# Properties

 Label 546.2.bm.b Level $546$ Weight $2$ Character orbit 546.bm Analytic conductor $4.360$ Analytic rank $0$ Dimension $20$ 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.bm (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 56 x^{18} + 1306 x^{16} + 16508 x^{14} + 123139 x^{12} + 552164 x^{10} + 1447090 x^{8} + 2035844 x^{6} + 1263505 x^{4} + 215520 x^{2} + 576$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 56 x^{18} + 1306 x^{16} + 16508 x^{14} + 123139 x^{12} + 552164 x^{10} + 1447090 x^{8} + 2035844 x^{6} + 1263505 x^{4} + 215520 x^{2} + 576$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3022 \nu^{18} + 132131 \nu^{16} + 2192815 \nu^{14} + 17182550 \nu^{12} + 65265673 \nu^{10} + 134404286 \nu^{8} + 345978835 \nu^{6} + 848977739 \nu^{4} + 501209410 \nu^{2} + 8506128$$$$)/ 137229096$$ $$\beta_{2}$$ $$=$$ $$($$$$-16974619 \nu^{19} + 3676068 \nu^{18} - 1283960340 \nu^{17} + 460750880 \nu^{16} - 38602727614 \nu^{15} + 18053043108 \nu^{14} - 604945859288 \nu^{13} + 340385291796 \nu^{12} - 5391398303493 \nu^{11} + 3535551559876 \nu^{10} - 27758334534728 \nu^{9} + 20962795318116 \nu^{8} - 79467120501562 \nu^{7} + 68761329525984 \nu^{6} - 112674210776124 \nu^{5} + 110553906440468 \nu^{4} - 56873843883391 \nu^{3} + 61065638669232 \nu^{2} + 1467907393416 \nu + 635189599008$$$$)/ 1762021592640$$ $$\beta_{3}$$ $$=$$ $$($$$$68783 \nu^{18} + 3326870 \nu^{16} + 64093833 \nu^{14} + 629361751 \nu^{12} + 3334559671 \nu^{10} + 9194428011 \nu^{8} + 11072929004 \nu^{6} + 1920888413 \nu^{4} - 1984961058 \nu^{2} + 610118280 \nu + 49665168$$$$)/ 1220236560$$ $$\beta_{4}$$ $$=$$ $$($$$$-68783 \nu^{18} - 3326870 \nu^{16} - 64093833 \nu^{14} - 629361751 \nu^{12} - 3334559671 \nu^{10} - 9194428011 \nu^{8} - 11072929004 \nu^{6} - 1920888413 \nu^{4} + 1984961058 \nu^{2} + 610118280 \nu - 49665168$$$$)/ 1220236560$$ $$\beta_{5}$$ $$=$$ $$($$$$14856982 \nu^{19} - 4850310 \nu^{18} + 790524490 \nu^{17} - 212070255 \nu^{16} + 17247293007 \nu^{15} - 3519468075 \nu^{14} + 199629861749 \nu^{13} - 27577992750 \nu^{12} + 1322768996024 \nu^{11} - 104751405165 \nu^{10} + 5042771893329 \nu^{9} - 215718879030 \nu^{8} + 10572724690426 \nu^{7} - 555296030175 \nu^{6} + 11171084080987 \nu^{5} - 1362609271095 \nu^{4} + 5268731003973 \nu^{3} - 804441103050 \nu^{2} + 395714733552 \nu - 13652335440$$$$)/ 440505398160$$ $$\beta_{6}$$ $$=$$ $$($$$$-29415555 \nu^{19} - 35398716 \nu^{18} - 1659317510 \nu^{17} - 1937602550 \nu^{16} - 39004638160 \nu^{15} - 44009978766 \nu^{14} - 497204747280 \nu^{13} - 540038980872 \nu^{12} - 3743049420175 \nu^{11} - 3900900559222 \nu^{10} - 16966527705440 \nu^{9} - 16905660839712 \nu^{8} - 45106099779540 \nu^{7} - 42604079503458 \nu^{6} - 64574560029530 \nu^{5} - 56102602880186 \nu^{4} - 39535638354895 \nu^{3} - 28630898868504 \nu^{2} - 4010123301240 \nu - 2399382291456$$$$)/ 881010796320$$ $$\beta_{7}$$ $$=$$ $$($$$$-29415555 \nu^{19} + 35398716 \nu^{18} - 1659317510 \nu^{17} + 1937602550 \nu^{16} - 39004638160 \nu^{15} + 44009978766 \nu^{14} - 497204747280 \nu^{13} + 540038980872 \nu^{12} - 3743049420175 \nu^{11} + 3900900559222 \nu^{10} - 16966527705440 \nu^{9} + 16905660839712 \nu^{8} - 45106099779540 \nu^{7} + 42604079503458 \nu^{6} - 64574560029530 \nu^{5} + 56102602880186 \nu^{4} - 39535638354895 \nu^{3} + 28630898868504 \nu^{2} - 4010123301240 \nu + 2399382291456$$$$)/ 881010796320$$ $$\beta_{8}$$ $$=$$ $$($$$$-81018389 \nu^{19} - 34968720 \nu^{18} - 4920211380 \nu^{17} - 1891987600 \nu^{16} - 125695121574 \nu^{15} - 42564285900 \nu^{14} - 1756829707708 \nu^{13} - 519074683140 \nu^{12} - 14608051723923 \nu^{11} - 3736505960960 \nu^{10} - 73424383345548 \nu^{9} - 16065764420820 \nu^{8} - 215842472125742 \nu^{7} - 39051161390160 \nu^{6} - 337345097373984 \nu^{5} - 44801530182100 \nu^{4} - 225383049182661 \nu^{3} - 10979716228260 \nu^{2} - 38407760456184 \nu + 4195815050880$$$$)/ 1762021592640$$ $$\beta_{9}$$ $$=$$ $$($$$$81018389 \nu^{19} - 34968720 \nu^{18} + 4920211380 \nu^{17} - 1891987600 \nu^{16} + 125695121574 \nu^{15} - 42564285900 \nu^{14} + 1756829707708 \nu^{13} - 519074683140 \nu^{12} + 14608051723923 \nu^{11} - 3736505960960 \nu^{10} + 73424383345548 \nu^{9} - 16065764420820 \nu^{8} + 215842472125742 \nu^{7} - 39051161390160 \nu^{6} + 337345097373984 \nu^{5} - 44801530182100 \nu^{4} + 225383049182661 \nu^{3} - 10979716228260 \nu^{2} + 38407760456184 \nu + 4195815050880$$$$)/ 1762021592640$$ $$\beta_{10}$$ $$=$$ $$($$$$-94807885 \nu^{19} - 59427928 \nu^{18} - 5289840320 \nu^{17} - 3162097960 \nu^{16} - 122970816790 \nu^{15} - 68989172028 \nu^{14} - 1551010693280 \nu^{13} - 798519446996 \nu^{12} - 11564236180015 \nu^{11} - 5291075984096 \nu^{10} - 51930495392480 \nu^{9} - 20171087573316 \nu^{8} - 136332666788530 \nu^{7} - 42290898761704 \nu^{6} - 190792879709240 \nu^{5} - 44684336323948 \nu^{4} - 114339799652545 \nu^{3} - 21074924015892 \nu^{2} - 17215230963000 \nu - 1582858934208$$$$)/ 1762021592640$$ $$\beta_{11}$$ $$=$$ $$($$$$94807885 \nu^{19} - 59427928 \nu^{18} + 5289840320 \nu^{17} - 3162097960 \nu^{16} + 122970816790 \nu^{15} - 68989172028 \nu^{14} + 1551010693280 \nu^{13} - 798519446996 \nu^{12} + 11564236180015 \nu^{11} - 5291075984096 \nu^{10} + 51930495392480 \nu^{9} - 20171087573316 \nu^{8} + 136332666788530 \nu^{7} - 42290898761704 \nu^{6} + 190792879709240 \nu^{5} - 44684336323948 \nu^{4} + 114339799652545 \nu^{3} - 21074924015892 \nu^{2} + 17215230963000 \nu - 1582858934208$$$$)/ 1762021592640$$ $$\beta_{12}$$ $$=$$ $$($$$$-344897 \nu^{19} - 19039100 \nu^{17} - 437128002 \nu^{15} - 5437184344 \nu^{13} - 39952824679 \nu^{11} - 177101468424 \nu^{9} - 462319287686 \nu^{7} - 657864772052 \nu^{5} - 428095530333 \nu^{3} - 82272045672 \nu - 2440473120$$$$)/ 4880946240$$ $$\beta_{13}$$ $$=$$ $$($$$$123568043 \nu^{19} - 7693436 \nu^{18} + 6859712900 \nu^{17} - 252356260 \nu^{16} + 157753029898 \nu^{15} - 977742396 \nu^{14} + 1949844438556 \nu^{13} + 58826777048 \nu^{12} + 14032943832661 \nu^{11} + 1024826425528 \nu^{10} + 59401464181676 \nu^{9} + 7322561212008 \nu^{8} + 141574139518154 \nu^{7} + 25844069280772 \nu^{6} + 168056538877328 \nu^{5} + 43033037004044 \nu^{4} + 68173843836067 \nu^{3} + 25759775744436 \nu^{2} - 8838421625112 \nu + 1824337931904$$$$)/ 1762021592640$$ $$\beta_{14}$$ $$=$$ $$($$$$131492007 \nu^{19} + 51179864 \nu^{18} + 7688673660 \nu^{17} + 2757223100 \nu^{16} + 188167076682 \nu^{15} + 62283517764 \nu^{14} + 2507500302684 \nu^{13} + 774344657368 \nu^{12} + 19803931915569 \nu^{11} + 5819128016788 \nu^{10} + 94442358003564 \nu^{9} + 26980040013768 \nu^{8} + 264922559713626 \nu^{7} + 73725442937132 \nu^{6} + 404673911407992 \nu^{5} + 102655008267884 \nu^{4} + 285433819851903 \nu^{3} + 50688962013936 \nu^{2} + 67044202893432 \nu + 3119514604704$$$$)/ 1762021592640$$ $$\beta_{15}$$ $$=$$ $$($$$$172641151 \nu^{19} - 3676068 \nu^{18} + 9295720300 \nu^{17} - 460750880 \nu^{16} + 207338905966 \nu^{15} - 18053043108 \nu^{14} + 2497075527272 \nu^{13} - 340385291796 \nu^{12} + 17737074056537 \nu^{11} - 3535551559876 \nu^{10} + 76102656250232 \nu^{9} - 20962795318116 \nu^{8} + 193198213075498 \nu^{7} - 68761329525984 \nu^{6} + 268911548642356 \nu^{5} - 110553906440468 \nu^{4} + 171805755421699 \nu^{3} - 61065638669232 \nu^{2} + 34136347726776 \nu - 635189599008$$$$)/ 1762021592640$$ $$\beta_{16}$$ $$=$$ $$($$$$-201918419 \nu^{19} + 170259720 \nu^{18} - 11398391100 \nu^{17} + 8870380840 \nu^{16} - 267792362934 \nu^{15} + 187918056900 \nu^{14} - 3403815370048 \nu^{13} + 2085640207380 \nu^{12} - 25447075375413 \nu^{11} + 12988633033160 \nu^{10} - 113794727769888 \nu^{9} + 44929184189460 \nu^{8} - 295724587957202 \nu^{7} + 79598602064640 \nu^{6} - 411689959278924 \nu^{5} + 60623708205580 \nu^{4} - 257661071860311 \nu^{3} + 18303873785700 \nu^{2} - 52016067402984 \nu + 3867753991680$$$$)/ 1762021592640$$ $$\beta_{17}$$ $$=$$ $$($$$$70734202 \nu^{19} + 51307110 \nu^{18} + 4079650620 \nu^{17} + 2690592110 \nu^{16} + 98371871127 \nu^{15} + 57620585700 \nu^{14} + 1290161269439 \nu^{13} + 651178722630 \nu^{12} + 10013781774834 \nu^{11} + 4181284748530 \nu^{10} + 46804777778859 \nu^{9} + 15248737152570 \nu^{8} + 127891765020736 \nu^{7} + 29662440863700 \nu^{6} + 187258764163227 \nu^{5} + 26356309596920 \nu^{4} + 120761030260743 \nu^{3} + 7320897503490 \nu^{2} + 22605956964792 \nu - 82015264800$$$$)/ 440505398160$$ $$\beta_{18}$$ $$=$$ $$($$$$349867935 \nu^{19} + 18978576 \nu^{18} + 19838226880 \nu^{17} + 1367677040 \nu^{16} + 468890923370 \nu^{15} + 39698937336 \nu^{14} + 6008355434520 \nu^{13} + 605238958872 \nu^{12} + 45401111928245 \nu^{11} + 5270273410432 \nu^{10} + 205774317577720 \nu^{9} + 26551812327192 \nu^{8} + 542829366020310 \nu^{7} + 74182962936288 \nu^{6} + 763523329994560 \nu^{5} + 101532544719416 \nu^{4} + 467947463340515 \nu^{3} + 48870394053144 \nu^{2} + 74540001435000 \nu + 2806319104416$$$$)/ 1762021592640$$ $$\beta_{19}$$ $$=$$ $$($$$$349867935 \nu^{19} - 18978576 \nu^{18} + 19838226880 \nu^{17} - 1367677040 \nu^{16} + 468890923370 \nu^{15} - 39698937336 \nu^{14} + 6008355434520 \nu^{13} - 605238958872 \nu^{12} + 45401111928245 \nu^{11} - 5270273410432 \nu^{10} + 205774317577720 \nu^{9} - 26551812327192 \nu^{8} + 542829366020310 \nu^{7} - 74182962936288 \nu^{6} + 763523329994560 \nu^{5} - 101532544719416 \nu^{4} + 467947463340515 \nu^{3} - 48870394053144 \nu^{2} + 74540001435000 \nu - 2806319104416$$$$)/ 1762021592640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{4} + \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{18} + \beta_{15} + \beta_{14} + \beta_{13} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2} - 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} - \beta_{16} + 2 \beta_{15} + 8 \beta_{12} + \beta_{9} + 2 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{19} + 16 \beta_{18} - \beta_{17} - \beta_{16} - 11 \beta_{15} - 13 \beta_{14} - 13 \beta_{13} - \beta_{11} - 10 \beta_{10} + 2 \beta_{9} + \beta_{8} - 12 \beta_{7} + 12 \beta_{6} - 13 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} - \beta_{1} + 48$$ $$\nu^{5}$$ $$=$$ $$-6 \beta_{19} - 8 \beta_{18} - 11 \beta_{17} + 11 \beta_{16} - 38 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 136 \beta_{12} + 14 \beta_{11} - 16 \beta_{10} - 17 \beta_{9} + 6 \beta_{8} - 30 \beta_{5} + 107 \beta_{4} + 105 \beta_{3} - 38 \beta_{2} - 15 \beta_{1} - 68$$ $$\nu^{6}$$ $$=$$ $$51 \beta_{19} - 235 \beta_{18} + 27 \beta_{17} + 27 \beta_{16} + 126 \beta_{15} + 184 \beta_{14} + 184 \beta_{13} + 35 \beta_{11} + 103 \beta_{10} - 30 \beta_{9} - 3 \beta_{8} + 137 \beta_{7} - 137 \beta_{6} + 160 \beta_{4} - 92 \beta_{3} - 126 \beta_{2} + 23 \beta_{1} - 559$$ $$\nu^{7}$$ $$=$$ $$142 \beta_{19} + 194 \beta_{18} + 114 \beta_{17} - 114 \beta_{16} + 596 \beta_{15} - 52 \beta_{14} + 52 \beta_{13} + 1968 \beta_{12} - 346 \beta_{11} + 398 \beta_{10} + 246 \beta_{9} - 132 \beta_{8} + 16 \beta_{7} + 16 \beta_{6} + 364 \beta_{5} - 1239 \beta_{4} - 1187 \beta_{3} + 596 \beta_{2} + 182 \beta_{1} + 984$$ $$\nu^{8}$$ $$=$$ $$-726 \beta_{19} + 3361 \beta_{18} - 534 \beta_{17} - 534 \beta_{16} - 1501 \beta_{15} - 2635 \beta_{14} - 2635 \beta_{13} - 858 \beta_{11} - 1225 \beta_{10} + 390 \beta_{9} - 144 \beta_{8} - 1615 \beta_{7} + 1615 \beta_{6} - 1963 \beta_{4} + 1596 \beta_{3} + 1501 \beta_{2} - 348 \beta_{1} + 7091$$ $$\nu^{9}$$ $$=$$ $$-2478 \beta_{19} - 3528 \beta_{18} - 1267 \beta_{17} + 1267 \beta_{16} - 8894 \beta_{15} + 1050 \beta_{14} - 1050 \beta_{13} - 27572 \beta_{12} + 6384 \beta_{11} - 7434 \beta_{10} - 3469 \beta_{9} + 2202 \beta_{8} - 414 \beta_{7} - 414 \beta_{6} - 4046 \beta_{5} + 15250 \beta_{4} + 14200 \beta_{3} - 8894 \beta_{2} - 2023 \beta_{1} - 13786$$ $$\nu^{10}$$ $$=$$ $$10101 \beta_{19} - 47776 \beta_{18} + 9241 \beta_{17} + 9241 \beta_{16} + 18425 \beta_{15} + 37675 \beta_{14} + 37675 \beta_{13} + 17053 \beta_{11} + 16228 \beta_{10} - 4970 \beta_{9} + 4271 \beta_{8} + 19926 \beta_{7} - 19926 \beta_{6} + 24331 \beta_{4} - 25156 \beta_{3} - 18425 \beta_{2} + 4453 \beta_{1} - 94044$$ $$\nu^{11}$$ $$=$$ $$38802 \beta_{19} + 57794 \beta_{18} + 15473 \beta_{17} - 15473 \beta_{16} + 130466 \beta_{15} - 18992 \beta_{14} + 18992 \beta_{13} + 384376 \beta_{12} - 106202 \beta_{11} + 125194 \beta_{10} + 49061 \beta_{9} - 33588 \beta_{8} + 7572 \beta_{7} + 7572 \beta_{6} + 42642 \beta_{5} - 196235 \beta_{4} - 177243 \beta_{3} + 130466 \beta_{2} + 21321 \beta_{1} + 192188$$ $$\nu^{12}$$ $$=$$ $$-141291 \beta_{19} + 679267 \beta_{18} - 149061 \beta_{17} - 149061 \beta_{16} - 232152 \beta_{15} - 537976 \beta_{14} - 537976 \beta_{13} - 301661 \beta_{11} - 227989 \beta_{10} + 63924 \beta_{9} - 85137 \beta_{8} - 256121 \beta_{7} + 256121 \beta_{6} - 306112 \beta_{4} + 379784 \beta_{3} + 232152 \beta_{2} - 51335 \beta_{1} + 1279681$$ $$\nu^{13}$$ $$=$$ $$-578836 \beta_{19} - 900656 \beta_{18} - 204786 \beta_{17} + 204786 \beta_{16} - 1899968 \beta_{15} + 321820 \beta_{14} - 321820 \beta_{13} - 5372220 \beta_{12} + 1680346 \beta_{11} - 2002166 \beta_{10} - 699558 \beta_{9} + 494772 \beta_{8} - 122758 \beta_{7} - 122758 \beta_{6} - 430204 \beta_{5} + 2606667 \beta_{4} + 2284847 \beta_{3} - 1899968 \beta_{2} - 215102 \beta_{1} - 2686110$$ $$\nu^{14}$$ $$=$$ $$1992900 \beta_{19} - 9676477 \beta_{18} + 2308476 \beta_{17} + 2308476 \beta_{16} + 2994721 \beta_{15} + 7683577 \beta_{14} + 7683577 \beta_{13} + 4982352 \beta_{11} + 3288217 \beta_{10} - 836676 \beta_{9} + 1471800 \beta_{8} + 3399157 \beta_{7} - 3399157 \beta_{6} + 3915121 \beta_{4} - 5609256 \beta_{3} - 2994721 \beta_{2} + 539772 \beta_{1} - 17694713$$ $$\nu^{15}$$ $$=$$ $$8433696 \beta_{19} + 13655280 \beta_{18} + 2859385 \beta_{17} - 2859385 \beta_{16} + 27567590 \beta_{15} - 5221584 \beta_{14} + 5221584 \beta_{13} + 75422000 \beta_{12} - 25830552 \beta_{11} + 31052136 \beta_{10} + 10045285 \beta_{9} - 7185900 \beta_{8} + 1892220 \beta_{7} + 1892220 \beta_{6} + 4132202 \beta_{5} - 35414506 \beta_{4} - 30192922 \beta_{3} + 27567590 \beta_{2} + 2066101 \beta_{1} + 37711000$$ $$\nu^{16}$$ $$=$$ $$-28294599 \beta_{19} + 138146536 \beta_{18} - 34870729 \beta_{17} - 34870729 \beta_{16} - 39444923 \beta_{15} - 109851937 \beta_{14} - 109851937 \beta_{13} - 78842209 \beta_{11} - 47880118 \beta_{10} + 11154710 \beta_{9} - 23716019 \beta_{8} - 46186800 \beta_{7} + 46186800 \beta_{6} - 50908225 \beta_{4} + 81870316 \beta_{3} + 39444923 \beta_{2} - 5061721 \beta_{1} + 247305396$$ $$\nu^{17}$$ $$=$$ $$-121462470 \beta_{19} - 203639060 \beta_{18} - 41125079 \beta_{17} + 41125079 \beta_{16} - 399127106 \beta_{15} + 82176590 \beta_{14} - 82176590 \beta_{13} - 1063737088 \beta_{12} + 389821850 \beta_{11} - 471998440 \beta_{10} - 144905081 \beta_{9} + 103780002 \beta_{8} - 28485816 \beta_{7} - 28485816 \beta_{6} - 36896838 \beta_{5} + 488888063 \beta_{4} + 406711473 \beta_{3} - 399127106 \beta_{2} - 18448419 \beta_{1} - 531868544$$ $$\nu^{18}$$ $$=$$ $$403541787 \beta_{19} - 1976053159 \beta_{18} + 518481447 \beta_{17} + 518481447 \beta_{16} + 528939894 \beta_{15} + 1572511372 \beta_{14} + 1572511372 \beta_{13} + 1213273331 \beta_{11} + 698641747 \beta_{10} - 151207974 \beta_{9} + 367273473 \beta_{8} + 638207825 \beta_{7} - 638207825 \beta_{6} + 672481396 \beta_{4} - 1187112980 \beta_{3} - 528939894 \beta_{2} + 38527355 \beta_{1} - 3482371615$$ $$\nu^{19}$$ $$=$$ $$1739867998 \beta_{19} + 3004912406 \beta_{18} + 599680470 \beta_{17} - 599680470 \beta_{16} + 5770381364 \beta_{15} - 1265044408 \beta_{14} + 1265044408 \beta_{13} + 15064234992 \beta_{12} - 5809688350 \beta_{11} + 7074732758 \beta_{10} + 2095342362 \beta_{9} - 1495661892 \beta_{8} + 423488572 \beta_{7} + 423488572 \beta_{6} + 286345084 \beta_{5} - 6826036659 \beta_{4} - 5560992251 \beta_{3} + 5770381364 \beta_{2} + 143172542 \beta_{1} + 7532117496$$

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$\beta_{12}$$ $$1$$ $$-\beta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
205.1
 2.55339i 1.91536i − 0.0521119i − 2.62249i − 3.79415i 2.99764i 1.77962i 1.05091i − 0.508531i − 3.31964i − 2.99764i − 1.77962i − 1.05091i 0.508531i 3.31964i − 2.55339i − 1.91536i 0.0521119i 2.62249i 3.79415i
1.00000i −0.500000 + 0.866025i −1.00000 −2.21130 1.27669i 0.866025 + 0.500000i 1.74192 1.99141i 1.00000i −0.500000 0.866025i −1.27669 + 2.21130i
205.2 1.00000i −0.500000 + 0.866025i −1.00000 −1.65875 0.957680i 0.866025 + 0.500000i −2.64520 0.0538842i 1.00000i −0.500000 0.866025i −0.957680 + 1.65875i
205.3 1.00000i −0.500000 + 0.866025i −1.00000 0.0451302 + 0.0260560i 0.866025 + 0.500000i −1.11788 + 2.39799i 1.00000i −0.500000 0.866025i 0.0260560 0.0451302i
205.4 1.00000i −0.500000 + 0.866025i −1.00000 2.27114 + 1.31124i 0.866025 + 0.500000i −0.116897 2.64317i 1.00000i −0.500000 0.866025i 1.31124 2.27114i
205.5 1.00000i −0.500000 + 0.866025i −1.00000 3.28583 + 1.89707i 0.866025 + 0.500000i 2.13806 + 1.55842i 1.00000i −0.500000 0.866025i 1.89707 3.28583i
205.6 1.00000i −0.500000 + 0.866025i −1.00000 −2.59603 1.49882i −0.866025 0.500000i 2.50724 0.844840i 1.00000i −0.500000 0.866025i 1.49882 2.59603i
205.7 1.00000i −0.500000 + 0.866025i −1.00000 −1.54119 0.889808i −0.866025 0.500000i −2.51386 + 0.824914i 1.00000i −0.500000 0.866025i 0.889808 1.54119i
205.8 1.00000i −0.500000 + 0.866025i −1.00000 −0.910115 0.525455i −0.866025 0.500000i 0.963812 + 2.46395i 1.00000i −0.500000 0.866025i 0.525455 0.910115i
205.9 1.00000i −0.500000 + 0.866025i −1.00000 0.440400 + 0.254265i −0.866025 0.500000i −0.212907 2.63717i 1.00000i −0.500000 0.866025i −0.254265 + 0.440400i
205.10 1.00000i −0.500000 + 0.866025i −1.00000 2.87489 + 1.65982i −0.866025 0.500000i −0.744278 2.53891i 1.00000i −0.500000 0.866025i −1.65982 + 2.87489i
277.1 1.00000i −0.500000 0.866025i −1.00000 −2.59603 + 1.49882i −0.866025 + 0.500000i 2.50724 + 0.844840i 1.00000i −0.500000 + 0.866025i 1.49882 + 2.59603i
277.2 1.00000i −0.500000 0.866025i −1.00000 −1.54119 + 0.889808i −0.866025 + 0.500000i −2.51386 0.824914i 1.00000i −0.500000 + 0.866025i 0.889808 + 1.54119i
277.3 1.00000i −0.500000 0.866025i −1.00000 −0.910115 + 0.525455i −0.866025 + 0.500000i 0.963812 2.46395i 1.00000i −0.500000 + 0.866025i 0.525455 + 0.910115i
277.4 1.00000i −0.500000 0.866025i −1.00000 0.440400 0.254265i −0.866025 + 0.500000i −0.212907 + 2.63717i 1.00000i −0.500000 + 0.866025i −0.254265 0.440400i
277.5 1.00000i −0.500000 0.866025i −1.00000 2.87489 1.65982i −0.866025 + 0.500000i −0.744278 + 2.53891i 1.00000i −0.500000 + 0.866025i −1.65982 2.87489i
277.6 1.00000i −0.500000 0.866025i −1.00000 −2.21130 + 1.27669i 0.866025 0.500000i 1.74192 + 1.99141i 1.00000i −0.500000 + 0.866025i −1.27669 2.21130i
277.7 1.00000i −0.500000 0.866025i −1.00000 −1.65875 + 0.957680i 0.866025 0.500000i −2.64520 + 0.0538842i 1.00000i −0.500000 + 0.866025i −0.957680 1.65875i
277.8 1.00000i −0.500000 0.866025i −1.00000 0.0451302 0.0260560i 0.866025 0.500000i −1.11788 2.39799i 1.00000i −0.500000 + 0.866025i 0.0260560 + 0.0451302i
277.9 1.00000i −0.500000 0.866025i −1.00000 2.27114 1.31124i 0.866025 0.500000i −0.116897 + 2.64317i 1.00000i −0.500000 + 0.866025i 1.31124 + 2.27114i
277.10 1.00000i −0.500000 0.866025i −1.00000 3.28583 1.89707i 0.866025 0.500000i 2.13806 1.55842i 1.00000i −0.500000 + 0.866025i 1.89707 + 3.28583i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 277.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bm.b yes 20
3.b odd 2 1 1638.2.dt.b 20
7.c even 3 1 546.2.bd.b 20
13.e even 6 1 546.2.bd.b 20
21.h odd 6 1 1638.2.cr.b 20
39.h odd 6 1 1638.2.cr.b 20
91.k even 6 1 inner 546.2.bm.b yes 20
273.bp odd 6 1 1638.2.dt.b 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bd.b 20 7.c even 3 1
546.2.bd.b 20 13.e even 6 1
546.2.bm.b yes 20 1.a even 1 1 trivial
546.2.bm.b yes 20 91.k even 6 1 inner
1638.2.cr.b 20 21.h odd 6 1
1638.2.cr.b 20 39.h odd 6 1
1638.2.dt.b 20 3.b odd 2 1
1638.2.dt.b 20 273.bp odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{20} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{10}$$
$3$ $$( 1 + T + T^{2} )^{10}$$
$5$ $$576 - 19296 T + 215448 T^{2} + 804 T^{3} - 631751 T^{4} + 10242 T^{5} + 1613768 T^{6} + 1609092 T^{7} + 250999 T^{8} - 392970 T^{9} - 139756 T^{10} + 72234 T^{11} + 40612 T^{12} - 5544 T^{13} - 5416 T^{14} + 318 T^{15} + 523 T^{16} - 28 T^{18} + T^{20}$$
$7$ $$282475249 + 57648010 T^{2} + 9882516 T^{3} + 5445468 T^{5} - 533022 T^{6} + 615342 T^{7} + 144060 T^{8} - 29022 T^{9} + 49779 T^{10} - 4146 T^{11} + 2940 T^{12} + 1794 T^{13} - 222 T^{14} + 324 T^{15} + 12 T^{17} + 10 T^{18} + T^{20}$$
$11$ $$11664 + 723168 T + 15541308 T^{2} + 36941832 T^{3} - 14750019 T^{4} - 116538912 T^{5} + 48769533 T^{6} + 390045456 T^{7} + 367260615 T^{8} + 42239232 T^{9} - 34276266 T^{10} - 6188778 T^{11} + 2388915 T^{12} + 468126 T^{13} - 95586 T^{14} - 18708 T^{15} + 3190 T^{16} + 456 T^{17} - 64 T^{18} - 6 T^{19} + T^{20}$$
$13$ $$137858491849 - 84835994984 T + 11420230094 T^{2} + 1602500588 T^{4} - 810903912 T^{5} + 216206770 T^{6} - 57223062 T^{7} + 14248052 T^{8} - 4797546 T^{9} + 1668859 T^{10} - 369042 T^{11} + 84308 T^{12} - 26046 T^{13} + 7570 T^{14} - 2184 T^{15} + 332 T^{16} + 14 T^{18} - 8 T^{19} + T^{20}$$
$17$ $$( 445776 + 881112 T + 257197 T^{2} - 268226 T^{3} - 76649 T^{4} + 18082 T^{5} + 5020 T^{6} - 458 T^{7} - 122 T^{8} + 4 T^{9} + T^{10} )^{2}$$
$19$ $$82791801 + 1229347692 T + 5973925365 T^{2} - 1645210116 T^{3} - 3337203564 T^{4} + 825292152 T^{5} + 1411678503 T^{6} - 31219992 T^{7} - 257424021 T^{8} - 10173600 T^{9} + 34053192 T^{10} + 4198140 T^{11} - 2388557 T^{12} - 434136 T^{13} + 116047 T^{14} + 33324 T^{15} - 324 T^{16} - 708 T^{17} - 11 T^{18} + 12 T^{19} + T^{20}$$
$23$ $$( -1526664 + 1284756 T + 392821 T^{2} - 321580 T^{3} - 55277 T^{4} + 24454 T^{5} + 3506 T^{6} - 746 T^{7} - 98 T^{8} + 8 T^{9} + T^{10} )^{2}$$
$29$ $$4446755856 + 2587072464 T + 6714550380 T^{2} + 1794205188 T^{3} + 6423437485 T^{4} + 1983931810 T^{5} + 2139531093 T^{6} + 308852330 T^{7} + 342008423 T^{8} + 27630990 T^{9} + 39039346 T^{10} + 651712 T^{11} + 2904741 T^{12} - 43622 T^{13} + 158578 T^{14} - 6930 T^{15} + 5486 T^{16} - 280 T^{17} + 126 T^{18} - 8 T^{19} + T^{20}$$
$31$ $$6038441240976 - 25465592637360 T + 36524591326332 T^{2} - 3063271642020 T^{3} - 5158616187483 T^{4} + 674889618888 T^{5} + 489337228113 T^{6} - 88008407760 T^{7} - 25169147237 T^{8} + 5728307004 T^{9} + 924777306 T^{10} - 273220776 T^{11} - 15569895 T^{12} + 7809312 T^{13} + 92294 T^{14} - 161766 T^{15} + 6030 T^{16} + 1656 T^{17} - 90 T^{18} - 12 T^{19} + T^{20}$$
$37$ $$222697212379849 + 162576562917472 T^{2} + 42593770820658 T^{4} + 5383670071644 T^{6} + 382705170207 T^{8} + 16509800310 T^{10} + 446221662 T^{12} + 7548246 T^{14} + 77025 T^{16} + 430 T^{18} + T^{20}$$
$41$ $$15776364816 - 436474904832 T + 4929779521980 T^{2} - 25025697537984 T^{3} + 46265965595325 T^{4} + 34959339742266 T^{5} + 6375258384279 T^{6} - 810043458114 T^{7} - 333795736931 T^{8} + 13258773096 T^{9} + 14162455032 T^{10} + 992259666 T^{11} - 209598111 T^{12} - 25835160 T^{13} + 2387210 T^{14} + 498858 T^{15} + 3336 T^{16} - 3348 T^{17} - 78 T^{18} + 18 T^{19} + T^{20}$$
$43$ $$237795695449 + 1016911206480 T + 3887472618188 T^{2} + 4132519425596 T^{3} + 5514682195653 T^{4} - 2006149690316 T^{5} + 4577886138092 T^{6} - 298438108128 T^{7} + 336705745006 T^{8} - 18166707864 T^{9} + 17632467897 T^{10} - 829265814 T^{11} + 437160679 T^{12} - 29923080 T^{13} + 8002964 T^{14} - 476222 T^{15} + 73323 T^{16} - 3970 T^{17} + 461 T^{18} - 18 T^{19} + T^{20}$$
$47$ $$20736 - 1804032 T + 64358064 T^{2} - 1047578832 T^{3} + 5693001705 T^{4} + 26012916126 T^{5} + 32256219024 T^{6} + 917593272 T^{7} - 5679268272 T^{8} - 293847768 T^{9} + 1006914654 T^{10} + 290645124 T^{11} + 7840231 T^{12} - 6894894 T^{13} - 453848 T^{14} + 137592 T^{15} + 15801 T^{16} - 930 T^{17} - 143 T^{18} + 6 T^{19} + T^{20}$$
$53$ $$274366440000 - 1949133132000 T + 15644025509400 T^{2} + 11332271801340 T^{3} + 16735444716681 T^{4} - 2867168914776 T^{5} + 2354297583006 T^{6} - 359979131448 T^{7} + 180906291486 T^{8} - 28187517942 T^{9} + 8599199922 T^{10} - 1044152802 T^{11} + 238360081 T^{12} - 25186344 T^{13} + 4582646 T^{14} - 380958 T^{15} + 52911 T^{16} - 3570 T^{17} + 401 T^{18} - 18 T^{19} + T^{20}$$
$59$ $$149267013030144 + 146991608856096 T^{2} + 49012427697745 T^{4} + 7704751805348 T^{6} + 639356701402 T^{8} + 29618626232 T^{10} + 791383507 T^{12} + 12376328 T^{14} + 110986 T^{16} + 524 T^{18} + T^{20}$$
$61$ $$979103150309376 - 1099787832852480 T + 1060413047341056 T^{2} - 419723503239168 T^{3} + 187530918322176 T^{4} - 51291111223296 T^{5} + 18200735631360 T^{6} - 4008649506816 T^{7} + 1052098661632 T^{8} - 161769683712 T^{9} + 33347501952 T^{10} - 3885048896 T^{11} + 709217136 T^{12} - 54647856 T^{13} + 8266236 T^{14} - 271620 T^{15} + 63429 T^{16} - 722 T^{17} + 339 T^{18} + 6 T^{19} + T^{20}$$
$67$ $$7928165051579536 + 11030653395993552 T + 4814637678874156 T^{2} - 418932389139924 T^{3} - 603040426475283 T^{4} + 4299829411782 T^{5} + 48548722721427 T^{6} - 2070697121562 T^{7} - 1786780129122 T^{8} + 109224463080 T^{9} + 46778253759 T^{10} - 4437882102 T^{11} - 589747665 T^{12} + 74932338 T^{13} + 5296335 T^{14} - 924420 T^{15} - 10242 T^{16} + 5592 T^{17} - 41 T^{18} - 24 T^{19} + T^{20}$$
$71$ $$39520987511184 + 128828683146672 T + 145736054578188 T^{2} + 18752704358868 T^{3} - 18184552336935 T^{4} - 3555915624576 T^{5} + 1799694719568 T^{6} + 548010578052 T^{7} - 42847957449 T^{8} - 27459410886 T^{9} + 321785424 T^{10} + 1010784312 T^{11} + 62001648 T^{12} - 17183394 T^{13} - 1557888 T^{14} + 218658 T^{15} + 26743 T^{16} - 1248 T^{17} - 196 T^{18} + 6 T^{19} + T^{20}$$
$73$ $$60267010291489 - 86841325045632 T + 32280697159660 T^{2} + 13588719392256 T^{3} - 8761200752052 T^{4} - 1991567797566 T^{5} + 1726468062312 T^{6} + 72314700540 T^{7} - 115709101224 T^{8} - 920469048 T^{9} + 5503562811 T^{10} - 224271192 T^{11} - 143584188 T^{12} + 9836364 T^{13} + 2676936 T^{14} - 303186 T^{15} - 16164 T^{16} + 3072 T^{17} + 64 T^{18} - 24 T^{19} + T^{20}$$
$79$ $$3052036952064 - 5370498256896 T + 8494374764736 T^{2} - 4685774101056 T^{3} + 3303114058953 T^{4} - 914034629826 T^{5} + 708924429612 T^{6} - 155318524812 T^{7} + 84433026483 T^{8} - 13614741630 T^{9} + 6568478568 T^{10} - 929428722 T^{11} + 278927496 T^{12} - 20808252 T^{13} + 5302260 T^{14} - 343674 T^{15} + 66339 T^{16} - 1932 T^{17} + 288 T^{18} + T^{20}$$
$83$ $$1010456721763575696 + 228715934125009992 T^{2} + 21326993360852929 T^{4} + 1078661212938182 T^{6} + 32921213441245 T^{8} + 637061161628 T^{10} + 7959084046 T^{12} + 63689870 T^{14} + 313564 T^{16} + 860 T^{18} + T^{20}$$
$89$ $$122705435257742736 + 31444010928906408 T^{2} + 3469872462023601 T^{4} + 216292357987632 T^{6} + 8394517985796 T^{8} + 210717030870 T^{10} + 3438638230 T^{12} + 35674120 T^{14} + 222417 T^{16} + 742 T^{18} + T^{20}$$
$97$ $$26392935453786729 + 58716345411481122 T + 16429854831338583 T^{2} - 60316529011346682 T^{3} + 29710167544881315 T^{4} - 3871309155902568 T^{5} - 732244620906468 T^{6} + 192807832814514 T^{7} + 13703055941259 T^{8} - 4800119544072 T^{9} - 280180487907 T^{10} + 78548350338 T^{11} + 6983452492 T^{12} - 623209398 T^{13} - 99348902 T^{14} + 689358 T^{15} + 968157 T^{16} + 88512 T^{17} + 3994 T^{18} + 96 T^{19} + T^{20}$$