# Properties

 Label 546.2.bk.c Level $546$ Weight $2$ Character orbit 546.bk Analytic conductor $4.360$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bk (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: $$x^{20} - 26 x^{18} + 431 x^{16} - 4370 x^{14} + 32381 x^{12} - 160412 x^{10} + 573820 x^{8} - 1203000 x^{6} + 1802736 x^{4} - 1479600 x^{2} + 810000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 26 x^{18} + 431 x^{16} - 4370 x^{14} + 32381 x^{12} - 160412 x^{10} + 573820 x^{8} - 1203000 x^{6} + 1802736 x^{4} - 1479600 x^{2} + 810000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$28019 \nu^{18} - 1353739 \nu^{16} + 26963794 \nu^{14} - 360314635 \nu^{12} + 3156128704 \nu^{10} - 20133182803 \nu^{8} + 83994238835 \nu^{6} - 237015654210 \nu^{4} + 297462412584 \nu^{2} - 250163334900$$$$)/ 50836113540$$ $$\beta_{3}$$ $$=$$ $$($$$$-44475185183309 \nu^{18} + 1111041078951634 \nu^{16} - 18084966383986879 \nu^{14} + 177039174291601030 \nu^{12} - 1277486247684027829 \nu^{10} + 5995841565690639808 \nu^{8} - 20576343745927842080 \nu^{6} + 37596543499851389100 \nu^{4} - 62198599614489017424 \nu^{2} + 50611365044085146400$$$$)/ 29788674347322896400$$ $$\beta_{4}$$ $$=$$ $$($$$$55035689845889 \nu^{18} - 1659354637685389 \nu^{16} + 29371204163944159 \nu^{14} - 326343275679756655 \nu^{12} + 2555747319795309709 \nu^{10} - 13673533059435858643 \nu^{8} + 50009087939965713230 \nu^{6} - 110374051077279571800 \nu^{4} + 127629265491923145804 \nu^{2} - 120812927249833066800$$$$)/ 14894337173661448200$$ $$\beta_{5}$$ $$=$$ $$($$$$-77692557753089 \nu^{18} + 2201273852695039 \nu^{16} - 38029896643673809 \nu^{14} + 407675637548107105 \nu^{12} - 3125003239762471459 \nu^{10} + 16145736567415122793 \nu^{8} - 57962640077800382180 \nu^{6} + 119363259988343929800 \nu^{4} - 135226324968492570804 \nu^{2} + 49459354207104522600$$$$)/ 14894337173661448200$$ $$\beta_{6}$$ $$=$$ $$($$$$33725347069901 \nu^{18} - 1050450717606226 \nu^{16} + 18602646637357231 \nu^{14} - 212361949031617270 \nu^{12} + 1699751992786405981 \nu^{10} - 9682130275915869712 \nu^{8} + 37905793462148708720 \nu^{6} - 99091515201679004700 \nu^{4} + 128260555625267837136 \nu^{2} - 106914810142896069600$$$$)/ 4964779057887149400$$ $$\beta_{7}$$ $$=$$ $$($$$$4648903677221 \nu^{18} - 113351516709967 \nu^{16} + 1811111136343567 \nu^{14} - 17177764265330041 \nu^{12} + 119069448244709965 \nu^{10} - 517102936163945677 \nu^{8} + 1560063479999681954 \nu^{6} - 1880244205907612040 \nu^{4} + 1589052741356731500 \nu^{2} + 347795170261852680$$$$)/ 595773486946457928$$ $$\beta_{8}$$ $$=$$ $$($$$$-322032565057271 \nu^{19} + 11859524449404796 \nu^{17} - 223809673072159051 \nu^{15} + 2765615661557949520 \nu^{13} - 23311059688117023001 \nu^{11} + 140959974009499429402 \nu^{9} - 572615928604122502970 \nu^{7} + 1557452785763658478500 \nu^{5} - 1990722852631793523456 \nu^{3} + 1668268939276286796600 \nu$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$44475185183309 \nu^{19} - 1111041078951634 \nu^{17} + 18084966383986879 \nu^{15} - 177039174291601030 \nu^{13} + 1277486247684027829 \nu^{11} - 5995841565690639808 \nu^{9} + 20576343745927842080 \nu^{7} - 37596543499851389100 \nu^{5} + 62198599614489017424 \nu^{3} - 50611365044085146400 \nu$$$$)/ 29788674347322896400$$ $$\beta_{10}$$ $$=$$ $$($$$$391520578290986 \nu^{19} - 13663579712282986 \nu^{17} + 255963980887869691 \nu^{15} - 3125402797329623920 \nu^{13} + 26310114572005382041 \nu^{11} - 158463344445233229832 \nu^{9} + 642899897243505005345 \nu^{7} - 1741811077249390971000 \nu^{5} + 2147122284551663438196 \nu^{3} - 2028007326225526886700 \nu$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$4118545123555 \nu^{18} - 99803503973213 \nu^{16} + 1594643307283613 \nu^{14} - 15078157122141251 \nu^{12} + 104838016251566135 \nu^{10} - 455297700833150903 \nu^{8} + 1392716921970335452 \nu^{6} - 1655513446327009560 \nu^{4} + 1399125800772778500 \nu^{2} + 641380419567834588$$$$)/ 148943371736614482$$ $$\beta_{12}$$ $$=$$ $$($$$$-1362283753645301 \nu^{19} - 416452024229730 \nu^{18} + 31187621397299176 \nu^{17} + 15714733167592230 \nu^{16} - 476098479545611831 \nu^{15} - 303950377849931130 \nu^{14} + 4133044441562248720 \nu^{13} + 3764662764445615350 \nu^{12} - 26033959536042673681 \nu^{11} - 32009754135684243630 \nu^{10} + 89558798367488156362 \nu^{9} + 191064641800777445010 \nu^{8} - 185219467494945898520 \nu^{7} - 782402826477440228100 \nu^{6} - 282627389544845872500 \nu^{5} + 2078415493939380735000 \nu^{4} + 594817423688173891464 \nu^{3} - 2961087135199821674280 \nu^{2} - 1476521828552909476800 \nu + 2343882760893636534000$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$1362283753645301 \nu^{19} - 416452024229730 \nu^{18} - 31187621397299176 \nu^{17} + 15714733167592230 \nu^{16} + 476098479545611831 \nu^{15} - 303950377849931130 \nu^{14} - 4133044441562248720 \nu^{13} + 3764662764445615350 \nu^{12} + 26033959536042673681 \nu^{11} - 32009754135684243630 \nu^{10} - 89558798367488156362 \nu^{9} + 191064641800777445010 \nu^{8} + 185219467494945898520 \nu^{7} - 782402826477440228100 \nu^{6} + 282627389544845872500 \nu^{5} + 2078415493939380735000 \nu^{4} - 594817423688173891464 \nu^{3} - 2961087135199821674280 \nu^{2} + 1476521828552909476800 \nu + 2343882760893636534000$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-92653087 \nu^{19} + 2400574562 \nu^{17} - 39527358797 \nu^{15} + 396804851990 \nu^{13} - 2892105219647 \nu^{11} + 13915828380644 \nu^{9} - 47126239541440 \nu^{7} + 86263392010500 \nu^{5} - 95924359183032 \nu^{3} + 47850783750000 \nu$$$$)/ 15250834062000$$ $$\beta_{15}$$ $$=$$ $$($$$$2998690871693449 \nu^{19} - 967658731840290 \nu^{18} - 76144716304401374 \nu^{17} + 28231091725823790 \nu^{16} + 1238839244655528719 \nu^{15} - 500944928333240490 \nu^{14} - 12156194106854016830 \nu^{13} + 5612779152631765050 \nu^{12} + 86712676767524493569 \nu^{11} - 45027385294991528490 \nu^{10} - 400084998044185615688 \nu^{9} + 249816636957105838230 \nu^{8} + 1296401186626868280580 \nu^{7} - 962278691674444731300 \nu^{6} - 2063027087120276338500 \nu^{5} + 2320112869313354712000 \nu^{4} + 2099488471871128366464 \nu^{3} - 2990045771704525060440 \nu^{2} - 261479454950447913600 \nu + 2624775274346553558000$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-1710663097424611 \nu^{19} + 46554198801770861 \nu^{17} - 781484664185157566 \nu^{15} + 8135775445241496545 \nu^{13} - 60859876066885217666 \nu^{11} + 308508409540763641457 \nu^{9} - 1093297994336058932095 \nu^{7} + 2287391743187547516000 \nu^{5} - 2608209440207473875696 \nu^{3} + 1438560026068388384700 \nu$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$1762133072739407 \nu^{19} - 46972217177414332 \nu^{17} + 784263510726092167 \nu^{15} - 8096518537167816340 \nu^{13} + 60644058024402744817 \nu^{11} - 308701664864232066934 \nu^{9} + 1124211059940359939990 \nu^{7} - 2511309937318507354500 \nu^{5} + 3587773588810568547552 \nu^{3} - 2954696334905271472200 \nu$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{18}$$ $$=$$ $$($$$$-1832909324721542 \nu^{19} - 483829365920145 \nu^{18} + 46405166244337942 \nu^{17} + 14115545862911895 \nu^{16} - 755056870207665952 \nu^{15} - 250472464166620245 \nu^{14} + 7405890860614016140 \nu^{13} + 2806389576315882525 \nu^{12} - 52937485241392455502 \nu^{11} - 22513692647495764245 \nu^{10} + 245011310764772606404 \nu^{9} + 124908318478552919115 \nu^{8} - 802523171407892955890 \nu^{7} - 481139345837222365650 \nu^{6} + 1313487619809023587500 \nu^{5} + 1160056434656677356000 \nu^{4} - 1516233733044231813912 \nu^{3} - 1495022885852262530220 \nu^{2} + 510324965305862554800 \nu + 1312387637173276779000$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{19}$$ $$=$$ $$($$$$41250852113590 \nu^{19} - 993597754957391 \nu^{17} + 15875534896050191 \nu^{15} - 149769573612355541 \nu^{13} + 1043719042266298445 \nu^{11} - 4532734376806038221 \nu^{9} + 14002554840084527197 \nu^{7} - 16481529987301654920 \nu^{5} + 13929052580397499500 \nu^{3} + 22129027378582254444 \nu$$$$)/ 4468301152098434460$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{13} + \beta_{12} - \beta_{7} + \beta_{4} - 4 \beta_{3} + 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{19} - \beta_{17} + \beta_{13} - \beta_{12} + 2 \beta_{10} + 6 \beta_{9} + 2 \beta_{8} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$10 \beta_{13} + 10 \beta_{12} - 10 \beta_{7} + \beta_{6} - 10 \beta_{5} - 26 \beta_{3} - 2 \beta_{2} - 10$$ $$\nu^{5}$$ $$=$$ $$11 \beta_{18} - 9 \beta_{17} - 24 \beta_{16} - 11 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + 57 \beta_{9} + 30 \beta_{8} + 2 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-15 \beta_{11} + 36 \beta_{7} - 88 \beta_{5} - 88 \beta_{4} - 378$$ $$\nu^{7}$$ $$=$$ $$73 \beta_{19} + 139 \beta_{18} - 242 \beta_{16} - 139 \beta_{15} - 114 \beta_{14} - 139 \beta_{13} + 139 \beta_{12} - 242 \beta_{10} + 139 \beta_{9} + 114 \beta_{8} - 312 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-6 \beta_{18} - 6 \beta_{15} - 760 \beta_{13} - 760 \beta_{12} - 169 \beta_{11} - 6 \beta_{9} + 1212 \beta_{7} - 169 \beta_{6} - 772 \beta_{4} + 1658 \beta_{3} + 452 \beta_{2} - 2430$$ $$\nu^{9}$$ $$=$$ $$603 \beta_{19} + 476 \beta_{18} + 603 \beta_{17} - 476 \beta_{15} - 1494 \beta_{14} - 923 \beta_{13} + 923 \beta_{12} - 2322 \beta_{10} - 2864 \beta_{9} - 2322 \beta_{8} - 2864 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$156 \beta_{18} + 156 \beta_{15} - 6712 \beta_{13} - 6712 \beta_{12} + 156 \beta_{9} + 6868 \beta_{7} - 1719 \beta_{6} + 6868 \beta_{5} + 13942 \beta_{3} + 5244 \beta_{2} + 6868$$ $$\nu^{11}$$ $$=$$ $$-8119 \beta_{18} + 5149 \beta_{17} + 21794 \beta_{16} + 8119 \beta_{15} + 5556 \beta_{13} - 5556 \beta_{12} - 37915 \beta_{9} - 38708 \beta_{8} - 5556 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$5172 \beta_{18} + 5172 \beta_{15} - 2586 \beta_{13} - 2586 \beta_{12} + 16645 \beta_{11} + 5172 \beta_{9} - 45032 \beta_{7} + 61888 \beta_{5} + 61888 \beta_{4} + 5172 \beta_{2} + 242914$$ $$\nu^{13}$$ $$=$$ $$-45243 \beta_{19} - 131323 \beta_{18} + 202098 \beta_{16} + 131323 \beta_{15} + 177870 \beta_{14} + 131323 \beta_{13} - 131323 \beta_{12} + 202098 \beta_{10} - 131323 \beta_{9} - 177870 \beta_{8} + 149076 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$35016 \beta_{18} + 35016 \beta_{15} + 492724 \beta_{13} + 492724 \beta_{12} + 156855 \beta_{11} + 35016 \beta_{9} - 984304 \beta_{7} + 156855 \beta_{6} + 562756 \beta_{4} - 1031110 \beta_{3} - 491580 \beta_{2} + 1593866$$ $$\nu^{15}$$ $$=$$ $$-405901 \beta_{19} - 631644 \beta_{18} - 405901 \beta_{17} + 631644 \beta_{15} + 1792674 \beta_{14} + 614563 \beta_{13} - 614563 \beta_{12} + 1860770 \beta_{10} + 1842912 \beta_{9} + 1860770 \beta_{8} + 1842912 \beta_{1}$$ $$\nu^{16}$$ $$=$$ $$-422982 \beta_{18} - 422982 \beta_{15} + 4724146 \beta_{13} + 4724146 \beta_{12} - 422982 \beta_{9} - 5147128 \beta_{7} + 1454869 \beta_{6} - 5147128 \beta_{5} - 9018098 \beta_{3} - 5548376 \beta_{2} - 5147128$$ $$\nu^{17}$$ $$=$$ $$5333051 \beta_{18} - 3692259 \beta_{17} - 17060442 \beta_{16} - 5333051 \beta_{15} - 6394340 \beta_{13} + 6394340 \beta_{12} + 28130007 \beta_{9} + 34652280 \beta_{8} + 6394340 \beta_{1}$$ $$\nu^{18}$$ $$=$$ $$-9507096 \beta_{18} - 9507096 \beta_{15} + 4753548 \beta_{13} + 4753548 \beta_{12} - 13368183 \beta_{11} - 9507096 \beta_{9} + 34821108 \beta_{7} - 47241916 \beta_{5} - 47241916 \beta_{4} - 9507096 \beta_{2} - 174041982$$ $$\nu^{19}$$ $$=$$ $$33873733 \beta_{19} + 109691851 \beta_{18} - 156041306 \beta_{16} - 109691851 \beta_{15} - 169590642 \beta_{14} - 109691851 \beta_{13} + 109691851 \beta_{12} - 156041306 \beta_{10} + 109691851 \beta_{9} + 169590642 \beta_{8} - 83963220 \beta_{1}$$

## Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1 + \beta_{3}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −2.38129 − 1.37484i −2.18313 − 1.26043i −0.935588 − 0.540162i 1.14595 + 0.661615i 2.62200 + 1.51381i −2.62200 − 1.51381i −1.14595 − 0.661615i 0.935588 + 0.540162i 2.18313 + 1.26043i 2.38129 + 1.37484i −2.38129 + 1.37484i −2.18313 + 1.26043i −0.935588 + 0.540162i 1.14595 − 0.661615i 2.62200 − 1.51381i −2.62200 + 1.51381i −1.14595 + 0.661615i 0.935588 − 0.540162i 2.18313 − 1.26043i 2.38129 − 1.37484i
−0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i −2.38129 + 1.37484i 1.00000i −0.588218 2.57953i 1.00000i −0.500000 0.866025i 1.37484 2.38129i
25.2 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i −2.18313 + 1.26043i 1.00000i 1.47350 + 2.19745i 1.00000i −0.500000 0.866025i 1.26043 2.18313i
25.3 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i −0.935588 + 0.540162i 1.00000i −1.55401 + 2.14127i 1.00000i −0.500000 0.866025i 0.540162 0.935588i
25.4 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 1.14595 0.661615i 1.00000i −2.62831 0.303302i 1.00000i −0.500000 0.866025i −0.661615 + 1.14595i
25.5 −0.866025 + 0.500000i 0.500000 0.866025i 0.500000 0.866025i 2.62200 1.51381i 1.00000i 2.43101 + 1.04411i 1.00000i −0.500000 0.866025i −1.51381 + 2.62200i
25.6 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −2.62200 + 1.51381i 1.00000i −2.43101 1.04411i 1.00000i −0.500000 0.866025i −1.51381 + 2.62200i
25.7 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i −1.14595 + 0.661615i 1.00000i 2.62831 + 0.303302i 1.00000i −0.500000 0.866025i −0.661615 + 1.14595i
25.8 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i 0.935588 0.540162i 1.00000i 1.55401 2.14127i 1.00000i −0.500000 0.866025i 0.540162 0.935588i
25.9 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i 2.18313 1.26043i 1.00000i −1.47350 2.19745i 1.00000i −0.500000 0.866025i 1.26043 2.18313i
25.10 0.866025 0.500000i 0.500000 0.866025i 0.500000 0.866025i 2.38129 1.37484i 1.00000i 0.588218 + 2.57953i 1.00000i −0.500000 0.866025i 1.37484 2.38129i
415.1 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −2.38129 1.37484i 1.00000i −0.588218 + 2.57953i 1.00000i −0.500000 + 0.866025i 1.37484 + 2.38129i
415.2 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −2.18313 1.26043i 1.00000i 1.47350 2.19745i 1.00000i −0.500000 + 0.866025i 1.26043 + 2.18313i
415.3 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −0.935588 0.540162i 1.00000i −1.55401 2.14127i 1.00000i −0.500000 + 0.866025i 0.540162 + 0.935588i
415.4 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 1.14595 + 0.661615i 1.00000i −2.62831 + 0.303302i 1.00000i −0.500000 + 0.866025i −0.661615 1.14595i
415.5 −0.866025 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 2.62200 + 1.51381i 1.00000i 2.43101 1.04411i 1.00000i −0.500000 + 0.866025i −1.51381 2.62200i
415.6 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −2.62200 1.51381i 1.00000i −2.43101 + 1.04411i 1.00000i −0.500000 + 0.866025i −1.51381 2.62200i
415.7 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i −1.14595 0.661615i 1.00000i 2.62831 0.303302i 1.00000i −0.500000 + 0.866025i −0.661615 1.14595i
415.8 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.935588 + 0.540162i 1.00000i 1.55401 + 2.14127i 1.00000i −0.500000 + 0.866025i 0.540162 + 0.935588i
415.9 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 2.18313 + 1.26043i 1.00000i −1.47350 + 2.19745i 1.00000i −0.500000 + 0.866025i 1.26043 + 2.18313i
415.10 0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 + 0.866025i 2.38129 + 1.37484i 1.00000i 0.588218 2.57953i 1.00000i −0.500000 + 0.866025i 1.37484 + 2.38129i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r 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.bk.c 20
3.b odd 2 1 1638.2.dm.e 20
7.c even 3 1 inner 546.2.bk.c 20
7.c even 3 1 3822.2.c.m 10
7.d odd 6 1 3822.2.c.n 10
13.b even 2 1 inner 546.2.bk.c 20
21.h odd 6 1 1638.2.dm.e 20
39.d odd 2 1 1638.2.dm.e 20
91.r even 6 1 inner 546.2.bk.c 20
91.r even 6 1 3822.2.c.m 10
91.s odd 6 1 3822.2.c.n 10
273.w odd 6 1 1638.2.dm.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.c 20 1.a even 1 1 trivial
546.2.bk.c 20 7.c even 3 1 inner
546.2.bk.c 20 13.b even 2 1 inner
546.2.bk.c 20 91.r even 6 1 inner
1638.2.dm.e 20 3.b odd 2 1
1638.2.dm.e 20 21.h odd 6 1
1638.2.dm.e 20 39.d odd 2 1
1638.2.dm.e 20 273.w odd 6 1
3822.2.c.m 10 7.c even 3 1
3822.2.c.m 10 91.r even 6 1
3822.2.c.n 10 7.d odd 6 1
3822.2.c.n 10 91.s 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

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{5}$$
$3$ $$( 1 - T + T^{2} )^{10}$$
$5$ $$810000 - 1479600 T^{2} + 1802736 T^{4} - 1203000 T^{6} + 573820 T^{8} - 160412 T^{10} + 32381 T^{12} - 4370 T^{14} + 431 T^{16} - 26 T^{18} + T^{20}$$
$7$ $$282475249 - 5764801 T^{2} + 352947 T^{4} - 840350 T^{6} + 34349 T^{8} + 8745 T^{10} + 701 T^{12} - 350 T^{14} + 3 T^{16} - T^{18} + T^{20}$$
$11$ $$31640625 - 123305625 T^{2} + 354653991 T^{4} - 463840638 T^{6} + 448256305 T^{8} - 49404427 T^{10} + 3711825 T^{12} - 157034 T^{14} + 4851 T^{16} - 85 T^{18} + T^{20}$$
$13$ $$( 371293 - 57122 T + 41743 T^{2} - 8788 T^{3} - 208 T^{4} - 724 T^{5} - 16 T^{6} - 52 T^{7} + 19 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$17$ $$( 2916 - 8748 T + 37854 T^{2} + 40338 T^{3} + 37801 T^{4} + 11991 T^{5} + 3408 T^{6} + 277 T^{7} + 60 T^{8} + 3 T^{9} + T^{10} )^{2}$$
$19$ $$207360000 - 243993600 T^{2} + 191699136 T^{4} - 81294000 T^{6} + 24711025 T^{8} - 4865779 T^{10} + 694806 T^{12} - 58775 T^{14} + 3414 T^{16} - 67 T^{18} + T^{20}$$
$23$ $$( 944784 - 104976 T + 239112 T^{2} + 64152 T^{3} + 44820 T^{4} + 7380 T^{5} + 2380 T^{6} + 308 T^{7} + 84 T^{8} + 8 T^{9} + T^{10} )^{2}$$
$29$ $$( 3969 + 567 T - 408 T^{2} - 64 T^{3} + 7 T^{4} + T^{5} )^{4}$$
$31$ $$1358954496 - 2179989504 T^{2} + 2693136384 T^{4} - 1126772736 T^{6} + 341934592 T^{8} - 38512432 T^{10} + 3032289 T^{12} - 137522 T^{14} + 4515 T^{16} - 82 T^{18} + T^{20}$$
$37$ $$2821109907456 - 3683115712512 T^{2} + 4093983378432 T^{4} - 874325788416 T^{6} + 142364357136 T^{8} - 6361498944 T^{10} + 199304064 T^{12} - 3330936 T^{14} + 40176 T^{16} - 240 T^{18} + T^{20}$$
$41$ $$( 4671995904 + 279444480 T^{2} + 6613924 T^{4} + 77256 T^{6} + 444 T^{8} + T^{10} )^{2}$$
$43$ $$( -1712 + 1112 T + 236 T^{2} - 66 T^{3} - 6 T^{4} + T^{5} )^{4}$$
$47$ $$3662186256 - 32972021568 T^{2} + 279673222716 T^{4} - 153162143736 T^{6} + 73568399173 T^{8} - 3469156159 T^{10} + 111394986 T^{12} - 2014835 T^{14} + 26622 T^{16} - 199 T^{18} + T^{20}$$
$53$ $$( 6395841 + 1191159 T + 1203093 T^{2} - 20892 T^{3} + 137797 T^{4} + 4583 T^{5} + 4821 T^{6} + 424 T^{7} + 153 T^{8} + 11 T^{9} + T^{10} )^{2}$$
$59$ $$2620828195700625 - 1764560824296525 T^{2} + 1073633490820251 T^{4} - 72298491657858 T^{6} + 3381721250481 T^{8} - 77998422651 T^{10} + 1280634961 T^{12} - 12599286 T^{14} + 89903 T^{16} - 369 T^{18} + T^{20}$$
$61$ $$( 12376324 + 7071180 T + 8694414 T^{2} - 3637234 T^{3} + 1446313 T^{4} - 215555 T^{5} + 30592 T^{6} - 1673 T^{7} + 188 T^{8} - 7 T^{9} + T^{10} )^{2}$$
$67$ $$982045424460816 - 4363258701831840 T^{2} + 19214042111950716 T^{4} - 759589890982992 T^{6} + 19356444653229 T^{8} - 295892316735 T^{10} + 3305019058 T^{12} - 24842131 T^{14} + 136998 T^{16} - 463 T^{18} + T^{20}$$
$71$ $$( 1913187600 + 195097896 T^{2} + 5966865 T^{4} + 78003 T^{6} + 459 T^{8} + T^{10} )^{2}$$
$73$ $$167961600000000 - 43752130560000 T^{2} + 7610913156096 T^{4} - 708667575552 T^{6} + 46962462736 T^{8} - 1979787472 T^{10} + 61038624 T^{12} - 1257512 T^{14} + 18876 T^{16} - 172 T^{18} + T^{20}$$
$79$ $$( 3154176 + 3125760 T + 5150656 T^{2} - 2599328 T^{3} + 1052944 T^{4} - 192620 T^{5} + 29353 T^{6} - 1994 T^{7} + 163 T^{8} - 2 T^{9} + T^{10} )^{2}$$
$83$ $$( 5817744 + 14479056 T^{2} + 2815908 T^{4} + 65089 T^{6} + 482 T^{8} + T^{10} )^{2}$$
$89$ $$36901294867020960000 - 2654583194115590400 T^{2} + 124336902279266496 T^{4} - 3346666880301504 T^{6} + 64801596203280 T^{8} - 811832311968 T^{10} + 7460763280 T^{12} - 46157096 T^{14} + 208140 T^{16} - 572 T^{18} + T^{20}$$
$97$ $$( 15148194084 + 2090930604 T^{2} + 40638568 T^{4} + 291069 T^{6} + 894 T^{8} + T^{10} )^{2}$$