# Properties

 Label 1805.2.a.u 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: $$x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19$$ x^9 - 12*x^7 - 4*x^6 + 48*x^5 + 27*x^4 - 72*x^3 - 51*x^2 + 27*x + 19 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{6} - \nu^{5} - 8\nu^{4} + 4\nu^{3} + 17\nu^{2} - 2\nu - 7$$ v^6 - v^5 - 8*v^4 + 4*v^3 + 17*v^2 - 2*v - 7 $$\beta_{4}$$ $$=$$ $$\nu^{7} - \nu^{6} - 8\nu^{5} + 4\nu^{4} + 17\nu^{3} - 2\nu^{2} - 7\nu$$ v^7 - v^6 - 8*v^5 + 4*v^4 + 17*v^3 - 2*v^2 - 7*v $$\beta_{5}$$ $$=$$ $$\nu^{8} - \nu^{7} - 10\nu^{6} + 6\nu^{5} + 33\nu^{4} - 10\nu^{3} - 41\nu^{2} + 4\nu + 14$$ v^8 - v^7 - 10*v^6 + 6*v^5 + 33*v^4 - 10*v^3 - 41*v^2 + 4*v + 14 $$\beta_{6}$$ $$=$$ $$2\nu^{8} - 3\nu^{7} - 19\nu^{6} + 21\nu^{5} + 60\nu^{4} - 42\nu^{3} - 72\nu^{2} + 21\nu + 22$$ 2*v^8 - 3*v^7 - 19*v^6 + 21*v^5 + 60*v^4 - 42*v^3 - 72*v^2 + 21*v + 22 $$\beta_{7}$$ $$=$$ $$-3\nu^{8} + 5\nu^{7} + 29\nu^{6} - 36\nu^{5} - 96\nu^{4} + 72\nu^{3} + 123\nu^{2} - 32\nu - 38$$ -3*v^8 + 5*v^7 + 29*v^6 - 36*v^5 - 96*v^4 + 72*v^3 + 123*v^2 - 32*v - 38 $$\beta_{8}$$ $$=$$ $$-4\nu^{8} + 6\nu^{7} + 39\nu^{6} - 42\nu^{5} - 129\nu^{4} + 81\nu^{3} + 165\nu^{2} - 33\nu - 53$$ -4*v^8 + 6*v^7 + 39*v^6 - 42*v^5 - 129*v^4 + 81*v^3 + 165*v^2 - 33*v - 53
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} - \beta_{5} + \beta_{2} + 4\beta _1 + 2$$ -b8 + b7 - b5 + b2 + 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$-\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + 6\beta_{2} + 7\beta _1 + 12$$ -b8 + 2*b7 + b6 - b4 - b3 + 6*b2 + 7*b1 + 12 $$\nu^{5}$$ $$=$$ $$-7\beta_{8} + 9\beta_{7} + 3\beta_{6} - 7\beta_{5} - \beta_{4} - 2\beta_{3} + 9\beta_{2} + 20\beta _1 + 16$$ -7*b8 + 9*b7 + 3*b6 - 7*b5 - b4 - 2*b3 + 9*b2 + 20*b1 + 16 $$\nu^{6}$$ $$=$$ $$-11\beta_{8} + 21\beta_{7} + 11\beta_{6} - 3\beta_{5} - 9\beta_{4} - 9\beta_{3} + 36\beta_{2} + 45\beta _1 + 60$$ -11*b8 + 21*b7 + 11*b6 - 3*b5 - 9*b4 - 9*b3 + 36*b2 + 45*b1 + 60 $$\nu^{7}$$ $$=$$ $$-46\beta_{8} + 68\beta_{7} + 31\beta_{6} - 42\beta_{5} - 12\beta_{4} - 21\beta_{3} + 69\beta_{2} + 118\beta _1 + 112$$ -46*b8 + 68*b7 + 31*b6 - 42*b5 - 12*b4 - 21*b3 + 69*b2 + 118*b1 + 112 $$\nu^{8}$$ $$=$$ $$- 91 \beta_{8} + 168 \beta_{7} + 90 \beta_{6} - 39 \beta_{5} - 63 \beta_{4} - 66 \beta_{3} + 228 \beta_{2} + 294 \beta _1 + 349$$ -91*b8 + 168*b7 + 90*b6 - 39*b5 - 63*b4 - 66*b3 + 228*b2 + 294*b1 + 349

## 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
 2.62224 1.81702 1.68361 0.719457 −0.593847 −1.19408 −1.46231 −1.57047 −2.02162
−2.62224 0.928776 4.87613 −1.00000 −2.43547 −3.83157 −7.54188 −2.13737 2.62224
1.2 −1.81702 −0.177104 1.30157 −1.00000 0.321803 1.07346 1.26906 −2.96863 1.81702
1.3 −1.68361 3.25202 0.834534 −1.00000 −5.47512 −0.548389 1.96219 7.57562 1.68361
1.4 −0.719457 1.23428 −1.48238 −1.00000 −0.888013 1.29194 2.50542 −1.47655 0.719457
1.5 0.593847 1.93003 −1.64735 −1.00000 1.14614 1.06052 −2.16596 0.725033 −0.593847
1.6 1.19408 −2.27318 −0.574177 −1.00000 −2.71435 −2.19649 −3.07377 2.16734 −1.19408
1.7 1.46231 −1.51036 0.138346 −1.00000 −2.20860 4.07172 −2.72231 −0.718827 −1.46231
1.8 1.57047 −2.28502 0.466387 −1.00000 −3.58856 −4.01337 −2.40850 2.22131 −1.57047
1.9 2.02162 1.90055 2.08694 −1.00000 3.84218 3.09218 0.175759 0.612075 −2.02162
 $$n$$: 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.u 9
5.b even 2 1 9025.2.a.cd 9
19.b odd 2 1 1805.2.a.t 9
19.f odd 18 2 95.2.k.b 18
57.j even 18 2 855.2.bs.b 18
95.d odd 2 1 9025.2.a.ce 9
95.o odd 18 2 475.2.l.b 18
95.r even 36 4 475.2.u.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.b 18 19.f odd 18 2
475.2.l.b 18 95.o odd 18 2
475.2.u.c 36 95.r even 36 4
855.2.bs.b 18 57.j even 18 2
1805.2.a.t 9 19.b odd 2 1
1805.2.a.u 9 1.a even 1 1 trivial
9025.2.a.cd 9 5.b even 2 1
9025.2.a.ce 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1805))$$:

 $$T_{2}^{9} - 12T_{2}^{7} + 4T_{2}^{6} + 48T_{2}^{5} - 27T_{2}^{4} - 72T_{2}^{3} + 51T_{2}^{2} + 27T_{2} - 19$$ T2^9 - 12*T2^7 + 4*T2^6 + 48*T2^5 - 27*T2^4 - 72*T2^3 + 51*T2^2 + 27*T2 - 19 $$T_{3}^{9} - 3T_{3}^{8} - 12T_{3}^{7} + 37T_{3}^{6} + 39T_{3}^{5} - 147T_{3}^{4} - 6T_{3}^{3} + 186T_{3}^{2} - 75T_{3} - 19$$ T3^9 - 3*T3^8 - 12*T3^7 + 37*T3^6 + 39*T3^5 - 147*T3^4 - 6*T3^3 + 186*T3^2 - 75*T3 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} - 12 T^{7} + 4 T^{6} + 48 T^{5} + \cdots - 19$$
$3$ $$T^{9} - 3 T^{8} - 12 T^{7} + 37 T^{6} + \cdots - 19$$
$5$ $$(T + 1)^{9}$$
$7$ $$T^{9} - 33 T^{7} + 10 T^{6} + \cdots - 343$$
$11$ $$T^{9} - 36 T^{7} - 40 T^{6} + 282 T^{5} + \cdots - 19$$
$13$ $$T^{9} + 3 T^{8} - 57 T^{7} + \cdots + 9937$$
$17$ $$T^{9} + 9 T^{8} - 12 T^{7} - 256 T^{6} + \cdots + 1$$
$19$ $$T^{9}$$
$23$ $$T^{9} - 114 T^{7} + 170 T^{6} + \cdots - 63197$$
$29$ $$T^{9} - 15 T^{8} - 54 T^{7} + \cdots - 702001$$
$31$ $$T^{9} - 30 T^{8} + 285 T^{7} + \cdots - 996623$$
$37$ $$T^{9} - 30 T^{8} + 267 T^{7} + \cdots - 27721$$
$41$ $$T^{9} - 18 T^{8} - 18 T^{7} + \cdots - 363977$$
$43$ $$T^{9} + 6 T^{8} - 99 T^{7} + \cdots - 10099$$
$47$ $$T^{9} - 21 T^{8} - 60 T^{7} + \cdots + 5721697$$
$53$ $$T^{9} + 9 T^{8} - 147 T^{7} + \cdots - 4387499$$
$59$ $$T^{9} - 27 T^{8} + 138 T^{7} + \cdots - 577711$$
$61$ $$T^{9} - 12 T^{8} - 219 T^{7} + \cdots + 1862369$$
$67$ $$T^{9} - 36 T^{8} + 183 T^{7} + \cdots + 60058259$$
$71$ $$T^{9} + 6 T^{8} - 177 T^{7} + \cdots + 92683$$
$73$ $$T^{9} + 9 T^{8} - 267 T^{7} + \cdots - 1023553$$
$79$ $$T^{9} - 45 T^{8} + 627 T^{7} + \cdots - 17803297$$
$83$ $$T^{9} - 171 T^{7} - 507 T^{6} + \cdots - 9829$$
$89$ $$T^{9} + 9 T^{8} - 474 T^{7} + \cdots - 11971$$
$97$ $$T^{9} - 45 T^{8} + \cdots - 191335897$$