# Properties

 Label 507.2.e.k 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{4} + (\beta_{3} - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (3 \beta_{5} - \beta_{4} - \beta_{3}) q^{7} + (\beta_{3} + \beta_{2}) q^{8} - \beta_{5} q^{9}+O(q^{10})$$ q + b1 * q^2 + (b5 - 1) * q^3 + (b5 + b4 + b3 - b2 + b1) * q^4 + (b3 - 1) * q^5 + (b2 - b1) * q^6 + (3*b5 - b4 - b3) * q^7 + (b3 + b2) * q^8 - b5 * q^9 $$q + \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{4} + (\beta_{3} - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (3 \beta_{5} - \beta_{4} - \beta_{3}) q^{7} + (\beta_{3} + \beta_{2}) q^{8} - \beta_{5} q^{9} + ( - \beta_{5} - 2 \beta_1 + 1) q^{10} + ( - 2 \beta_{5} - 4 \beta_{4} - 3 \beta_1 + 2) q^{11} + ( - \beta_{3} + \beta_{2} - 1) q^{12} + (4 \beta_{2} - 1) q^{14} + ( - \beta_{5} + \beta_{4} + 1) q^{15} + (3 \beta_{4} + 2 \beta_1) q^{16} + (3 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{17} - \beta_{2} q^{18} + (3 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{19} + (\beta_{2} - \beta_1) q^{20} + (\beta_{3} - 3) q^{21} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{22} + (3 \beta_{5} + 5 \beta_{4} + 2 \beta_1 - 3) q^{23} + (\beta_{4} - \beta_1) q^{24} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{25} + q^{27} + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_1 - 2) q^{28} + ( - 3 \beta_{5} - 2 \beta_{4} + \beta_1 + 3) q^{29} + (\beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{30} + ( - 2 \beta_{3} + 5 \beta_{2} - 5) q^{31} + (\beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{32} + (2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{33} + (\beta_{3} + \beta_{2}) q^{34} + ( - 5 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1) q^{35} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{36} + ( - 4 \beta_{5} + \beta_{4} + \beta_1 + 4) q^{37} + ( - 3 \beta_{3} + 5 \beta_{2} + 4) q^{38} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{40} + \beta_1 q^{41} + ( - \beta_{5} - 4 \beta_1 + 1) q^{42} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{43} + (3 \beta_{3} + \beta_{2} + 6) q^{44} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{45} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{46} + (3 \beta_{3} - 9 \beta_{2} + 7) q^{47} + ( - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{48} + (4 \beta_{5} - 6 \beta_{4} + \beta_1 - 4) q^{49} + (3 \beta_{5} - \beta_{4} - \beta_1 - 3) q^{50} + ( - \beta_{3} + \beta_{2} - 3) q^{51} + (2 \beta_{3} + \beta_{2} - 4) q^{53} + \beta_1 q^{54} + ( - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_1 + 3) q^{55} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{56} + ( - \beta_{3} - 3 \beta_{2} - 3) q^{57} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{58} + ( - 8 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{59} - \beta_{2} q^{60} + (7 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{61} + ( - 3 \beta_{5} + 5 \beta_{4} + 2 \beta_1 + 3) q^{62} + ( - 3 \beta_{5} + \beta_{4} + 3) q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} + 1) q^{64} + (3 \beta_{3} - 5 \beta_{2} + 1) q^{66} + ( - 4 \beta_{5} - 3 \beta_{4} - 4 \beta_1 + 4) q^{67} + (4 \beta_{5} + 3 \beta_{4} + 2 \beta_1 - 4) q^{68} + ( - 3 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{69} + ( - \beta_{3} - 8 \beta_{2} + 5) q^{70} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{71} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{72} + ( - \beta_{3} + 2 \beta_{2} - 7) q^{73} + (\beta_{4} + \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{74} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_1 + 2) q^{75} + (4 \beta_{5} + 7 \beta_{4} + 6 \beta_1 - 4) q^{76} + (10 \beta_{3} - 8 \beta_{2} + 1) q^{77} + (5 \beta_{3} - 4 \beta_{2}) q^{79} + (4 \beta_{5} - 3 \beta_{4} - \beta_1 - 4) q^{80} + (\beta_{5} - 1) q^{81} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{82} + (3 \beta_{3} + \beta_{2} + 6) q^{83} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{84} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{85} + ( - 4 \beta_{3} + 5 \beta_{2} + 2) q^{86} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{87} + ( - 6 \beta_{5} - 5 \beta_{4} - 6 \beta_1 + 6) q^{88} + (2 \beta_{5} - \beta_{4} + 2 \beta_1 - 2) q^{89} + (2 \beta_{2} - 1) q^{90} + ( - 6 \beta_{3} + \beta_{2} - 8) q^{92} + ( - 5 \beta_{5} - 2 \beta_{4} - 5 \beta_1 + 5) q^{93} + (6 \beta_{5} - 9 \beta_{4} - 5 \beta_1 - 6) q^{94} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 7 \beta_1) q^{95} + ( - 4 \beta_{3} + 3 \beta_{2} - 1) q^{96} + ( - 3 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{97} + ( - 7 \beta_{5} + \beta_{4} + \beta_{3} + 9 \beta_{2} - 9 \beta_1) q^{98} + ( - 4 \beta_{3} + 3 \beta_{2} - 2) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b5 - 1) * q^3 + (b5 + b4 + b3 - b2 + b1) * q^4 + (b3 - 1) * q^5 + (b2 - b1) * q^6 + (3*b5 - b4 - b3) * q^7 + (b3 + b2) * q^8 - b5 * q^9 + (-b5 - 2*b1 + 1) * q^10 + (-2*b5 - 4*b4 - 3*b1 + 2) * q^11 + (-b3 + b2 - 1) * q^12 + (4*b2 - 1) * q^14 + (-b5 + b4 + 1) * q^15 + (3*b4 + 2*b1) * q^16 + (3*b5 + b4 + b3 - b2 + b1) * q^17 - b2 * q^18 + (3*b5 + b4 + b3 + 3*b2 - 3*b1) * q^19 + (b2 - b1) * q^20 + (b3 - 3) * q^21 + (-b5 - 3*b4 - 3*b3 + 5*b2 - 5*b1) * q^22 + (3*b5 + 5*b4 + 2*b1 - 3) * q^23 + (b4 - b1) * q^24 + (-2*b3 - b2 - 2) * q^25 + q^27 + (2*b5 + 2*b4 + 3*b1 - 2) * q^28 + (-3*b5 - 2*b4 + b1 + 3) * q^29 + (b5 - 2*b2 + 2*b1) * q^30 + (-2*b3 + 5*b2 - 5) * q^31 + (b5 + 4*b4 + 4*b3 - 3*b2 + 3*b1) * q^32 + (2*b5 + 4*b4 + 4*b3 - 3*b2 + 3*b1) * q^33 + (b3 + b2) * q^34 + (-5*b5 + 4*b4 + 4*b3 + b2 - b1) * q^35 + (-b5 - b4 - b1 + 1) * q^36 + (-4*b5 + b4 + b1 + 4) * q^37 + (-3*b3 + 5*b2 + 4) * q^38 + (-b3 - 3*b2 + 3) * q^40 + b1 * q^41 + (-b5 - 4*b1 + 1) * q^42 + (-b5 - 2*b4 - 2*b3 + 4*b2 - 4*b1) * q^43 + (3*b3 + b2 + 6) * q^44 + (b5 - b4 - b3) * q^45 + (3*b5 + 2*b4 + 2*b3 - 4*b2 + 4*b1) * q^46 + (3*b3 - 9*b2 + 7) * q^47 + (-3*b4 - 3*b3 + 2*b2 - 2*b1) * q^48 + (4*b5 - 6*b4 + b1 - 4) * q^49 + (3*b5 - b4 - b1 - 3) * q^50 + (-b3 + b2 - 3) * q^51 + (2*b3 + b2 - 4) * q^53 + b1 * q^54 + (-3*b5 + 2*b4 + 2*b1 + 3) * q^55 + (-3*b5 + 3*b4 + 3*b3 + 5*b2 - 5*b1) * q^56 + (-b3 - 3*b2 - 3) * q^57 + (-3*b5 + b4 + b3 - 2*b2 + 2*b1) * q^58 + (-8*b5 - 6*b4 - 6*b3 + 4*b2 - 4*b1) * q^59 - b2 * q^60 + (7*b5 + 3*b4 + 3*b3 - 5*b2 + 5*b1) * q^61 + (-3*b5 + 5*b4 + 2*b1 + 3) * q^62 + (-3*b5 + b4 + 3) * q^63 + (-3*b3 - 2*b2 + 1) * q^64 + (3*b3 - 5*b2 + 1) * q^66 + (-4*b5 - 3*b4 - 4*b1 + 4) * q^67 + (4*b5 + 3*b4 + 2*b1 - 4) * q^68 + (-3*b5 - 5*b4 - 5*b3 + 2*b2 - 2*b1) * q^69 + (-b3 - 8*b2 + 5) * q^70 + (-b5 - 2*b4 - 2*b3 - 5*b2 + 5*b1) * q^71 + (-b4 - b3 - b2 + b1) * q^72 + (-b3 + 2*b2 - 7) * q^73 + (b4 + b3 - 6*b2 + 6*b1) * q^74 + (-2*b5 - 2*b4 + b1 + 2) * q^75 + (4*b5 + 7*b4 + 6*b1 - 4) * q^76 + (10*b3 - 8*b2 + 1) * q^77 + (5*b3 - 4*b2) * q^79 + (4*b5 - 3*b4 - b1 - 4) * q^80 + (b5 - 1) * q^81 + (-b5 + b4 + b3 - b2 + b1) * q^82 + (3*b3 + b2 + 6) * q^83 + (-2*b5 - 2*b4 - 2*b3 + 3*b2 - 3*b1) * q^84 + (-2*b5 + 2*b4 + 2*b3 + b2 - b1) * q^85 + (-4*b3 + 5*b2 + 2) * q^86 + (3*b5 + 2*b4 + 2*b3 + b2 - b1) * q^87 + (-6*b5 - 5*b4 - 6*b1 + 6) * q^88 + (2*b5 - b4 + 2*b1 - 2) * q^89 + (2*b2 - 1) * q^90 + (-6*b3 + b2 - 8) * q^92 + (-5*b5 - 2*b4 - 5*b1 + 5) * q^93 + (6*b5 - 9*b4 - 5*b1 - 6) * q^94 + (2*b5 + 2*b4 + 2*b3 - 7*b2 + 7*b1) * q^95 + (-4*b3 + 3*b2 - 1) * q^96 + (-3*b5 - 10*b4 - 10*b3 + 4*b2 - 4*b1) * q^97 + (-7*b5 + b4 + b3 + 9*b2 - 9*b1) * q^98 + (-4*b3 + 3*b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} - 3 q^{3} + q^{4} - 8 q^{5} + q^{6} + 10 q^{7} - 3 q^{9}+O(q^{10})$$ 6 * q + q^2 - 3 * q^3 + q^4 - 8 * q^5 + q^6 + 10 * q^7 - 3 * q^9 $$6 q + q^{2} - 3 q^{3} + q^{4} - 8 q^{5} + q^{6} + 10 q^{7} - 3 q^{9} + q^{10} - q^{11} - 2 q^{12} + 2 q^{14} + 4 q^{15} + 5 q^{16} + 7 q^{17} - 2 q^{18} + 11 q^{19} + q^{20} - 20 q^{21} + 5 q^{22} - 2 q^{23} - 10 q^{25} + 6 q^{27} - q^{28} + 8 q^{29} + q^{30} - 16 q^{31} - 4 q^{32} - q^{33} - 18 q^{35} + q^{36} + 14 q^{37} + 40 q^{38} + 14 q^{40} + q^{41} - q^{42} + 3 q^{43} + 32 q^{44} + 4 q^{45} + 3 q^{46} + 18 q^{47} + 5 q^{48} - 17 q^{49} - 11 q^{50} - 14 q^{51} - 26 q^{53} + q^{54} + 13 q^{55} - 7 q^{56} - 22 q^{57} - 12 q^{58} - 14 q^{59} - 2 q^{60} + 13 q^{61} + 16 q^{62} + 10 q^{63} + 8 q^{64} - 10 q^{66} + 5 q^{67} - 7 q^{68} - 2 q^{69} + 16 q^{70} - 6 q^{71} - 36 q^{73} - 7 q^{74} + 5 q^{75} + q^{76} - 30 q^{77} - 18 q^{79} - 16 q^{80} - 3 q^{81} - 5 q^{82} + 32 q^{83} - q^{84} - 7 q^{85} + 30 q^{86} + 8 q^{87} + 7 q^{88} - 5 q^{89} - 2 q^{90} - 34 q^{92} + 8 q^{93} - 32 q^{94} - 3 q^{95} + 8 q^{96} + 5 q^{97} - 13 q^{98} + 2 q^{99}+O(q^{100})$$ 6 * q + q^2 - 3 * q^3 + q^4 - 8 * q^5 + q^6 + 10 * q^7 - 3 * q^9 + q^10 - q^11 - 2 * q^12 + 2 * q^14 + 4 * q^15 + 5 * q^16 + 7 * q^17 - 2 * q^18 + 11 * q^19 + q^20 - 20 * q^21 + 5 * q^22 - 2 * q^23 - 10 * q^25 + 6 * q^27 - q^28 + 8 * q^29 + q^30 - 16 * q^31 - 4 * q^32 - q^33 - 18 * q^35 + q^36 + 14 * q^37 + 40 * q^38 + 14 * q^40 + q^41 - q^42 + 3 * q^43 + 32 * q^44 + 4 * q^45 + 3 * q^46 + 18 * q^47 + 5 * q^48 - 17 * q^49 - 11 * q^50 - 14 * q^51 - 26 * q^53 + q^54 + 13 * q^55 - 7 * q^56 - 22 * q^57 - 12 * q^58 - 14 * q^59 - 2 * q^60 + 13 * q^61 + 16 * q^62 + 10 * q^63 + 8 * q^64 - 10 * q^66 + 5 * q^67 - 7 * q^68 - 2 * q^69 + 16 * q^70 - 6 * q^71 - 36 * q^73 - 7 * q^74 + 5 * q^75 + q^76 - 30 * q^77 - 18 * q^79 - 16 * q^80 - 3 * q^81 - 5 * q^82 + 32 * q^83 - q^84 - 7 * q^85 + 30 * q^86 + 8 * q^87 + 7 * q^88 - 5 * q^89 - 2 * q^90 - 34 * q^92 + 8 * q^93 - 32 * q^94 - 3 * q^95 + 8 * q^96 + 5 * q^97 - 13 * q^98 + 2 * 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
 −0.623490 + 1.07992i 0.222521 − 0.385418i 0.900969 − 1.56052i −0.623490 − 1.07992i 0.222521 + 0.385418i 0.900969 + 1.56052i
−0.623490 + 1.07992i −0.500000 + 0.866025i 0.222521 + 0.385418i −2.80194 −0.623490 1.07992i 2.40097 + 4.15860i −3.04892 −0.500000 0.866025i 1.74698 3.02586i
22.2 0.222521 0.385418i −0.500000 + 0.866025i 0.900969 + 1.56052i 0.246980 0.222521 + 0.385418i 0.876510 + 1.51816i 1.69202 −0.500000 0.866025i 0.0549581 0.0951903i
22.3 0.900969 1.56052i −0.500000 + 0.866025i −0.623490 1.07992i −1.44504 0.900969 + 1.56052i 1.72252 + 2.98349i 1.35690 −0.500000 0.866025i −1.30194 + 2.25502i
484.1 −0.623490 1.07992i −0.500000 0.866025i 0.222521 0.385418i −2.80194 −0.623490 + 1.07992i 2.40097 4.15860i −3.04892 −0.500000 + 0.866025i 1.74698 + 3.02586i
484.2 0.222521 + 0.385418i −0.500000 0.866025i 0.900969 1.56052i 0.246980 0.222521 0.385418i 0.876510 1.51816i 1.69202 −0.500000 + 0.866025i 0.0549581 + 0.0951903i
484.3 0.900969 + 1.56052i −0.500000 0.866025i −0.623490 + 1.07992i −1.44504 0.900969 1.56052i 1.72252 2.98349i 1.35690 −0.500000 + 0.866025i −1.30194 2.25502i
 $$n$$: 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.k 6
13.b even 2 1 507.2.e.j 6
13.c even 3 1 507.2.a.j 3
13.c even 3 1 inner 507.2.e.k 6
13.d odd 4 2 507.2.j.h 12
13.e even 6 1 507.2.a.k yes 3
13.e even 6 1 507.2.e.j 6
13.f odd 12 2 507.2.b.g 6
13.f odd 12 2 507.2.j.h 12
39.h odd 6 1 1521.2.a.p 3
39.i odd 6 1 1521.2.a.q 3
39.k even 12 2 1521.2.b.m 6
52.i odd 6 1 8112.2.a.cf 3
52.j odd 6 1 8112.2.a.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.c even 3 1
507.2.a.k yes 3 13.e even 6 1
507.2.b.g 6 13.f odd 12 2
507.2.e.j 6 13.b even 2 1
507.2.e.j 6 13.e even 6 1
507.2.e.k 6 1.a even 1 1 trivial
507.2.e.k 6 13.c even 3 1 inner
507.2.j.h 12 13.d odd 4 2
507.2.j.h 12 13.f odd 12 2
1521.2.a.p 3 39.h odd 6 1
1521.2.a.q 3 39.i odd 6 1
1521.2.b.m 6 39.k even 12 2
8112.2.a.by 3 52.j odd 6 1
8112.2.a.cf 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 $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{6} - T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} - 2T_{2} + 1$$ T2^6 - T2^5 + 3*T2^4 + 5*T2^2 - 2*T2 + 1 $$T_{5}^{3} + 4T_{5}^{2} + 3T_{5} - 1$$ T5^3 + 4*T5^2 + 3*T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1$$
$3$ $$(T^{2} + T + 1)^{3}$$
$5$ $$(T^{3} + 4 T^{2} + 3 T - 1)^{2}$$
$7$ $$T^{6} - 10 T^{5} + 69 T^{4} + \cdots + 841$$
$11$ $$T^{6} + T^{5} + 31 T^{4} + 56 T^{3} + \cdots + 1849$$
$13$ $$T^{6}$$
$17$ $$T^{6} - 7 T^{5} + 35 T^{4} - 84 T^{3} + \cdots + 49$$
$19$ $$T^{6} - 11 T^{5} + 111 T^{4} + \cdots + 12769$$
$23$ $$T^{6} + 2 T^{5} + 47 T^{4} + \cdots + 6889$$
$29$ $$T^{6} - 8 T^{5} + 59 T^{4} + \cdots + 1849$$
$31$ $$(T^{3} + 8 T^{2} - 23 T - 197)^{2}$$
$37$ $$T^{6} - 14 T^{5} + 133 T^{4} + \cdots + 8281$$
$41$ $$T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1$$
$43$ $$T^{6} - 3 T^{5} + 34 T^{4} + 133 T^{3} + \cdots + 841$$
$47$ $$(T^{3} - 9 T^{2} - 120 T + 911)^{2}$$
$53$ $$(T^{3} + 13 T^{2} + 40 T + 29)^{2}$$
$59$ $$T^{6} + 14 T^{5} + 196 T^{4} + \cdots + 3136$$
$61$ $$T^{6} - 13 T^{5} + 157 T^{4} + \cdots + 49729$$
$67$ $$T^{6} - 5 T^{5} + 47 T^{4} + \cdots + 9409$$
$71$ $$T^{6} + 6 T^{5} + 115 T^{4} + \cdots + 212521$$
$73$ $$(T^{3} + 18 T^{2} + 101 T + 167)^{2}$$
$79$ $$(T^{3} + 9 T^{2} - 22 T - 169)^{2}$$
$83$ $$(T^{3} - 16 T^{2} + 55 T + 43)^{2}$$
$89$ $$T^{6} + 5 T^{5} + 33 T^{4} - 42 T^{3} + \cdots + 1$$
$97$ $$T^{6} - 5 T^{5} + 194 T^{4} + \cdots + 1413721$$