7\nu + 2$$ 2*v^8 - 7*v^7 - 7*v^6 + 30*v^5 + 5*v^4 - 34*v^3 - v^2 + 7*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} + 2\beta _1 + 1$$ b6 + b4 + 2*b1 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{8} - \beta_{7} + 3\beta_{6} + 2\beta_{4} + \beta_{3} + 8\beta_1$$ b8 - b7 + 3*b6 + 2*b4 + b3 + 8*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{8} - 2\beta_{7} + 12\beta_{6} + 8\beta_{4} + 3\beta_{3} + 23\beta _1 + 2$$ 3*b8 - 2*b7 + 12*b6 + 8*b4 + 3*b3 + 23*b1 + 2 $$\nu^{5}$$ $$=$$ $$12\beta_{8} - 9\beta_{7} + 38\beta_{6} - \beta_{5} + 23\beta_{4} + 13\beta_{3} + 77\beta _1 + 1$$ 12*b8 - 9*b7 + 38*b6 - b5 + 23*b4 + 13*b3 + 77*b1 + 1 $$\nu^{6}$$ $$=$$ $$38\beta_{8} - 25\beta_{7} + 128\beta_{6} - 3\beta_{5} + 77\beta_{4} + 41\beta_{3} + \beta_{2} + 242\beta _1 + 7$$ 38*b8 - 25*b7 + 128*b6 - 3*b5 + 77*b4 + 41*b3 + b2 + 242*b1 + 7 $$\nu^{7}$$ $$=$$ $$128\beta_{8} - 87\beta_{7} + 411\beta_{6} - 13\beta_{5} + 242\beta_{4} + 142\beta_{3} + 3\beta_{2} + 786\beta _1 + 12$$ 128*b8 - 87*b7 + 411*b6 - 13*b5 + 242*b4 + 142*b3 + 3*b2 + 786*b1 + 12 $$\nu^{8}$$ $$=$$ $$411 \beta_{8} - 269 \beta_{7} + 1338 \beta_{6} - 41 \beta_{5} + 786 \beta_{4} + 455 \beta_{3} + 14 \beta_{2} + 2519 \beta _1 + 46$$ 411*b8 - 269*b7 + 1338*b6 - 41*b5 + 786*b4 + 455*b3 + 14*b2 + 2519*b1 + 46

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.22274 1.63278 1.30799 0.789016 0.309891 −0.256961 −1.13237 −1.28997 −1.58312
−2.22274 1.03700 2.94057 1.00000 −2.30498 −2.02508 −2.09064 −1.92463 −2.22274
1.2 −0.632780 1.91964 −1.59959 1.00000 −1.21471 4.08895 2.27775 0.685005 −0.632780
1.3 −0.307988 −1.64392 −1.90514 1.00000 0.506308 0.0891959 1.20274 −0.297533 −0.307988
1.4 0.210984 0.0798955 −1.95549 1.00000 0.0168566 −1.68723 −0.834543 −2.99362 0.210984
1.5 0.690109 −0.694850 −1.52375 1.00000 −0.479522 −2.33464 −2.43177 −2.51718 0.690109
1.6 1.25696 3.01225 −0.420048 1.00000 3.78628 3.72392 −3.04191 6.07366 1.25696
1.7 2.13237 2.23040 2.54700 1.00000 4.75604 1.48562 1.16642 1.97468 2.13237
1.8 2.28997 3.30730 3.24395 1.00000 7.57362 −2.93910 2.84861 7.93825 2.28997
1.9 2.58312 −0.247720 4.67249 1.00000 −0.639888 −0.401640 6.90335 −2.93864 2.58312
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.a.v 9
5.b even 2 1 9025.2.a.cc 9
19.b odd 2 1 1805.2.a.s 9
19.e even 9 2 95.2.k.a 18
57.l odd 18 2 855.2.bs.c 18
95.d odd 2 1 9025.2.a.cf 9
95.p even 18 2 475.2.l.c 18
95.q odd 36 4 475.2.u.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a 18 19.e even 9 2
475.2.l.c 18 95.p even 18 2
475.2.u.b 36 95.q odd 36 4
855.2.bs.c 18 57.l odd 18 2
1805.2.a.s 9 19.b odd 2 1
1805.2.a.v 9 1.a even 1 1 trivial
9025.2.a.cc 9 5.b even 2 1
9025.2.a.cf 9 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1805))$$:

 $$T_{2}^{9} - 6T_{2}^{8} + 6T_{2}^{7} + 26T_{2}^{6} - 60T_{2}^{5} + 21T_{2}^{4} + 30T_{2}^{3} - 15T_{2}^{2} - 3T_{2} + 1$$ T2^9 - 6*T2^8 + 6*T2^7 + 26*T2^6 - 60*T2^5 + 21*T2^4 + 30*T2^3 - 15*T2^2 - 3*T2 + 1 $$T_{3}^{9} - 9T_{3}^{8} + 24T_{3}^{7} + T_{3}^{6} - 87T_{3}^{5} + 81T_{3}^{4} + 44T_{3}^{3} - 48T_{3}^{2} - 9T_{3} + 1$$ T3^9 - 9*T3^8 + 24*T3^7 + T3^6 - 87*T3^5 + 81*T3^4 + 44*T3^3 - 48*T3^2 - 9*T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} - 6 T^{8} + 6 T^{7} + 26 T^{6} + \cdots + 1$$
$3$ $$T^{9} - 9 T^{8} + 24 T^{7} + T^{6} + \cdots + 1$$
$5$ $$(T - 1)^{9}$$
$7$ $$T^{9} - 27 T^{7} - 24 T^{6} + 213 T^{5} + \cdots + 19$$
$11$ $$T^{9} - 60 T^{7} - 52 T^{6} + \cdots + 773$$
$13$ $$T^{9} - 9 T^{8} - 15 T^{7} + 358 T^{6} + \cdots + 53$$
$17$ $$T^{9} + 9 T^{8} - 18 T^{7} + \cdots + 1691$$
$19$ $$T^{9}$$
$23$ $$T^{9} + 12 T^{8} - 36 T^{7} + \cdots - 45667$$
$29$ $$T^{9} - 9 T^{8} - 66 T^{7} + \cdots - 210403$$
$31$ $$T^{9} - 18 T^{8} + 21 T^{7} + \cdots + 216991$$
$37$ $$T^{9} - 18 T^{8} - 15 T^{7} + \cdots - 11125$$
$41$ $$T^{9} - 6 T^{8} - 78 T^{7} + 377 T^{6} + \cdots + 361$$
$43$ $$T^{9} + 12 T^{8} - 153 T^{7} + \cdots + 1591019$$
$47$ $$T^{9} - 15 T^{8} - 108 T^{7} + \cdots + 1425943$$
$53$ $$T^{9} - 15 T^{8} - 159 T^{7} + \cdots - 211859$$
$59$ $$T^{9} - 21 T^{8} + 6 T^{7} + \cdots + 733771$$
$61$ $$T^{9} + 12 T^{8} - 123 T^{7} + \cdots + 61561$$
$67$ $$T^{9} - 60 T^{8} + 1467 T^{7} + \cdots - 6493589$$
$71$ $$T^{9} + 18 T^{8} - 225 T^{7} + \cdots - 53239843$$
$73$ $$T^{9} - 27 T^{8} - 33 T^{7} + \cdots - 24197203$$
$79$ $$T^{9} - 15 T^{8} - 141 T^{7} + \cdots + 3833$$
$83$ $$T^{9} - 507 T^{7} + \cdots + 189817057$$
$89$ $$T^{9} + 39 T^{8} + 438 T^{7} + \cdots + 957419$$
$97$ $$T^{9} - 15 T^{8} - 243 T^{7} + \cdots - 1914625$$