# Properties

 Label 120.2.k.b Level $120$ Weight $2$ Character orbit 120.k Analytic conductor $0.958$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 120.k (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.958204824255$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4$$ (v^5 + 3*v^3 - 4*v^2 + 2*v - 8) / 4 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 6\nu + 8 ) / 4$$ (-v^5 + 2*v^4 - 3*v^3 + 6*v^2 - 6*v + 8) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 3\nu^{3} + 3\nu^{2} - 2\nu + 6 ) / 2$$ (-v^5 + v^4 - 3*v^3 + 3*v^2 - 2*v + 6) / 2 $$\beta_{4}$$ $$=$$ $$( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 6\nu - 12 ) / 4$$ (3*v^5 - 2*v^4 + 5*v^3 - 6*v^2 + 6*v - 12) / 4 $$\beta_{5}$$ $$=$$ $$( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 10\nu^{2} - 2\nu + 12 ) / 4$$ (-3*v^5 + 2*v^4 - 5*v^3 + 10*v^2 - 2*v + 12) / 4
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2$$ (b5 + b4 + b3 - b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2$$ (b5 + b4 - b3 + b2 - b1 - 1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2$$ (b5 - b4 - 3*b3 + b2 + b1 + 3) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} + 7\beta _1 + 3 ) / 2$$ (b5 + b4 + 3*b3 + b2 + 7*b1 + 3) / 2 $$\nu^{5}$$ $$=$$ $$( -\beta_{5} + 5\beta_{4} + 3\beta_{3} + 3\beta_{2} - \beta _1 + 1 ) / 2$$ (-b5 + 5*b4 + 3*b3 + 3*b2 - b1 + 1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 −0.671462 + 1.24464i −0.671462 − 1.24464i 0.264658 + 1.38923i 0.264658 − 1.38923i 1.40680 + 0.144584i 1.40680 − 0.144584i
−0.671462 1.24464i 1.00000i −1.09828 + 1.67146i 1.00000i 1.24464 0.671462i 4.68585 2.81783 + 0.244644i −1.00000 1.24464 0.671462i
61.2 −0.671462 + 1.24464i 1.00000i −1.09828 1.67146i 1.00000i 1.24464 + 0.671462i 4.68585 2.81783 0.244644i −1.00000 1.24464 + 0.671462i
61.3 0.264658 1.38923i 1.00000i −1.85991 0.735342i 1.00000i −1.38923 0.264658i 0.941367 −1.51380 + 2.38923i −1.00000 −1.38923 0.264658i
61.4 0.264658 + 1.38923i 1.00000i −1.85991 + 0.735342i 1.00000i −1.38923 + 0.264658i 0.941367 −1.51380 2.38923i −1.00000 −1.38923 + 0.264658i
61.5 1.40680 0.144584i 1.00000i 1.95819 0.406803i 1.00000i 0.144584 + 1.40680i −3.62721 2.69597 0.855416i −1.00000 0.144584 + 1.40680i
61.6 1.40680 + 0.144584i 1.00000i 1.95819 + 0.406803i 1.00000i 0.144584 1.40680i −3.62721 2.69597 + 0.855416i −1.00000 0.144584 1.40680i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.k.b 6
3.b odd 2 1 360.2.k.f 6
4.b odd 2 1 480.2.k.b 6
5.b even 2 1 600.2.k.c 6
5.c odd 4 1 600.2.d.e 6
5.c odd 4 1 600.2.d.f 6
8.b even 2 1 inner 120.2.k.b 6
8.d odd 2 1 480.2.k.b 6
12.b even 2 1 1440.2.k.f 6
15.d odd 2 1 1800.2.k.p 6
15.e even 4 1 1800.2.d.q 6
15.e even 4 1 1800.2.d.r 6
16.e even 4 1 3840.2.a.bp 3
16.e even 4 1 3840.2.a.bq 3
16.f odd 4 1 3840.2.a.bo 3
16.f odd 4 1 3840.2.a.br 3
20.d odd 2 1 2400.2.k.c 6
20.e even 4 1 2400.2.d.e 6
20.e even 4 1 2400.2.d.f 6
24.f even 2 1 1440.2.k.f 6
24.h odd 2 1 360.2.k.f 6
40.e odd 2 1 2400.2.k.c 6
40.f even 2 1 600.2.k.c 6
40.i odd 4 1 600.2.d.e 6
40.i odd 4 1 600.2.d.f 6
40.k even 4 1 2400.2.d.e 6
40.k even 4 1 2400.2.d.f 6
60.h even 2 1 7200.2.k.p 6
60.l odd 4 1 7200.2.d.q 6
60.l odd 4 1 7200.2.d.r 6
120.i odd 2 1 1800.2.k.p 6
120.m even 2 1 7200.2.k.p 6
120.q odd 4 1 7200.2.d.q 6
120.q odd 4 1 7200.2.d.r 6
120.w even 4 1 1800.2.d.q 6
120.w even 4 1 1800.2.d.r 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 1.a even 1 1 trivial
120.2.k.b 6 8.b even 2 1 inner
360.2.k.f 6 3.b odd 2 1
360.2.k.f 6 24.h odd 2 1
480.2.k.b 6 4.b odd 2 1
480.2.k.b 6 8.d odd 2 1
600.2.d.e 6 5.c odd 4 1
600.2.d.e 6 40.i odd 4 1
600.2.d.f 6 5.c odd 4 1
600.2.d.f 6 40.i odd 4 1
600.2.k.c 6 5.b even 2 1
600.2.k.c 6 40.f even 2 1
1440.2.k.f 6 12.b even 2 1
1440.2.k.f 6 24.f even 2 1
1800.2.d.q 6 15.e even 4 1
1800.2.d.q 6 120.w even 4 1
1800.2.d.r 6 15.e even 4 1
1800.2.d.r 6 120.w even 4 1
1800.2.k.p 6 15.d odd 2 1
1800.2.k.p 6 120.i odd 2 1
2400.2.d.e 6 20.e even 4 1
2400.2.d.e 6 40.k even 4 1
2400.2.d.f 6 20.e even 4 1
2400.2.d.f 6 40.k even 4 1
2400.2.k.c 6 20.d odd 2 1
2400.2.k.c 6 40.e odd 2 1
3840.2.a.bo 3 16.f odd 4 1
3840.2.a.bp 3 16.e even 4 1
3840.2.a.bq 3 16.e even 4 1
3840.2.a.br 3 16.f odd 4 1
7200.2.d.q 6 60.l odd 4 1
7200.2.d.q 6 120.q odd 4 1
7200.2.d.r 6 60.l odd 4 1
7200.2.d.r 6 120.q odd 4 1
7200.2.k.p 6 60.h even 2 1
7200.2.k.p 6 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{3} - 2T_{7}^{2} - 16T_{7} + 16$$ acting on $$S_{2}^{\mathrm{new}}(120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + 3 T^{4} - 6 T^{3} + \cdots + 8$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$(T^{2} + 1)^{3}$$
$7$ $$(T^{3} - 2 T^{2} - 16 T + 16)^{2}$$
$11$ $$T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096$$
$13$ $$T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 256$$
$17$ $$(T^{3} - 6 T^{2} - 16 T + 32)^{2}$$
$19$ $$T^{6} + 40 T^{4} + 272 T^{2} + \cdots + 256$$
$23$ $$(T^{3} + 4 T^{2} - 12 T - 16)^{2}$$
$29$ $$(T^{2} + 4)^{3}$$
$31$ $$(T^{3} + 6 T^{2} - 16 T - 64)^{2}$$
$37$ $$T^{6} + 136 T^{4} + 5648 T^{2} + \cdots + 65536$$
$41$ $$(T^{3} + 10 T^{2} - 36 T - 232)^{2}$$
$43$ $$T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384$$
$47$ $$(T^{3} - 4 T^{2} - 92 T + 496)^{2}$$
$53$ $$(T^{2} + 4)^{3}$$
$59$ $$T^{6} + 80 T^{4} + 576 T^{2} + \cdots + 1024$$
$61$ $$T^{6} + 256 T^{4} + 17408 T^{2} + \cdots + 262144$$
$67$ $$(T^{2} + 16)^{3}$$
$71$ $$(T^{3} + 4 T^{2} - 112 T + 64)^{2}$$
$73$ $$(T + 6)^{6}$$
$79$ $$(T^{3} - 18 T^{2} + 80 T - 64)^{2}$$
$83$ $$T^{6} + 224 T^{4} + 8448 T^{2} + \cdots + 65536$$
$89$ $$(T^{3} + 14 T^{2} - 4 T - 184)^{2}$$
$97$ $$(T^{3} - 18 T^{2} - 4 T + 328)^{2}$$