# Properties

 Label 8281.2.a.cp Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1241179138$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1$$ x^12 - 16*x^10 + 88*x^8 - 197*x^6 + 172*x^4 - 36*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{10} + 88x^{8} - 197x^{6} + 172x^{4} - 36x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 5\nu^{10} - 66\nu^{8} + 242\nu^{6} - 127\nu^{4} - 248\nu^{2} + 32 ) / 11$$ (5*v^10 - 66*v^8 + 242*v^6 - 127*v^4 - 248*v^2 + 32) / 11 $$\beta_{3}$$ $$=$$ $$( -3\nu^{10} + 44\nu^{8} - 209\nu^{6} + 360\nu^{4} - 223\nu^{2} + 27 ) / 11$$ (-3*v^10 + 44*v^8 - 209*v^6 + 360*v^4 - 223*v^2 + 27) / 11 $$\beta_{4}$$ $$=$$ $$( -3\nu^{11} + 44\nu^{9} - 209\nu^{7} + 360\nu^{5} - 223\nu^{3} + 27\nu ) / 11$$ (-3*v^11 + 44*v^9 - 209*v^7 + 360*v^5 - 223*v^3 + 27*v) / 11 $$\beta_{5}$$ $$=$$ $$( -2\nu^{11} + 22\nu^{9} - 33\nu^{7} - 244\nu^{5} + 559\nu^{3} - 191\nu ) / 11$$ (-2*v^11 + 22*v^9 - 33*v^7 - 244*v^5 + 559*v^3 - 191*v) / 11 $$\beta_{6}$$ $$=$$ $$( -8\nu^{10} + 110\nu^{8} - 451\nu^{6} + 476\nu^{4} + 91\nu^{2} - 27 ) / 11$$ (-8*v^10 + 110*v^8 - 451*v^6 + 476*v^4 + 91*v^2 - 27) / 11 $$\beta_{7}$$ $$=$$ $$\nu^{9} - 14\nu^{7} + 60\nu^{5} - 76\nu^{3} + 12\nu$$ v^9 - 14*v^7 + 60*v^5 - 76*v^3 + 12*v $$\beta_{8}$$ $$=$$ $$( 10\nu^{10} - 143\nu^{8} + 638\nu^{6} - 903\nu^{4} + 263\nu^{2} + 9 ) / 11$$ (10*v^10 - 143*v^8 + 638*v^6 - 903*v^4 + 263*v^2 + 9) / 11 $$\beta_{9}$$ $$=$$ $$( 10\nu^{10} - 143\nu^{8} + 638\nu^{6} - 903\nu^{4} + 274\nu^{2} - 24 ) / 11$$ (10*v^10 - 143*v^8 + 638*v^6 - 903*v^4 + 274*v^2 - 24) / 11 $$\beta_{10}$$ $$=$$ $$( -8\nu^{11} + 121\nu^{9} - 605\nu^{7} + 1147\nu^{5} - 822\nu^{3} + 171\nu ) / 11$$ (-8*v^11 + 121*v^9 - 605*v^7 + 1147*v^5 - 822*v^3 + 171*v) / 11 $$\beta_{11}$$ $$=$$ $$( -13\nu^{11} + 198\nu^{9} - 1001\nu^{7} + 1923\nu^{5} - 1333\nu^{3} + 194\nu ) / 11$$ (-13*v^11 + 198*v^9 - 1001*v^7 + 1923*v^5 - 1333*v^3 + 194*v) / 11
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{8} + 3$$ b9 - b8 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{4} + 5\beta_1$$ -b11 + b10 + b7 + b5 + b4 + 5*b1 $$\nu^{4}$$ $$=$$ $$6\beta_{9} - 6\beta_{8} - \beta_{6} + \beta_{3} - \beta_{2} + 16$$ 6*b9 - 6*b8 - b6 + b3 - b2 + 16 $$\nu^{5}$$ $$=$$ $$-9\beta_{11} + 10\beta_{10} + 8\beta_{7} + 8\beta_{5} + 7\beta_{4} + 29\beta_1$$ -9*b11 + 10*b10 + 8*b7 + 8*b5 + 7*b4 + 29*b1 $$\nu^{6}$$ $$=$$ $$35\beta_{9} - 37\beta_{8} - 11\beta_{6} + 6\beta_{3} - 10\beta_{2} + 94$$ 35*b9 - 37*b8 - 11*b6 + 6*b3 - 10*b2 + 94 $$\nu^{7}$$ $$=$$ $$-67\beta_{11} + 79\beta_{10} + 57\beta_{7} + 58\beta_{5} + 41\beta_{4} + 176\beta_1$$ -67*b11 + 79*b10 + 57*b7 + 58*b5 + 41*b4 + 176*b1 $$\nu^{8}$$ $$=$$ $$205\beta_{9} - 234\beta_{8} - 95\beta_{6} + 25\beta_{3} - 79\beta_{2} + 574$$ 205*b9 - 234*b8 - 95*b6 + 25*b3 - 79*b2 + 574 $$\nu^{9}$$ $$=$$ $$-474\beta_{11} + 582\beta_{10} + 395\beta_{7} + 408\beta_{5} + 230\beta_{4} + 1092\beta_1$$ -474*b11 + 582*b10 + 395*b7 + 408*b5 + 230*b4 + 1092*b1 $$\nu^{10}$$ $$=$$ $$1214\beta_{9} - 1500\beta_{8} - 747\beta_{6} + 65\beta_{3} - 582\beta_{2} + 3576$$ 1214*b9 - 1500*b8 - 747*b6 + 65*b3 - 582*b2 + 3576 $$\nu^{11}$$ $$=$$ $$-3290\beta_{11} + 4158\beta_{10} + 2708\beta_{7} + 2829\beta_{5} + 1279\beta_{4} + 6872\beta_1$$ -3290*b11 + 4158*b10 + 2708*b7 + 2829*b5 + 1279*b4 + 6872*b1

## 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
 −2.58860 −2.30327 −1.37905 −1.34523 −0.499987 −0.180824 0.180824 0.499987 1.34523 1.37905 2.30327 2.58860
−2.58860 0.518466 4.70085 −1.61205 −1.34210 0 −6.99143 −2.73119 4.17296
1.2 −2.30327 −1.47336 3.30504 −0.847292 3.39354 0 −3.00585 −0.829208 1.95154
1.3 −1.37905 2.88120 −0.0982074 −0.805948 −3.97334 0 2.89354 5.30133 1.11145
1.4 −1.34523 2.05010 −0.190366 −3.56778 −2.75785 0 2.94654 1.20292 4.79947
1.5 −0.499987 0.849601 −1.75001 1.04248 −0.424789 0 1.87496 −2.27818 −0.521224
1.6 −0.180824 −1.82601 −1.96730 −2.68664 0.330186 0 0.717383 0.334323 0.485809
1.7 0.180824 −1.82601 −1.96730 2.68664 −0.330186 0 −0.717383 0.334323 0.485809
1.8 0.499987 0.849601 −1.75001 −1.04248 0.424789 0 −1.87496 −2.27818 −0.521224
1.9 1.34523 2.05010 −0.190366 3.56778 2.75785 0 −2.94654 1.20292 4.79947
1.10 1.37905 2.88120 −0.0982074 0.805948 3.97334 0 −2.89354 5.30133 1.11145
1.11 2.30327 −1.47336 3.30504 0.847292 −3.39354 0 3.00585 −0.829208 1.95154
1.12 2.58860 0.518466 4.70085 1.61205 1.34210 0 6.99143 −2.73119 4.17296
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$13$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.cp 12
7.b odd 2 1 8281.2.a.co 12
7.c even 3 2 1183.2.e.j 24
13.b even 2 1 inner 8281.2.a.cp 12
13.f odd 12 2 637.2.q.g 12
91.b odd 2 1 8281.2.a.co 12
91.r even 6 2 1183.2.e.j 24
91.w even 12 2 637.2.u.g 12
91.x odd 12 2 91.2.k.b 12
91.ba even 12 2 637.2.k.i 12
91.bc even 12 2 637.2.q.i 12
91.bd odd 12 2 91.2.u.b yes 12
273.bv even 12 2 819.2.bm.f 12
273.bw even 12 2 819.2.do.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 91.x odd 12 2
91.2.u.b yes 12 91.bd odd 12 2
637.2.k.i 12 91.ba even 12 2
637.2.q.g 12 13.f odd 12 2
637.2.q.i 12 91.bc even 12 2
637.2.u.g 12 91.w even 12 2
819.2.bm.f 12 273.bv even 12 2
819.2.do.e 12 273.bw even 12 2
1183.2.e.j 24 7.c even 3 2
1183.2.e.j 24 91.r even 6 2
8281.2.a.co 12 7.b odd 2 1
8281.2.a.co 12 91.b odd 2 1
8281.2.a.cp 12 1.a even 1 1 trivial
8281.2.a.cp 12 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{12} - 16T_{2}^{10} + 88T_{2}^{8} - 197T_{2}^{6} + 172T_{2}^{4} - 36T_{2}^{2} + 1$$ T2^12 - 16*T2^10 + 88*T2^8 - 197*T2^6 + 172*T2^4 - 36*T2^2 + 1 $$T_{3}^{6} - 3T_{3}^{5} - 5T_{3}^{4} + 16T_{3}^{3} + 4T_{3}^{2} - 19T_{3} + 7$$ T3^6 - 3*T3^5 - 5*T3^4 + 16*T3^3 + 4*T3^2 - 19*T3 + 7 $$T_{5}^{12} - 25T_{5}^{10} + 201T_{5}^{8} - 636T_{5}^{6} + 878T_{5}^{4} - 539T_{5}^{2} + 121$$ T5^12 - 25*T5^10 + 201*T5^8 - 636*T5^6 + 878*T5^4 - 539*T5^2 + 121 $$T_{11}^{12} - 62T_{11}^{10} + 1355T_{11}^{8} - 13284T_{11}^{6} + 61227T_{11}^{4} - 122593T_{11}^{2} + 85849$$ T11^12 - 62*T11^10 + 1355*T11^8 - 13284*T11^6 + 61227*T11^4 - 122593*T11^2 + 85849 $$T_{17}^{6} - 17T_{17}^{5} + 96T_{17}^{4} - 198T_{17}^{3} + 56T_{17}^{2} + 146T_{17} + 19$$ T17^6 - 17*T17^5 + 96*T17^4 - 198*T17^3 + 56*T17^2 + 146*T17 + 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 16 T^{10} + 88 T^{8} - 197 T^{6} + \cdots + 1$$
$3$ $$(T^{6} - 3 T^{5} - 5 T^{4} + 16 T^{3} + 4 T^{2} + \cdots + 7)^{2}$$
$5$ $$T^{12} - 25 T^{10} + 201 T^{8} + \cdots + 121$$
$7$ $$T^{12}$$
$11$ $$T^{12} - 62 T^{10} + 1355 T^{8} + \cdots + 85849$$
$13$ $$T^{12}$$
$17$ $$(T^{6} - 17 T^{5} + 96 T^{4} - 198 T^{3} + \cdots + 19)^{2}$$
$19$ $$T^{12} - 79 T^{10} + 1984 T^{8} + \cdots + 1$$
$23$ $$(T^{6} - 3 T^{5} - 50 T^{4} + 259 T^{3} + \cdots + 793)^{2}$$
$29$ $$(T^{6} - T^{5} - 86 T^{4} + 190 T^{3} + \cdots + 4009)^{2}$$
$31$ $$T^{12} - 232 T^{10} + \cdots + 241274089$$
$37$ $$T^{12} - 147 T^{10} + 7164 T^{8} + \cdots + 123201$$
$41$ $$T^{12} - 354 T^{10} + \cdots + 389707081$$
$43$ $$(T^{6} - 11 T^{5} - 49 T^{4} + 816 T^{3} + \cdots + 20467)^{2}$$
$47$ $$T^{12} - 41 T^{10} + 509 T^{8} + \cdots + 121$$
$53$ $$(T^{6} - 8 T^{5} - 38 T^{4} + 404 T^{3} + \cdots - 17)^{2}$$
$59$ $$T^{12} - 553 T^{10} + \cdots + 35582408689$$
$61$ $$(T^{6} + 5 T^{5} - 75 T^{4} - 354 T^{3} + \cdots + 1777)^{2}$$
$67$ $$T^{12} - 439 T^{10} + \cdots + 5708255809$$
$71$ $$T^{12} - 446 T^{10} + \cdots + 639230089$$
$73$ $$T^{12} - 400 T^{10} + \cdots + 484396081$$
$79$ $$(T^{6} - 35 T^{5} + 344 T^{4} + \cdots + 255121)^{2}$$
$83$ $$T^{12} - 463 T^{10} + \cdots + 402363481$$
$89$ $$T^{12} - 430 T^{10} + \cdots + 145033849$$
$97$ $$T^{12} - 361 T^{10} + 29979 T^{8} + \cdots + 1681$$