# Properties

 Label 406.2.e.a Level $406$ Weight $2$ Character orbit 406.e Analytic conductor $3.242$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$406 = 2 \cdot 7 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 406.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.24192632206$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.3118758597603.1 Defining polynomial: $$x^{10} - 4x^{8} - 16x^{6} - 34x^{5} + 43x^{4} + 155x^{3} + 199x^{2} + 124x + 43$$ x^10 - 4*x^8 - 16*x^6 - 34*x^5 + 43*x^4 + 155*x^3 + 199*x^2 + 124*x + 43 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} - 1) q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{3} + 1) q^{6} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_1) q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - 3) q^{9}+O(q^{10})$$ q + (b4 - 1) * q^2 + (-b4 - b3) * q^3 - b4 * q^4 + (b9 - b8 + b7 + b6 + b4 + b2 - 1) * q^5 + (b6 + b3 + 1) * q^6 + (b9 - b8 + b7 + b3 - b1) * q^7 + q^8 + (b9 - b7 - b6 + b5 + 3*b4 - 3) * q^9 $$q + (\beta_{4} - 1) q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} - \beta_{4} q^{4} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{3} + 1) q^{6} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_1) q^{7} + q^{8} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - 3) q^{9} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{10} + (\beta_{9} + \beta_{4} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{4} - 1) q^{12} + (\beta_{6} + \beta_{5} + \beta_{3} - \beta_1 + 2) q^{13} + (\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1) q^{14} + ( - \beta_{9} - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{15} + (\beta_{4} - 1) q^{16} + ( - \beta_{8} - 2 \beta_{4} - \beta_1) q^{17} + ( - \beta_{9} + \beta_{8} - 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{18} + ( - 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{19} + ( - \beta_{9} - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{20} + ( - 3 \beta_{9} + \beta_{8} - \beta_{4} - 2 \beta_{2} + 3) q^{21} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_1 - 1) q^{22} + ( - 2 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{2}) q^{23} + ( - \beta_{4} - \beta_{3}) q^{24} + (\beta_{9} + \beta_{8} - \beta_{4} + 4 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - 2) q^{26} + ( - 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + \cdots + 5) q^{27}+ \cdots + (3 \beta_{9} - \beta_{8} + 4 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + \cdots - 7) q^{99}+O(q^{100})$$ q + (b4 - 1) * q^2 + (-b4 - b3) * q^3 - b4 * q^4 + (b9 - b8 + b7 + b6 + b4 + b2 - 1) * q^5 + (b6 + b3 + 1) * q^6 + (b9 - b8 + b7 + b3 - b1) * q^7 + q^8 + (b9 - b7 - b6 + b5 + 3*b4 - 3) * q^9 + (b8 - b4 + b3) * q^10 + (b9 + b4 - b2 + b1) * q^11 + (-b6 + b4 - 1) * q^12 + (b6 + b5 + b3 - b1 + 2) * q^13 + (b8 - b7 - b6 - b5 - b3 + b1) * q^14 + (-b9 - b7 + 2*b6 + 2*b5 + 2*b3 - b2 - 2*b1 - 2) * q^15 + (b4 - 1) * q^16 + (-b8 - 2*b4 - b1) * q^17 + (-b9 + b8 - 3*b4 - b3 + b2 - b1) * q^18 + (-2*b9 + 2*b8 - 2*b7 + b6 - 2*b2) * q^19 + (-b9 - b7 - b6 - b3 - b2 + 1) * q^20 + (-3*b9 + b8 - b4 - 2*b2 + 3) * q^21 + (-b9 + b8 - 2*b7 + b5 - b1 - 1) * q^22 + (-2*b9 + b8 + b5 - b2) * q^23 + (-b4 - b3) * q^24 + (b9 + b8 - b4 + 4*b3 - b2 - 2*b1) * q^25 + (-b6 - b5 + 2*b4 - 2) * q^26 + (-2*b8 + 2*b7 + 3*b6 + b5 + 3*b3 - 2*b2 - b1 + 5) * q^27 + (-b9 + b6 + b5) * q^28 - q^29 + (b9 - b8 + b7 - 2*b6 - 2*b5 - 2*b4 + b2 + 2) * q^30 + (2*b9 - 2*b8 - b4 + b3 - 2*b2 - b1) * q^31 - b4 * q^32 + (b9 - b8 + b7 + 3*b6 + b5 + b4 + b2 - 1) * q^33 + (b9 + b7 - b5 + b2 + b1 + 2) * q^34 + (-b9 + b8 - 3*b7 - 3*b6 - 3*b3 - 3*b2 + b1) * q^35 + (-b8 + b7 + b6 - b5 + b3 - b2 + b1 + 3) * q^36 + (2*b9 - 2*b7 - 2*b6 - b5) * q^37 + (-2*b8 + b3) * q^38 + (-b9 + 2*b8 - 5*b4 - b3 + b2 - b1) * q^39 + (b9 - b8 + b7 + b6 + b4 + b2 - 1) * q^40 + (2*b9 - b8 + 3*b7 - 2*b5 + b2 + 2*b1 + 6) * q^41 + (2*b9 - 3*b8 + 3*b4 - b2 - 2) * q^42 + (-3*b8 + 3*b7 + b6 - b5 + b3 - 3*b2 + b1 + 2) * q^43 + (-b8 + 2*b7 - b5 - b4 + b2 + 1) * q^44 + (-3*b9 + 3*b8 - 3*b4 + 3*b2 - b1) * q^45 + (b9 - 2*b8 - b2 - b1) * q^46 + (2*b9 - 2*b8 + 2*b7 + 3*b6 + 2*b4 + 2*b2 - 2) * q^47 + (b6 + b3 + 1) * q^48 + (-b9 + 3*b8 - 3*b7 - b6 + 2*b5 + 4*b4 - 2*b1 - 5) * q^49 + (-2*b9 + b8 - 3*b7 - 4*b6 - 2*b5 - 4*b3 - b2 + 2*b1 + 1) * q^50 + (-3*b9 + 2*b8 - b7 - 2*b6 + b5 + 2*b4 - 2*b2 - 2) * q^51 + (-2*b4 - b3 + b1) * q^52 + (-2*b8 - b4 - b3 + 2*b1) * q^53 + (2*b9 - 2*b7 - 3*b6 - b5 + 5*b4 - 5) * q^54 + (-3*b9 + 4*b8 - 7*b7 + b6 + 3*b5 + b3 + b2 - 3*b1 - 2) * q^55 + (b9 - b8 + b7 + b3 - b1) * q^56 + (2*b9 + 3*b8 - b7 - 2*b6 - b5 - 2*b3 + 5*b2 + b1 - 9) * q^57 + (-b4 + 1) * q^58 + (3*b4 + 2*b3 + b1) * q^59 + (b8 + 2*b4 - 2*b3 + 2*b1) * q^60 + (-2*b9 + 2*b8 - 2*b7 + 2*b6 + 2*b5 + 3*b4 - 2*b2 - 3) * q^61 + (2*b8 - 2*b7 - b6 - b5 - b3 + 2*b2 + b1 + 1) * q^62 + (2*b9 + b8 + 3*b7 - 2*b6 - b5 - 2*b4 - 5*b3 + 6*b2 + 2*b1 - 1) * q^63 + q^64 + (b9 - 2*b8 + 3*b7 - b6 - 2*b5 + 2*b2) * q^65 + (b8 - b4 + 3*b3 - b1) * q^66 + (b9 - 5*b8 + b4 - b2 - b1) * q^67 + (-b9 + b8 - b7 + b5 + 2*b4 - b2 - 2) * q^68 + (3*b9 + 2*b8 + b7 - 2*b6 - 3*b5 - 2*b3 + 5*b2 + 3*b1 + 2) * q^69 + (-b8 + 3*b6 + b5 + 2*b2 - b1) * q^70 + (4*b9 + 4*b7 + b6 + b3 + 4*b2 - 4) * q^71 + (b9 - b7 - b6 + b5 + 3*b4 - 3) * q^72 + (3*b9 - 2*b8 + 3*b4 - 3*b2 + 3*b1) * q^73 + (-2*b9 + 2*b8 - 2*b3 + 2*b2 + b1) * q^74 + (-4*b9 + b8 + 2*b7 - 3*b6 - 4*b5 - 9*b4 - b2 + 9) * q^75 + (2*b9 + 2*b7 - b6 - b3 + 2*b2) * q^76 + (-2*b9 - b8 - 3*b6 - 2*b5 - b4 - 3*b3 + 2*b2 + 3) * q^77 + (-b9 - b8 + b6 - b5 + b3 - 2*b2 + b1 + 5) * q^78 + (-2*b9 + 2*b7 + 2*b6 - b5 - 5*b4 + 5) * q^79 + (b8 - b4 + b3) * q^80 + (-2*b9 + 5*b8 - 5*b4 - 5*b3 + 2*b2 + 4*b1) * q^81 + (-b9 + 2*b8 - 3*b7 + 2*b5 + 6*b4 - 2*b2 - 6) * q^82 + (-b9 + b8 - 2*b7 + 2*b6 + 2*b5 + 2*b3 - 2*b1 + 5) * q^83 + (b9 + 2*b8 - 2*b4 + 3*b2 - 1) * q^84 + (-b9 - 2*b8 + b7 - 2*b6 - 2*b3 - 3*b2 - 1) * q^85 + (3*b9 - 3*b7 - b6 + b5 + 2*b4 - 2) * q^86 + (b4 + b3) * q^87 + (b9 + b4 - b2 + b1) * q^88 + (6*b9 - 5*b8 + 4*b7 + 2*b6 + 2*b5 + 4*b4 + 5*b2 - 4) * q^89 + (-3*b8 + 3*b7 - b5 - 3*b2 + b1 + 3) * q^90 + (4*b9 - 3*b8 + 3*b7 + b6 + b5 + 5*b4 + 2*b3 - 2*b1 - 1) * q^91 + (b9 + b8 - b5 + 2*b2 + b1) * q^92 + (-6*b9 + 3*b8 + 2*b6 + 3*b5 + 2*b4 - 3*b2 - 2) * q^93 + (2*b8 - 2*b4 + 3*b3) * q^94 + (-2*b9 + b4 + 3*b3 + 2*b2 - 2*b1) * q^95 + (-b6 + b4 - 1) * q^96 + (-3*b8 + 3*b7 + b6 - b5 + b3 - 3*b2 + b1 + 11) * q^97 + (-2*b9 - b8 + b7 - 2*b5 - 5*b4 - b3 + 1) * q^98 + (3*b9 - b8 + 4*b7 + 3*b6 + b5 + 3*b3 + 2*b2 - b1 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - 7 q^{5} + 6 q^{6} - 3 q^{7} + 10 q^{8} - 8 q^{9}+O(q^{10})$$ 10 * q - 5 * q^2 - 3 * q^3 - 5 * q^4 - 7 * q^5 + 6 * q^6 - 3 * q^7 + 10 * q^8 - 8 * q^9 $$10 q - 5 q^{2} - 3 q^{3} - 5 q^{4} - 7 q^{5} + 6 q^{6} - 3 q^{7} + 10 q^{8} - 8 q^{9} - 7 q^{10} - 3 q^{12} + 20 q^{13} + 3 q^{14} - 20 q^{15} - 5 q^{16} - 8 q^{17} - 8 q^{18} - 2 q^{19} + 14 q^{20} + 19 q^{21} - q^{23} - 3 q^{24} - 12 q^{25} - 10 q^{26} + 30 q^{27} - 10 q^{29} + 10 q^{30} - 11 q^{31} - 5 q^{32} - 9 q^{33} + 16 q^{34} + 10 q^{35} + 16 q^{36} + 8 q^{37} - 2 q^{38} - 18 q^{39} - 7 q^{40} + 46 q^{41} - 8 q^{42} - 6 q^{43} - 4 q^{45} - q^{46} - 16 q^{47} + 6 q^{48} - 11 q^{49} + 24 q^{50} - 7 q^{51} - 10 q^{52} - 7 q^{53} - 15 q^{54} + 12 q^{55} - 3 q^{56} - 68 q^{57} + 5 q^{58} + 9 q^{59} + 10 q^{60} - 15 q^{61} + 22 q^{62} - 3 q^{63} + 10 q^{64} - 5 q^{65} - 9 q^{66} + 4 q^{67} - 8 q^{68} + 28 q^{69} + 4 q^{70} - 44 q^{71} - 8 q^{72} + 8 q^{74} + 34 q^{75} + 4 q^{76} + 39 q^{77} + 36 q^{78} + 13 q^{79} - 7 q^{80} - 17 q^{81} - 23 q^{82} + 56 q^{83} - 11 q^{84} - 14 q^{85} + 3 q^{86} + 3 q^{87} - 17 q^{89} + 8 q^{90} + 6 q^{91} + 2 q^{92} - 17 q^{93} - 16 q^{94} + 9 q^{95} - 3 q^{96} + 84 q^{97} - 20 q^{98} - 84 q^{99}+O(q^{100})$$ 10 * q - 5 * q^2 - 3 * q^3 - 5 * q^4 - 7 * q^5 + 6 * q^6 - 3 * q^7 + 10 * q^8 - 8 * q^9 - 7 * q^10 - 3 * q^12 + 20 * q^13 + 3 * q^14 - 20 * q^15 - 5 * q^16 - 8 * q^17 - 8 * q^18 - 2 * q^19 + 14 * q^20 + 19 * q^21 - q^23 - 3 * q^24 - 12 * q^25 - 10 * q^26 + 30 * q^27 - 10 * q^29 + 10 * q^30 - 11 * q^31 - 5 * q^32 - 9 * q^33 + 16 * q^34 + 10 * q^35 + 16 * q^36 + 8 * q^37 - 2 * q^38 - 18 * q^39 - 7 * q^40 + 46 * q^41 - 8 * q^42 - 6 * q^43 - 4 * q^45 - q^46 - 16 * q^47 + 6 * q^48 - 11 * q^49 + 24 * q^50 - 7 * q^51 - 10 * q^52 - 7 * q^53 - 15 * q^54 + 12 * q^55 - 3 * q^56 - 68 * q^57 + 5 * q^58 + 9 * q^59 + 10 * q^60 - 15 * q^61 + 22 * q^62 - 3 * q^63 + 10 * q^64 - 5 * q^65 - 9 * q^66 + 4 * q^67 - 8 * q^68 + 28 * q^69 + 4 * q^70 - 44 * q^71 - 8 * q^72 + 8 * q^74 + 34 * q^75 + 4 * q^76 + 39 * q^77 + 36 * q^78 + 13 * q^79 - 7 * q^80 - 17 * q^81 - 23 * q^82 + 56 * q^83 - 11 * q^84 - 14 * q^85 + 3 * q^86 + 3 * q^87 - 17 * q^89 + 8 * q^90 + 6 * q^91 + 2 * q^92 - 17 * q^93 - 16 * q^94 + 9 * q^95 - 3 * q^96 + 84 * q^97 - 20 * q^98 - 84 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4x^{8} - 16x^{6} - 34x^{5} + 43x^{4} + 155x^{3} + 199x^{2} + 124x + 43$$ :

 $$\beta_{1}$$ $$=$$ $$( 244169 \nu^{9} - 750258 \nu^{8} + 198422 \nu^{7} + 3041052 \nu^{6} - 8027063 \nu^{5} + 7504426 \nu^{4} + 14486388 \nu^{3} - 35106720 \nu^{2} + \cdots - 17035541 ) / 27779803$$ (244169*v^9 - 750258*v^8 + 198422*v^7 + 3041052*v^6 - 8027063*v^5 + 7504426*v^4 + 14486388*v^3 - 35106720*v^2 - 10216212*v - 17035541) / 27779803 $$\beta_{2}$$ $$=$$ $$( - 313192 \nu^{9} + 3118568 \nu^{8} - 2157819 \nu^{7} - 10818671 \nu^{6} + 20747939 \nu^{5} - 49339027 \nu^{4} - 63283058 \nu^{3} + \cdots + 139659043 ) / 27779803$$ (-313192*v^9 + 3118568*v^8 - 2157819*v^7 - 10818671*v^6 + 20747939*v^5 - 49339027*v^4 - 63283058*v^3 + 175712874*v^2 + 186951717*v + 139659043) / 27779803 $$\beta_{3}$$ $$=$$ $$( - 893358 \nu^{9} + 475480 \nu^{8} + 5106016 \nu^{7} - 5222834 \nu^{6} + 14656748 \nu^{5} + 26556357 \nu^{4} - 84060972 \nu^{3} - 113795607 \nu^{2} + \cdots + 29238847 ) / 27779803$$ (-893358*v^9 + 475480*v^8 + 5106016*v^7 - 5222834*v^6 + 14656748*v^5 + 26556357*v^4 - 84060972*v^3 - 113795607*v^2 - 25063761*v + 29238847) / 27779803 $$\beta_{4}$$ $$=$$ $$( - 5726 \nu^{9} + 8728 \nu^{8} + 15695 \nu^{7} - 30701 \nu^{6} + 113887 \nu^{5} + 47565 \nu^{4} - 438521 \nu^{3} - 328616 \nu^{2} - 105832 \nu + 163662 ) / 139597$$ (-5726*v^9 + 8728*v^8 + 15695*v^7 - 30701*v^6 + 113887*v^5 + 47565*v^4 - 438521*v^3 - 328616*v^2 - 105832*v + 163662) / 139597 $$\beta_{5}$$ $$=$$ $$( 1616901 \nu^{9} - 715422 \nu^{8} - 6561850 \nu^{7} + 4318132 \nu^{6} - 26461030 \nu^{5} - 45525517 \nu^{4} + 100658881 \nu^{3} + 204814762 \nu^{2} + \cdots + 75110912 ) / 27779803$$ (1616901*v^9 - 715422*v^8 - 6561850*v^7 + 4318132*v^6 - 26461030*v^5 - 45525517*v^4 + 100658881*v^3 + 204814762*v^2 + 195068629*v + 75110912) / 27779803 $$\beta_{6}$$ $$=$$ $$( - 1768918 \nu^{9} + 516157 \nu^{8} + 5805510 \nu^{7} + 1407113 \nu^{6} + 25732292 \nu^{5} + 50029236 \nu^{4} - 65805547 \nu^{3} + \cdots - 138763851 ) / 27779803$$ (-1768918*v^9 + 516157*v^8 + 5805510*v^7 + 1407113*v^6 + 25732292*v^5 + 50029236*v^4 - 65805547*v^3 - 283984231*v^2 - 307071937*v - 138763851) / 27779803 $$\beta_{7}$$ $$=$$ $$( - 1850159 \nu^{9} - 1056286 \nu^{8} + 10198817 \nu^{7} + 514287 \nu^{6} + 22231484 \nu^{5} + 89090025 \nu^{4} - 99565695 \nu^{3} + \cdots - 199826103 ) / 27779803$$ (-1850159*v^9 - 1056286*v^8 + 10198817*v^7 + 514287*v^6 + 22231484*v^5 + 89090025*v^4 - 99565695*v^3 - 351561857*v^2 - 351513099*v - 199826103) / 27779803 $$\beta_{8}$$ $$=$$ $$( 2396972 \nu^{9} - 3681779 \nu^{8} - 7578548 \nu^{7} + 14487182 \nu^{6} - 48162975 \nu^{5} - 21260146 \nu^{4} + 196378044 \nu^{3} + 144963467 \nu^{2} + \cdots - 77176853 ) / 27779803$$ (2396972*v^9 - 3681779*v^8 - 7578548*v^7 + 14487182*v^6 - 48162975*v^5 - 21260146*v^4 + 196378044*v^3 + 144963467*v^2 + 42482805*v - 77176853) / 27779803 $$\beta_{9}$$ $$=$$ $$( 2611328 \nu^{9} - 2640778 \nu^{8} - 9404480 \nu^{7} + 11479543 \nu^{6} - 50294265 \nu^{5} - 42008212 \nu^{4} + 188098184 \nu^{3} + 228513885 \nu^{2} + \cdots + 13015032 ) / 27779803$$ (2611328*v^9 - 2640778*v^8 - 9404480*v^7 + 11479543*v^6 - 50294265*v^5 - 42008212*v^4 + 188098184*v^3 + 228513885*v^2 + 204385111*v + 13015032) / 27779803
 $$\nu$$ $$=$$ $$( \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 - 1 ) / 3$$ (b9 + b8 + b7 + b6 + 2*b5 + 2*b4 + 2*b3 + b2 - b1 - 1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{4} - 2\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 3$$ (-b9 - b8 + 2*b7 - b6 + 4*b5 + b4 - 2*b3 - b2 - 2*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 5 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 10 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 11 \beta _1 + 4 ) / 3$$ (5*b9 - 7*b8 + 5*b7 + 5*b6 + 10*b5 - 11*b4 + 4*b3 + 2*b2 - 11*b1 + 4) / 3 $$\nu^{4}$$ $$=$$ $$8\beta_{9} - 6\beta_{8} + 7\beta_{7} + 2\beta_{6} + 2\beta_{5} - 7\beta_{4} - \beta_{3} + 6\beta_{2} - \beta _1 + 13$$ 8*b9 - 6*b8 + 7*b7 + 2*b6 + 2*b5 - 7*b4 - b3 + 6*b2 - b1 + 13 $$\nu^{5}$$ $$=$$ $$( 32 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} + 26 \beta_{6} + 31 \beta_{5} - 20 \beta_{4} + 25 \beta_{3} + 53 \beta_{2} - 2 \beta _1 + 61 ) / 3$$ (32*b9 + 2*b8 + 41*b7 + 26*b6 + 31*b5 - 20*b4 + 25*b3 + 53*b2 - 2*b1 + 61) / 3 $$\nu^{6}$$ $$=$$ $$( \beta_{9} - 2 \beta_{8} + 64 \beta_{7} + 16 \beta_{6} + 113 \beta_{5} + 29 \beta_{4} - 7 \beta_{3} + 31 \beta_{2} + 11 \beta _1 + 53 ) / 3$$ (b9 - 2*b8 + 64*b7 + 16*b6 + 113*b5 + 29*b4 - 7*b3 + 31*b2 + 11*b1 + 53) / 3 $$\nu^{7}$$ $$=$$ $$( - 2 \beta_{9} - 143 \beta_{8} + 121 \beta_{7} + 55 \beta_{6} + 302 \beta_{5} - 202 \beta_{4} + 17 \beta_{3} + 10 \beta_{2} - 127 \beta _1 + 23 ) / 3$$ (-2*b9 - 143*b8 + 121*b7 + 55*b6 + 302*b5 - 202*b4 + 17*b3 + 10*b2 - 127*b1 + 23) / 3 $$\nu^{8}$$ $$=$$ $$124 \beta_{9} - 213 \beta_{8} + 134 \beta_{7} + 56 \beta_{6} + 131 \beta_{5} - 275 \beta_{4} - 47 \beta_{3} + 81 \beta_{2} - 33 \beta _1 + 184$$ 124*b9 - 213*b8 + 134*b7 + 56*b6 + 131*b5 - 275*b4 - 47*b3 + 81*b2 - 33*b1 + 184 $$\nu^{9}$$ $$=$$ $$( 1048 \beta_{9} - 890 \beta_{8} + 1075 \beta_{7} + 544 \beta_{6} + 455 \beta_{5} - 1864 \beta_{4} + 113 \beta_{3} + 1321 \beta_{2} + 353 \beta _1 + 1898 ) / 3$$ (1048*b9 - 890*b8 + 1075*b7 + 544*b6 + 455*b5 - 1864*b4 + 113*b3 + 1321*b2 + 353*b1 + 1898) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/406\mathbb{Z}\right)^\times$$.

 $$n$$ $$59$$ $$379$$ $$\chi(n)$$ $$-1 + \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
233.1
 −0.359001 + 0.701254i 0.522109 + 2.12798i −0.676693 − 0.583217i 2.31940 − 0.319028i −1.80582 − 0.194943i −0.359001 − 0.701254i 0.522109 − 2.12798i −0.676693 + 0.583217i 2.31940 + 0.319028i −1.80582 + 0.194943i
−0.500000 + 0.866025i −1.61748 2.80156i −0.500000 0.866025i −0.572197 + 0.991074i 3.23497 0.469216 + 2.60381i 1.00000 −3.73251 + 6.46489i −0.572197 0.991074i
233.2 −0.500000 + 0.866025i −1.20348 2.08450i −0.500000 0.866025i 1.10394 1.91209i 2.40697 −1.36646 2.26557i 1.00000 −1.39675 + 2.41924i 1.10394 + 1.91209i
233.3 −0.500000 + 0.866025i −0.257545 0.446080i −0.500000 0.866025i −1.84343 + 3.19291i 0.515089 −2.63485 + 0.239979i 1.00000 1.36734 2.36831i −1.84343 3.19291i
233.4 −0.500000 + 0.866025i 0.357289 + 0.618843i −0.500000 0.866025i −0.116584 + 0.201930i −0.714579 2.36799 + 1.18009i 1.00000 1.24469 2.15586i −0.116584 0.201930i
233.5 −0.500000 + 0.866025i 1.22122 + 2.11522i −0.500000 0.866025i −2.07173 + 3.58835i −2.44245 −0.335903 2.62434i 1.00000 −1.48277 + 2.56824i −2.07173 3.58835i
291.1 −0.500000 0.866025i −1.61748 + 2.80156i −0.500000 + 0.866025i −0.572197 0.991074i 3.23497 0.469216 2.60381i 1.00000 −3.73251 6.46489i −0.572197 + 0.991074i
291.2 −0.500000 0.866025i −1.20348 + 2.08450i −0.500000 + 0.866025i 1.10394 + 1.91209i 2.40697 −1.36646 + 2.26557i 1.00000 −1.39675 2.41924i 1.10394 1.91209i
291.3 −0.500000 0.866025i −0.257545 + 0.446080i −0.500000 + 0.866025i −1.84343 3.19291i 0.515089 −2.63485 0.239979i 1.00000 1.36734 + 2.36831i −1.84343 + 3.19291i
291.4 −0.500000 0.866025i 0.357289 0.618843i −0.500000 + 0.866025i −0.116584 0.201930i −0.714579 2.36799 1.18009i 1.00000 1.24469 + 2.15586i −0.116584 + 0.201930i
291.5 −0.500000 0.866025i 1.22122 2.11522i −0.500000 + 0.866025i −2.07173 3.58835i −2.44245 −0.335903 + 2.62434i 1.00000 −1.48277 2.56824i −2.07173 + 3.58835i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 291.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 406.2.e.a 10
7.c even 3 1 inner 406.2.e.a 10
7.c even 3 1 2842.2.a.z 5
7.d odd 6 1 2842.2.a.x 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
406.2.e.a 10 1.a even 1 1 trivial
406.2.e.a 10 7.c even 3 1 inner
2842.2.a.x 5 7.d odd 6 1
2842.2.a.z 5 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} + 3 T_{3}^{9} + 16 T_{3}^{8} + 17 T_{3}^{7} + 100 T_{3}^{6} + 104 T_{3}^{5} + 382 T_{3}^{4} - 16 T_{3}^{3} + 169 T_{3}^{2} + 42 T_{3} + 49$$ acting on $$S_{2}^{\mathrm{new}}(406, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{5}$$
$3$ $$T^{10} + 3 T^{9} + 16 T^{8} + 17 T^{7} + \cdots + 49$$
$5$ $$T^{10} + 7 T^{9} + 43 T^{8} + 112 T^{7} + \cdots + 81$$
$7$ $$T^{10} + 3 T^{9} + 10 T^{8} + \cdots + 16807$$
$11$ $$T^{10} + 33 T^{8} - 108 T^{7} + \cdots + 81$$
$13$ $$(T^{5} - 10 T^{4} + 28 T^{3} + 7 T^{2} + \cdots + 103)^{2}$$
$17$ $$T^{10} + 8 T^{9} + 55 T^{8} + 116 T^{7} + \cdots + 9$$
$19$ $$T^{10} + 2 T^{9} + 67 T^{8} + \cdots + 2634129$$
$23$ $$T^{10} + T^{9} + 40 T^{8} - 233 T^{7} + \cdots + 1089$$
$29$ $$(T + 1)^{10}$$
$31$ $$T^{10} + 11 T^{9} + 145 T^{8} + \cdots + 149769$$
$37$ $$T^{10} - 8 T^{9} + 111 T^{8} + \cdots + 121801$$
$41$ $$(T^{5} - 23 T^{4} + 132 T^{3} + 295 T^{2} + \cdots + 7113)^{2}$$
$43$ $$(T^{5} + 3 T^{4} - 163 T^{3} - 379 T^{2} + \cdots + 16843)^{2}$$
$47$ $$T^{10} + 16 T^{9} + 247 T^{8} + \cdots + 3969$$
$53$ $$T^{10} + 7 T^{9} + 106 T^{8} + \cdots + 33489$$
$59$ $$T^{10} - 9 T^{9} + 120 T^{8} + \cdots + 431649$$
$61$ $$T^{10} + 15 T^{9} + 235 T^{8} + \cdots + 28376929$$
$67$ $$T^{10} - 4 T^{9} + 190 T^{8} + \cdots + 131813361$$
$71$ $$(T^{5} + 22 T^{4} + 51 T^{3} - 1142 T^{2} + \cdots + 18189)^{2}$$
$73$ $$T^{10} + 247 T^{8} - 452 T^{7} + \cdots + 116281$$
$79$ $$T^{10} - 13 T^{9} + 202 T^{8} + \cdots + 2442969$$
$83$ $$(T^{5} - 28 T^{4} + 213 T^{3} + \cdots + 25851)^{2}$$
$89$ $$T^{10} + 17 T^{9} + \cdots + 26494398441$$
$97$ $$(T^{5} - 42 T^{4} + 539 T^{3} + \cdots + 15007)^{2}$$