# Properties

 Label 1014.2.i.g Level $1014$ Weight $2$ Character orbit 1014.i Analytic conductor $8.097$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.17213603549184.1 Defining polynomial: $$x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1$$ x^12 - 5*x^10 + 19*x^8 - 28*x^6 + 31*x^4 - 6*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559$$ (-25*v^11 + 95*v^9 - 361*v^7 + 155*v^5 - 30*v^3 - 1563*v) / 559 $$\beta_{3}$$ $$=$$ $$( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559$$ (25*v^10 - 95*v^8 + 361*v^6 - 155*v^4 + 30*v^2 + 1004) / 559 $$\beta_{4}$$ $$=$$ $$( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43$$ (3*v^10 - 20*v^8 + 76*v^6 - 139*v^4 + 124*v^2 - 24) / 43 $$\beta_{5}$$ $$=$$ $$( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559$$ (45*v^10 - 171*v^8 + 538*v^6 - 279*v^4 + 54*v^2 + 242) / 559 $$\beta_{6}$$ $$=$$ $$( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559$$ (70*v^11 - 266*v^9 + 899*v^7 - 434*v^5 + 84*v^3 + 1246*v) / 559 $$\beta_{7}$$ $$=$$ $$( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559$$ (114*v^10 - 545*v^8 + 2071*v^6 - 2831*v^4 + 3379*v^2 - 95) / 559 $$\beta_{8}$$ $$=$$ $$( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559$$ (114*v^11 - 545*v^9 + 2071*v^7 - 2831*v^5 + 3379*v^3 - 95*v) / 559 $$\beta_{9}$$ $$=$$ $$( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559$$ (-128*v^10 + 710*v^8 - 2698*v^6 + 4483*v^4 - 4402*v^2 + 852) / 559 $$\beta_{10}$$ $$=$$ $$( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559$$ (-242*v^11 + 1255*v^9 - 4769*v^7 + 7314*v^5 - 7781*v^3 + 1506*v) / 559 $$\beta_{11}$$ $$=$$ $$( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559$$ (-317*v^11 + 1540*v^9 - 5852*v^7 + 8338*v^5 - 9548*v^3 + 1848*v) / 559
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1$$ b9 + b7 + b4 - b3 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + 3\beta_{8} + \beta_{2}$$ b11 + 3*b8 + b2 $$\nu^{4}$$ $$=$$ $$3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2$$ 3*b9 + 2*b7 + 4*b4 - 2 $$\nu^{5}$$ $$=$$ $$4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1$$ 4*b11 - b10 + 9*b8 - 9*b1 $$\nu^{6}$$ $$=$$ $$-5\beta_{5} + 9\beta_{3} - 14$$ -5*b5 + 9*b3 - 14 $$\nu^{7}$$ $$=$$ $$-5\beta_{6} - 14\beta_{2} - 28\beta_1$$ -5*b6 - 14*b2 - 28*b1 $$\nu^{8}$$ $$=$$ $$-28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28$$ -28*b9 - 14*b7 - 19*b5 - 47*b4 + 28*b3 - 28 $$\nu^{9}$$ $$=$$ $$-47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2}$$ -47*b11 + 19*b10 - 89*b8 - 19*b6 - 47*b2 $$\nu^{10}$$ $$=$$ $$-89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42$$ -89*b9 - 42*b7 - 155*b4 + 42 $$\nu^{11}$$ $$=$$ $$-155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1$$ -155*b11 + 66*b10 - 286*b8 + 286*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −1.07992 − 0.623490i 0.385418 + 0.222521i 1.56052 + 0.900969i −1.56052 − 0.900969i −0.385418 − 0.222521i 1.07992 + 0.623490i 1.56052 − 0.900969i 0.385418 − 0.222521i −1.07992 + 0.623490i 1.07992 − 0.623490i −0.385418 + 0.222521i −1.56052 + 0.900969i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.692021i 0.866025 + 0.500000i 0.309081 + 0.178448i 1.00000i −0.500000 + 0.866025i 0.346011 + 0.599308i
361.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.356896i 0.866025 + 0.500000i −3.50647 2.02446i 1.00000i −0.500000 + 0.866025i 0.178448 + 0.309081i
361.3 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 4.04892i 0.866025 + 0.500000i 0.599308 + 0.346011i 1.00000i −0.500000 + 0.866025i −2.02446 3.50647i
361.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 4.04892i −0.866025 0.500000i −0.599308 0.346011i 1.00000i −0.500000 + 0.866025i −2.02446 3.50647i
361.5 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.356896i −0.866025 0.500000i 3.50647 + 2.02446i 1.00000i −0.500000 + 0.866025i 0.178448 + 0.309081i
361.6 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.692021i −0.866025 0.500000i −0.309081 0.178448i 1.00000i −0.500000 + 0.866025i 0.346011 + 0.599308i
823.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 4.04892i 0.866025 0.500000i 0.599308 0.346011i 1.00000i −0.500000 0.866025i −2.02446 + 3.50647i
823.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.356896i 0.866025 0.500000i −3.50647 + 2.02446i 1.00000i −0.500000 0.866025i 0.178448 0.309081i
823.3 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.692021i 0.866025 0.500000i 0.309081 0.178448i 1.00000i −0.500000 0.866025i 0.346011 0.599308i
823.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.692021i −0.866025 + 0.500000i −0.309081 + 0.178448i 1.00000i −0.500000 0.866025i 0.346011 0.599308i
823.5 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.356896i −0.866025 + 0.500000i 3.50647 2.02446i 1.00000i −0.500000 0.866025i 0.178448 0.309081i
823.6 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 4.04892i −0.866025 + 0.500000i −0.599308 + 0.346011i 1.00000i −0.500000 0.866025i −2.02446 + 3.50647i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 823.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.g 12
13.b even 2 1 inner 1014.2.i.g 12
13.c even 3 1 1014.2.b.g 6
13.c even 3 1 inner 1014.2.i.g 12
13.d odd 4 1 1014.2.e.k 6
13.d odd 4 1 1014.2.e.m 6
13.e even 6 1 1014.2.b.g 6
13.e even 6 1 inner 1014.2.i.g 12
13.f odd 12 1 1014.2.a.m 3
13.f odd 12 1 1014.2.a.o yes 3
13.f odd 12 1 1014.2.e.k 6
13.f odd 12 1 1014.2.e.m 6
39.h odd 6 1 3042.2.b.r 6
39.i odd 6 1 3042.2.b.r 6
39.k even 12 1 3042.2.a.bd 3
39.k even 12 1 3042.2.a.be 3
52.l even 12 1 8112.2.a.bz 3
52.l even 12 1 8112.2.a.ce 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 13.f odd 12 1
1014.2.a.o yes 3 13.f odd 12 1
1014.2.b.g 6 13.c even 3 1
1014.2.b.g 6 13.e even 6 1
1014.2.e.k 6 13.d odd 4 1
1014.2.e.k 6 13.f odd 12 1
1014.2.e.m 6 13.d odd 4 1
1014.2.e.m 6 13.f odd 12 1
1014.2.i.g 12 1.a even 1 1 trivial
1014.2.i.g 12 13.b even 2 1 inner
1014.2.i.g 12 13.c even 3 1 inner
1014.2.i.g 12 13.e even 6 1 inner
3042.2.a.bd 3 39.k even 12 1
3042.2.a.be 3 39.k even 12 1
3042.2.b.r 6 39.h odd 6 1
3042.2.b.r 6 39.i odd 6 1
8112.2.a.bz 3 52.l even 12 1
8112.2.a.ce 3 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1014, [\chi])$$:

 $$T_{5}^{6} + 17T_{5}^{4} + 10T_{5}^{2} + 1$$ T5^6 + 17*T5^4 + 10*T5^2 + 1 $$T_{7}^{12} - 17T_{7}^{10} + 279T_{7}^{8} - 168T_{7}^{6} + 83T_{7}^{4} - 10T_{7}^{2} + 1$$ T7^12 - 17*T7^10 + 279*T7^8 - 168*T7^6 + 83*T7^4 - 10*T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - T^{2} + 1)^{3}$$
$3$ $$(T^{2} + T + 1)^{6}$$
$5$ $$(T^{6} + 17 T^{4} + 10 T^{2} + 1)^{2}$$
$7$ $$T^{12} - 17 T^{10} + 279 T^{8} - 168 T^{6} + \cdots + 1$$
$11$ $$T^{12} - 33 T^{10} + 859 T^{8} + \cdots + 28561$$
$13$ $$T^{12}$$
$17$ $$(T^{6} - 12 T^{5} + 124 T^{4} + \cdots + 10816)^{2}$$
$19$ $$T^{12} - 80 T^{10} + 4864 T^{8} + \cdots + 16777216$$
$23$ $$(T^{6} - 16 T^{5} + 180 T^{4} + \cdots + 10816)^{2}$$
$29$ $$(T^{6} + 13 T^{5} + 157 T^{4} + \cdots + 49729)^{2}$$
$31$ $$(T^{6} + 125 T^{4} + 1006 T^{2} + \cdots + 841)^{2}$$
$37$ $$T^{12} - 104 T^{10} + 10608 T^{8} + \cdots + 4096$$
$41$ $$T^{12} - 84 T^{10} + 5488 T^{8} + \cdots + 9834496$$
$43$ $$(T^{6} + 8 T^{5} + 108 T^{4} + \cdots + 118336)^{2}$$
$47$ $$(T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096)^{2}$$
$53$ $$(T^{3} - 15 T^{2} - 72 T + 1247)^{4}$$
$59$ $$T^{12} - 41 T^{10} + 1515 T^{8} + \cdots + 28561$$
$61$ $$(T^{6} - 10 T^{5} + 76 T^{4} - 224 T^{3} + \cdots + 64)^{2}$$
$67$ $$T^{12} - 404 T^{10} + \cdots + 1529041063936$$
$71$ $$T^{12} - 180 T^{10} + \cdots + 116985856$$
$73$ $$(T^{6} + 69 T^{4} + 614 T^{2} + 169)^{2}$$
$79$ $$(T^{3} + 5 T^{2} - 204 T - 1469)^{4}$$
$83$ $$(T^{6} + 497 T^{4} + 70854 T^{2} + \cdots + 2181529)^{2}$$
$89$ $$T^{12} - 52 T^{10} + 2288 T^{8} + \cdots + 4096$$
$97$ $$T^{12} - 77 T^{10} + 5635 T^{8} + \cdots + 2401$$