# Properties

 Label 546.2.g.c Level $546$ Weight $2$ Character orbit 546.g Analytic conductor $4.360$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 4 x^{10} - 8 x^{8} + 26 x^{7} - 50 x^{6} + 78 x^{5} - 72 x^{4} + 324 x^{2} - 486 x + 729$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 4 x^{10} - 8 x^{8} + 26 x^{7} - 50 x^{6} + 78 x^{5} - 72 x^{4} + 324 x^{2} - 486 x + 729$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{10} - 5 \nu^{9} + 10 \nu^{8} + 15 \nu^{7} + 19 \nu^{6} - 4 \nu^{5} - 47 \nu^{4} - 69 \nu^{3} - 90 \nu^{2} - 81 \nu - 405$$$$)/648$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{10} + 29 \nu^{9} - 58 \nu^{8} + 87 \nu^{7} - 79 \nu^{6} - 32 \nu^{5} + 173 \nu^{4} - 783 \nu^{3} + 1566 \nu^{2} - 2349 \nu + 1701$$$$)/648$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 17 \nu^{10} - 16 \nu^{9} - 3 \nu^{8} + 53 \nu^{7} - 92 \nu^{6} + 215 \nu^{5} - 387 \nu^{4} + 288 \nu^{3} + 297 \nu^{2} - 891 \nu + 2916$$$$)/972$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{11} - 29 \nu^{10} + 58 \nu^{9} - 87 \nu^{8} + 79 \nu^{7} + 32 \nu^{6} - 173 \nu^{5} + 783 \nu^{4} - 1566 \nu^{3} + 2349 \nu^{2} - 1701 \nu$$$$)/1944$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{10} + 4 \nu^{9} - 8 \nu^{7} + 26 \nu^{6} - 50 \nu^{5} + 78 \nu^{4} - 72 \nu^{3} + 324 \nu - 486$$$$)/243$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} + \nu^{10} - 2 \nu^{9} - 7 \nu^{8} + 5 \nu^{7} - 12 \nu^{6} + 15 \nu^{5} - 31 \nu^{4} + 42 \nu^{3} + 177 \nu^{2} - 243 \nu + 216$$$$)/216$$ $$\beta_{8}$$ $$=$$ $$($$$$7 \nu^{11} - 59 \nu^{10} + 208 \nu^{9} - 387 \nu^{8} + 601 \nu^{7} - 538 \nu^{6} - 53 \nu^{5} + 2193 \nu^{4} - 5328 \nu^{3} + 10881 \nu^{2} - 13851 \nu + 12150$$$$)/1944$$ $$\beta_{9}$$ $$=$$ $$($$$$7 \nu^{11} - 17 \nu^{10} + 43 \nu^{9} - 84 \nu^{8} + 88 \nu^{7} - 10 \nu^{6} - 257 \nu^{5} + 633 \nu^{4} - 1377 \nu^{3} + 1674 \nu^{2} - 1620 \nu + 972$$$$)/972$$ $$\beta_{10}$$ $$=$$ $$($$$$5 \nu^{11} - 10 \nu^{10} + 29 \nu^{9} - 45 \nu^{8} + 50 \nu^{7} - 59 \nu^{6} - 79 \nu^{5} + 354 \nu^{4} - 783 \nu^{3} + 1323 \nu^{2} - 1134 \nu + 1701$$$$)/648$$ $$\beta_{11}$$ $$=$$ $$($$$$-5 \nu^{11} + 10 \nu^{10} + \nu^{9} - 15 \nu^{8} + 70 \nu^{7} - 103 \nu^{6} + 163 \nu^{5} - 222 \nu^{4} - 231 \nu^{3} + 693 \nu^{2} - 1998 \nu + 1701$$$$)/648$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 5 \beta_{3} - 2 \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{11} - 2 \beta_{10} - 2 \beta_{7} - 3 \beta_{6} + 10 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 2$$ $$\nu^{6}$$ $$=$$ $$-8 \beta_{10} + 6 \beta_{6} + 12 \beta_{5} + 6 \beta_{4} - 6 \beta_{1} + 15$$ $$\nu^{7}$$ $$=$$ $$-12 \beta_{11} - 12 \beta_{10} + 8 \beta_{8} - 6 \beta_{7} - 14 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} + 14 \beta_{2} + 9 \beta_{1} - 6$$ $$\nu^{8}$$ $$=$$ $$21 \beta_{10} - 25 \beta_{9} + 6 \beta_{8} + \beta_{7} + 17 \beta_{6} - 22 \beta_{5} - 10 \beta_{4} + 33 \beta_{2} - 8 \beta_{1} + 16$$ $$\nu^{9}$$ $$=$$ $$-8 \beta_{11} - 27 \beta_{10} - 21 \beta_{9} + 5 \beta_{8} - 40 \beta_{7} + 37 \beta_{6} - 10 \beta_{5} + 21 \beta_{4} + 43 \beta_{3} - 46 \beta_{2} - 5 \beta_{1} - 9$$ $$\nu^{10}$$ $$=$$ $$57 \beta_{11} + 33 \beta_{9} + 8 \beta_{8} + 9 \beta_{7} + 95 \beta_{6} - 40 \beta_{5} + 31 \beta_{4} - 57 \beta_{3} - 48 \beta_{2} - 48 \beta_{1} - 25$$ $$\nu^{11}$$ $$=$$ $$-72 \beta_{11} + 48 \beta_{10} - 24 \beta_{8} - 48 \beta_{7} - 27 \beta_{6} + 48 \beta_{5} + 120 \beta_{4} + 24 \beta_{3} - 88 \beta_{1} - 216$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0.146987 + 1.72580i 0.684481 + 1.59106i 1.64187 + 0.551597i −1.73194 + 0.0198536i 1.07299 − 1.35967i −0.814390 − 1.52865i 0.146987 − 1.72580i 0.684481 − 1.59106i 1.64187 − 0.551597i −1.73194 − 0.0198536i 1.07299 + 1.35967i −0.814390 + 1.52865i
1.00000i −1.72580 0.146987i −1.00000 3.83276 −0.146987 + 1.72580i −0.655092 2.56337i 1.00000i 2.95679 + 0.507343i 3.83276i
209.2 1.00000i −1.59106 0.684481i −1.00000 −4.42062 −0.684481 + 1.59106i −2.43879 + 1.02583i 1.00000i 2.06297 + 2.17811i 4.42062i
209.3 1.00000i −0.551597 1.64187i −1.00000 0.124244 −1.64187 + 0.551597i 1.46370 2.20399i 1.00000i −2.39148 + 1.81130i 0.124244i
209.4 1.00000i −0.0198536 + 1.73194i −1.00000 1.44804 1.73194 + 0.0198536i −2.59654 + 0.507916i 1.00000i −2.99921 0.0687703i 1.44804i
209.5 1.00000i 1.35967 1.07299i −1.00000 −0.745238 −1.07299 1.35967i −2.35778 1.20037i 1.00000i 0.697395 2.91781i 0.745238i
209.6 1.00000i 1.52865 + 0.814390i −1.00000 1.76081 0.814390 1.52865i 2.58450 0.566014i 1.00000i 1.67354 + 2.48983i 1.76081i
209.7 1.00000i −1.72580 + 0.146987i −1.00000 3.83276 −0.146987 1.72580i −0.655092 + 2.56337i 1.00000i 2.95679 0.507343i 3.83276i
209.8 1.00000i −1.59106 + 0.684481i −1.00000 −4.42062 −0.684481 1.59106i −2.43879 1.02583i 1.00000i 2.06297 2.17811i 4.42062i
209.9 1.00000i −0.551597 + 1.64187i −1.00000 0.124244 −1.64187 0.551597i 1.46370 + 2.20399i 1.00000i −2.39148 1.81130i 0.124244i
209.10 1.00000i −0.0198536 1.73194i −1.00000 1.44804 1.73194 0.0198536i −2.59654 0.507916i 1.00000i −2.99921 + 0.0687703i 1.44804i
209.11 1.00000i 1.35967 + 1.07299i −1.00000 −0.745238 −1.07299 + 1.35967i −2.35778 + 1.20037i 1.00000i 0.697395 + 2.91781i 0.745238i
209.12 1.00000i 1.52865 0.814390i −1.00000 1.76081 0.814390 + 1.52865i 2.58450 + 0.566014i 1.00000i 1.67354 2.48983i 1.76081i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.12 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.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.g.c 12
3.b odd 2 1 546.2.g.d yes 12
7.b odd 2 1 546.2.g.d yes 12
21.c even 2 1 inner 546.2.g.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.g.c 12 1.a even 1 1 trivial
546.2.g.c 12 21.c even 2 1 inner
546.2.g.d yes 12 3.b odd 2 1
546.2.g.d yes 12 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 2 T_{5}^{5} - 18 T_{5}^{4} + 46 T_{5}^{3} - 7 T_{5}^{2} - 32 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{6}$$
$3$ $$729 + 486 T + 42 T^{5} + 58 T^{6} + 14 T^{7} + 2 T^{11} + T^{12}$$
$5$ $$( 4 - 32 T - 7 T^{2} + 46 T^{3} - 18 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$7$ $$117649 + 134456 T + 45619 T^{2} - 2744 T^{3} - 4753 T^{4} - 1008 T^{5} - 166 T^{6} - 144 T^{7} - 97 T^{8} - 8 T^{9} + 19 T^{10} + 8 T^{11} + T^{12}$$
$11$ $$9709456 + 4446712 T^{2} + 819657 T^{4} + 77388 T^{6} + 3918 T^{8} + 100 T^{10} + T^{12}$$
$13$ $$( 1 + T^{2} )^{6}$$
$17$ $$( 3032 + 2252 T - 43 T^{2} - 310 T^{3} - 42 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$19$ $$1893376 + 1586176 T^{2} + 456400 T^{4} + 60360 T^{6} + 3865 T^{8} + 110 T^{10} + T^{12}$$
$23$ $$96983104 + 39686240 T^{2} + 5661977 T^{4} + 371940 T^{6} + 11950 T^{8} + 180 T^{10} + T^{12}$$
$29$ $$565504 + 1063424 T^{2} + 577760 T^{4} + 88832 T^{6} + 5345 T^{8} + 126 T^{10} + T^{12}$$
$31$ $$378535936 + 161841152 T^{2} + 21072128 T^{4} + 1078432 T^{6} + 25065 T^{8} + 262 T^{10} + T^{12}$$
$37$ $$( -12743 - 19992 T + 2931 T^{2} + 1056 T^{3} - 133 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$41$ $$( 1024 - 1792 T - 2268 T^{2} - 628 T^{3} - 7 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$43$ $$( 68 + 116 T - 143 T^{2} - 280 T^{3} - 54 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$47$ $$( 16000 + 44400 T - 16075 T^{2} + 278 T^{3} + 326 T^{4} - 34 T^{5} + T^{6} )^{2}$$
$53$ $$18969001984 + 2815459328 T^{2} + 162919424 T^{4} + 4607488 T^{6} + 65232 T^{8} + 424 T^{10} + T^{12}$$
$59$ $$( 512 + 2560 T + 96 T^{2} - 784 T^{3} - 132 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$61$ $$2567651584 + 575957216 T^{2} + 46525953 T^{4} + 1751172 T^{6} + 32646 T^{8} + 292 T^{10} + T^{12}$$
$67$ $$( 15376 - 10912 T - 632 T^{2} + 1384 T^{3} - 191 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$71$ $$195552256 + 356571648 T^{2} + 40476368 T^{4} + 1709128 T^{6} + 33185 T^{8} + 298 T^{10} + T^{12}$$
$73$ $$117948286096 + 13874690344 T^{2} + 561470513 T^{4} + 10711276 T^{6} + 105254 T^{8} + 516 T^{10} + T^{12}$$
$79$ $$( -64 + 512 T - 592 T^{2} + 152 T^{3} + 45 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$83$ $$( 22912 - 34496 T + 7952 T^{2} + 2000 T^{3} - 144 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$89$ $$( -41344 + 62528 T - 29104 T^{2} + 3952 T^{3} - 16 T^{4} - 24 T^{5} + T^{6} )^{2}$$
$97$ $$1435500544 + 577363968 T^{2} + 57298064 T^{4} + 2354680 T^{6} + 44321 T^{8} + 358 T^{10} + T^{12}$$