# Properties

 Label 429.2.i.d Level $429$ Weight $2$ Character orbit 429.i Analytic conductor $3.426$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.3118758597603.1 Defining polynomial: $$x^{10} - 4 x^{8} - 16 x^{6} - 34 x^{5} + 43 x^{4} + 155 x^{3} + 199 x^{2} + 124 x + 43$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$244169 \nu^{9} - 750258 \nu^{8} + 198422 \nu^{7} + 3041052 \nu^{6} - 8027063 \nu^{5} + 7504426 \nu^{4} + 14486388 \nu^{3} - 35106720 \nu^{2} - 10216212 \nu - 17035541$$$$)/27779803$$ $$\beta_{2}$$ $$=$$ $$($$$$-313192 \nu^{9} + 3118568 \nu^{8} - 2157819 \nu^{7} - 10818671 \nu^{6} + 20747939 \nu^{5} - 49339027 \nu^{4} - 63283058 \nu^{3} + 175712874 \nu^{2} + 186951717 \nu + 139659043$$$$)/27779803$$ $$\beta_{3}$$ $$=$$ $$($$$$-893358 \nu^{9} + 475480 \nu^{8} + 5106016 \nu^{7} - 5222834 \nu^{6} + 14656748 \nu^{5} + 26556357 \nu^{4} - 84060972 \nu^{3} - 113795607 \nu^{2} - 25063761 \nu + 29238847$$$$)/27779803$$ $$\beta_{4}$$ $$=$$ $$($$$$-5726 \nu^{9} + 8728 \nu^{8} + 15695 \nu^{7} - 30701 \nu^{6} + 113887 \nu^{5} + 47565 \nu^{4} - 438521 \nu^{3} - 328616 \nu^{2} - 105832 \nu + 163662$$$$)/139597$$ $$\beta_{5}$$ $$=$$ $$($$$$1616901 \nu^{9} - 715422 \nu^{8} - 6561850 \nu^{7} + 4318132 \nu^{6} - 26461030 \nu^{5} - 45525517 \nu^{4} + 100658881 \nu^{3} + 204814762 \nu^{2} + 195068629 \nu + 75110912$$$$)/27779803$$ $$\beta_{6}$$ $$=$$ $$($$$$-1768918 \nu^{9} + 516157 \nu^{8} + 5805510 \nu^{7} + 1407113 \nu^{6} + 25732292 \nu^{5} + 50029236 \nu^{4} - 65805547 \nu^{3} - 283984231 \nu^{2} - 307071937 \nu - 138763851$$$$)/27779803$$ $$\beta_{7}$$ $$=$$ $$($$$$-1850159 \nu^{9} - 1056286 \nu^{8} + 10198817 \nu^{7} + 514287 \nu^{6} + 22231484 \nu^{5} + 89090025 \nu^{4} - 99565695 \nu^{3} - 351561857 \nu^{2} - 351513099 \nu - 199826103$$$$)/27779803$$ $$\beta_{8}$$ $$=$$ $$($$$$2396972 \nu^{9} - 3681779 \nu^{8} - 7578548 \nu^{7} + 14487182 \nu^{6} - 48162975 \nu^{5} - 21260146 \nu^{4} + 196378044 \nu^{3} + 144963467 \nu^{2} + 42482805 \nu - 77176853$$$$)/27779803$$ $$\beta_{9}$$ $$=$$ $$($$$$2611328 \nu^{9} - 2640778 \nu^{8} - 9404480 \nu^{7} + 11479543 \nu^{6} - 50294265 \nu^{5} - 42008212 \nu^{4} + 188098184 \nu^{3} + 228513885 \nu^{2} + 204385111 \nu + 13015032$$$$)/27779803$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_{1} - 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + 4 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 10 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 11 \beta_{1} + 4$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{9} - 6 \beta_{8} + 7 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 7 \beta_{4} - \beta_{3} + 6 \beta_{2} - \beta_{1} + 13$$ $$\nu^{5}$$ $$=$$ $$($$$$32 \beta_{9} + 2 \beta_{8} + 41 \beta_{7} + 26 \beta_{6} + 31 \beta_{5} - 20 \beta_{4} + 25 \beta_{3} + 53 \beta_{2} - 2 \beta_{1} + 61$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{9} - 2 \beta_{8} + 64 \beta_{7} + 16 \beta_{6} + 113 \beta_{5} + 29 \beta_{4} - 7 \beta_{3} + 31 \beta_{2} + 11 \beta_{1} + 53$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{9} - 143 \beta_{8} + 121 \beta_{7} + 55 \beta_{6} + 302 \beta_{5} - 202 \beta_{4} + 17 \beta_{3} + 10 \beta_{2} - 127 \beta_{1} + 23$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$124 \beta_{9} - 213 \beta_{8} + 134 \beta_{7} + 56 \beta_{6} + 131 \beta_{5} - 275 \beta_{4} - 47 \beta_{3} + 81 \beta_{2} - 33 \beta_{1} + 184$$ $$\nu^{9}$$ $$=$$ $$($$$$1048 \beta_{9} - 890 \beta_{8} + 1075 \beta_{7} + 544 \beta_{6} + 455 \beta_{5} - 1864 \beta_{4} + 113 \beta_{3} + 1321 \beta_{2} + 353 \beta_{1} + 1898$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 −0.359001 + 0.701254i −0.676693 − 0.583217i −1.80582 − 0.194943i 0.522109 + 2.12798i 2.31940 − 0.319028i −0.359001 − 0.701254i −0.676693 + 0.583217i −1.80582 + 0.194943i 0.522109 − 2.12798i 2.31940 + 0.319028i
−1.18968 + 2.06059i 0.500000 0.866025i −1.83068 3.17083i 2.23497 1.18968 + 2.06059i −1.82822 3.16657i 3.95297 −0.500000 0.866025i −2.65890 + 4.60534i
100.2 −1.10097 + 1.90694i 0.500000 0.866025i −1.42428 2.46692i −0.484911 1.10097 + 1.90694i 0.958152 + 1.65957i 1.86848 −0.500000 0.866025i 0.533873 0.924696i
100.3 0.149489 0.258923i 0.500000 0.866025i 0.955306 + 1.65464i −3.44245 −0.149489 0.258923i −2.46991 4.27802i 1.16919 −0.500000 0.866025i −0.514608 + 0.891327i
100.4 0.900458 1.55964i 0.500000 0.866025i −0.621650 1.07673i 1.40697 −0.900458 1.55964i 0.888568 + 1.53904i 1.36275 −0.500000 0.866025i 1.26692 2.19437i
100.5 1.24071 2.14896i 0.500000 0.866025i −2.07870 3.60041i −1.71458 −1.24071 2.14896i −1.04859 1.81621i −5.35339 −0.500000 0.866025i −2.12729 + 3.68457i
133.1 −1.18968 2.06059i 0.500000 + 0.866025i −1.83068 + 3.17083i 2.23497 1.18968 2.06059i −1.82822 + 3.16657i 3.95297 −0.500000 + 0.866025i −2.65890 4.60534i
133.2 −1.10097 1.90694i 0.500000 + 0.866025i −1.42428 + 2.46692i −0.484911 1.10097 1.90694i 0.958152 1.65957i 1.86848 −0.500000 + 0.866025i 0.533873 + 0.924696i
133.3 0.149489 + 0.258923i 0.500000 + 0.866025i 0.955306 1.65464i −3.44245 −0.149489 + 0.258923i −2.46991 + 4.27802i 1.16919 −0.500000 + 0.866025i −0.514608 0.891327i
133.4 0.900458 + 1.55964i 0.500000 + 0.866025i −0.621650 + 1.07673i 1.40697 −0.900458 + 1.55964i 0.888568 1.53904i 1.36275 −0.500000 + 0.866025i 1.26692 + 2.19437i
133.5 1.24071 + 2.14896i 0.500000 + 0.866025i −2.07870 + 3.60041i −1.71458 −1.24071 + 2.14896i −1.04859 + 1.81621i −5.35339 −0.500000 + 0.866025i −2.12729 3.68457i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 133.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 429.2.i.d 10
13.c even 3 1 inner 429.2.i.d 10
13.c even 3 1 5577.2.a.r 5
13.e even 6 1 5577.2.a.s 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.i.d 10 1.a even 1 1 trivial
429.2.i.d 10 13.c even 3 1 inner
5577.2.a.r 5 13.c even 3 1
5577.2.a.s 5 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$:

 $$T_{2}^{10} + \cdots$$ $$T_{7}^{10} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$49 - 168 T + 583 T^{2} - 116 T^{3} + 241 T^{4} - 17 T^{5} + 76 T^{6} - 2 T^{7} + 10 T^{8} + T^{10}$$
$3$ $$( 1 - T + T^{2} )^{5}$$
$5$ $$( 9 + 16 T - 10 T^{2} - 9 T^{3} + 2 T^{4} + T^{5} )^{2}$$
$7$ $$16641 - 2193 T + 7900 T^{2} + 1003 T^{3} + 2578 T^{4} + 367 T^{5} + 430 T^{6} + 118 T^{7} + 49 T^{8} + 7 T^{9} + T^{10}$$
$11$ $$( 1 + T + T^{2} )^{5}$$
$13$ $$371293 - 257049 T + 68107 T^{2} - 6253 T^{3} - 1703 T^{4} + 827 T^{5} - 131 T^{6} - 37 T^{7} + 31 T^{8} - 9 T^{9} + T^{10}$$
$17$ $$337561 + 374745 T + 549655 T^{2} - 63524 T^{3} + 98242 T^{4} + 13501 T^{5} + 5374 T^{6} + 241 T^{7} + 82 T^{8} + 3 T^{9} + T^{10}$$
$19$ $$9 + 387 T + 16281 T^{2} + 15426 T^{3} + 13260 T^{4} + 2883 T^{5} + 1050 T^{6} + 177 T^{7} + 58 T^{8} + 7 T^{9} + T^{10}$$
$23$ $$1929321 + 2107113 T + 1679017 T^{2} + 654614 T^{3} + 202330 T^{4} + 36017 T^{5} + 6526 T^{6} + 797 T^{7} + 130 T^{8} + 11 T^{9} + T^{10}$$
$29$ $$25190361 + 11955258 T + 5688981 T^{2} + 956502 T^{3} + 238719 T^{4} + 14835 T^{5} + 6828 T^{6} + 198 T^{7} + 100 T^{8} - 2 T^{9} + T^{10}$$
$31$ $$( -381 - 281 T + 833 T^{2} - 81 T^{3} - 10 T^{4} + T^{5} )^{2}$$
$37$ $$42849 + 36018 T + 35865 T^{2} + 18900 T^{3} + 13752 T^{4} + 6552 T^{5} + 3018 T^{6} + 801 T^{7} + 168 T^{8} + 15 T^{9} + T^{10}$$
$41$ $$9 + 30 T + 88 T^{2} + 94 T^{3} + 112 T^{4} + 7 T^{5} + 79 T^{6} + 10 T^{7} + 13 T^{8} - 2 T^{9} + T^{10}$$
$43$ $$729 - 9477 T + 117369 T^{2} - 77922 T^{3} + 60534 T^{4} + 3483 T^{5} + 2682 T^{6} + 159 T^{7} + 88 T^{8} + 7 T^{9} + T^{10}$$
$47$ $$( -1057 - 789 T + 169 T^{2} + 71 T^{3} - 18 T^{4} + T^{5} )^{2}$$
$53$ $$( -19467 - 1884 T + 1689 T^{2} - 87 T^{3} - 15 T^{4} + T^{5} )^{2}$$
$59$ $$4712685201 - 996646182 T + 279695920 T^{2} - 19611130 T^{3} + 4348402 T^{4} - 202501 T^{5} + 51499 T^{6} - 1012 T^{7} + 265 T^{8} - 4 T^{9} + T^{10}$$
$61$ $$21609 - 44100 T + 127926 T^{2} + 72990 T^{3} + 69006 T^{4} + 4383 T^{5} + 3537 T^{6} - 306 T^{7} + 211 T^{8} - 14 T^{9} + T^{10}$$
$67$ $$597529 + 156919 T + 213588 T^{2} + 96963 T^{3} + 72270 T^{4} + 23319 T^{5} + 7146 T^{6} + 906 T^{7} + 117 T^{8} - 5 T^{9} + T^{10}$$
$71$ $$202806081 - 118385433 T + 47060901 T^{2} - 11415942 T^{3} + 2157474 T^{4} - 280845 T^{5} + 31038 T^{6} - 2433 T^{7} + 220 T^{8} - 13 T^{9} + T^{10}$$
$73$ $$( 7071 - 4172 T - 133 T^{2} + 228 T^{3} - 28 T^{4} + T^{5} )^{2}$$
$79$ $$( 2239 - 2020 T + 445 T^{2} + 34 T^{3} - 16 T^{4} + T^{5} )^{2}$$
$83$ $$( -1701 + 1059 T + 387 T^{2} - 63 T^{3} - 12 T^{4} + T^{5} )^{2}$$
$89$ $$3969 - 16821 T + 121374 T^{2} + 177867 T^{3} + 704538 T^{4} - 213894 T^{5} + 79032 T^{6} + 48 T^{7} + 309 T^{8} - 6 T^{9} + T^{10}$$
$97$ $$8695003009 + 1040077038 T + 304098685 T^{2} + 20094404 T^{3} + 5361448 T^{4} + 322196 T^{5} + 55918 T^{6} + 1847 T^{7} + 304 T^{8} + 9 T^{9} + T^{10}$$