# Properties

 Label 21.6.c.a Level $21$ Weight $6$ Character orbit 21.c Analytic conductor $3.368$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 21.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.36806021607$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 484 x^{10} + 194194 x^{8} - 39867800 x^{6} + 5398720873 x^{4} - 310089434788 x^{2} + 9371104623076$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} -\beta_{4} q^{3} + ( -16 - \beta_{1} ) q^{4} + \beta_{7} q^{5} + ( -\beta_{7} + \beta_{8} ) q^{6} + ( 9 - \beta_{2} + \beta_{4} ) q^{7} + ( 19 \beta_{5} + \beta_{10} ) q^{8} + ( -40 - 2 \beta_{1} + \beta_{2} - 7 \beta_{5} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{2} -\beta_{4} q^{3} + ( -16 - \beta_{1} ) q^{4} + \beta_{7} q^{5} + ( -\beta_{7} + \beta_{8} ) q^{6} + ( 9 - \beta_{2} + \beta_{4} ) q^{7} + ( 19 \beta_{5} + \beta_{10} ) q^{8} + ( -40 - 2 \beta_{1} + \beta_{2} - 7 \beta_{5} + \beta_{11} ) q^{9} + ( 2 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{10} + ( -\beta_{2} - \beta_{3} - 14 \beta_{5} - \beta_{10} - 2 \beta_{11} ) q^{11} + ( -3 \beta_{2} - 3 \beta_{3} + 12 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{12} + ( \beta_{2} + \beta_{3} - 7 \beta_{4} - \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{13} + ( \beta_{2} + \beta_{3} + 4 \beta_{4} - 21 \beta_{5} - \beta_{6} + 10 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{14} + ( 115 + 8 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 47 \beta_{5} - 3 \beta_{10} - \beta_{11} ) q^{15} + ( 396 + 21 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{16} + ( -13 \beta_{4} + \beta_{6} - 5 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} ) q^{17} + ( -310 - 11 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + 122 \beta_{5} + 3 \beta_{10} - 2 \beta_{11} ) q^{18} + ( 6 \beta_{2} + 6 \beta_{3} + 43 \beta_{4} + 5 \beta_{6} ) q^{19} + ( 43 \beta_{4} - 4 \beta_{6} - 31 \beta_{7} + 20 \beta_{8} + 5 \beta_{9} ) q^{20} + ( 335 + 28 \beta_{1} + \beta_{2} - 3 \beta_{3} - 16 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} - 9 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{21} + ( -720 - 40 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{22} + ( 2 \beta_{2} + 2 \beta_{3} + 100 \beta_{5} + \beta_{10} + 4 \beta_{11} ) q^{23} + ( -9 \beta_{2} - 9 \beta_{3} + 21 \beta_{4} + 3 \beta_{6} + 58 \beta_{7} - 7 \beta_{8} - 6 \beta_{9} ) q^{24} + ( 641 + 20 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{25} + ( -117 \beta_{4} + 18 \beta_{6} + 11 \beta_{7} + 12 \beta_{8} + 3 \beta_{9} ) q^{26} + ( -9 \beta_{2} - 9 \beta_{3} + 24 \beta_{4} - 15 \beta_{6} + 6 \beta_{7} + 12 \beta_{9} ) q^{27} + ( -664 - 63 \beta_{1} + 10 \beta_{2} + 14 \beta_{3} - 199 \beta_{4} - 14 \beta_{6} - 7 \beta_{7} + 14 \beta_{8} - 7 \beta_{9} ) q^{28} + ( -\beta_{2} - \beta_{3} - 122 \beta_{5} - 2 \beta_{10} - 2 \beta_{11} ) q^{29} + ( -2270 - 145 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} - 335 \beta_{5} - 3 \beta_{10} - 10 \beta_{11} ) q^{30} + ( -11 \beta_{2} - 11 \beta_{3} + 132 \beta_{4} + 10 \beta_{6} ) q^{31} + ( 8 \beta_{2} + 8 \beta_{3} - 619 \beta_{5} + 3 \beta_{10} + 16 \beta_{11} ) q^{32} + ( 18 \beta_{2} + 18 \beta_{3} + 9 \beta_{4} + 39 \beta_{6} - 73 \beta_{7} + 16 \beta_{8} - 6 \beta_{9} ) q^{33} + ( 6 \beta_{2} + 6 \beta_{3} - 314 \beta_{4} - 28 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} ) q^{34} + ( -10 \beta_{2} - 10 \beta_{3} + 163 \beta_{4} + 644 \beta_{5} - 25 \beta_{6} + 54 \beta_{7} - 16 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} - 20 \beta_{11} ) q^{35} + ( 4512 + 168 \beta_{1} + 12 \beta_{3} + 591 \beta_{5} + 21 \beta_{10} + 12 \beta_{11} ) q^{36} + ( 2238 + 192 \beta_{1} + 20 \beta_{2} - 20 \beta_{3} ) q^{37} + ( -93 \beta_{4} + 12 \beta_{6} - 45 \beta_{7} - 12 \beta_{8} - 3 \beta_{9} ) q^{38} + ( -1357 - 8 \beta_{1} - 20 \beta_{2} + 39 \beta_{3} + 677 \beta_{5} - 24 \beta_{10} + 19 \beta_{11} ) q^{39} + ( -38 \beta_{2} - 38 \beta_{3} + 879 \beta_{4} + 70 \beta_{6} + 11 \beta_{7} - 22 \beta_{8} + 11 \beta_{9} ) q^{40} + ( 355 \beta_{4} - 55 \beta_{6} + 39 \beta_{7} - 40 \beta_{8} - 10 \beta_{9} ) q^{41} + ( -322 + 91 \beta_{1} - 11 \beta_{2} + 3 \beta_{3} - 51 \beta_{4} - 1267 \beta_{5} - 45 \beta_{6} - 40 \beta_{7} - 5 \beta_{8} + 18 \beta_{9} - 24 \beta_{10} - 8 \beta_{11} ) q^{42} + ( -6092 - 112 \beta_{1} + 36 \beta_{2} - 36 \beta_{3} ) q^{43} + ( -20 \beta_{2} - 20 \beta_{3} + 1528 \beta_{5} - 4 \beta_{10} - 40 \beta_{11} ) q^{44} + ( 63 \beta_{2} + 63 \beta_{3} - 96 \beta_{4} + 24 \beta_{6} - 105 \beta_{7} + 36 \beta_{8} + 24 \beta_{9} ) q^{45} + ( 4900 + 118 \beta_{1} + 16 \beta_{2} - 16 \beta_{3} ) q^{46} + ( -650 \beta_{4} + 86 \beta_{6} + 100 \beta_{7} - 64 \beta_{8} - 16 \beta_{9} ) q^{47} + ( 51 \beta_{2} + 51 \beta_{3} - 368 \beta_{4} - 53 \beta_{6} - 9 \beta_{7} + 5 \beta_{8} - 53 \beta_{9} ) q^{48} + ( 651 - 140 \beta_{1} - 35 \beta_{2} - 105 \beta_{3} - 238 \beta_{4} - 42 \beta_{6} + 28 \beta_{7} - 56 \beta_{8} + 28 \beta_{9} ) q^{49} + ( 10 \beta_{2} + 10 \beta_{3} - 1461 \beta_{5} - 30 \beta_{10} + 20 \beta_{11} ) q^{50} + ( 1962 - 72 \beta_{1} - 45 \beta_{2} + 81 \beta_{3} - 1548 \beta_{5} + 27 \beta_{10} + 36 \beta_{11} ) q^{51} + ( 30 \beta_{2} + 30 \beta_{3} + 47 \beta_{4} + 10 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{52} + ( -15 \beta_{2} - 15 \beta_{3} - 2710 \beta_{5} + 26 \beta_{10} - 30 \beta_{11} ) q^{53} + ( -36 \beta_{2} - 36 \beta_{3} + 429 \beta_{4} + 102 \beta_{6} + 276 \beta_{7} - 63 \beta_{8} + 21 \beta_{9} ) q^{54} + ( -70 \beta_{2} - 70 \beta_{3} - 904 \beta_{4} - 84 \beta_{6} - 40 \beta_{7} + 80 \beta_{8} - 40 \beta_{9} ) q^{55} + ( 36 \beta_{2} + 36 \beta_{3} + 571 \beta_{4} + 2149 \beta_{5} - 64 \beta_{6} - 291 \beta_{7} + 164 \beta_{8} + 41 \beta_{9} + 27 \beta_{10} + 72 \beta_{11} ) q^{56} + ( 8715 - 246 \beta_{1} - 75 \beta_{2} + 18 \beta_{3} + 399 \beta_{5} - 36 \beta_{10} - 57 \beta_{11} ) q^{57} + ( -5900 - 182 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -109 \beta_{4} + 43 \beta_{6} - 40 \beta_{7} + 256 \beta_{8} + 64 \beta_{9} ) q^{59} + ( -12648 - 141 \beta_{1} + 12 \beta_{2} - 72 \beta_{3} + 6009 \beta_{5} + 63 \beta_{10} - 60 \beta_{11} ) q^{60} + ( 53 \beta_{2} + 53 \beta_{3} + 509 \beta_{4} + 75 \beta_{6} - 70 \beta_{7} + 140 \beta_{8} - 70 \beta_{9} ) q^{61} + ( -2 \beta_{4} - 22 \beta_{6} + 370 \beta_{7} - 208 \beta_{8} - 52 \beta_{9} ) q^{62} + ( -10048 - 56 \beta_{1} + 43 \beta_{2} + 87 \beta_{3} - 147 \beta_{4} - 1981 \beta_{5} - 24 \beta_{6} + 240 \beta_{7} + 72 \beta_{8} - 24 \beta_{9} + 60 \beta_{10} + 13 \beta_{11} ) q^{63} + ( -16636 + 91 \beta_{1} - 60 \beta_{2} + 60 \beta_{3} ) q^{64} + ( 71 \beta_{2} + 71 \beta_{3} + 1886 \beta_{5} - 24 \beta_{10} + 142 \beta_{11} ) q^{65} + ( -138 \beta_{2} - 138 \beta_{3} + 476 \beta_{4} - 8 \beta_{6} - 228 \beta_{7} - 64 \beta_{8} - 8 \beta_{9} ) q^{66} + ( 6832 - 512 \beta_{1} + 74 \beta_{2} - 74 \beta_{3} ) q^{67} + ( -330 \beta_{4} + 60 \beta_{6} - 654 \beta_{7} + 120 \beta_{8} + 30 \beta_{9} ) q^{68} + ( -27 \beta_{2} - 27 \beta_{3} - 39 \beta_{4} - 81 \beta_{6} + 179 \beta_{7} - 116 \beta_{8} + 18 \beta_{9} ) q^{69} + ( 30380 + 826 \beta_{1} + 28 \beta_{2} + 252 \beta_{3} - 833 \beta_{4} - 56 \beta_{6} + 21 \beta_{7} - 42 \beta_{8} + 21 \beta_{9} ) q^{70} + ( -35 \beta_{2} - 35 \beta_{3} + 1998 \beta_{5} + 129 \beta_{10} - 70 \beta_{11} ) q^{71} + ( 18676 + 905 \beta_{1} + 116 \beta_{2} - 156 \beta_{3} - 6632 \beta_{5} - 84 \beta_{10} - 40 \beta_{11} ) q^{72} + ( -120 \beta_{2} - 120 \beta_{3} + 1148 \beta_{4} + 52 \beta_{6} + 112 \beta_{7} - 224 \beta_{8} + 112 \beta_{9} ) q^{73} + ( -40 \beta_{2} - 40 \beta_{3} - 8478 \beta_{5} - 152 \beta_{10} - 80 \beta_{11} ) q^{74} + ( 45 \beta_{2} + 45 \beta_{3} - 631 \beta_{4} - 60 \beta_{6} - 30 \beta_{7} - 60 \beta_{9} ) q^{75} + ( 126 \beta_{2} + 126 \beta_{3} + 1073 \beta_{4} + 118 \beta_{6} + 9 \beta_{7} - 18 \beta_{8} + 9 \beta_{9} ) q^{76} + ( -40 \beta_{2} - 40 \beta_{3} - 797 \beta_{4} + 1624 \beta_{5} + 89 \beta_{6} - 155 \beta_{7} - 232 \beta_{8} - 58 \beta_{9} - 156 \beta_{10} - 80 \beta_{11} ) q^{77} + ( 33086 - 215 \beta_{1} + 250 \beta_{2} - 132 \beta_{3} + 929 \beta_{5} - 51 \beta_{10} + 118 \beta_{11} ) q^{78} + ( -20438 - 704 \beta_{1} - 121 \beta_{2} + 121 \beta_{3} ) q^{79} + ( 401 \beta_{4} - 116 \beta_{6} + 1071 \beta_{7} - 548 \beta_{8} - 137 \beta_{9} ) q^{80} + ( -20985 + 780 \beta_{1} + 33 \beta_{2} - 9 \beta_{3} + 6312 \beta_{5} - 36 \beta_{10} + 24 \beta_{11} ) q^{81} + ( 158 \beta_{2} + 158 \beta_{3} - 1478 \beta_{4} - 140 \beta_{6} + 126 \beta_{7} - 252 \beta_{8} + 126 \beta_{9} ) q^{82} + ( 329 \beta_{4} + \beta_{6} - 696 \beta_{7} + 448 \beta_{8} + 112 \beta_{9} ) q^{83} + ( -50208 - 1155 \beta_{1} - 45 \beta_{2} - 285 \beta_{3} + 384 \beta_{4} - 3255 \beta_{5} + 47 \beta_{6} - 225 \beta_{7} - 71 \beta_{8} + 47 \beta_{9} - 9 \beta_{10} + 60 \beta_{11} ) q^{84} + ( -27180 + 1584 \beta_{1} + 36 \beta_{2} - 36 \beta_{3} ) q^{85} + ( -72 \beta_{2} - 72 \beta_{3} + 10876 \beta_{5} + 184 \beta_{10} - 144 \beta_{11} ) q^{86} + ( 27 \beta_{2} + 27 \beta_{3} - 12 \beta_{4} + 36 \beta_{6} - 220 \beta_{7} + 112 \beta_{8} ) q^{87} + ( 49280 + 272 \beta_{1} - 376 \beta_{2} + 376 \beta_{3} ) q^{88} + ( 2169 \beta_{4} - 333 \beta_{6} + 951 \beta_{7} - 216 \beta_{8} - 54 \beta_{9} ) q^{89} + ( -342 \beta_{2} - 342 \beta_{3} + 1875 \beta_{4} + 402 \beta_{6} - 861 \beta_{7} + 252 \beta_{8} + 159 \beta_{9} ) q^{90} + ( 26050 - 448 \beta_{1} + 128 \beta_{2} - 42 \beta_{3} + 2399 \beta_{4} + 287 \beta_{6} - 224 \beta_{7} + 448 \beta_{8} - 224 \beta_{9} ) q^{91} + ( 32 \beta_{2} + 32 \beta_{3} - 5446 \beta_{5} - 54 \beta_{10} + 64 \beta_{11} ) q^{92} + ( 32840 + 796 \beta_{1} - 35 \beta_{2} - 33 \beta_{3} + 476 \beta_{5} + 66 \beta_{10} - 68 \beta_{11} ) q^{93} + ( 328 \beta_{2} + 328 \beta_{3} - 4194 \beta_{4} - 224 \beta_{6} - 358 \beta_{7} + 716 \beta_{8} - 358 \beta_{9} ) q^{94} + ( 165 \beta_{2} + 165 \beta_{3} - 3498 \beta_{5} + 234 \beta_{10} + 330 \beta_{11} ) q^{95} + ( -99 \beta_{2} - 99 \beta_{3} - 177 \beta_{4} - 327 \beta_{6} - 342 \beta_{7} + 543 \beta_{8} + 78 \beta_{9} ) q^{96} + ( 218 \beta_{2} + 218 \beta_{3} + 574 \beta_{4} + 34 \beta_{6} + 212 \beta_{7} - 424 \beta_{8} + 212 \beta_{9} ) q^{97} + ( -70 \beta_{2} - 70 \beta_{3} - 714 \beta_{4} + 5089 \beta_{5} + 84 \beta_{6} + 1414 \beta_{7} - 168 \beta_{8} - 42 \beta_{9} + 210 \beta_{10} - 140 \beta_{11} ) q^{98} + ( 58568 - 1784 \beta_{1} - 113 \beta_{2} + 111 \beta_{3} + 1418 \beta_{5} + 147 \beta_{10} - 2 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 196 q^{4} + 112 q^{7} - 492 q^{9} + O(q^{10})$$ $$12 q - 196 q^{4} + 112 q^{7} - 492 q^{9} + 1392 q^{15} + 4868 q^{16} - 3804 q^{18} + 4116 q^{21} - 8752 q^{22} + 7812 q^{25} - 8204 q^{28} - 27876 q^{30} + 54864 q^{36} + 27464 q^{37} - 16080 q^{39} - 3444 q^{42} - 73840 q^{43} + 59144 q^{46} + 6972 q^{49} + 23760 q^{51} + 103968 q^{57} - 71512 q^{58} - 152676 q^{60} - 120624 q^{63} - 198788 q^{64} + 79344 q^{67} + 368760 q^{70} + 226644 q^{72} + 394644 q^{78} - 247104 q^{79} - 248868 q^{81} - 608076 q^{84} - 320112 q^{85} + 595456 q^{88} + 310128 q^{91} + 397272 q^{93} + 696576 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 484 x^{10} + 194194 x^{8} - 39867800 x^{6} + 5398720873 x^{4} - 310089434788 x^{2} + 9371104623076$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-349147121 \nu^{10} + 169990205666 \nu^{8} - 61997289967221 \nu^{6} + 7560470698384632 \nu^{4} - 415570181119723252 \nu^{2} - 143613946297148622848$$$$)/ 3947826063466678260$$ $$\beta_{2}$$ $$=$$ $$($$$$577504845699127 \nu^{11} - 1004579622171450745 \nu^{10} + 374336704079137293 \nu^{9} + 395907946641998307965 \nu^{8} - 131158409145645034223 \nu^{7} - 152029727478765806402495 \nu^{6} + 70229486585879462216711 \nu^{5} + 24861901393476176772532055 \nu^{4} - 10626952373201584498113336 \nu^{3} - 2612427275855053931633362360 \nu^{2} + 951404568144139405042041596 \nu + 53073780740985490469756474940$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{3}$$ $$=$$ $$($$$$577504845699127 \nu^{11} - 248013272115908985 \nu^{10} + 374336704079137293 \nu^{9} + 184885180116586887805 \nu^{8} - 131158409145645034223 \nu^{7} - 67677206516921055969215 \nu^{6} + 70229486585879462216711 \nu^{5} + 17111887660893465735726935 \nu^{4} - 10626952373201584498113336 \nu^{3} - 2245231200830831428337587640 \nu^{2} + 951404568144139405042041596 \nu + 121195116196042929085505820220$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$577504845699127 \nu^{11} - 48998272303591527 \nu^{10} + 374336704079137293 \nu^{9} + 21732933775471801827 \nu^{8} - 131158409145645034223 \nu^{7} - 9301668115511901228897 \nu^{6} + 70229486585879462216711 \nu^{5} + 1759912673667006372196809 \nu^{4} - 10626952373201584498113336 \nu^{3} - 239098800392450457417176904 \nu^{2} + 951404568144139405042041596 \nu + 8677691999895854207596704324$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$3044261635951 \nu^{11} - 1261215390376736 \nu^{9} + 513466320948530936 \nu^{7} - 91823119593034019372 \nu^{5} + 13245030916416474157507 \nu^{3} - 346601680658050067655342 \nu$$$$)/$$$$86\!\cdots\!40$$ $$\beta_{6}$$ $$=$$ $$($$$$-6352553302690397 \nu^{11} - 342987906125140689 \nu^{10} - 4117703744870510223 \nu^{9} + 152130536428302612789 \nu^{8} + 1442742500602095376453 \nu^{7} - 65111676808583308602279 \nu^{6} - 772524352444674084383821 \nu^{5} + 12319388715669044605377663 \nu^{4} + 116896476105217429479246696 \nu^{3} - 1673691602747153201920238328 \nu^{2} - 10465450249585533455462457556 \nu + 60743843999270979453176930268$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-4098481027121609 \nu^{11} + 2311563543697803789 \nu^{9} - 919842678122588551919 \nu^{7} + 211269798280779715972103 \nu^{5} - 30971319860817288804044088 \nu^{3} + 2809702495623861085900177148 \nu$$$$)/$$$$22\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$-12186479224947065 \nu^{11} + 236477664280405513 \nu^{10} + 5833740869340325245 \nu^{9} - 118764019037705983053 \nu^{8} - 2581807210901607796415 \nu^{7} + 44136265893726143171503 \nu^{6} + 601389429685657730452055 \nu^{5} - 9023312510876450545262791 \nu^{4} - 93658837586405340329632440 \nu^{3} + 900360183782671323450941496 \nu^{2} + 8737324822460963612976696380 \nu - 32126049935398466323987159036$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$-17908510932748293 \nu^{11} - 798915840210847471 \nu^{10} + 5921344539047631033 \nu^{9} + 409857274824408526731 \nu^{8} - 2930453838121103386323 \nu^{7} - 148640059228368868999321 \nu^{6} + 569550803052117642309771 \nu^{5} + 30813512022504783064460737 \nu^{4} - 93494178331571349556836696 \nu^{3} - 2884144333953333921552235272 \nu^{2} + 9001030949241786839026913676 \nu + 102471123741906302673158523172$$$$)/$$$$44\!\cdots\!80$$ $$\beta_{10}$$ $$=$$ $$($$$$-461224318369921 \nu^{11} + 205734379002992841 \nu^{9} - 85356242717080477261 \nu^{7} + 15657685747090694023087 \nu^{5} - 2096997695238490684479882 \nu^{3} + 55913380571522714673251392 \nu$$$$)/$$$$17\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$-195276234019366967 \nu^{11} + 626296447143679865 \nu^{10} + 85383222741115034867 \nu^{9} - 290396563379292597885 \nu^{8} - 32741541475383854341137 \nu^{7} + 109853466997843431185855 \nu^{6} + 5929568022904152143378169 \nu^{5} - 20986894527184821254129495 \nu^{4} - 646780893199580469509813704 \nu^{3} + 2428829238342942679985475000 \nu^{2} + 17308395451973405549861733124 \nu - 87134448468514209777631147580$$$$)/$$$$44\!\cdots\!80$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$9 \beta_{7} + \beta_{6} + 54 \beta_{5} - 7 \beta_{4}$$$$)/54$$ $$\nu^{2}$$ $$=$$ $$($$$$8 \beta_{9} - 16 \beta_{8} + 8 \beta_{7} + 48 \beta_{4} - 15 \beta_{3} - 9 \beta_{2} + 6 \beta_{1} + 1452$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$108 \beta_{11} + 108 \beta_{10} + 102 \beta_{9} + 408 \beta_{8} - 573 \beta_{7} + 55 \beta_{6} + 23274 \beta_{5} - 79 \beta_{4} + 54 \beta_{3} + 54 \beta_{2}$$$$)/54$$ $$\nu^{4}$$ $$=$$ $$($$$$1632 \beta_{9} - 3264 \beta_{8} + 1632 \beta_{7} - 112 \beta_{6} + 7264 \beta_{4} - 2337 \beta_{3} - 1263 \beta_{2} - 13938 \beta_{1} - 457392$$$$)/18$$ $$\nu^{5}$$ $$=$$ $$($$$$32796 \beta_{11} - 31860 \beta_{10} + 55794 \beta_{9} + 223176 \beta_{8} - 619233 \beta_{7} - 46901 \beta_{6} + 437814 \beta_{5} + 495689 \beta_{4} + 16398 \beta_{3} + 16398 \beta_{2}$$$$)/54$$ $$\nu^{6}$$ $$=$$ $$($$$$-135680 \beta_{9} + 271360 \beta_{8} - 135680 \beta_{7} - 418040 \beta_{6} - 6240880 \beta_{4} + 632367 \beta_{3} + 603033 \beta_{2} - 3604350 \beta_{1} - 145767480$$$$)/18$$ $$\nu^{7}$$ $$=$$ $$($$$$6058116 \beta_{11} - 27571428 \beta_{10} + 1628442 \beta_{9} + 6513768 \beta_{8} - 48032865 \beta_{7} - 11392613 \beta_{6} - 1592631846 \beta_{5} + 84633617 \beta_{4} + 3029058 \beta_{3} + 3029058 \beta_{2}$$$$)/54$$ $$\nu^{8}$$ $$=$$ $$($$$$-105542016 \beta_{9} + 211084032 \beta_{8} - 105542016 \beta_{7} - 126681632 \beta_{6} - 2218838848 \beta_{4} + 328880259 \beta_{3} + 179834589 \beta_{2} + 618614982 \beta_{1} + 11513807496$$$$)/18$$ $$\nu^{9}$$ $$=$$ $$($$$$-115352964 \beta_{11} - 3406638924 \beta_{10} - 5179046022 \beta_{9} - 20716184088 \beta_{8} + 43372281303 \beta_{7} - 185035565 \beta_{6} - 261332288442 \beta_{5} - 14241889111 \beta_{4} - 57676482 \beta_{3} - 57676482 \beta_{2}$$$$)/54$$ $$\nu^{10}$$ $$=$$ $$($$$$-1475567008 \beta_{9} + 2951134016 \beta_{8} - 1475567008 \beta_{7} + 10127357464 \beta_{6} + 128051078384 \beta_{4} + 15082797423 \beta_{3} - 36159644727 \beta_{2} + 428719815474 \beta_{1} + 12452817789624$$$$)/18$$ $$\nu^{11}$$ $$=$$ $$($$$$-550266690924 \beta_{11} + 1808171237268 \beta_{10} - 1181191737750 \beta_{9} - 4724766951000 \beta_{8} + 10910349756111 \beta_{7} + 304797136363 \beta_{6} + 93761288872362 \beta_{5} - 5677155167791 \beta_{4} - 275133345462 \beta_{3} - 275133345462 \beta_{2}$$$$)/54$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1
 −13.7499 + 10.2458i 13.7499 + 10.2458i −12.2117 + 5.57294i 12.2117 + 5.57294i 6.98672 + 2.99433i −6.98672 + 2.99433i 6.98672 − 2.99433i −6.98672 − 2.99433i −12.2117 − 5.57294i 12.2117 − 5.57294i −13.7499 − 10.2458i 13.7499 − 10.2458i
10.2458i −4.60012 + 14.8943i −72.9763 −73.2992 152.604 + 47.1319i 55.7250 117.054i 419.835i −200.678 137.031i 751.009i
20.2 10.2458i 4.60012 14.8943i −72.9763 73.2992 −152.604 47.1319i 55.7250 + 117.054i 419.835i −200.678 137.031i 751.009i
20.3 5.57294i −13.5283 7.74498i 0.942331 −46.2136 −43.1623 + 75.3925i −120.146 + 48.7024i 183.586i 123.031 + 209.553i 257.545i
20.4 5.57294i 13.5283 + 7.74498i 0.942331 46.2136 43.1623 75.3925i −120.146 48.7024i 183.586i 123.031 + 209.553i 257.545i
20.5 2.99433i −9.94100 + 12.0074i 23.0340 61.8023 35.9539 + 29.7666i 92.4210 + 90.9140i 164.790i −45.3530 238.730i 185.056i
20.6 2.99433i 9.94100 12.0074i 23.0340 −61.8023 −35.9539 29.7666i 92.4210 90.9140i 164.790i −45.3530 238.730i 185.056i
20.7 2.99433i −9.94100 12.0074i 23.0340 61.8023 35.9539 29.7666i 92.4210 90.9140i 164.790i −45.3530 + 238.730i 185.056i
20.8 2.99433i 9.94100 + 12.0074i 23.0340 −61.8023 −35.9539 + 29.7666i 92.4210 + 90.9140i 164.790i −45.3530 + 238.730i 185.056i
20.9 5.57294i −13.5283 + 7.74498i 0.942331 −46.2136 −43.1623 75.3925i −120.146 48.7024i 183.586i 123.031 209.553i 257.545i
20.10 5.57294i 13.5283 7.74498i 0.942331 46.2136 43.1623 + 75.3925i −120.146 + 48.7024i 183.586i 123.031 209.553i 257.545i
20.11 10.2458i −4.60012 14.8943i −72.9763 −73.2992 152.604 47.1319i 55.7250 + 117.054i 419.835i −200.678 + 137.031i 751.009i
20.12 10.2458i 4.60012 + 14.8943i −72.9763 73.2992 −152.604 + 47.1319i 55.7250 117.054i 419.835i −200.678 + 137.031i 751.009i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 20.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.c.a 12
3.b odd 2 1 inner 21.6.c.a 12
4.b odd 2 1 336.6.k.d 12
7.b odd 2 1 inner 21.6.c.a 12
12.b even 2 1 336.6.k.d 12
21.c even 2 1 inner 21.6.c.a 12
28.d even 2 1 336.6.k.d 12
84.h odd 2 1 336.6.k.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.c.a 12 1.a even 1 1 trivial
21.6.c.a 12 3.b odd 2 1 inner
21.6.c.a 12 7.b odd 2 1 inner
21.6.c.a 12 21.c even 2 1 inner
336.6.k.d 12 4.b odd 2 1
336.6.k.d 12 12.b even 2 1
336.6.k.d 12 28.d even 2 1
336.6.k.d 12 84.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 29232 + 4480 T^{2} + 145 T^{4} + T^{6} )^{2}$$
$3$ $$205891132094649 + 857748962646 T^{2} + 5460556275 T^{4} + 20094156 T^{6} + 92475 T^{8} + 246 T^{10} + T^{12}$$
$5$ $$( -43827584976 + 40153428 T^{2} - 11328 T^{4} + T^{6} )^{2}$$
$7$ $$( 4747561509943 - 15818613944 T - 2941225 T^{2} + 3067792 T^{3} - 175 T^{4} - 56 T^{5} + T^{6} )^{2}$$
$11$ $$( 1749955777855488 + 56962463488 T^{2} + 516844 T^{4} + T^{6} )^{2}$$
$13$ $$( 545717977444800 + 141513292836 T^{2} + 1411356 T^{4} + T^{6} )^{2}$$
$17$ $$( -1468438189270246656 + 4402713595392 T^{2} - 3963492 T^{4} + T^{6} )^{2}$$
$19$ $$( 246289511999088 + 79283815092 T^{2} + 4732776 T^{4} + T^{6} )^{2}$$
$23$ $$( 979568294804713152 + 3173893314928 T^{2} + 3199348 T^{4} + T^{6} )^{2}$$
$29$ $$( 232486319948698368 + 1625659185808 T^{2} + 3085432 T^{4} + T^{6} )^{2}$$
$31$ $$( 4364932849705156608 + 168820061594112 T^{2} + 33732156 T^{4} + T^{6} )^{2}$$
$37$ $$( 403376576936 - 85907668 T - 6866 T^{2} + T^{3} )^{4}$$
$41$ $$( -$$$$19\!\cdots\!96$$$$+ 17968431810970368 T^{2} - 335428548 T^{4} + T^{6} )^{2}$$
$43$ $$( -624577978816 + 2096624 T + 18460 T^{2} + T^{3} )^{4}$$
$47$ $$( -$$$$52\!\cdots\!76$$$$+ 434267430602467392 T^{2} - 1163949936 T^{4} + T^{6} )^{2}$$
$53$ $$($$$$19\!\cdots\!00$$$$+ 430660046188993936 T^{2} + 1357212856 T^{4} + T^{6} )^{2}$$
$59$ $$( -$$$$10\!\cdots\!44$$$$+ 3434612455856955492 T^{2} - 3338458908 T^{4} + T^{6} )^{2}$$
$61$ $$($$$$35\!\cdots\!72$$$$+ 469149207610188132 T^{2} + 1690298364 T^{4} + T^{6} )^{2}$$
$67$ $$( 14656583988736 - 935181408 T - 19836 T^{2} + T^{3} )^{4}$$
$71$ $$($$$$30\!\cdots\!72$$$$+ 5825406061652122624 T^{2} + 4881486316 T^{4} + T^{6} )^{2}$$
$73$ $$($$$$48\!\cdots\!52$$$$+ 6039885976049636352 T^{2} + 6528000384 T^{4} + T^{6} )^{2}$$
$79$ $$( -53317412578496 - 465562020 T + 61776 T^{2} + T^{3} )^{4}$$
$83$ $$( -$$$$16\!\cdots\!00$$$$+ 9861651897257145636 T^{2} - 13411696956 T^{4} + T^{6} )^{2}$$
$89$ $$( -$$$$16\!\cdots\!04$$$$+ 95396036142002706432 T^{2} - 17303041044 T^{4} + T^{6} )^{2}$$
$97$ $$($$$$29\!\cdots\!12$$$$+ 97394897895013919808 T^{2} + 20156995056 T^{4} + T^{6} )^{2}$$