# Properties

 Label 177.3.c.a Level $177$ Weight $3$ Character orbit 177.c Analytic conductor $4.823$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 177.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.82290067918$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + 5962230 x^{6} + 6517782 x^{4} + 3428244 x^{2} + 570861$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{8} q^{3} + ( -2 + \beta_{2} ) q^{4} -\beta_{6} q^{5} + \beta_{9} q^{6} + \beta_{14} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{8} q^{3} + ( -2 + \beta_{2} ) q^{4} -\beta_{6} q^{5} + \beta_{9} q^{6} + \beta_{14} q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} + ( \beta_{1} - \beta_{3} + \beta_{7} ) q^{10} + ( \beta_{7} + \beta_{10} ) q^{11} + ( -1 + 2 \beta_{8} + \beta_{15} ) q^{12} + ( \beta_{1} + \beta_{9} + \beta_{11} ) q^{13} + ( -2 \beta_{1} - \beta_{10} + \beta_{13} + \beta_{19} ) q^{14} + ( 1 - \beta_{16} ) q^{15} + ( -\beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{12} ) q^{16} + ( 1 + \beta_{2} - \beta_{14} + \beta_{16} + \beta_{18} ) q^{17} + 3 \beta_{1} q^{18} + ( -2 - \beta_{2} + 2 \beta_{5} - 2 \beta_{6} - \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{19} + ( -8 + 2 \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{12} + \beta_{16} - \beta_{18} ) q^{20} + ( \beta_{5} + \beta_{14} - \beta_{18} ) q^{21} + ( 3 - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} + 2 \beta_{8} - 2 \beta_{12} + 3 \beta_{14} - \beta_{15} - 2 \beta_{18} ) q^{22} + ( -2 \beta_{1} - \beta_{4} - \beta_{10} - \beta_{13} ) q^{23} + ( -\beta_{3} - \beta_{9} + \beta_{19} ) q^{24} + ( 4 - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} - 4 \beta_{8} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} - 2 \beta_{16} + 2 \beta_{18} ) q^{25} + ( -8 + 2 \beta_{2} - \beta_{5} + \beta_{6} + 7 \beta_{8} + \beta_{12} - \beta_{14} + 3 \beta_{15} - 2 \beta_{16} + \beta_{18} ) q^{26} -3 \beta_{8} q^{27} + ( 10 - 4 \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{12} - 3 \beta_{14} + 2 \beta_{18} ) q^{28} + ( -4 + 3 \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{14} - 2 \beta_{15} + \beta_{16} ) q^{29} + ( \beta_{1} + \beta_{4} + \beta_{9} - \beta_{11} - \beta_{19} ) q^{30} + ( \beta_{1} + 2 \beta_{3} - \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{17} - \beta_{19} ) q^{31} + ( -\beta_{1} + \beta_{3} + \beta_{4} - 4 \beta_{9} + \beta_{10} - \beta_{13} ) q^{32} + ( \beta_{1} + \beta_{4} - \beta_{9} + \beta_{10} - \beta_{17} ) q^{33} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{17} ) q^{34} + ( -1 - 3 \beta_{5} + \beta_{6} + 5 \beta_{8} - \beta_{12} + 2 \beta_{14} - \beta_{15} + 2 \beta_{16} - 2 \beta_{18} ) q^{35} + ( -6 + 3 \beta_{2} ) q^{36} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{19} ) q^{37} + ( -4 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{7} + \beta_{13} + \beta_{19} ) q^{38} + ( 4 \beta_{1} - \beta_{7} + 2 \beta_{9} - \beta_{11} + \beta_{13} ) q^{39} + ( -9 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{17} + \beta_{19} ) q^{40} + ( \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} + 2 \beta_{16} ) q^{41} + ( -\beta_{1} + \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{13} + \beta_{17} + \beta_{19} ) q^{42} + ( -2 \beta_{1} + \beta_{10} + \beta_{13} - 2 \beta_{17} ) q^{43} + ( 4 \beta_{1} - \beta_{4} - 3 \beta_{7} - 2 \beta_{10} + \beta_{13} + 2 \beta_{17} + 2 \beta_{19} ) q^{44} -3 \beta_{6} q^{45} + ( 9 - 5 \beta_{2} + 5 \beta_{5} - \beta_{6} + \beta_{12} + 2 \beta_{14} - 2 \beta_{16} ) q^{46} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{7} + 6 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{17} - 2 \beta_{19} ) q^{47} + ( -6 + 3 \beta_{2} - 2 \beta_{5} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{48} + ( 13 - 4 \beta_{2} + \beta_{5} - 3 \beta_{6} - 4 \beta_{8} + \beta_{12} + \beta_{14} - 4 \beta_{15} + 2 \beta_{16} - 2 \beta_{18} ) q^{49} + ( 9 \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{7} + 6 \beta_{9} + 3 \beta_{10} - 2 \beta_{11} - 2 \beta_{17} - 3 \beta_{19} ) q^{50} + ( -2 + 3 \beta_{6} - 2 \beta_{8} + \beta_{12} - 3 \beta_{14} + 2 \beta_{15} + \beta_{18} ) q^{51} + ( -7 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} - 8 \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{17} ) q^{52} + ( -2 - \beta_{2} + \beta_{5} + \beta_{6} - 6 \beta_{8} + 3 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} - 2 \beta_{16} - \beta_{18} ) q^{53} + 3 \beta_{9} q^{54} + ( 14 \beta_{1} + \beta_{3} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - \beta_{17} - 5 \beta_{19} ) q^{55} + ( 14 \beta_{1} - 4 \beta_{3} + \beta_{4} + 2 \beta_{10} - 2 \beta_{17} + \beta_{19} ) q^{56} + ( 3 \beta_{8} + 3 \beta_{12} + 2 \beta_{15} - 2 \beta_{16} ) q^{57} + ( -7 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{7} + 9 \beta_{9} + \beta_{11} - 2 \beta_{13} - 3 \beta_{19} ) q^{58} + ( -9 - 5 \beta_{1} + 7 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + 3 \beta_{15} - \beta_{18} - \beta_{19} ) q^{59} + ( -3 + 3 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} + 7 \beta_{8} - \beta_{12} + \beta_{15} + 2 \beta_{16} - \beta_{18} ) q^{60} + ( -7 \beta_{1} + \beta_{4} - \beta_{7} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} + 3 \beta_{13} + 2 \beta_{17} - \beta_{19} ) q^{61} + ( -3 \beta_{2} + \beta_{5} - 3 \beta_{6} - 12 \beta_{8} + \beta_{12} - 4 \beta_{14} + 2 \beta_{15} + 5 \beta_{18} ) q^{62} + 3 \beta_{14} q^{63} + ( 11 - 6 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} - 10 \beta_{8} - 2 \beta_{12} + 2 \beta_{14} - 6 \beta_{15} + 2 \beta_{16} ) q^{64} + ( 4 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{7} - 4 \beta_{9} - \beta_{10} - 4 \beta_{11} + 2 \beta_{13} - 3 \beta_{19} ) q^{65} + ( -6 + 3 \beta_{2} - 3 \beta_{5} - 2 \beta_{8} - \beta_{12} + 3 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} - 4 \beta_{18} ) q^{66} + ( -8 \beta_{1} - 5 \beta_{3} + \beta_{4} + 2 \beta_{7} - 4 \beta_{9} + \beta_{10} - 3 \beta_{13} - \beta_{19} ) q^{67} + ( 22 - 5 \beta_{2} + 5 \beta_{5} - 3 \beta_{6} + \beta_{8} - 3 \beta_{12} + 9 \beta_{14} - 3 \beta_{15} - \beta_{16} - 3 \beta_{18} ) q^{68} + ( \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{17} - \beta_{19} ) q^{69} + ( \beta_{3} - 5 \beta_{4} - 4 \beta_{7} - 2 \beta_{9} - 7 \beta_{10} + 2 \beta_{11} + \beta_{13} + 2 \beta_{17} + 3 \beta_{19} ) q^{70} + ( 5 - 7 \beta_{2} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 3 \beta_{12} - \beta_{14} + 6 \beta_{15} + 2 \beta_{18} ) q^{71} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{72} + ( 12 \beta_{1} - \beta_{7} + 3 \beta_{9} + 4 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{17} - \beta_{19} ) q^{73} + ( 3 + 7 \beta_{2} + 2 \beta_{5} - 7 \beta_{6} - 8 \beta_{8} + 4 \beta_{12} + \beta_{14} - 2 \beta_{15} - 5 \beta_{16} + \beta_{18} ) q^{74} + ( 12 - 3 \beta_{2} + 4 \beta_{5} - 6 \beta_{6} - 3 \beta_{8} - \beta_{12} - 2 \beta_{14} - 3 \beta_{15} + 3 \beta_{16} + \beta_{18} ) q^{75} + ( 7 + 4 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{14} - 5 \beta_{15} + 6 \beta_{16} + 2 \beta_{18} ) q^{76} + ( -10 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{7} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{17} + 5 \beta_{19} ) q^{77} + ( -23 + 6 \beta_{2} + 2 \beta_{5} - 6 \beta_{6} + 7 \beta_{8} - \beta_{12} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{78} + ( -21 + 3 \beta_{2} + 4 \beta_{5} + 2 \beta_{6} - 16 \beta_{8} + \beta_{14} - 4 \beta_{15} + 6 \beta_{16} - 2 \beta_{18} ) q^{79} + ( 17 - 2 \beta_{2} + \beta_{5} + 8 \beta_{6} - 11 \beta_{8} + \beta_{12} - 4 \beta_{14} + \beta_{15} - 4 \beta_{16} + 2 \beta_{18} ) q^{80} + 9 q^{81} + ( 4 \beta_{1} + 4 \beta_{7} - 6 \beta_{9} + 5 \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{82} + ( 8 \beta_{1} + 4 \beta_{3} + \beta_{4} - 2 \beta_{7} - 8 \beta_{9} + 4 \beta_{11} + \beta_{19} ) q^{83} + ( 5 + 3 \beta_{2} - 3 \beta_{5} - 10 \beta_{8} + 4 \beta_{12} - 9 \beta_{14} + \beta_{16} + 4 \beta_{18} ) q^{84} + ( -19 + 3 \beta_{2} - 4 \beta_{5} + 26 \beta_{8} + 2 \beta_{12} - 4 \beta_{14} + 4 \beta_{15} - 4 \beta_{16} + 2 \beta_{18} ) q^{85} + ( 13 - 3 \beta_{2} + 6 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} - 6 \beta_{12} + 7 \beta_{14} - 2 \beta_{15} + \beta_{16} - 9 \beta_{18} ) q^{86} + ( 7 - 6 \beta_{2} - 2 \beta_{5} + 3 \beta_{6} + 5 \beta_{8} - 2 \beta_{12} - 2 \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{87} + ( -12 + \beta_{2} - 4 \beta_{5} - 2 \beta_{6} - 6 \beta_{8} + \beta_{12} - 3 \beta_{14} - 3 \beta_{15} - 8 \beta_{16} + 6 \beta_{18} ) q^{88} + ( -14 \beta_{1} + 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} + 10 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - \beta_{13} - 2 \beta_{17} - 3 \beta_{19} ) q^{89} + ( 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{7} ) q^{90} + ( -8 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - \beta_{7} + 3 \beta_{9} - 4 \beta_{10} + \beta_{11} + 3 \beta_{13} - \beta_{17} + 4 \beta_{19} ) q^{91} + ( 8 \beta_{1} - 5 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} ) q^{92} + ( -9 \beta_{1} - 4 \beta_{3} - \beta_{4} - \beta_{7} + 2 \beta_{9} - 3 \beta_{10} + \beta_{13} + 2 \beta_{19} ) q^{93} + ( 10 + \beta_{5} - 5 \beta_{6} + 28 \beta_{8} + 3 \beta_{12} - 6 \beta_{14} + 10 \beta_{15} + 4 \beta_{16} + 8 \beta_{18} ) q^{94} + ( 36 - 9 \beta_{2} - 4 \beta_{5} + 15 \beta_{6} - 14 \beta_{8} - 4 \beta_{14} - 4 \beta_{15} - 5 \beta_{16} ) q^{95} + ( -10 \beta_{1} - \beta_{4} + 3 \beta_{7} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{17} + 2 \beta_{19} ) q^{96} + ( -12 \beta_{1} + 5 \beta_{3} + 3 \beta_{4} - 4 \beta_{7} - 10 \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{13} - \beta_{19} ) q^{97} + ( 25 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{7} + 14 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{17} - \beta_{19} ) q^{98} + ( 3 \beta_{7} + 3 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 40q^{4} - 8q^{7} + 60q^{9} + O(q^{10})$$ $$20q - 40q^{4} - 8q^{7} + 60q^{9} - 24q^{12} + 24q^{15} - 8q^{16} + 16q^{17} - 60q^{19} - 164q^{20} + 40q^{22} + 100q^{25} - 156q^{26} + 200q^{28} - 60q^{29} - 32q^{35} - 120q^{36} + 28q^{41} + 180q^{46} - 96q^{48} + 284q^{49} - 24q^{51} - 8q^{53} + 24q^{57} - 152q^{59} - 72q^{60} - 8q^{62} - 24q^{63} + 204q^{64} - 120q^{66} + 384q^{68} + 92q^{71} + 104q^{74} + 240q^{75} + 120q^{76} - 468q^{78} - 420q^{79} + 376q^{80} + 180q^{81} + 168q^{84} - 348q^{85} + 232q^{86} + 144q^{87} - 212q^{88} + 152q^{94} + 788q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + 5962230 x^{6} + 6517782 x^{4} + 3428244 x^{2} + 570861$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 6$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 9 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$24659 \nu^{19} + 525507 \nu^{17} - 8932411 \nu^{15} - 381955669 \nu^{13} - 4643278510 \nu^{11} - 26218006538 \nu^{9} - 69374899519 \nu^{7} - 65989625577 \nu^{5} + 29168824437 \nu^{3} + 66292360677 \nu$$$$)/ 1272772800$$ $$\beta_{5}$$ $$=$$ $$($$$$-30094 \nu^{18} - 1782837 \nu^{16} - 43396174 \nu^{14} - 556908571 \nu^{12} - 4008507490 \nu^{10} - 15799048592 \nu^{8} - 30267136996 \nu^{6} - 19599735393 \nu^{4} + 4066529058 \nu^{2} + 3502798593$$$$)/ 424257600$$ $$\beta_{6}$$ $$=$$ $$($$$$-224699 \nu^{18} - 12416727 \nu^{16} - 286167029 \nu^{14} - 3569199791 \nu^{12} - 26102535290 \nu^{10} - 113015025382 \nu^{8} - 277256257841 \nu^{6} - 345908981403 \nu^{4} - 180467025957 \nu^{2} - 27178002297$$$$)/ 2545545600$$ $$\beta_{7}$$ $$=$$ $$($$$$224699 \nu^{19} + 12416727 \nu^{17} + 286167029 \nu^{15} + 3569199791 \nu^{13} + 26102535290 \nu^{11} + 113015025382 \nu^{9} + 277256257841 \nu^{7} + 345908981403 \nu^{5} + 183012571557 \nu^{3} + 47542367097 \nu$$$$)/ 2545545600$$ $$\beta_{8}$$ $$=$$ $$($$$$101093 \nu^{18} + 5880439 \nu^{16} + 142869803 \nu^{14} + 1879285587 \nu^{12} + 14501957730 \nu^{10} + 66552078674 \nu^{8} + 176366820787 \nu^{6} + 251067857371 \nu^{4} + 163972337499 \nu^{2} + 31714629129$$$$)/ 848515200$$ $$\beta_{9}$$ $$=$$ $$($$$$-101093 \nu^{19} - 5880439 \nu^{17} - 142869803 \nu^{15} - 1879285587 \nu^{13} - 14501957730 \nu^{11} - 66552078674 \nu^{9} - 176366820787 \nu^{7} - 251067857371 \nu^{5} - 163972337499 \nu^{3} - 31714629129 \nu$$$$)/ 848515200$$ $$\beta_{10}$$ $$=$$ $$($$$$-533161 \nu^{19} - 29389353 \nu^{17} - 671121631 \nu^{15} - 8215396849 \nu^{13} - 58270361110 \nu^{11} - 242014514498 \nu^{9} - 571953485299 \nu^{7} - 738822733317 \nu^{5} - 525855487023 \nu^{3} - 204166271583 \nu$$$$)/ 2545545600$$ $$\beta_{11}$$ $$=$$ $$($$$$288077 \nu^{19} + 17226546 \nu^{17} + 434018867 \nu^{15} + 5992261568 \nu^{13} + 49357376120 \nu^{11} + 247369943086 \nu^{9} + 737013687893 \nu^{7} + 1217773094394 \nu^{5} + 963279349011 \nu^{3} + 252139008006 \nu$$$$)/ 1272772800$$ $$\beta_{12}$$ $$=$$ $$($$$$650693 \nu^{18} + 37002339 \nu^{16} + 883008803 \nu^{14} + 11503461887 \nu^{12} + 89063236730 \nu^{10} + 417533205874 \nu^{8} + 1153854360587 \nu^{6} + 1732172167671 \nu^{4} + 1155608132499 \nu^{2} + 182480565429$$$$)/ 2545545600$$ $$\beta_{13}$$ $$=$$ $$($$$$243091 \nu^{19} + 14075643 \nu^{17} + 341817061 \nu^{15} + 4524039619 \nu^{13} + 35488858210 \nu^{11} + 168058138838 \nu^{9} + 468566188369 \nu^{7} + 713155586127 \nu^{5} + 487322342613 \nu^{3} + 69057391773 \nu$$$$)/ 848515200$$ $$\beta_{14}$$ $$=$$ $$($$$$-455827 \nu^{18} - 27087171 \nu^{16} - 671585917 \nu^{14} - 8996141443 \nu^{12} - 70454145970 \nu^{10} - 326343085286 \nu^{8} - 864618265393 \nu^{6} - 1207876631319 \nu^{4} - 746157263661 \nu^{2} - 127003783581$$$$)/ 1272772800$$ $$\beta_{15}$$ $$=$$ $$($$$$-421417 \nu^{18} - 24288891 \nu^{16} - 584638807 \nu^{14} - 7622076103 \nu^{12} - 58375869370 \nu^{10} - 266635464506 \nu^{8} - 706529064703 \nu^{6} - 1011477345999 \nu^{4} - 668977183431 \nu^{2} - 131729208501$$$$)/ 848515200$$ $$\beta_{16}$$ $$=$$ $$($$$$-447991 \nu^{18} - 24915743 \nu^{16} - 575077561 \nu^{14} - 7131169119 \nu^{12} - 51379348410 \nu^{10} - 217347655438 \nu^{8} - 521670314669 \nu^{6} - 657771247427 \nu^{4} - 373332993513 \nu^{2} - 64472580273$$$$)/ 848515200$$ $$\beta_{17}$$ $$=$$ $$($$$$596207 \nu^{19} + 34251811 \nu^{17} + 822056297 \nu^{15} + 10690895763 \nu^{13} + 81679588770 \nu^{11} + 371406113726 \nu^{9} + 970768221613 \nu^{7} + 1328785187479 \nu^{5} + 763909813401 \nu^{3} + 86604679221 \nu$$$$)/ 848515200$$ $$\beta_{18}$$ $$=$$ $$($$$$-2097083 \nu^{18} - 119728059 \nu^{16} - 2851123493 \nu^{14} - 36718884347 \nu^{12} - 277206592130 \nu^{10} - 1243217830294 \nu^{8} - 3207001892297 \nu^{6} - 4379269314351 \nu^{4} - 2634572355669 \nu^{2} - 421529033349$$$$)/ 2545545600$$ $$\beta_{19}$$ $$=$$ $$($$$$-90587 \nu^{19} - 5241276 \nu^{17} - 126656027 \nu^{15} - 1657491608 \nu^{13} - 12735217820 \nu^{11} - 58286462566 \nu^{9} - 154453690883 \nu^{7} - 220585114764 \nu^{5} - 145005710091 \nu^{3} - 27510621786 \nu$$$$)/ 106064400$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 9 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{12} + 2 \beta_{8} - \beta_{6} + \beta_{5} - 13 \beta_{2} + 56$$ $$\nu^{5}$$ $$=$$ $$-\beta_{13} + \beta_{10} - 4 \beta_{9} + \beta_{4} - 15 \beta_{3} + 95 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{16} - 6 \beta_{15} + 2 \beta_{14} + 18 \beta_{12} - 50 \beta_{8} + 16 \beta_{6} - 22 \beta_{5} + 158 \beta_{2} - 597$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{19} + 20 \beta_{13} + 2 \beta_{11} - 24 \beta_{10} + 106 \beta_{9} + 2 \beta_{7} - 24 \beta_{4} + 198 \beta_{3} - 1075 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-2 \beta_{18} - 48 \beta_{16} + 158 \beta_{15} - 38 \beta_{14} - 266 \beta_{12} + 950 \beta_{8} - 208 \beta_{6} + 362 \beta_{5} - 1941 \beta_{2} + 6756$$ $$\nu^{9}$$ $$=$$ $$72 \beta_{19} + 2 \beta_{17} - 304 \beta_{13} - 48 \beta_{11} + 398 \beta_{10} - 2004 \beta_{9} - 60 \beta_{7} + 410 \beta_{4} - 2573 \beta_{3} + 12663 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$104 \beta_{18} + 794 \beta_{16} - 2944 \beta_{15} + 470 \beta_{14} + 3739 \beta_{12} - 15914 \beta_{8} + 2589 \beta_{6} - 5355 \beta_{5} + 24235 \beta_{2} - 79374$$ $$\nu^{11}$$ $$=$$ $$-1680 \beta_{19} - 104 \beta_{17} + 4209 \beta_{13} + 794 \beta_{11} - 5721 \beta_{10} + 33046 \beta_{9} + 1254 \beta_{7} - 6149 \beta_{4} + 33507 \beta_{3} - 153405 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-2984 \beta_{18} - 11276 \beta_{16} + 47820 \beta_{15} - 4256 \beta_{14} - 51676 \beta_{12} + 248084 \beta_{8} - 31872 \beta_{6} + 75380 \beta_{5} - 306844 \beta_{2} + 958629$$ $$\nu^{13}$$ $$=$$ $$32288 \beta_{19} + 2984 \beta_{17} - 55932 \beta_{13} - 11276 \beta_{11} + 76652 \beta_{10} - 507324 \beta_{9} - 22788 \beta_{7} + 86656 \beta_{4} - 438212 \beta_{3} + 1897953 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$65740 \beta_{18} + 147808 \beta_{16} - 724052 \beta_{15} + 19600 \beta_{14} + 709288 \beta_{12} - 3698012 \beta_{8} + 390356 \beta_{6} - 1034040 \beta_{5} + 3928549 \beta_{2} - 11827082$$ $$\nu^{15}$$ $$=$$ $$-556644 \beta_{19} - 65740 \beta_{17} + 728888 \beta_{13} + 147808 \beta_{11} - 987900 \beta_{10} + 7465688 \beta_{9} + 384672 \beta_{7} - 1181848 \beta_{4} + 5752245 \beta_{3} - 23863777 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$-1250416 \beta_{18} - 1846204 \beta_{16} + 10525640 \beta_{15} + 286768 \beta_{14} - 9701529 \beta_{12} + 53525778 \beta_{8} - 4759917 \beta_{6} + 13984329 \beta_{5} - 50739657 \beta_{2} + 148356592$$ $$\nu^{17}$$ $$=$$ $$8966204 \beta_{19} + 1250416 \beta_{17} - 9414761 \beta_{13} - 1846204 \beta_{11} + 12447145 \beta_{10} - 106942352 \beta_{9} - 6192028 \beta_{7} + 15830533 \beta_{4} - 75726743 \beta_{3} + 303804087 \beta_{1}$$ $$\nu^{18}$$ $$=$$ $$21697080 \beta_{18} + 22328214 \beta_{16} - 149178658 \beta_{15} - 11158094 \beta_{14} + 132391686 \beta_{12} - 759348102 \beta_{8} + 57801920 \beta_{6} - 187612562 \beta_{5} + 659833362 \beta_{2} - 1885103977$$ $$\nu^{19}$$ $$=$$ $$-138008538 \beta_{19} - 21697080 \beta_{17} + 121233592 \beta_{13} + 22328214 \beta_{11} - 154757388 \beta_{10} + 1504560110 \beta_{9} + 96286846 \beta_{7} - 209940776 \beta_{4} + 999205626 \beta_{3} - 3905050051 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$$.

 $$n$$ $$61$$ $$119$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
58.1
 − 3.65759i − 3.38526i − 3.07256i − 2.96307i − 2.72775i − 1.86235i − 1.61360i − 1.39995i − 1.08628i − 0.537675i 0.537675i 1.08628i 1.39995i 1.61360i 1.86235i 2.72775i 2.96307i 3.07256i 3.38526i 3.65759i
3.65759i −1.73205 −9.37798 0.395869 6.33514i −6.20847 19.6705i 3.00000 1.44793i
58.2 3.38526i 1.73205 −7.45997 9.27487 5.86344i −9.96030 11.7129i 3.00000 31.3978i
58.3 3.07256i 1.73205 −5.44061 −6.03179 5.32183i −9.30133 4.42637i 3.00000 18.5330i
58.4 2.96307i 1.73205 −4.77980 3.25089 5.13219i 11.8224 2.31061i 3.00000 9.63262i
58.5 2.72775i −1.73205 −3.44061 5.71516 4.72460i 8.69147 1.52587i 3.00000 15.5895i
58.6 1.86235i −1.73205 0.531642 −9.04540 3.22569i 1.29987 8.43952i 3.00000 16.8457i
58.7 1.61360i 1.73205 1.39629 −6.36659 2.79484i 6.66564 8.70746i 3.00000 10.2731i
58.8 1.39995i −1.73205 2.04015 0.273484 2.42478i −10.8938 8.45589i 3.00000 0.382863i
58.9 1.08628i 1.73205 2.82000 3.33671 1.88149i −1.22637 7.40842i 3.00000 3.62460i
58.10 0.537675i −1.73205 3.71091 −0.803210 0.931281i 5.11098 4.14596i 3.00000 0.431866i
58.11 0.537675i −1.73205 3.71091 −0.803210 0.931281i 5.11098 4.14596i 3.00000 0.431866i
58.12 1.08628i 1.73205 2.82000 3.33671 1.88149i −1.22637 7.40842i 3.00000 3.62460i
58.13 1.39995i −1.73205 2.04015 0.273484 2.42478i −10.8938 8.45589i 3.00000 0.382863i
58.14 1.61360i 1.73205 1.39629 −6.36659 2.79484i 6.66564 8.70746i 3.00000 10.2731i
58.15 1.86235i −1.73205 0.531642 −9.04540 3.22569i 1.29987 8.43952i 3.00000 16.8457i
58.16 2.72775i −1.73205 −3.44061 5.71516 4.72460i 8.69147 1.52587i 3.00000 15.5895i
58.17 2.96307i 1.73205 −4.77980 3.25089 5.13219i 11.8224 2.31061i 3.00000 9.63262i
58.18 3.07256i 1.73205 −5.44061 −6.03179 5.32183i −9.30133 4.42637i 3.00000 18.5330i
58.19 3.38526i 1.73205 −7.45997 9.27487 5.86344i −9.96030 11.7129i 3.00000 31.3978i
58.20 3.65759i −1.73205 −9.37798 0.395869 6.33514i −6.20847 19.6705i 3.00000 1.44793i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 58.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.3.c.a 20
3.b odd 2 1 531.3.c.c 20
59.b odd 2 1 inner 177.3.c.a 20
177.d even 2 1 531.3.c.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.3.c.a 20 1.a even 1 1 trivial
177.3.c.a 20 59.b odd 2 1 inner
531.3.c.c 20 3.b odd 2 1
531.3.c.c 20 177.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(177, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$570861 + 3428244 T^{2} + 6517782 T^{4} + 5962230 T^{6} + 3057025 T^{8} + 941588 T^{10} + 179593 T^{12} + 21286 T^{14} + 1522 T^{16} + 60 T^{18} + T^{20}$$
$3$ $$( -3 + T^{2} )^{10}$$
$5$ $$( 17368 - 93688 T + 65182 T^{2} + 191628 T^{3} - 95136 T^{4} - 7402 T^{5} + 6659 T^{6} + 54 T^{7} - 150 T^{8} + T^{10} )^{2}$$
$7$ $$( -34966256 + 1344176 T + 24344288 T^{2} - 1881452 T^{3} - 1483963 T^{4} + 82008 T^{5} + 32992 T^{6} - 1058 T^{7} - 308 T^{8} + 4 T^{9} + T^{10} )^{2}$$
$11$ $$21\!\cdots\!64$$$$+$$$$15\!\cdots\!92$$$$T^{2} + 10531291720390688232 T^{4} + 289175251270957944 T^{6} + 4251629142774373 T^{8} + 37444001597150 T^{10} + 207403924959 T^{12} + 729050892 T^{14} + 1578275 T^{16} + 1918 T^{18} + T^{20}$$
$13$ $$74249342697749076 + 9438920898713759448 T^{2} + 1343720508336647784 T^{4} + 69808489946223828 T^{6} + 1698690675911197 T^{8} + 21947540614750 T^{10} + 160953805727 T^{12} + 683525256 T^{14} + 1639431 T^{16} + 2030 T^{18} + T^{20}$$
$17$ $$( -139536176 + 175543536 T + 535260560 T^{2} + 125118476 T^{3} - 14345539 T^{4} - 2626790 T^{5} + 190838 T^{6} + 14580 T^{7} - 1074 T^{8} - 8 T^{9} + T^{10} )^{2}$$
$19$ $$( -726592156400 + 78018051920 T + 20555393124 T^{2} - 2033596512 T^{3} - 178170556 T^{4} + 14708444 T^{5} + 741617 T^{6} - 39038 T^{7} - 1513 T^{8} + 30 T^{9} + T^{10} )^{2}$$
$23$ $$13\!\cdots\!84$$$$+$$$$55\!\cdots\!28$$$$T^{2} +$$$$98\!\cdots\!72$$$$T^{4} + 97528187369536664832 T^{6} + 594545970389902096 T^{8} + 2318963636778328 T^{10} + 5827423674581 T^{12} + 9259830150 T^{14} + 8867727 T^{16} + 4622 T^{18} + T^{20}$$
$29$ $$( 14765383626976 - 4091986296992 T + 212037367534 T^{2} + 27838496048 T^{3} - 2667239570 T^{4} - 8475458 T^{5} + 5681991 T^{6} - 56400 T^{7} - 4031 T^{8} + 30 T^{9} + T^{10} )^{2}$$
$31$ $$2899712607570282816 + 57172612248279279648 T^{2} +$$$$20\!\cdots\!72$$$$T^{4} + 17297393662994364552 T^{6} + 492887486038552600 T^{8} + 5365602181023436 T^{10} + 18652028595125 T^{12} + 29138273862 T^{14} + 22286007 T^{16} + 7898 T^{18} + T^{20}$$
$37$ $$35\!\cdots\!24$$$$+$$$$13\!\cdots\!40$$$$T^{2} +$$$$35\!\cdots\!24$$$$T^{4} +$$$$33\!\cdots\!80$$$$T^{6} +$$$$12\!\cdots\!33$$$$T^{8} + 231131957654067664 T^{10} + 249314990959292 T^{12} + 159007532622 T^{14} + 59294976 T^{16} + 11936 T^{18} + T^{20}$$
$41$ $$( -103416495474416 + 6146968905504 T + 1558908734776 T^{2} - 66053629560 T^{3} - 7322777103 T^{4} + 159660940 T^{5} + 13506226 T^{6} - 30568 T^{7} - 6986 T^{8} - 14 T^{9} + T^{10} )^{2}$$
$43$ $$49\!\cdots\!84$$$$+$$$$13\!\cdots\!08$$$$T^{2} +$$$$87\!\cdots\!76$$$$T^{4} +$$$$26\!\cdots\!20$$$$T^{6} +$$$$41\!\cdots\!57$$$$T^{8} + 3940436221151054426 T^{10} + 2275046763140583 T^{12} + 813241440828 T^{14} + 174668363 T^{16} + 20506 T^{18} + T^{20}$$
$47$ $$63\!\cdots\!36$$$$+$$$$55\!\cdots\!32$$$$T^{2} +$$$$13\!\cdots\!08$$$$T^{4} +$$$$54\!\cdots\!72$$$$T^{6} +$$$$99\!\cdots\!72$$$$T^{8} + 9571241567268414656 T^{10} + 5238062152435653 T^{12} + 1639608854022 T^{14} + 289078187 T^{16} + 26662 T^{18} + T^{20}$$
$53$ $$( -7275670131459200 - 264424198764480 T + 53382200628046 T^{2} + 157475665032 T^{3} - 80837475516 T^{4} + 94666078 T^{5} + 48851959 T^{6} - 62458 T^{7} - 12470 T^{8} + 4 T^{9} + T^{10} )^{2}$$
$59$ $$26\!\cdots\!01$$$$+$$$$11\!\cdots\!92$$$$T +$$$$39\!\cdots\!82$$$$T^{2} +$$$$90\!\cdots\!44$$$$T^{3} +$$$$19\!\cdots\!65$$$$T^{4} +$$$$36\!\cdots\!40$$$$T^{5} +$$$$74\!\cdots\!60$$$$T^{6} +$$$$13\!\cdots\!20$$$$T^{7} +$$$$26\!\cdots\!34$$$$T^{8} +$$$$43\!\cdots\!80$$$$T^{9} + 7781753992934864756 T^{10} + 126179142543212480 T^{11} + 2196829919034594 T^{12} + 32861469591920 T^{13} + 508852605960 T^{14} + 7084877840 T^{15} + 109357965 T^{16} + 1459304 T^{17} + 18302 T^{18} + 152 T^{19} + T^{20}$$
$61$ $$71\!\cdots\!64$$$$+$$$$32\!\cdots\!40$$$$T^{2} +$$$$63\!\cdots\!24$$$$T^{4} +$$$$69\!\cdots\!96$$$$T^{6} +$$$$45\!\cdots\!00$$$$T^{8} +$$$$19\!\cdots\!04$$$$T^{10} + 51354057757150117 T^{12} + 8515661826814 T^{14} + 841833991 T^{16} + 45074 T^{18} + T^{20}$$
$67$ $$12\!\cdots\!00$$$$+$$$$70\!\cdots\!00$$$$T^{2} +$$$$12\!\cdots\!56$$$$T^{4} +$$$$10\!\cdots\!16$$$$T^{6} +$$$$47\!\cdots\!28$$$$T^{8} +$$$$12\!\cdots\!12$$$$T^{10} + 219093891162623381 T^{12} + 23744577768024 T^{14} + 1595382498 T^{16} + 60680 T^{18} + T^{20}$$
$71$ $$( -242741808356634 + 5014241770536 T + 8396764593174 T^{2} + 45307709514 T^{3} - 61275105165 T^{4} - 1458630294 T^{5} + 55021489 T^{6} + 578186 T^{7} - 14361 T^{8} - 46 T^{9} + T^{10} )^{2}$$
$73$ $$35\!\cdots\!16$$$$+$$$$10\!\cdots\!20$$$$T^{2} +$$$$13\!\cdots\!08$$$$T^{4} +$$$$92\!\cdots\!48$$$$T^{6} +$$$$40\!\cdots\!60$$$$T^{8} +$$$$11\!\cdots\!96$$$$T^{10} + 199313923109977037 T^{12} + 22826961864642 T^{14} + 1598636283 T^{16} + 61790 T^{18} + T^{20}$$
$79$ $$( -37223524784297456 - 24724419876669088 T - 1984690747543448 T^{2} - 12637689996320 T^{3} + 1232472545725 T^{4} + 17864880674 T^{5} - 171657777 T^{6} - 4158228 T^{7} - 7397 T^{8} + 210 T^{9} + T^{10} )^{2}$$
$83$ $$33\!\cdots\!96$$$$+$$$$76\!\cdots\!48$$$$T^{2} +$$$$47\!\cdots\!28$$$$T^{4} +$$$$50\!\cdots\!56$$$$T^{6} +$$$$18\!\cdots\!29$$$$T^{8} +$$$$24\!\cdots\!32$$$$T^{10} + 94054789608167452 T^{12} + 16219368709762 T^{14} + 1412869528 T^{16} + 60392 T^{18} + T^{20}$$
$89$ $$14\!\cdots\!16$$$$+$$$$71\!\cdots\!24$$$$T^{2} +$$$$13\!\cdots\!76$$$$T^{4} +$$$$12\!\cdots\!12$$$$T^{6} +$$$$65\!\cdots\!76$$$$T^{8} +$$$$20\!\cdots\!16$$$$T^{10} + 397811690058591829 T^{12} + 44368887587086 T^{14} + 2721427483 T^{16} + 83718 T^{18} + T^{20}$$
$97$ $$60\!\cdots\!36$$$$+$$$$63\!\cdots\!08$$$$T^{2} +$$$$12\!\cdots\!28$$$$T^{4} +$$$$68\!\cdots\!04$$$$T^{6} +$$$$17\!\cdots\!52$$$$T^{8} +$$$$27\!\cdots\!68$$$$T^{10} + 2574099252043990549 T^{12} + 153476771298280 T^{14} + 5613172642 T^{16} + 114744 T^{18} + T^{20}$$