# Properties

 Label 210.2.r.b Level 210 Weight 2 Character orbit 210.r Analytic conductor 1.677 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 210.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} - 864 x^{3} + 891 x^{2} - 972 x + 729$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{10} - 11 \nu^{9} + 32 \nu^{8} - 64 \nu^{7} + 120 \nu^{6} - 237 \nu^{5} + 360 \nu^{4} - 576 \nu^{3} + 864 \nu^{2} - 891 \nu + 972$$$$)/243$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{11} + \nu^{10} - 11 \nu^{9} + 35 \nu^{8} - 10 \nu^{7} + 123 \nu^{6} - 183 \nu^{5} + 63 \nu^{4} - 630 \nu^{3} + 351 \nu^{2} - 243 \nu + 1701$$$$)/324$$ $$\beta_{4}$$ $$=$$ $$($$$$7 \nu^{11} - 16 \nu^{10} + 74 \nu^{9} - 191 \nu^{8} + 343 \nu^{7} - 810 \nu^{6} + 1290 \nu^{5} - 1971 \nu^{4} + 3843 \nu^{3} - 4158 \nu^{2} + 5184 \nu - 6075$$$$)/972$$ $$\beta_{5}$$ $$=$$ $$($$$$28 \nu^{11} - 67 \nu^{10} + 191 \nu^{9} - 545 \nu^{8} + 802 \nu^{7} - 1821 \nu^{6} + 3243 \nu^{5} - 4005 \nu^{4} + 8082 \nu^{3} - 8721 \nu^{2} + 7695 \nu - 10935$$$$)/972$$ $$\beta_{6}$$ $$=$$ $$($$$$-16 \nu^{11} + 31 \nu^{10} - 98 \nu^{9} + 284 \nu^{8} - 400 \nu^{7} + 969 \nu^{6} - 1668 \nu^{5} + 2070 \nu^{4} - 4302 \nu^{3} + 4455 \nu^{2} - 4212 \nu + 5832$$$$)/486$$ $$\beta_{7}$$ $$=$$ $$($$$$25 \nu^{11} - 79 \nu^{10} + 227 \nu^{9} - 578 \nu^{8} + 1027 \nu^{7} - 1971 \nu^{6} + 3495 \nu^{5} - 5130 \nu^{4} + 8487 \nu^{3} - 10071 \nu^{2} + 9801 \nu - 8748$$$$)/972$$ $$\beta_{8}$$ $$=$$ $$($$$$-15 \nu^{11} + 32 \nu^{10} - 98 \nu^{9} + 289 \nu^{8} - 415 \nu^{7} + 998 \nu^{6} - 1734 \nu^{5} + 2157 \nu^{4} - 4635 \nu^{3} + 4878 \nu^{2} - 4644 \nu + 6885$$$$)/324$$ $$\beta_{9}$$ $$=$$ $$($$$$5 \nu^{11} - 13 \nu^{10} + 39 \nu^{9} - 110 \nu^{8} + 171 \nu^{7} - 377 \nu^{6} + 675 \nu^{5} - 894 \nu^{4} + 1719 \nu^{3} - 1917 \nu^{2} + 1809 \nu - 2268$$$$)/108$$ $$\beta_{10}$$ $$=$$ $$($$$$26 \nu^{11} - 56 \nu^{10} + 193 \nu^{9} - 538 \nu^{8} + 812 \nu^{7} - 1920 \nu^{6} + 3255 \nu^{5} - 4356 \nu^{4} + 8766 \nu^{3} - 9558 \nu^{2} + 9801 \nu - 12636$$$$)/486$$ $$\beta_{11}$$ $$=$$ $$($$$$11 \nu^{11} - 26 \nu^{10} + 76 \nu^{9} - 208 \nu^{8} + 317 \nu^{7} - 708 \nu^{6} + 1230 \nu^{5} - 1638 \nu^{4} + 3123 \nu^{3} - 3348 \nu^{2} + 3240 \nu - 3888$$$$)/162$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{3} + \beta_{2} + 2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{11} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + 5 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{1} + 5$$ $$\nu^{5}$$ $$=$$ $$-\beta_{11} - 4 \beta_{10} + 5 \beta_{9} + 5 \beta_{8} - \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{11} - 5 \beta_{10} + 13 \beta_{9} - 6 \beta_{8} + 9 \beta_{7} - 2 \beta_{6} - 26 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 8 \beta_{1} - 7$$ $$\nu^{7}$$ $$=$$ $$-14 \beta_{11} - 11 \beta_{10} - 5 \beta_{9} - 16 \beta_{8} + 23 \beta_{7} - 19 \beta_{6} - 9 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} - 8$$ $$\nu^{8}$$ $$=$$ $$10 \beta_{11} - 18 \beta_{10} - 10 \beta_{9} + 13 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 43 \beta_{5} + 47 \beta_{4} - 23 \beta_{3} - 24 \beta_{2} - 12 \beta_{1} + 58$$ $$\nu^{9}$$ $$=$$ $$3 \beta_{11} + 22 \beta_{10} + 21 \beta_{9} + 43 \beta_{8} + 13 \beta_{7} + 63 \beta_{6} + 26 \beta_{5} - 11 \beta_{4} - 60 \beta_{3} + 8 \beta_{2} + 13 \beta_{1} + 39$$ $$\nu^{10}$$ $$=$$ $$-81 \beta_{11} - 10 \beta_{10} + 60 \beta_{9} + 18 \beta_{8} + 81 \beta_{7} - 157 \beta_{6} - 121 \beta_{5} - 23 \beta_{4} - 46 \beta_{3} + 19 \beta_{2} + 13 \beta_{1} - 53$$ $$\nu^{11}$$ $$=$$ $$16 \beta_{11} - 94 \beta_{10} - 48 \beta_{9} - 74 \beta_{8} + 106 \beta_{7} - 170 \beta_{6} - 222 \beta_{5} - 12 \beta_{4} - 104 \beta_{3} - 149 \beta_{2} - 26 \beta_{1} + 68$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$1 + \beta_{6}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.73138 − 0.0481063i −0.384890 − 1.68874i −1.21252 + 1.23685i −0.111613 + 1.72845i 0.312065 − 1.70371i 1.66557 + 0.475255i 1.73138 + 0.0481063i −0.384890 + 1.68874i −1.21252 − 1.23685i −0.111613 − 1.72845i 0.312065 + 1.70371i 1.66557 − 0.475255i
−0.866025 + 0.500000i −1.52347 + 0.824030i 0.500000 0.866025i −0.500000 0.866025i 0.907353 1.47537i 2.27338 1.35342i 1.00000i 1.64195 2.51078i 0.866025 + 0.500000i
101.2 −0.866025 + 0.500000i −0.511048 1.65494i 0.500000 0.866025i −0.500000 0.866025i 1.27005 + 1.17770i 2.63608 + 0.226058i 1.00000i −2.47766 + 1.69151i 0.866025 + 0.500000i
101.3 −0.866025 + 0.500000i 1.66850 + 0.464886i 0.500000 0.866025i −0.500000 0.866025i −1.67740 + 0.431645i −0.311378 + 2.62736i 1.00000i 2.56776 + 1.55132i 0.866025 + 0.500000i
101.4 0.866025 0.500000i −0.960885 1.44108i 0.500000 0.866025i −0.500000 0.866025i −1.55269 0.767566i −1.91871 1.82168i 1.00000i −1.15340 + 2.76942i −0.866025 0.500000i
101.5 0.866025 0.500000i 1.12211 + 1.31942i 0.500000 0.866025i −0.500000 0.866025i 1.63149 + 0.581597i 1.26546 2.32349i 1.00000i −0.481739 + 2.96107i −0.866025 0.500000i
101.6 0.866025 0.500000i 1.20480 1.24437i 0.500000 0.866025i −0.500000 0.866025i 0.421203 1.68006i 0.0551777 + 2.64518i 1.00000i −0.0969112 2.99843i −0.866025 0.500000i
131.1 −0.866025 0.500000i −1.52347 0.824030i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.907353 + 1.47537i 2.27338 + 1.35342i 1.00000i 1.64195 + 2.51078i 0.866025 0.500000i
131.2 −0.866025 0.500000i −0.511048 + 1.65494i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.27005 1.17770i 2.63608 0.226058i 1.00000i −2.47766 1.69151i 0.866025 0.500000i
131.3 −0.866025 0.500000i 1.66850 0.464886i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.67740 0.431645i −0.311378 2.62736i 1.00000i 2.56776 1.55132i 0.866025 0.500000i
131.4 0.866025 + 0.500000i −0.960885 + 1.44108i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.55269 + 0.767566i −1.91871 + 1.82168i 1.00000i −1.15340 2.76942i −0.866025 + 0.500000i
131.5 0.866025 + 0.500000i 1.12211 1.31942i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.63149 0.581597i 1.26546 + 2.32349i 1.00000i −0.481739 2.96107i −0.866025 + 0.500000i
131.6 0.866025 + 0.500000i 1.20480 + 1.24437i 0.500000 + 0.866025i −0.500000 + 0.866025i 0.421203 + 1.68006i 0.0551777 2.64518i 1.00000i −0.0969112 + 2.99843i −0.866025 + 0.500000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.r.b yes 12
3.b odd 2 1 210.2.r.a 12
5.b even 2 1 1050.2.s.f 12
5.c odd 4 1 1050.2.u.e 12
5.c odd 4 1 1050.2.u.h 12
7.c even 3 1 1470.2.b.a 12
7.d odd 6 1 210.2.r.a 12
7.d odd 6 1 1470.2.b.b 12
15.d odd 2 1 1050.2.s.g 12
15.e even 4 1 1050.2.u.f 12
15.e even 4 1 1050.2.u.g 12
21.g even 6 1 inner 210.2.r.b yes 12
21.g even 6 1 1470.2.b.a 12
21.h odd 6 1 1470.2.b.b 12
35.i odd 6 1 1050.2.s.g 12
35.k even 12 1 1050.2.u.f 12
35.k even 12 1 1050.2.u.g 12
105.p even 6 1 1050.2.s.f 12
105.w odd 12 1 1050.2.u.e 12
105.w odd 12 1 1050.2.u.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.r.a 12 3.b odd 2 1
210.2.r.a 12 7.d odd 6 1
210.2.r.b yes 12 1.a even 1 1 trivial
210.2.r.b yes 12 21.g even 6 1 inner
1050.2.s.f 12 5.b even 2 1
1050.2.s.f 12 105.p even 6 1
1050.2.s.g 12 15.d odd 2 1
1050.2.s.g 12 35.i odd 6 1
1050.2.u.e 12 5.c odd 4 1
1050.2.u.e 12 105.w odd 12 1
1050.2.u.f 12 15.e even 4 1
1050.2.u.f 12 35.k even 12 1
1050.2.u.g 12 15.e even 4 1
1050.2.u.g 12 35.k even 12 1
1050.2.u.h 12 5.c odd 4 1
1050.2.u.h 12 105.w odd 12 1
1470.2.b.a 12 7.c even 3 1
1470.2.b.a 12 21.g even 6 1
1470.2.b.b 12 7.d odd 6 1
1470.2.b.b 12 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 16 T^{4} - 18 T^{5} + 6 T^{6} - 54 T^{7} + 144 T^{8} - 108 T^{9} + 162 T^{10} - 486 T^{11} + 729 T^{12}$$
$5$ $$( 1 + T + T^{2} )^{6}$$
$7$ $$1 - 8 T + 39 T^{2} - 152 T^{3} + 483 T^{4} - 1360 T^{5} + 3642 T^{6} - 9520 T^{7} + 23667 T^{8} - 52136 T^{9} + 93639 T^{10} - 134456 T^{11} + 117649 T^{12}$$
$11$ $$1 - 12 T + 104 T^{2} - 672 T^{3} + 3656 T^{4} - 16524 T^{5} + 63276 T^{6} - 198252 T^{7} + 468664 T^{8} - 516768 T^{9} - 2129576 T^{10} + 16995156 T^{11} - 68897930 T^{12} + 186946716 T^{13} - 257678696 T^{14} - 687818208 T^{15} + 6861709624 T^{16} - 31928682852 T^{17} + 112097293836 T^{18} - 322006013604 T^{19} + 783696068936 T^{20} - 1584540848352 T^{21} + 2697492158504 T^{22} - 3423740047332 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 - 24 T^{2} + 632 T^{4} - 13172 T^{6} + 225768 T^{8} - 3379208 T^{10} + 49928150 T^{12} - 571086152 T^{14} + 6448159848 T^{16} - 63578728148 T^{18} + 515541815672 T^{20} - 3308603804376 T^{22} + 23298085122481 T^{24}$$
$17$ $$1 + 12 T + 10 T^{2} - 240 T^{3} + 929 T^{4} + 10248 T^{5} - 21642 T^{6} - 175188 T^{7} + 650254 T^{8} + 1893108 T^{9} - 17461318 T^{10} - 24663192 T^{11} + 233538997 T^{12} - 419274264 T^{13} - 5046320902 T^{14} + 9300839604 T^{15} + 54309864334 T^{16} - 248741908116 T^{17} - 522385268298 T^{18} + 4205150720904 T^{19} + 6480478662689 T^{20} - 28461090359280 T^{21} + 20159939004490 T^{22} + 411262755691596 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 + 24 T^{2} - 316 T^{4} + 144 T^{5} - 5636 T^{6} - 33888 T^{7} + 181380 T^{8} - 941760 T^{9} + 1187824 T^{10} + 9129072 T^{11} - 54892570 T^{12} + 173452368 T^{13} + 428804464 T^{14} - 6459531840 T^{15} + 23637622980 T^{16} - 83910042912 T^{17} - 265150585316 T^{18} + 128717530416 T^{19} - 5366805920956 T^{20} + 147145590187224 T^{22} + 2213314919066161 T^{24}$$
$23$ $$1 - 24 T + 367 T^{2} - 4200 T^{3} + 39535 T^{4} - 321816 T^{5} + 2347952 T^{6} - 15699792 T^{7} + 97859137 T^{8} - 573159912 T^{9} + 3169289665 T^{10} - 16552915872 T^{11} + 81697151198 T^{12} - 380717065056 T^{13} + 1676554232785 T^{14} - 6973636649304 T^{15} + 27384998757217 T^{16} - 101049246340656 T^{17} + 347581161649328 T^{18} - 1095727306051752 T^{19} + 3096024803084335 T^{20} - 7564841178144600 T^{21} + 15203529615409183 T^{22} - 22867434189934248 T^{23} + 21914624432020321 T^{24}$$
$29$ $$1 - 158 T^{2} + 12415 T^{4} - 625726 T^{6} + 22774147 T^{8} - 673287380 T^{10} + 19008498890 T^{12} - 566234686580 T^{14} + 16107721464307 T^{16} - 372196417356046 T^{18} + 6210559216910815 T^{20} - 66471742861431758 T^{22} + 353814783205469041 T^{24}$$
$31$ $$1 - 12 T + 162 T^{2} - 1368 T^{3} + 10769 T^{4} - 77664 T^{5} + 554878 T^{6} - 3968772 T^{7} + 27631230 T^{8} - 178188132 T^{9} + 1059815698 T^{10} - 6109630032 T^{11} + 33269854325 T^{12} - 189398530992 T^{13} + 1018482885778 T^{14} - 5308402640412 T^{15} + 25518021160830 T^{16} - 113622572872572 T^{17} + 492456267505918 T^{18} - 2136739662316704 T^{19} + 9184783582202129 T^{20} - 36169403115797928 T^{21} + 132779782490889762 T^{22} - 304901722756857972 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 + 8 T - 104 T^{2} - 832 T^{3} + 6788 T^{4} + 46312 T^{5} - 326044 T^{6} - 1583912 T^{7} + 13285604 T^{8} + 33469952 T^{9} - 514631840 T^{10} - 373683016 T^{11} + 19047573830 T^{12} - 13826271592 T^{13} - 704530988960 T^{14} + 1695353478656 T^{15} + 24899360878244 T^{16} - 109834725619784 T^{17} - 836539701295996 T^{18} + 4396485093783496 T^{19} + 23842710533215748 T^{20} - 108128167509504064 T^{21} - 500092774731456296 T^{22} + 1423340974235683304 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$( 1 - 2 T + 101 T^{2} - 150 T^{3} + 6062 T^{4} - 7442 T^{5} + 295913 T^{6} - 305122 T^{7} + 10190222 T^{8} - 10338150 T^{9} + 285401861 T^{10} - 231712402 T^{11} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 + 131 T^{2} - 196 T^{3} + 7479 T^{4} - 24892 T^{5} + 314798 T^{6} - 1070356 T^{7} + 13828671 T^{8} - 15583372 T^{9} + 447862931 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$1 + 16 T + 32 T^{2} - 544 T^{3} + 428 T^{4} + 39000 T^{5} + 312596 T^{6} + 2377608 T^{7} + 2268284 T^{8} - 67449856 T^{9} + 407948184 T^{10} + 5931851872 T^{11} + 27380961750 T^{12} + 278797037984 T^{13} + 901157538456 T^{14} - 7002846399488 T^{15} + 11068502337404 T^{16} + 545292523403256 T^{17} + 3369539594984084 T^{18} + 19758301698057000 T^{19} + 10191230691233708 T^{20} - 608806977367905248 T^{21} + 1683172231546561568 T^{22} + 39554547441344196848 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 - 48 T + 1272 T^{2} - 24192 T^{3} + 372104 T^{4} - 4941024 T^{5} + 58639988 T^{6} - 632702016 T^{7} + 6267491832 T^{8} - 57492176256 T^{9} + 491936676296 T^{10} - 3943130126256 T^{11} + 29642342340470 T^{12} - 208985896691568 T^{13} + 1381850123715464 T^{14} - 8559262724464512 T^{15} + 49453525218051192 T^{16} - 264593131503213888 T^{17} + 1299717870632226452 T^{18} - 5804275935001973088 T^{19} + 23167079840829073544 T^{20} - 79827880812877201536 T^{21} +$$$$22\!\cdots\!28$$$$T^{22} -$$$$44\!\cdots\!56$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + 12 T - 62 T^{2} - 192 T^{3} + 13661 T^{4} + 4488 T^{5} - 291210 T^{6} + 10035204 T^{7} + 17708314 T^{8} - 439140108 T^{9} + 4538367434 T^{10} + 27175198584 T^{11} - 197381258843 T^{12} + 1603336716456 T^{13} + 15798057037754 T^{14} - 90190156240932 T^{15} + 214578033439354 T^{16} + 7174411185021996 T^{17} - 12283393201595610 T^{18} + 11169067863867672 T^{19} + 2005850608112629181 T^{20} - 1663295197181748288 T^{21} - 31689238704639766862 T^{22} +$$$$36\!\cdots\!08$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 + 30 T + 633 T^{2} + 9990 T^{3} + 134066 T^{4} + 1612650 T^{5} + 18045535 T^{6} + 190552410 T^{7} + 1899056448 T^{8} + 17814782910 T^{9} + 157346567941 T^{10} + 1317498445782 T^{11} + 10515537066752 T^{12} + 80367405192702 T^{13} + 585486579308461 T^{14} + 4043617239694710 T^{15} + 26294033629032768 T^{16} + 160939860632635410 T^{17} + 929712718744528135 T^{18} + 5068144234509265650 T^{19} + 25701432624293474546 T^{20} +$$$$11\!\cdots\!90$$$$T^{21} +$$$$45\!\cdots\!33$$$$T^{22} +$$$$13\!\cdots\!30$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 4 T - 339 T^{2} - 1212 T^{3} + 65398 T^{4} + 197860 T^{5} - 8949237 T^{6} - 19422588 T^{7} + 965230072 T^{8} + 1219402116 T^{9} - 85058357911 T^{10} - 32989889052 T^{11} + 6246904001980 T^{12} - 2210322566484 T^{13} - 381826968662479 T^{14} + 366751038614508 T^{15} + 19450467973710712 T^{16} - 26222923701716916 T^{17} - 809533500666955053 T^{18} + 1199172398229208780 T^{19} + 26556013976849208118 T^{20} - 32974319688309475764 T^{21} -$$$$61\!\cdots\!11$$$$T^{22} +$$$$48\!\cdots\!32$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 - 300 T^{2} + 49610 T^{4} - 6300412 T^{6} + 667955583 T^{8} - 59469293656 T^{10} + 4533955360172 T^{12} - 299784709319896 T^{14} + 16973874197365023 T^{16} - 807084566019275452 T^{18} + 32035832685102203210 T^{20} -$$$$97\!\cdots\!00$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 + 158 T^{2} + 8161 T^{4} - 137736 T^{5} + 413890 T^{6} - 21056400 T^{7} + 34702222 T^{8} - 1012533192 T^{9} + 9774206798 T^{10} - 39652148664 T^{11} + 1268896234901 T^{12} - 2894606852472 T^{13} + 52086748026542 T^{14} - 393892624752264 T^{15} + 985482063591502 T^{16} - 43651424690845200 T^{17} + 62635722918754210 T^{18} - 1521624482426344392 T^{19} + 6581520809947595041 T^{20} +$$$$67\!\cdots\!42$$$$T^{22} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 4 T - 254 T^{2} + 1984 T^{3} + 51521 T^{4} - 543784 T^{5} - 2546434 T^{6} + 103559324 T^{7} - 209408530 T^{8} - 8333233172 T^{9} + 86160957682 T^{10} + 365142325960 T^{11} - 8321189358475 T^{12} + 28846243750840 T^{13} + 537730536893362 T^{14} - 4108608949889708 T^{15} - 8156479205590930 T^{16} + 318657880590314276 T^{17} - 619006161712162114 T^{18} - 10442778444129485656 T^{19} + 78162962995195929281 T^{20} +$$$$23\!\cdots\!96$$$$T^{21} -$$$$24\!\cdots\!54$$$$T^{22} +$$$$29\!\cdots\!16$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$( 1 - 20 T + 635 T^{2} - 8628 T^{3} + 148691 T^{4} - 1458128 T^{5} + 17076098 T^{6} - 121024624 T^{7} + 1024332299 T^{8} - 4933378236 T^{9} + 30136033835 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} )^{2}$$
$89$ $$1 + 26 T + 101 T^{2} - 1838 T^{3} + 1058 T^{4} + 234822 T^{5} - 191269 T^{6} - 17352042 T^{7} - 36283792 T^{8} + 94067578 T^{9} - 6641487 T^{10} + 7507089026 T^{11} - 13384168368 T^{12} + 668130923314 T^{13} - 52607218527 T^{14} + 66314726395082 T^{15} - 2276526422057872 T^{16} - 96894834089544858 T^{17} - 95057114540819509 T^{18} + 10386490522837910838 T^{19} + 4164910956432801698 T^{20} -$$$$64\!\cdots\!42$$$$T^{21} +$$$$31\!\cdots\!01$$$$T^{22} +$$$$72\!\cdots\!14$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 484 T^{2} + 116170 T^{4} - 18891668 T^{6} + 2452131295 T^{8} - 279695423560 T^{10} + 28680693351980 T^{12} - 2631654240276040 T^{14} + 217085420463948895 T^{16} - 15736230570413031572 T^{18} +$$$$91\!\cdots\!70$$$$T^{20} -$$$$35\!\cdots\!16$$$$T^{22} +$$$$69\!\cdots\!41$$$$T^{24}$$