# Properties

 Label 210.2.j.b Level 210 Weight 2 Character orbit 210.j 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.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 16 x^{10} + 86 x^{8} + 196 x^{6} + 185 x^{4} + 60 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{8} + \nu^{7} + 13 \nu^{6} + 12 \nu^{5} + 47 \nu^{4} + 35 \nu^{3} + 53 \nu^{2} + 22 \nu + 10$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{8} - \nu^{7} + 13 \nu^{6} - 12 \nu^{5} + 47 \nu^{4} - 35 \nu^{3} + 53 \nu^{2} - 22 \nu + 10$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} - 15 \nu^{9} + 15 \nu^{8} - 73 \nu^{7} + 71 \nu^{6} - 151 \nu^{5} + 127 \nu^{4} - 152 \nu^{3} + 82 \nu^{2} - 84 \nu + 24$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} + 15 \nu^{9} + 15 \nu^{8} + 73 \nu^{7} + 71 \nu^{6} + 151 \nu^{5} + 127 \nu^{4} + 152 \nu^{3} + 82 \nu^{2} + 84 \nu + 24$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{11} + \nu^{10} - 45 \nu^{9} + 15 \nu^{8} - 217 \nu^{7} + 69 \nu^{6} - 421 \nu^{5} + 103 \nu^{4} - 298 \nu^{3} + 4 \nu^{2} - 32 \nu - 28$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{11} + \nu^{10} + 45 \nu^{9} + 15 \nu^{8} + 217 \nu^{7} + 69 \nu^{6} + 421 \nu^{5} + 103 \nu^{4} + 298 \nu^{3} + 4 \nu^{2} + 32 \nu - 28$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{11} + 31 \nu^{9} + 157 \nu^{7} + 321 \nu^{5} + 243 \nu^{3} + 46 \nu$$$$)/8$$ $$\beta_{8}$$ $$=$$ $$($$$$3 \nu^{11} - 3 \nu^{10} + 49 \nu^{9} - 43 \nu^{8} + 271 \nu^{7} - 191 \nu^{6} + 633 \nu^{5} - 327 \nu^{4} + 588 \nu^{3} - 192 \nu^{2} + 140 \nu - 12$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{11} + 3 \nu^{10} - 47 \nu^{9} + 47 \nu^{8} - 243 \nu^{7} + 241 \nu^{6} - 515 \nu^{5} + 491 \nu^{4} - 412 \nu^{3} + 342 \nu^{2} - 92 \nu + 48$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} + 3 \nu^{10} + 49 \nu^{9} + 43 \nu^{8} + 271 \nu^{7} + 191 \nu^{6} + 633 \nu^{5} + 327 \nu^{4} + 588 \nu^{3} + 192 \nu^{2} + 140 \nu + 12$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$3 \nu^{11} + 3 \nu^{10} + 47 \nu^{9} + 47 \nu^{8} + 243 \nu^{7} + 241 \nu^{6} + 515 \nu^{5} + 491 \nu^{4} + 412 \nu^{3} + 342 \nu^{2} + 92 \nu + 48$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_{1} - 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{11} + 5 \beta_{10} + 3 \beta_{9} + 5 \beta_{8} - 14 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 7 \beta_{4} - 7 \beta_{3} + 12 \beta_{2} - 12 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-13 \beta_{11} + 11 \beta_{10} - 13 \beta_{9} - 11 \beta_{8} + 9 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 24 \beta_{2} + 24 \beta_{1} + 42$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$11 \beta_{11} - 33 \beta_{10} - 11 \beta_{9} - 33 \beta_{8} + 110 \beta_{7} - 35 \beta_{6} + 35 \beta_{5} - 49 \beta_{4} + 49 \beta_{3} - 86 \beta_{2} + 86 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$117 \beta_{11} - 93 \beta_{10} + 117 \beta_{9} + 93 \beta_{8} - 77 \beta_{6} - 77 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 210 \beta_{2} - 210 \beta_{1} - 322$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-49 \beta_{11} + 243 \beta_{10} + 49 \beta_{9} + 243 \beta_{8} - 874 \beta_{7} + 267 \beta_{6} - 267 \beta_{5} + 365 \beta_{4} - 365 \beta_{3} + 648 \beta_{2} - 648 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-963 \beta_{11} + 745 \beta_{10} - 963 \beta_{9} - 745 \beta_{8} + 631 \beta_{6} + 631 \beta_{5} + 23 \beta_{4} + 23 \beta_{3} + 1716 \beta_{2} + 1716 \beta_{1} + 2510$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$269 \beta_{11} - 1863 \beta_{10} - 269 \beta_{9} - 1863 \beta_{8} + 6930 \beta_{7} - 2089 \beta_{6} + 2089 \beta_{5} - 2811 \beta_{4} + 2811 \beta_{3} - 5006 \beta_{2} + 5006 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$7707 \beta_{11} - 5887 \beta_{10} + 7707 \beta_{9} + 5887 \beta_{8} - 5059 \beta_{6} - 5059 \beta_{5} - 385 \beta_{4} - 385 \beta_{3} - 13714 \beta_{2} - 13714 \beta_{1} - 19678$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-1747 \beta_{11} + 14513 \beta_{10} + 1747 \beta_{9} + 14513 \beta_{8} - 54798 \beta_{7} + 16453 \beta_{6} - 16453 \beta_{5} + 21955 \beta_{4} - 21955 \beta_{3} + 39116 \beta_{2} - 39116 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 0.297931i − 1.69093i 2.80721i 1.12212i − 0.678294i − 1.85804i − 0.297931i 1.69093i − 2.80721i − 1.12212i 0.678294i 1.85804i
−0.707107 0.707107i −1.10458 + 1.33413i 1.00000i −0.953972 2.02236i 1.72443 0.162311i −0.707107 + 0.707107i 0.707107 0.707107i −0.559788 2.94731i −0.755464 + 2.10458i
113.2 −0.707107 0.707107i 1.27508 + 1.17225i 1.00000i −1.37462 + 1.76364i −0.0727133 1.73052i −0.707107 + 0.707107i 0.707107 0.707107i 0.251664 + 2.98943i 2.21908 0.275081i
113.3 −0.707107 0.707107i 1.53661 0.799269i 1.00000i 1.91438 1.15549i −1.65172 0.521378i −0.707107 + 0.707107i 0.707107 0.707107i 1.72234 2.45633i −2.17073 0.536610i
113.4 0.707107 + 0.707107i −0.931481 + 1.46025i 1.00000i 2.16244 + 0.569088i −1.69121 + 0.373900i 0.707107 0.707107i −0.707107 + 0.707107i −1.26469 2.72040i 1.12667 + 1.93148i
113.5 0.707107 + 0.707107i −0.430811 1.67762i 1.00000i 2.22680 0.203331i 0.881625 1.49088i 0.707107 0.707107i −0.707107 + 0.707107i −2.62880 + 1.44547i 1.71837 + 1.43081i
113.6 0.707107 + 0.707107i 1.65519 + 0.510256i 1.00000i −1.97503 + 1.04846i 0.809587 + 1.53120i 0.707107 0.707107i −0.707107 + 0.707107i 2.47928 + 1.68914i −2.13793 0.655185i
197.1 −0.707107 + 0.707107i −1.10458 1.33413i 1.00000i −0.953972 + 2.02236i 1.72443 + 0.162311i −0.707107 0.707107i 0.707107 + 0.707107i −0.559788 + 2.94731i −0.755464 2.10458i
197.2 −0.707107 + 0.707107i 1.27508 1.17225i 1.00000i −1.37462 1.76364i −0.0727133 + 1.73052i −0.707107 0.707107i 0.707107 + 0.707107i 0.251664 2.98943i 2.21908 + 0.275081i
197.3 −0.707107 + 0.707107i 1.53661 + 0.799269i 1.00000i 1.91438 + 1.15549i −1.65172 + 0.521378i −0.707107 0.707107i 0.707107 + 0.707107i 1.72234 + 2.45633i −2.17073 + 0.536610i
197.4 0.707107 0.707107i −0.931481 1.46025i 1.00000i 2.16244 0.569088i −1.69121 0.373900i 0.707107 + 0.707107i −0.707107 0.707107i −1.26469 + 2.72040i 1.12667 1.93148i
197.5 0.707107 0.707107i −0.430811 + 1.67762i 1.00000i 2.22680 + 0.203331i 0.881625 + 1.49088i 0.707107 + 0.707107i −0.707107 0.707107i −2.62880 1.44547i 1.71837 1.43081i
197.6 0.707107 0.707107i 1.65519 0.510256i 1.00000i −1.97503 1.04846i 0.809587 1.53120i 0.707107 + 0.707107i −0.707107 0.707107i 2.47928 1.68914i −2.13793 + 0.655185i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.j.b yes 12
3.b odd 2 1 210.2.j.a 12
5.b even 2 1 1050.2.j.d 12
5.c odd 4 1 210.2.j.a 12
5.c odd 4 1 1050.2.j.c 12
15.d odd 2 1 1050.2.j.c 12
15.e even 4 1 inner 210.2.j.b yes 12
15.e even 4 1 1050.2.j.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.j.a 12 3.b odd 2 1
210.2.j.a 12 5.c odd 4 1
210.2.j.b yes 12 1.a even 1 1 trivial
210.2.j.b yes 12 15.e even 4 1 inner
1050.2.j.c 12 5.c odd 4 1
1050.2.j.c 12 15.d odd 2 1
1050.2.j.d 12 5.b even 2 1
1050.2.j.d 12 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{3}$$
$3$ $$1 - 4 T + 8 T^{2} - 20 T^{3} + 57 T^{4} - 104 T^{5} + 160 T^{6} - 312 T^{7} + 513 T^{8} - 540 T^{9} + 648 T^{10} - 972 T^{11} + 729 T^{12}$$
$5$ $$1 - 4 T - 2 T^{2} + 4 T^{3} + 89 T^{4} - 88 T^{5} - 288 T^{6} - 440 T^{7} + 2225 T^{8} + 500 T^{9} - 1250 T^{10} - 12500 T^{11} + 15625 T^{12}$$
$7$ $$( 1 + T^{4} )^{3}$$
$11$ $$1 - 56 T^{2} + 1358 T^{4} - 19128 T^{6} + 195023 T^{8} - 1950544 T^{10} + 21368868 T^{12} - 236015824 T^{14} + 2855331743 T^{16} - 33886418808 T^{18} + 291099360398 T^{20} - 1452495777656 T^{22} + 3138428376721 T^{24}$$
$13$ $$1 - 32 T^{3} - 14 T^{4} + 272 T^{5} + 512 T^{6} - 5520 T^{7} - 4301 T^{8} + 109488 T^{9} + 220800 T^{10} + 406064 T^{11} - 6852572 T^{12} + 5278832 T^{13} + 37315200 T^{14} + 240545136 T^{15} - 122840861 T^{16} - 2049537360 T^{17} + 2471326208 T^{18} + 17067596624 T^{19} - 11420230094 T^{20} - 339343979936 T^{21} + 23298085122481 T^{24}$$
$17$ $$1 + 28 T + 392 T^{2} + 3716 T^{3} + 26626 T^{4} + 151924 T^{5} + 720808 T^{6} + 3009708 T^{7} + 12126655 T^{8} + 52482936 T^{9} + 248165456 T^{10} + 1175330632 T^{11} + 5125539580 T^{12} + 19980620744 T^{13} + 71719816784 T^{14} + 257848664568 T^{15} + 1012830352255 T^{16} + 4273354971756 T^{17} + 17398552835752 T^{18} + 62340292556852 T^{19} + 185736517624066 T^{20} + 440672549062852 T^{21} + 790269608976008 T^{22} + 959613096613724 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 - 104 T^{2} + 6018 T^{4} - 249384 T^{6} + 7954675 T^{8} - 203686352 T^{10} + 4270353988 T^{12} - 73530773072 T^{14} + 1036661200675 T^{16} - 11732489987304 T^{18} + 102207082380738 T^{20} - 637630890811304 T^{22} + 2213314919066161 T^{24}$$
$23$ $$1 - 24 T + 288 T^{2} - 2648 T^{3} + 22078 T^{4} - 163240 T^{5} + 1065248 T^{6} - 6483944 T^{7} + 37525599 T^{8} - 203370768 T^{9} + 1048113088 T^{10} - 5277774096 T^{11} + 25830292740 T^{12} - 121388804208 T^{13} + 554451823552 T^{14} - 2474412134256 T^{15} + 10501201149759 T^{16} - 41732887576792 T^{17} + 157694934685472 T^{18} - 555803705968280 T^{19} + 1728949933033918 T^{20} - 4769452247554024 T^{21} + 11930835229530912 T^{22} - 22867434189934248 T^{23} + 21914624432020321 T^{24}$$
$29$ $$( 1 + 4 T + 114 T^{2} + 548 T^{3} + 6631 T^{4} + 29256 T^{5} + 241692 T^{6} + 848424 T^{7} + 5576671 T^{8} + 13365172 T^{9} + 80630034 T^{10} + 82044596 T^{11} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 + 4 T + 28 T^{2} + 76 T^{3} + 807 T^{4} + 1288 T^{5} + 32184 T^{6} + 39928 T^{7} + 775527 T^{8} + 2264116 T^{9} + 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12} )^{2}$$
$37$ $$1 + 20 T + 200 T^{2} + 1052 T^{3} + 3634 T^{4} + 30620 T^{5} + 438952 T^{6} + 3655156 T^{7} + 16552975 T^{8} + 43779560 T^{9} + 283860944 T^{10} + 4234271992 T^{11} + 34611552220 T^{12} + 156668063704 T^{13} + 388605632336 T^{14} + 2217566052680 T^{15} + 31022940178975 T^{16} + 253462980492292 T^{17} + 1126230738683368 T^{18} + 2906814077812460 T^{19} + 12764350335548914 T^{20} + 136719750264421004 T^{21} + 961716874483569800 T^{22} + 3558352435589208260 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 - 212 T^{2} + 22434 T^{4} - 1643716 T^{6} + 97594575 T^{8} - 5014736296 T^{10} + 222514140252 T^{12} - 8429771713576 T^{14} + 275778943846575 T^{16} - 7807822342599556 T^{18} + 179133812590100514 T^{20} - 2845603773752309012 T^{22} + 22563490300366186081 T^{24}$$
$43$ $$1 - 8 T + 32 T^{2} - 184 T^{3} + 5014 T^{4} - 25096 T^{5} + 57248 T^{6} + 105352 T^{7} + 1869663 T^{8} + 20409392 T^{9} - 214625984 T^{10} + 2505871696 T^{11} - 14493786700 T^{12} + 107752482928 T^{13} - 396843444416 T^{14} + 1622689529744 T^{15} + 6392005734063 T^{16} + 15487633486936 T^{17} + 361885391829152 T^{18} - 6821559864341272 T^{19} + 58604636191891414 T^{20} - 92477040596379112 T^{21} + 691567434025095968 T^{22} - 7434349915769781656 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 16 T + 128 T^{2} + 48 T^{3} - 3930 T^{4} + 2832 T^{5} + 549504 T^{6} + 4835248 T^{7} + 9727055 T^{8} - 102211168 T^{9} - 484788992 T^{10} + 7463204576 T^{11} + 87433873108 T^{12} + 350770615072 T^{13} - 1070898883328 T^{14} - 10611870095264 T^{15} + 47464925469455 T^{16} + 1108939986406736 T^{17} + 5923221940146816 T^{18} + 1434756677151216 T^{19} - 93578356580720730 T^{20} + 53718262708932816 T^{21} + 6732688926186246272 T^{22} + 39554547441344196848 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 - 24 T + 288 T^{2} - 2968 T^{3} + 27318 T^{4} - 189144 T^{5} + 1076384 T^{6} - 4855704 T^{7} - 9792929 T^{8} + 439179408 T^{9} - 4825144512 T^{10} + 45292675216 T^{11} - 372958615948 T^{12} + 2400511786448 T^{13} - 13553830934208 T^{14} + 65383712724816 T^{15} - 77270920208849 T^{16} - 2030633528142072 T^{17} + 23857363689477536 T^{18} - 222189563833329528 T^{19} + 1700810222657559798 T^{20} - 9793698340468730744 T^{21} + 50367591465267758112 T^{22} -$$$$22\!\cdots\!28$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$( 1 + 16 T + 270 T^{2} + 3160 T^{3} + 33193 T^{4} + 284568 T^{5} + 2445376 T^{6} + 16789512 T^{7} + 115544833 T^{8} + 648997640 T^{9} + 3271687470 T^{10} + 11438788784 T^{11} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 + 178 T^{2} + 664 T^{3} + 12681 T^{4} + 121240 T^{5} + 672800 T^{6} + 7395640 T^{7} + 47186001 T^{8} + 150715384 T^{9} + 2464559698 T^{10} + 51520374361 T^{12} )^{2}$$
$67$ $$1 + 64 T^{3} + 2390 T^{4} - 9024 T^{5} + 2048 T^{6} - 508672 T^{7} + 20286015 T^{8} + 4116224 T^{9} + 3266560 T^{10} - 440818304 T^{11} + 142523500468 T^{12} - 29534826368 T^{13} + 14663587840 T^{14} + 1238007878912 T^{15} + 408785942872815 T^{16} - 686770838427904 T^{17} + 185258766682112 T^{18} - 54691861526434752 T^{19} + 970501749360371990 T^{20} + 1741218201362876608 T^{21} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 - 440 T^{2} + 91446 T^{4} - 12419864 T^{6} + 1305779695 T^{8} - 116298170672 T^{10} + 8924106004404 T^{12} - 586259078357552 T^{14} + 33182057065617295 T^{16} - 1590988104660206744 T^{18} + 59051577418299860406 T^{20} -$$$$14\!\cdots\!40$$$$T^{22} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 + 24 T + 288 T^{2} + 2936 T^{3} + 13126 T^{4} - 142696 T^{5} - 2894944 T^{6} - 37628616 T^{7} - 323587345 T^{8} - 985106832 T^{9} + 7253083968 T^{10} + 194887191344 T^{11} + 2401164656404 T^{12} + 14226764968112 T^{13} + 38651684465472 T^{14} - 383223304464144 T^{15} - 9189311407860145 T^{16} - 78006814913505288 T^{17} - 438104110389982816 T^{18} - 1576419579081065512 T^{19} + 10585595166201707206 T^{20} +$$$$17\!\cdots\!68$$$$T^{21} +$$$$12\!\cdots\!12$$$$T^{22} +$$$$75\!\cdots\!48$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 - 508 T^{2} + 135562 T^{4} - 24649964 T^{6} + 3381685343 T^{8} - 366753014104 T^{10} + 32137748903980 T^{12} - 2288905561023064 T^{14} + 131716918026362783 T^{16} - 5992097027444251244 T^{18} +$$$$20\!\cdots\!82$$$$T^{20} -$$$$48\!\cdots\!08$$$$T^{22} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 24 T + 288 T^{2} - 3784 T^{3} + 70066 T^{4} - 929320 T^{5} + 9284000 T^{6} - 107957496 T^{7} + 1400664563 T^{8} - 14071243520 T^{9} + 124154702400 T^{10} - 1299362373632 T^{11} + 13371946678180 T^{12} - 107847077011456 T^{13} + 855301744833600 T^{14} - 8045754118570240 T^{15} + 66473188444178723 T^{16} - 425248964460509928 T^{17} + 3035314426357796000 T^{18} - 25218074905680163640 T^{19} +$$$$15\!\cdots\!06$$$$T^{20} -$$$$70\!\cdots\!52$$$$T^{21} +$$$$44\!\cdots\!12$$$$T^{22} -$$$$30\!\cdots\!08$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$( 1 + 24 T + 358 T^{2} + 5272 T^{3} + 62767 T^{4} + 696208 T^{5} + 7360436 T^{6} + 61962512 T^{7} + 497177407 T^{8} + 3716596568 T^{9} + 22461722278 T^{10} + 134017426776 T^{11} + 496981290961 T^{12} )^{2}$$
$97$ $$1 - 8 T + 32 T^{2} + 1240 T^{3} + 7270 T^{4} - 77896 T^{5} + 1159328 T^{6} + 23481432 T^{7} - 27667953 T^{8} + 164181296 T^{9} + 23294478656 T^{10} + 157379895024 T^{11} - 641510857644 T^{12} + 15265849817328 T^{13} + 219177749674304 T^{14} + 149843835964208 T^{15} - 2449423985831793 T^{16} + 201643046305608024 T^{17} + 965687768530327712 T^{18} - 6293863167707090248 T^{19} + 56978142231120506470 T^{20} +$$$$94\!\cdots\!80$$$$T^{21} +$$$$23\!\cdots\!68$$$$T^{22} -$$$$57\!\cdots\!24$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$