# Properties

 Label 483.2.i.f Level $483$ Weight $2$ Character orbit 483.i Analytic conductor $3.857$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$483 = 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 483.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.85677441763$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} + 8 x^{10} - 3 x^{9} + 42 x^{8} - 15 x^{7} + 101 x^{6} + 6 x^{5} + 150 x^{4} - 28 x^{3} + 40 x^{2} + 8 x + 4$$ x^12 - x^11 + 8*x^10 - 3*x^9 + 42*x^8 - 15*x^7 + 101*x^6 + 6*x^5 + 150*x^4 - 28*x^3 + 40*x^2 + 8*x + 4 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 8 x^{10} - 3 x^{9} + 42 x^{8} - 15 x^{7} + 101 x^{6} + 6 x^{5} + 150 x^{4} - 28 x^{3} + 40 x^{2} + 8 x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 102623 \nu^{11} - 165571 \nu^{10} - 222384 \nu^{9} - 1691067 \nu^{8} - 1870676 \nu^{7} - 7686949 \nu^{6} + 7998165 \nu^{5} - 19046634 \nu^{4} + \cdots - 1558388 ) / 32857894$$ (-102623*v^11 - 165571*v^10 - 222384*v^9 - 1691067*v^8 - 1870676*v^7 - 7686949*v^6 + 7998165*v^5 - 19046634*v^4 + 17156370*v^3 - 13809438*v^2 + 125186588*v - 1558388) / 32857894 $$\beta_{3}$$ $$=$$ $$( - 387955 \nu^{11} - 1012381 \nu^{10} + 78220 \nu^{9} - 9799615 \nu^{8} + 1245446 \nu^{7} - 45151945 \nu^{6} + 57288245 \nu^{5} - 109780220 \nu^{4} + \cdots - 9835940 ) / 65715788$$ (-387955*v^11 - 1012381*v^10 + 78220*v^9 - 9799615*v^8 + 1245446*v^7 - 45151945*v^6 + 57288245*v^5 - 109780220*v^4 + 110445090*v^3 - 86675710*v^2 + 287420068*v - 9835940) / 65715788 $$\beta_{4}$$ $$=$$ $$( 389597 \nu^{11} - 492220 \nu^{10} + 2951205 \nu^{9} - 1391175 \nu^{8} + 14672007 \nu^{7} - 7714631 \nu^{6} + 31662348 \nu^{5} + 10335747 \nu^{4} + \cdots + 62587576 ) / 32857894$$ (389597*v^11 - 492220*v^10 + 2951205*v^9 - 1391175*v^8 + 14672007*v^7 - 7714631*v^6 + 31662348*v^5 + 10335747*v^4 + 39392916*v^3 + 6247654*v^2 + 1774442*v + 62587576) / 32857894 $$\beta_{5}$$ $$=$$ $$( 782053 \nu^{11} - 392456 \nu^{10} + 5764204 \nu^{9} + 605046 \nu^{8} + 31455051 \nu^{7} + 2941212 \nu^{6} + 71272722 \nu^{5} + 36354666 \nu^{4} + \cdots + 8030866 ) / 32857894$$ (782053*v^11 - 392456*v^10 + 5764204*v^9 + 605046*v^8 + 31455051*v^7 + 2941212*v^6 + 71272722*v^5 + 36354666*v^4 + 127643697*v^3 + 17495432*v^2 + 4671880*v + 8030866) / 32857894 $$\beta_{6}$$ $$=$$ $$( - 2035745 \nu^{11} - 2108749 \nu^{10} - 12760488 \nu^{9} - 25032941 \nu^{8} - 80460962 \nu^{7} - 131297395 \nu^{6} - 186501797 \nu^{5} + \cdots - 38350448 ) / 65715788$$ (-2035745*v^11 - 2108749*v^10 - 12760488*v^9 - 25032941*v^8 - 80460962*v^7 - 131297395*v^6 - 186501797*v^5 - 371196906*v^4 - 460140910*v^3 - 480064098*v^2 - 162121760*v - 38350448) / 65715788 $$\beta_{7}$$ $$=$$ $$( - 4015433 \nu^{11} + 5579539 \nu^{10} - 32908376 \nu^{9} + 23574707 \nu^{8} - 167438094 \nu^{7} + 123141597 \nu^{6} - 399676309 \nu^{5} + \cdots + 42936084 ) / 65715788$$ (-4015433*v^11 + 5579539*v^10 - 32908376*v^9 + 23574707*v^8 - 167438094*v^7 + 123141597*v^6 - 399676309*v^5 + 118452846*v^4 - 529605618*v^3 + 367719518*v^2 - 125626456*v + 42936084) / 65715788 $$\beta_{8}$$ $$=$$ $$( - 3745062 \nu^{11} + 2035588 \nu^{10} - 28003566 \nu^{9} - 2006303 \nu^{8} - 150105166 \nu^{7} - 10377711 \nu^{6} - 339284118 \nu^{5} + \cdots - 39617942 ) / 32857894$$ (-3745062*v^11 + 2035588*v^10 - 28003566*v^9 - 2006303*v^8 - 150105166*v^7 - 10377711*v^6 - 339284118*v^5 - 170000221*v^4 - 528040606*v^3 - 82481982*v^2 - 22081378*v - 39617942) / 32857894 $$\beta_{9}$$ $$=$$ $$( - 3881187 \nu^{11} + 6414399 \nu^{10} - 34645649 \nu^{9} + 33305993 \nu^{8} - 178419829 \nu^{7} + 168715213 \nu^{6} - 469846846 \nu^{5} + \cdots + 53681036 ) / 32857894$$ (-3881187*v^11 + 6414399*v^10 - 34645649*v^9 + 33305993*v^8 - 178419829*v^7 + 168715213*v^6 - 469846846*v^5 + 251390220*v^4 - 652666392*v^3 + 461865932*v^2 - 356135402*v + 53681036) / 32857894 $$\beta_{10}$$ $$=$$ $$( 12046299 \nu^{11} - 16738617 \nu^{10} + 98725128 \nu^{9} - 70724121 \nu^{8} + 502314282 \nu^{7} - 369424791 \nu^{6} + 1199028927 \nu^{5} + \cdots + 68339112 ) / 65715788$$ (12046299*v^11 - 16738617*v^10 + 98725128*v^9 - 70724121*v^8 + 502314282*v^7 - 369424791*v^6 + 1199028927*v^5 - 355358538*v^4 + 1588816854*v^3 - 1037442766*v^2 + 376879368*v + 68339112) / 65715788 $$\beta_{11}$$ $$=$$ $$( - 6206906 \nu^{11} + 4290544 \nu^{10} - 46524859 \nu^{9} + 1905036 \nu^{8} - 245693958 \nu^{7} + 6891144 \nu^{6} - 550437684 \nu^{5} + \cdots - 130991922 ) / 32857894$$ (-6206906*v^11 + 4290544*v^10 - 46524859*v^9 + 1905036*v^8 - 245693958*v^7 + 6891144*v^6 - 550437684*v^5 - 257725548*v^4 - 835966001*v^3 - 129075616*v^2 - 34888064*v - 130991922) / 32857894
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3\beta_{7} - 3$$ b10 + 3*b7 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + 4\beta_{5} + \beta_{2} - 1$$ b7 + b6 + 4*b5 + b2 - 1 $$\nu^{4}$$ $$=$$ $$-5\beta_{10} - \beta_{9} - 12\beta_{7} + 5\beta_{4} - \beta_{3} - \beta _1 + 5$$ -5*b10 - b9 - 12*b7 + 5*b4 - b3 - b1 + 5 $$\nu^{5}$$ $$=$$ $$2\beta_{8} - 7\beta_{7} - 7\beta_{6} - 18\beta_{5} - 2\beta_{3} - 18\beta _1 + 7$$ 2*b8 - 7*b7 - 7*b6 - 18*b5 - 2*b3 - 18*b1 + 7 $$\nu^{6}$$ $$=$$ $$-7\beta_{11} + 9\beta_{8} - 2\beta_{7} - 2\beta_{6} - 9\beta_{5} - 23\beta_{4} - 2\beta_{2} + 29$$ -7*b11 + 9*b8 - 2*b7 - 2*b6 - 9*b5 - 23*b4 - 2*b2 + 29 $$\nu^{7}$$ $$=$$ $$2\beta_{10} + 2\beta_{9} + 2\beta_{7} - 2\beta_{4} + 18\beta_{3} - 39\beta_{2} + 84\beta _1 - 2$$ 2*b10 + 2*b9 + 2*b7 - 2*b4 + 18*b3 - 39*b2 + 84*b1 - 2 $$\nu^{8}$$ $$=$$ $$39 \beta_{11} + 105 \beta_{10} + 39 \beta_{9} - 59 \beta_{8} + 236 \beta_{7} + 22 \beta_{6} + 61 \beta_{5} + 59 \beta_{3} + 61 \beta _1 - 236$$ 39*b11 + 105*b10 + 39*b9 - 59*b8 + 236*b7 + 22*b6 + 61*b5 + 59*b3 + 61*b1 - 236 $$\nu^{9}$$ $$=$$ $$22\beta_{11} - 120\beta_{8} + 203\beta_{7} + 203\beta_{6} + 400\beta_{5} + 24\beta_{4} + 203\beta_{2} - 205$$ 22*b11 - 120*b8 + 203*b7 + 203*b6 + 400*b5 + 24*b4 + 203*b2 - 205 $$\nu^{10}$$ $$=$$ $$-483\beta_{10} - 203\beta_{9} - 938\beta_{7} + 483\beta_{4} - 345\beta_{3} + 166\beta_{2} - 373\beta _1 + 483$$ -483*b10 - 203*b9 - 938*b7 + 483*b4 - 345*b3 + 166*b2 - 373*b1 + 483 $$\nu^{11}$$ $$=$$ $$- 166 \beta_{11} - 194 \beta_{10} - 166 \beta_{9} + 714 \beta_{8} - 1257 \beta_{7} - 1031 \beta_{6} - 1932 \beta_{5} - 714 \beta_{3} - 1932 \beta _1 + 1257$$ -166*b11 - 194*b10 - 166*b9 + 714*b8 - 1257*b7 - 1031*b6 - 1932*b5 - 714*b3 - 1932*b1 + 1257

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/483\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$346$$ $$442$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 −1.02170 + 1.76964i −0.682743 + 1.18255i −0.144978 + 0.251110i 0.296764 − 0.514011i 0.916322 − 1.58712i 1.13633 − 1.96819i −1.02170 − 1.76964i −0.682743 − 1.18255i −0.144978 − 0.251110i 0.296764 + 0.514011i 0.916322 + 1.58712i 1.13633 + 1.96819i
−1.02170 1.76964i 0.500000 0.866025i −1.08774 + 1.88403i −0.971745 1.68311i −2.04340 −1.16166 + 2.37709i 0.358585 −0.500000 0.866025i −1.98566 + 3.43927i
277.2 −0.682743 1.18255i 0.500000 0.866025i 0.0677246 0.117302i 0.663106 + 1.14853i −1.36549 2.35341 1.20891i −2.91593 −0.500000 0.866025i 0.905461 1.56830i
277.3 −0.144978 0.251110i 0.500000 0.866025i 0.957963 1.65924i −1.16311 2.01456i −0.289957 2.35341 1.20891i −1.13545 −0.500000 0.866025i −0.337250 + 0.584135i
277.4 0.296764 + 0.514011i 0.500000 0.866025i 0.823862 1.42697i 1.57560 + 2.72902i 0.593528 −1.69175 2.03420i 2.16503 −0.500000 0.866025i −0.935165 + 1.61975i
277.5 0.916322 + 1.58712i 0.500000 0.866025i −0.679293 + 1.17657i 0.471745 + 0.817086i 1.83264 −1.16166 + 2.37709i 1.17548 −0.500000 0.866025i −0.864540 + 1.49743i
277.6 1.13633 + 1.96819i 0.500000 0.866025i −1.58251 + 2.74099i −2.07560 3.59505i 2.27267 −1.69175 2.03420i −2.64772 −0.500000 0.866025i 4.71716 8.17036i
415.1 −1.02170 + 1.76964i 0.500000 + 0.866025i −1.08774 1.88403i −0.971745 + 1.68311i −2.04340 −1.16166 2.37709i 0.358585 −0.500000 + 0.866025i −1.98566 3.43927i
415.2 −0.682743 + 1.18255i 0.500000 + 0.866025i 0.0677246 + 0.117302i 0.663106 1.14853i −1.36549 2.35341 + 1.20891i −2.91593 −0.500000 + 0.866025i 0.905461 + 1.56830i
415.3 −0.144978 + 0.251110i 0.500000 + 0.866025i 0.957963 + 1.65924i −1.16311 + 2.01456i −0.289957 2.35341 + 1.20891i −1.13545 −0.500000 + 0.866025i −0.337250 0.584135i
415.4 0.296764 0.514011i 0.500000 + 0.866025i 0.823862 + 1.42697i 1.57560 2.72902i 0.593528 −1.69175 + 2.03420i 2.16503 −0.500000 + 0.866025i −0.935165 1.61975i
415.5 0.916322 1.58712i 0.500000 + 0.866025i −0.679293 1.17657i 0.471745 0.817086i 1.83264 −1.16166 2.37709i 1.17548 −0.500000 + 0.866025i −0.864540 1.49743i
415.6 1.13633 1.96819i 0.500000 + 0.866025i −1.58251 2.74099i −2.07560 + 3.59505i 2.27267 −1.69175 + 2.03420i −2.64772 −0.500000 + 0.866025i 4.71716 + 8.17036i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.i.f 12
7.c even 3 1 inner 483.2.i.f 12
7.c even 3 1 3381.2.a.bc 6
7.d odd 6 1 3381.2.a.bd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.f 12 1.a even 1 1 trivial
483.2.i.f 12 7.c even 3 1 inner
3381.2.a.bc 6 7.c even 3 1
3381.2.a.bd 6 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(483, [\chi])$$:

 $$T_{2}^{12} - T_{2}^{11} + 8 T_{2}^{10} - 3 T_{2}^{9} + 42 T_{2}^{8} - 15 T_{2}^{7} + 101 T_{2}^{6} + 6 T_{2}^{5} + 150 T_{2}^{4} - 28 T_{2}^{3} + 40 T_{2}^{2} + 8 T_{2} + 4$$ T2^12 - T2^11 + 8*T2^10 - 3*T2^9 + 42*T2^8 - 15*T2^7 + 101*T2^6 + 6*T2^5 + 150*T2^4 - 28*T2^3 + 40*T2^2 + 8*T2 + 4 $$T_{5}^{12} + 3 T_{5}^{11} + 24 T_{5}^{10} + 25 T_{5}^{9} + 278 T_{5}^{8} + 283 T_{5}^{7} + 1647 T_{5}^{6} + 58 T_{5}^{5} + 4044 T_{5}^{4} - 1540 T_{5}^{3} + 8748 T_{5}^{2} - 5180 T_{5} + 5476$$ T5^12 + 3*T5^11 + 24*T5^10 + 25*T5^9 + 278*T5^8 + 283*T5^7 + 1647*T5^6 + 58*T5^5 + 4044*T5^4 - 1540*T5^3 + 8748*T5^2 - 5180*T5 + 5476 $$T_{11}^{12} - 14 T_{11}^{11} + 144 T_{11}^{10} - 776 T_{11}^{9} + 3360 T_{11}^{8} - 7584 T_{11}^{7} + 19520 T_{11}^{6} - 24576 T_{11}^{5} + 86016 T_{11}^{4} - 53248 T_{11}^{3} + 98304 T_{11}^{2} + 32768 T_{11} + 65536$$ T11^12 - 14*T11^11 + 144*T11^10 - 776*T11^9 + 3360*T11^8 - 7584*T11^7 + 19520*T11^6 - 24576*T11^5 + 86016*T11^4 - 53248*T11^3 + 98304*T11^2 + 32768*T11 + 65536

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{11} + 8 T^{10} - 3 T^{9} + 42 T^{8} + \cdots + 4$$
$3$ $$(T^{2} - T + 1)^{6}$$
$5$ $$T^{12} + 3 T^{11} + 24 T^{10} + \cdots + 5476$$
$7$ $$(T^{6} + T^{5} + 2 T^{4} - 23 T^{3} + 14 T^{2} + \cdots + 343)^{2}$$
$11$ $$T^{12} - 14 T^{11} + 144 T^{10} + \cdots + 65536$$
$13$ $$(T^{6} - 31 T^{4} + 52 T^{3} + 47 T^{2} + \cdots - 1)^{2}$$
$17$ $$T^{12} + 15 T^{11} + 164 T^{10} + \cdots + 324$$
$19$ $$T^{12} - T^{11} + 66 T^{10} - 249 T^{9} + \cdots + 16$$
$23$ $$(T^{2} - T + 1)^{6}$$
$29$ $$(T^{6} + 6 T^{5} - 66 T^{4} - 414 T^{3} + \cdots - 4768)^{2}$$
$31$ $$T^{12} - 11 T^{11} + 120 T^{10} + \cdots + 5184$$
$37$ $$T^{12} - 5 T^{11} + 64 T^{10} + \cdots + 179776$$
$41$ $$(T^{6} - 18 T^{5} - 26 T^{4} + 2198 T^{3} + \cdots + 52928)^{2}$$
$43$ $$(T^{6} + 37 T^{5} + 503 T^{4} + \cdots - 19796)^{2}$$
$47$ $$T^{12} + 3 T^{11} + 122 T^{10} + \cdots + 22733824$$
$53$ $$T^{12} - 15 T^{11} + \cdots + 5465349184$$
$59$ $$T^{12} - 2 T^{11} + \cdots + 5805220864$$
$61$ $$T^{12} - 12 T^{11} + \cdots + 9929723904$$
$67$ $$T^{12} - 10 T^{11} + \cdots + 1142372401$$
$71$ $$(T^{6} + 21 T^{5} + 77 T^{4} - 773 T^{3} + \cdots - 3208)^{2}$$
$73$ $$T^{12} - 8 T^{11} + \cdots + 3064618881$$
$79$ $$T^{12} - 17 T^{11} + \cdots + 14531820304$$
$83$ $$(T^{6} - 12 T^{5} - 80 T^{4} + 1222 T^{3} + \cdots - 2224)^{2}$$
$89$ $$T^{12} + 18 T^{11} + 352 T^{10} + \cdots + 27290176$$
$97$ $$(T^{6} + 2 T^{5} - 228 T^{4} - 680 T^{3} + \cdots + 1152)^{2}$$