# Properties

 Label 420.2.l.f Level $420$ Weight $2$ Character orbit 420.l Analytic conductor $3.354$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.386672896.3 Defining polynomial: $$x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16$$ x^8 - x^6 - 2*x^5 + 2*x^4 - 4*x^3 - 4*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{5} + 2\nu^{4} + 2\nu^{3} - 4\nu ) / 8$$ (v^7 - v^5 + 2*v^4 + 2*v^3 - 4*v) / 8 $$\beta_{3}$$ $$=$$ $$( \nu^{7} - \nu^{5} - 2\nu^{4} + 2\nu^{3} - 4\nu^{2} - 4\nu ) / 8$$ (v^7 - v^5 - 2*v^4 + 2*v^3 - 4*v^2 - 4*v) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + \nu^{5} - 2\nu^{4} - 2\nu^{3} + 8\nu^{2} + 4\nu ) / 8$$ (-v^7 + v^5 - 2*v^4 - 2*v^3 + 8*v^2 + 4*v) / 8 $$\beta_{5}$$ $$=$$ $$( \nu^{6} - \nu^{4} + 2\nu^{3} + 2\nu^{2} - 4\nu - 8 ) / 4$$ (v^6 - v^4 + 2*v^3 + 2*v^2 - 4*v - 8) / 4 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + 2\nu^{6} - \nu^{5} - 2\nu^{3} - 4\nu^{2} - 12\nu ) / 8$$ (v^7 + 2*v^6 - v^5 - 2*v^3 - 4*v^2 - 12*v) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 2\nu^{6} + 3\nu^{5} + 2\nu^{3} - 4\nu^{2} - 12\nu - 24 ) / 8$$ (v^7 + 2*v^6 + 3*v^5 + 2*v^3 - 4*v^2 - 12*v - 24) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ b4 + b2 $$\nu^{3}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{4} + 2$$ -b6 + b5 - b4 + 2 $$\nu^{4}$$ $$=$$ $$-\beta_{4} - 2\beta_{3} + \beta_{2}$$ -b4 - 2*b3 + b2 $$\nu^{5}$$ $$=$$ $$2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4$$ 2*b7 - b6 - b5 + b4 + 4 $$\nu^{6}$$ $$=$$ $$2\beta_{6} + 2\beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 4\beta _1 + 4$$ 2*b6 + 2*b5 - b4 - 2*b3 - b2 + 4*b1 + 4 $$\nu^{7}$$ $$=$$ $$2\beta_{7} + \beta_{6} - 3\beta_{5} + 5\beta_{4} + 4\beta_{3} + 6\beta_{2} + 4\beta_1$$ 2*b7 + b6 - 3*b5 + 5*b4 + 4*b3 + 6*b2 + 4*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 1.40961 − 0.114062i 1.40961 + 0.114062i 0.621372 − 1.27039i 0.621372 + 1.27039i −0.835949 − 1.14070i −0.835949 + 1.14070i −1.19503 − 0.756243i −1.19503 + 0.756243i
−1.40961 0.114062i −1.47398 + 0.909606i 1.97398 + 0.321565i 1.00000 2.00000i 2.18148 1.11406i 1.00000 −2.74586 0.678435i 1.34523 2.68148i −1.63773 + 2.70515i
239.2 −1.40961 + 0.114062i −1.47398 0.909606i 1.97398 0.321565i 1.00000 + 2.00000i 2.18148 + 1.11406i 1.00000 −2.74586 + 0.678435i 1.34523 + 2.68148i −1.63773 2.70515i
239.3 −0.621372 1.27039i 1.72779 + 0.121372i −1.22779 + 1.57877i 1.00000 2.00000i −0.919412 2.27039i 1.00000 2.76858 + 0.578773i 2.97054 + 0.419412i −3.16216 0.0276478i
239.4 −0.621372 + 1.27039i 1.72779 0.121372i −1.22779 1.57877i 1.00000 + 2.00000i −0.919412 + 2.27039i 1.00000 2.76858 0.578773i 2.97054 0.419412i −3.16216 + 0.0276478i
239.5 0.835949 1.14070i 1.10238 + 1.33595i −0.602380 1.90713i 1.00000 + 2.00000i 2.44545 0.140697i 1.00000 −2.67901 0.907128i −0.569517 + 2.94545i 3.11734 + 0.531200i
239.6 0.835949 + 1.14070i 1.10238 1.33595i −0.602380 + 1.90713i 1.00000 2.00000i 2.44545 + 0.140697i 1.00000 −2.67901 + 0.907128i −0.569517 2.94545i 3.11734 0.531200i
239.7 1.19503 0.756243i −0.356193 1.69503i 0.856193 1.80747i 1.00000 2.00000i −1.70752 1.75624i 1.00000 −0.343707 2.80747i −2.74625 + 1.20752i −0.317456 3.14630i
239.8 1.19503 + 0.756243i −0.356193 + 1.69503i 0.856193 + 1.80747i 1.00000 + 2.00000i −1.70752 + 1.75624i 1.00000 −0.343707 + 2.80747i −2.74625 1.20752i −0.317456 + 3.14630i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.l.f yes 8
3.b odd 2 1 420.2.l.e yes 8
4.b odd 2 1 420.2.l.d yes 8
5.b even 2 1 420.2.l.c 8
12.b even 2 1 420.2.l.c 8
15.d odd 2 1 420.2.l.d yes 8
20.d odd 2 1 420.2.l.e yes 8
60.h even 2 1 inner 420.2.l.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.l.c 8 5.b even 2 1
420.2.l.c 8 12.b even 2 1
420.2.l.d yes 8 4.b odd 2 1
420.2.l.d yes 8 15.d odd 2 1
420.2.l.e yes 8 3.b odd 2 1
420.2.l.e yes 8 20.d odd 2 1
420.2.l.f yes 8 1.a even 1 1 trivial
420.2.l.f yes 8 60.h 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 $$S_{2}^{\mathrm{new}}(420, [\chi])$$:

 $$T_{11}^{4} - 2T_{11}^{3} - 11T_{11}^{2} + 16T_{11} + 16$$ T11^4 - 2*T11^3 - 11*T11^2 + 16*T11 + 16 $$T_{17}^{4} + 2T_{17}^{3} - 35T_{17}^{2} - 52T_{17} + 100$$ T17^4 + 2*T17^3 - 35*T17^2 - 52*T17 + 100 $$T_{43}^{4} + 4T_{43}^{3} - 56T_{43}^{2} - 192T_{43} - 128$$ T43^4 + 4*T43^3 - 56*T43^2 - 192*T43 - 128

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 16$$
$3$ $$T^{8} - 2 T^{7} + T^{6} - 2 T^{5} + \cdots + 81$$
$5$ $$(T^{2} - 2 T + 5)^{4}$$
$7$ $$(T - 1)^{8}$$
$11$ $$(T^{4} - 2 T^{3} - 11 T^{2} + 16 T + 16)^{2}$$
$13$ $$T^{8} + 70 T^{6} + 1321 T^{4} + \cdots + 6400$$
$17$ $$(T^{4} + 2 T^{3} - 35 T^{2} - 52 T + 100)^{2}$$
$19$ $$T^{8} + 88 T^{6} + 1104 T^{4} + \cdots + 1024$$
$23$ $$T^{8} + 96 T^{6} + 2656 T^{4} + \cdots + 6400$$
$29$ $$T^{8} + 70 T^{6} + 1321 T^{4} + \cdots + 6400$$
$31$ $$T^{8} + 192 T^{6} + 11488 T^{4} + \cdots + 12544$$
$37$ $$T^{8} + 136 T^{6} + 3984 T^{4} + \cdots + 4096$$
$41$ $$T^{8} + 104 T^{6} + 3472 T^{4} + \cdots + 65536$$
$43$ $$(T^{4} + 4 T^{3} - 56 T^{2} - 192 T - 128)^{2}$$
$47$ $$T^{8} + 22 T^{6} + 145 T^{4} + \cdots + 16$$
$53$ $$(T^{4} - 200 T^{2} - 256 T + 5392)^{2}$$
$59$ $$(T^{4} - 4 T^{3} - 140 T^{2} + 1152 T - 2240)^{2}$$
$61$ $$(T^{4} + 8 T^{3} - 148 T^{2} - 1232 T - 1312)^{2}$$
$67$ $$(T^{4} + 12 T^{3} - 124 T^{2} - 2112 T - 6848)^{2}$$
$71$ $$(T + 8)^{8}$$
$73$ $$(T^{2} + 64)^{4}$$
$79$ $$T^{8} + 502 T^{6} + 66289 T^{4} + \cdots + 753424$$
$83$ $$T^{8} + 432 T^{6} + \cdots + 29073664$$
$89$ $$T^{8} + 352 T^{6} + 26176 T^{4} + \cdots + 6553600$$
$97$ $$T^{8} + 694 T^{6} + \cdots + 16777216$$