# Properties

 Label 507.2.e.l Level $507$ Weight $2$ Character orbit 507.e Analytic conductor $4.048$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$$ (-v^5 + 3*v^4 - 9*v^3 + 5*v^2 - 2*v + 6) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$$ (-3*v^5 + 9*v^4 - 14*v^3 + 15*v^2 - 6*v + 18) / 13 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$$ (-4*v^5 - v^4 - 10*v^3 - 6*v^2 - 34*v - 2) / 13 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$$ (-6*v^5 + 5*v^4 - 15*v^3 - 9*v^2 - 25*v + 10) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ -b5 + b4 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_{2}$$ b3 - 3*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$$ 2*b5 - 3*b4 - 4*b1 - 2 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$$ b5 - 4*b4 - 4*b3 + 9*b2 - 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.900969 − 1.56052i 0.222521 − 0.385418i −0.623490 + 1.07992i 0.900969 + 1.56052i 0.222521 + 0.385418i −0.623490 − 1.07992i
−1.02446 + 1.77441i 0.500000 0.866025i −1.09903 1.90358i −3.35690 1.02446 + 1.77441i 1.12349 + 1.94594i 0.405813 −0.500000 0.866025i 3.43900 5.95652i
22.2 1.17845 2.04113i 0.500000 0.866025i −1.77748 3.07868i −3.69202 −1.17845 2.04113i −0.400969 0.694498i −3.66487 −0.500000 0.866025i −4.35086 + 7.53590i
22.3 1.34601 2.33136i 0.500000 0.866025i −2.62349 4.54402i 1.04892 −1.34601 2.33136i 0.277479 + 0.480608i −8.74094 −0.500000 0.866025i 1.41185 2.44540i
484.1 −1.02446 1.77441i 0.500000 + 0.866025i −1.09903 + 1.90358i −3.35690 1.02446 1.77441i 1.12349 1.94594i 0.405813 −0.500000 + 0.866025i 3.43900 + 5.95652i
484.2 1.17845 + 2.04113i 0.500000 + 0.866025i −1.77748 + 3.07868i −3.69202 −1.17845 + 2.04113i −0.400969 + 0.694498i −3.66487 −0.500000 + 0.866025i −4.35086 7.53590i
484.3 1.34601 + 2.33136i 0.500000 + 0.866025i −2.62349 + 4.54402i 1.04892 −1.34601 + 2.33136i 0.277479 0.480608i −8.74094 −0.500000 + 0.866025i 1.41185 + 2.44540i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 484.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.l 6
13.b even 2 1 507.2.e.i 6
13.c even 3 1 507.2.a.i 3
13.c even 3 1 inner 507.2.e.l 6
13.d odd 4 2 507.2.j.i 12
13.e even 6 1 507.2.a.l yes 3
13.e even 6 1 507.2.e.i 6
13.f odd 12 2 507.2.b.f 6
13.f odd 12 2 507.2.j.i 12
39.h odd 6 1 1521.2.a.n 3
39.i odd 6 1 1521.2.a.s 3
39.k even 12 2 1521.2.b.k 6
52.i odd 6 1 8112.2.a.cp 3
52.j odd 6 1 8112.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.c even 3 1
507.2.a.l yes 3 13.e even 6 1
507.2.b.f 6 13.f odd 12 2
507.2.e.i 6 13.b even 2 1
507.2.e.i 6 13.e even 6 1
507.2.e.l 6 1.a even 1 1 trivial
507.2.e.l 6 13.c even 3 1 inner
507.2.j.i 12 13.d odd 4 2
507.2.j.i 12 13.f odd 12 2
1521.2.a.n 3 39.h odd 6 1
1521.2.a.s 3 39.i odd 6 1
1521.2.b.k 6 39.k even 12 2
8112.2.a.cg 3 52.j odd 6 1
8112.2.a.cp 3 52.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{6} - 3T_{2}^{5} + 13T_{2}^{4} - 14T_{2}^{3} + 55T_{2}^{2} - 52T_{2} + 169$$ T2^6 - 3*T2^5 + 13*T2^4 - 14*T2^3 + 55*T2^2 - 52*T2 + 169 $$T_{5}^{3} + 6T_{5}^{2} + 5T_{5} - 13$$ T5^3 + 6*T5^2 + 5*T5 - 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + 13 T^{4} - 14 T^{3} + \cdots + 169$$
$3$ $$(T^{2} - T + 1)^{3}$$
$5$ $$(T^{3} + 6 T^{2} + 5 T - 13)^{2}$$
$7$ $$T^{6} - 2 T^{5} + 5 T^{4} + 3 T^{2} + \cdots + 1$$
$11$ $$T^{6} - 5 T^{5} + 33 T^{4} + \cdots + 1681$$
$13$ $$T^{6}$$
$17$ $$T^{6} - T^{5} + 17 T^{4} + 42 T^{3} + \cdots + 169$$
$19$ $$T^{6} + 7 T^{5} + 35 T^{4} + 84 T^{3} + \cdots + 49$$
$23$ $$T^{6} + 49 T^{4} + 182 T^{3} + \cdots + 8281$$
$29$ $$T^{6} - 2 T^{5} + 19 T^{4} - 28 T^{3} + \cdots + 841$$
$31$ $$(T^{3} - 16 T^{2} + 41 T + 197)^{2}$$
$37$ $$T^{6} - 22 T^{5} + 325 T^{4} + \cdots + 142129$$
$41$ $$T^{6} - 11 T^{5} + 97 T^{4} + \cdots + 841$$
$43$ $$T^{6} - 15 T^{5} + 178 T^{4} + \cdots + 1681$$
$47$ $$(T^{3} + 7 T^{2} + 14 T + 7)^{2}$$
$53$ $$(T^{3} + 17 T^{2} + 66 T - 41)^{2}$$
$59$ $$T^{6} + 6 T^{5} + 52 T^{4} + \cdots + 10816$$
$61$ $$T^{6} - 13 T^{5} + 185 T^{4} + \cdots + 27889$$
$67$ $$T^{6} - 11 T^{5} + 167 T^{4} + \cdots + 1681$$
$71$ $$T^{6} + 91 T^{4} - 406 T^{3} + \cdots + 41209$$
$73$ $$(T^{3} + 6 T^{2} - 135 T - 923)^{2}$$
$79$ $$(T^{3} - 3 T^{2} - 18 T + 27)^{2}$$
$83$ $$(T^{3} + 12 T^{2} + 41 T + 43)^{2}$$
$89$ $$T^{6} - T^{5} + 101 T^{4} + \cdots + 12769$$
$97$ $$T^{6} + 5 T^{5} + 306 T^{4} + \cdots + 2679769$$