# Properties

 Label 462.2.c.b Level $462$ Weight $2$ Character orbit 462.c Analytic conductor $3.689$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 16x^{10} + 94x^{8} + 246x^{6} + 277x^{4} + 114x^{2} + 9$$ x^12 + 16*x^10 + 94*x^8 + 246*x^6 + 277*x^4 + 114*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 16x^{10} + 94x^{8} + 246x^{6} + 277x^{4} + 114x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 8\nu^{4} + 17\nu^{2} + 9 ) / 2$$ (v^6 + 8*v^4 + 17*v^2 + 9) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{11} + 13\nu^{9} + 55\nu^{7} + 72\nu^{5} - 29\nu^{3} - 60\nu ) / 18$$ (v^11 + 13*v^9 + 55*v^7 + 72*v^5 - 29*v^3 - 60*v) / 18 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 10\nu^{5} + 29\nu^{3} + 2\nu^{2} + 21\nu + 6 ) / 2$$ (v^7 + 10*v^5 + 29*v^3 + 2*v^2 + 21*v + 6) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 10\nu^{5} + 29\nu^{3} - 2\nu^{2} + 21\nu - 6 ) / 2$$ (v^7 + 10*v^5 + 29*v^3 - 2*v^2 + 21*v - 6) / 2 $$\beta_{5}$$ $$=$$ $$( - \nu^{11} - 16 \nu^{9} + 3 \nu^{8} - 94 \nu^{7} + 33 \nu^{6} - 243 \nu^{5} + 111 \nu^{4} - 253 \nu^{3} + 114 \nu^{2} - 63 \nu + 27 ) / 12$$ (-v^11 - 16*v^9 + 3*v^8 - 94*v^7 + 33*v^6 - 243*v^5 + 111*v^4 - 253*v^3 + 114*v^2 - 63*v + 27) / 12 $$\beta_{6}$$ $$=$$ $$( \nu^{11} + 16 \nu^{9} + 3 \nu^{8} + 94 \nu^{7} + 33 \nu^{6} + 243 \nu^{5} + 111 \nu^{4} + 253 \nu^{3} + 114 \nu^{2} + 63 \nu + 27 ) / 12$$ (v^11 + 16*v^9 + 3*v^8 + 94*v^7 + 33*v^6 + 243*v^5 + 111*v^4 + 253*v^3 + 114*v^2 + 63*v + 27) / 12 $$\beta_{7}$$ $$=$$ $$( \nu^{10} + 16\nu^{8} + 91\nu^{6} + 219\nu^{4} + 202\nu^{2} + 42 ) / 6$$ (v^10 + 16*v^8 + 91*v^6 + 219*v^4 + 202*v^2 + 42) / 6 $$\beta_{8}$$ $$=$$ $$( - \nu^{11} + 2 \nu^{10} - 16 \nu^{9} + 29 \nu^{8} - 94 \nu^{7} + 149 \nu^{6} - 243 \nu^{5} + 315 \nu^{4} - 253 \nu^{3} + 230 \nu^{2} - 75 \nu + 33 ) / 12$$ (-v^11 + 2*v^10 - 16*v^9 + 29*v^8 - 94*v^7 + 149*v^6 - 243*v^5 + 315*v^4 - 253*v^3 + 230*v^2 - 75*v + 33) / 12 $$\beta_{9}$$ $$=$$ $$( \nu^{11} + 2 \nu^{10} + 16 \nu^{9} + 29 \nu^{8} + 94 \nu^{7} + 149 \nu^{6} + 243 \nu^{5} + 315 \nu^{4} + 253 \nu^{3} + 230 \nu^{2} + 75 \nu + 33 ) / 12$$ (v^11 + 2*v^10 + 16*v^9 + 29*v^8 + 94*v^7 + 149*v^6 + 243*v^5 + 315*v^4 + 253*v^3 + 230*v^2 + 75*v + 33) / 12 $$\beta_{10}$$ $$=$$ $$( \nu^{11} + 16\nu^{9} + 94\nu^{7} + 243\nu^{5} + 259\nu^{3} + 93\nu ) / 6$$ (v^11 + 16*v^9 + 94*v^7 + 243*v^5 + 259*v^3 + 93*v) / 6 $$\beta_{11}$$ $$=$$ $$( 7\nu^{11} + 109\nu^{9} + 610\nu^{7} + 1458\nu^{5} + 1354\nu^{3} + 345\nu ) / 18$$ (7*v^11 + 109*v^9 + 610*v^7 + 1458*v^5 + 1354*v^3 + 345*v) / 18
 $$\nu$$ $$=$$ $$( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} ) / 2$$ (b9 - b8 - b6 + b5) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} + \beta_{3} - 6 ) / 2$$ (-b4 + b3 - 6) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{10} - 5\beta_{9} + 5\beta_{8} + 3\beta_{6} - 3\beta_{5} ) / 2$$ (2*b10 - 5*b9 + 5*b8 + 3*b6 - 3*b5) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} - \beta_{5} + 5\beta_{4} - 5\beta_{3} + 26 ) / 2$$ (-b9 - b8 + 2*b7 - b6 - b5 + 5*b4 - 5*b3 + 26) / 2 $$\nu^{5}$$ $$=$$ $$( 2 \beta_{11} - 14 \beta_{10} + 23 \beta_{9} - 23 \beta_{8} - 13 \beta_{6} + 13 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} ) / 2$$ (2*b11 - 14*b10 + 23*b9 - 23*b8 - 13*b6 + 13*b5 + b4 + b3 - 2*b2) / 2 $$\nu^{6}$$ $$=$$ $$( 8\beta_{9} + 8\beta_{8} - 16\beta_{7} + 8\beta_{6} + 8\beta_{5} - 23\beta_{4} + 23\beta_{3} + 4\beta _1 - 124 ) / 2$$ (8*b9 + 8*b8 - 16*b7 + 8*b6 + 8*b5 - 23*b4 + 23*b3 + 4*b1 - 124) / 2 $$\nu^{7}$$ $$=$$ $$- 10 \beta_{11} + 41 \beta_{10} - 53 \beta_{9} + 53 \beta_{8} + 32 \beta_{6} - 32 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 10 \beta_{2}$$ -10*b11 + 41*b10 - 53*b9 + 53*b8 + 32*b6 - 32*b5 - 4*b4 - 4*b3 + 10*b2 $$\nu^{8}$$ $$=$$ $$( - 51 \beta_{9} - 51 \beta_{8} + 102 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 106 \beta_{4} - 106 \beta_{3} - 44 \beta _1 + 612 ) / 2$$ (-51*b9 - 51*b8 + 102*b7 - 47*b6 - 47*b5 + 106*b4 - 106*b3 - 44*b1 + 612) / 2 $$\nu^{9}$$ $$=$$ $$( 146 \beta_{11} - 456 \beta_{10} + 496 \beta_{9} - 496 \beta_{8} - 328 \beta_{6} + 328 \beta_{5} + 47 \beta_{4} + 47 \beta_{3} - 158 \beta_{2} ) / 2$$ (146*b11 - 456*b10 + 496*b9 - 496*b8 - 328*b6 + 328*b5 + 47*b4 + 47*b3 - 158*b2) / 2 $$\nu^{10}$$ $$=$$ $$( 307 \beta_{9} + 307 \beta_{8} - 602 \beta_{7} + 243 \beta_{6} + 243 \beta_{5} - 496 \beta_{4} + 496 \beta_{3} + 340 \beta _1 - 3074 ) / 2$$ (307*b9 + 307*b8 - 602*b7 + 243*b6 + 243*b5 - 496*b4 + 496*b3 + 340*b1 - 3074) / 2 $$\nu^{11}$$ $$=$$ $$( - 942 \beta_{11} + 2484 \beta_{10} - 2359 \beta_{9} + 2359 \beta_{8} + 1707 \beta_{6} - 1707 \beta_{5} - 243 \beta_{4} - 243 \beta_{3} + 1134 \beta_{2} ) / 2$$ (-942*b11 + 2484*b10 - 2359*b9 + 2359*b8 + 1707*b6 - 1707*b5 - 243*b4 - 243*b3 + 1134*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 2.18803i − 2.18803i 0.822410i − 0.822410i 1.18147i − 1.18147i 2.33939i − 2.33939i − 0.319443i 0.319443i 1.88826i − 1.88826i
1.00000 −1.64459 0.543441i 1.00000 0.118610i −1.64459 0.543441i 1.00000i 1.00000 2.40934 + 1.78747i 0.118610i
197.2 1.00000 −1.64459 + 0.543441i 1.00000 0.118610i −1.64459 + 0.543441i 1.00000i 1.00000 2.40934 1.78747i 0.118610i
197.3 1.00000 −0.742446 1.56486i 1.00000 3.23163i −0.742446 1.56486i 1.00000i 1.00000 −1.89755 + 2.32364i 3.23163i
197.4 1.00000 −0.742446 + 1.56486i 1.00000 3.23163i −0.742446 + 1.56486i 1.00000i 1.00000 −1.89755 2.32364i 3.23163i
197.5 1.00000 0.482127 1.66360i 1.00000 0.885172i 0.482127 1.66360i 1.00000i 1.00000 −2.53511 1.60413i 0.885172i
197.6 1.00000 0.482127 + 1.66360i 1.00000 0.885172i 0.482127 + 1.66360i 1.00000i 1.00000 −2.53511 + 1.60413i 0.885172i
197.7 1.00000 0.806630 1.53276i 1.00000 3.86104i 0.806630 1.53276i 1.00000i 1.00000 −1.69870 2.47274i 3.86104i
197.8 1.00000 0.806630 + 1.53276i 1.00000 3.86104i 0.806630 + 1.53276i 1.00000i 1.00000 −1.69870 + 2.47274i 3.86104i
197.9 1.00000 1.37401 1.05456i 1.00000 3.92876i 1.37401 1.05456i 1.00000i 1.00000 0.775791 2.89796i 3.92876i
197.10 1.00000 1.37401 + 1.05456i 1.00000 3.92876i 1.37401 + 1.05456i 1.00000i 1.00000 0.775791 + 2.89796i 3.92876i
197.11 1.00000 1.72427 0.163987i 1.00000 1.55439i 1.72427 0.163987i 1.00000i 1.00000 2.94622 0.565516i 1.55439i
197.12 1.00000 1.72427 + 0.163987i 1.00000 1.55439i 1.72427 + 0.163987i 1.00000i 1.00000 2.94622 + 0.565516i 1.55439i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.c.b yes 12
3.b odd 2 1 462.2.c.a 12
11.b odd 2 1 462.2.c.a 12
33.d even 2 1 inner 462.2.c.b yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.c.a 12 3.b odd 2 1
462.2.c.a 12 11.b odd 2 1
462.2.c.b yes 12 1.a even 1 1 trivial
462.2.c.b yes 12 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{6} - 2T_{17}^{5} - 50T_{17}^{4} + 64T_{17}^{3} + 348T_{17}^{2} - 328T_{17} - 184$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{12}$$
$3$ $$T^{12} - 4 T^{11} + 8 T^{10} - 8 T^{9} + \cdots + 729$$
$5$ $$T^{12} + 44 T^{10} + 680 T^{8} + \cdots + 64$$
$7$ $$(T^{2} + 1)^{6}$$
$11$ $$T^{12} - 8 T^{11} + 54 T^{10} + \cdots + 1771561$$
$13$ $$T^{12} + 60 T^{10} + 1108 T^{8} + \cdots + 64$$
$17$ $$(T^{6} - 2 T^{5} - 50 T^{4} + 64 T^{3} + \cdots - 184)^{2}$$
$19$ $$T^{12} + 188 T^{10} + \cdots + 12222016$$
$23$ $$T^{12} + 120 T^{10} + 3280 T^{8} + \cdots + 36864$$
$29$ $$(T^{6} + 4 T^{5} - 84 T^{4} - 144 T^{3} + \cdots - 320)^{2}$$
$31$ $$(T^{6} - 6 T^{5} - 82 T^{4} + 128 T^{3} + \cdots + 5288)^{2}$$
$37$ $$(T^{6} - 96 T^{4} + 128 T^{3} + 2192 T^{2} + \cdots + 3008)^{2}$$
$41$ $$(T^{6} + 10 T^{5} - 130 T^{4} + \cdots + 21832)^{2}$$
$43$ $$T^{12} + 320 T^{10} + 27872 T^{8} + \cdots + 1478656$$
$47$ $$T^{12} + 344 T^{10} + \cdots + 22353984$$
$53$ $$T^{12} + 208 T^{10} + 13344 T^{8} + \cdots + 4096$$
$59$ $$T^{12} + 196 T^{10} + 15076 T^{8} + \cdots + 2005056$$
$61$ $$T^{12} + 332 T^{10} + \cdots + 9424526400$$
$67$ $$(T^{6} - 12 T^{5} - 112 T^{4} + \cdots - 46784)^{2}$$
$71$ $$T^{12} + 616 T^{10} + \cdots + 40771686400$$
$73$ $$T^{12} + 280 T^{10} + \cdots + 318551104$$
$79$ $$T^{12} + 232 T^{10} + 15344 T^{8} + \cdots + 2166784$$
$83$ $$(T^{6} + 22 T^{5} + 124 T^{4} - 32 T^{3} + \cdots - 136)^{2}$$
$89$ $$T^{12} + 248 T^{10} + 21104 T^{8} + \cdots + 7573504$$
$97$ $$(T^{6} + 24 T^{5} + 52 T^{4} + \cdots - 121152)^{2}$$