Properties

Label 105.3.o.a
Level 105
Weight 3
Character orbit 105.o
Analytic conductor 2.861
Analytic rank 0
Dimension 16
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 105.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.86104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( -\beta_{3} - \beta_{9} + \beta_{13} ) q^{3} + ( -1 - \beta_{1} + \beta_{10} ) q^{4} + ( -\beta_{2} + \beta_{6} - \beta_{12} - \beta_{15} ) q^{5} + ( -5 - \beta_{8} - \beta_{11} ) q^{6} + ( 2 \beta_{5} + \beta_{7} + 3 \beta_{9} - \beta_{13} ) q^{7} + ( -\beta_{2} + \beta_{14} ) q^{8} + ( \beta_{1} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{9} +O(q^{10})\) \( q + \beta_{6} q^{2} + ( -\beta_{3} - \beta_{9} + \beta_{13} ) q^{3} + ( -1 - \beta_{1} + \beta_{10} ) q^{4} + ( -\beta_{2} + \beta_{6} - \beta_{12} - \beta_{15} ) q^{5} + ( -5 - \beta_{8} - \beta_{11} ) q^{6} + ( 2 \beta_{5} + \beta_{7} + 3 \beta_{9} - \beta_{13} ) q^{7} + ( -\beta_{2} + \beta_{14} ) q^{8} + ( \beta_{1} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} ) q^{9} + ( -5 - 5 \beta_{1} - 5 \beta_{5} + 5 \beta_{7} ) q^{10} + ( \beta_{4} + 3 \beta_{8} + 3 \beta_{12} + \beta_{14} - \beta_{15} ) q^{11} + ( -\beta_{2} - \beta_{4} + \beta_{5} - 4 \beta_{6} + 3 \beta_{13} - \beta_{15} ) q^{12} + ( -8 \beta_{7} - 3 \beta_{9} ) q^{13} + ( \beta_{2} - 3 \beta_{4} + 3 \beta_{8} + \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{14} + ( -5 - \beta_{2} + \beta_{3} + \beta_{6} + 5 \beta_{7} - 4 \beta_{8} ) q^{15} + ( -4 \beta_{1} + 2 \beta_{10} + 2 \beta_{11} ) q^{16} + ( -\beta_{3} - 2 \beta_{4} - 2 \beta_{14} - 2 \beta_{15} ) q^{17} + ( 7 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{18} + ( -6 \beta_{1} + \beta_{10} + \beta_{11} ) q^{19} + ( \beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} + 5 \beta_{14} ) q^{20} + ( 5 + 11 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{8} + 3 \beta_{10} + \beta_{11} - 2 \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{21} + ( 5 \beta_{7} + 5 \beta_{9} ) q^{22} + ( 4 \beta_{2} + 4 \beta_{4} + 3 \beta_{6} + 4 \beta_{15} ) q^{23} + ( -\beta_{4} - 6 \beta_{8} - 2 \beta_{10} - 6 \beta_{12} - \beta_{14} + \beta_{15} ) q^{24} + ( -5 \beta_{5} + 5 \beta_{7} - 5 \beta_{9} - 5 \beta_{10} + 5 \beta_{13} ) q^{25} + ( -8 \beta_{2} + 8 \beta_{4} - 5 \beta_{12} - 8 \beta_{15} ) q^{26} + ( 4 \beta_{2} + 5 \beta_{3} + 5 \beta_{6} - 4 \beta_{7} - 5 \beta_{9} - 4 \beta_{14} ) q^{27} + ( 2 \beta_{5} - 13 \beta_{7} + 3 \beta_{9} - 8 \beta_{13} ) q^{28} + ( 4 \beta_{2} + \beta_{8} + 4 \beta_{14} ) q^{29} + ( -5 \beta_{1} + 5 \beta_{2} - 5 \beta_{4} - 5 \beta_{6} + 5 \beta_{12} - 5 \beta_{13} + 5 \beta_{15} ) q^{30} + ( 43 + 43 \beta_{1} + 2 \beta_{10} ) q^{31} + ( -10 \beta_{3} + 6 \beta_{4} + 6 \beta_{14} + 6 \beta_{15} ) q^{32} + ( 4 \beta_{2} + 4 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} + 5 \beta_{13} + 4 \beta_{15} ) q^{33} + ( -5 + 3 \beta_{11} ) q^{34} + ( 4 \beta_{2} - 9 \beta_{3} + \beta_{6} + 6 \beta_{8} - 5 \beta_{12} + 5 \beta_{15} ) q^{35} + ( 31 - 2 \beta_{2} - 4 \beta_{8} + 3 \beta_{11} - 2 \beta_{14} ) q^{36} + ( 17 \beta_{5} - 6 \beta_{13} ) q^{37} + ( -9 \beta_{3} + \beta_{4} + \beta_{14} + \beta_{15} ) q^{38} + ( -4 - 4 \beta_{1} + 8 \beta_{4} - 5 \beta_{8} - 3 \beta_{10} - 5 \beta_{12} + 8 \beta_{14} - 8 \beta_{15} ) q^{39} + ( -15 \beta_{1} + 5 \beta_{5} - 5 \beta_{10} - 5 \beta_{11} + 5 \beta_{13} ) q^{40} + ( -\beta_{2} + 19 \beta_{8} - \beta_{14} ) q^{41} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 19 \beta_{5} - \beta_{6} - 6 \beta_{7} - 4 \beta_{9} - \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{42} + ( -17 \beta_{7} - 16 \beta_{9} ) q^{43} + ( \beta_{2} - \beta_{4} + 12 \beta_{12} + \beta_{15} ) q^{44} + ( -10 - 10 \beta_{1} + \beta_{3} - 10 \beta_{4} + \beta_{8} + 10 \beta_{9} + \beta_{12} - 10 \beta_{13} - 10 \beta_{14} + \beta_{15} ) q^{45} + ( -15 - 15 \beta_{1} + 11 \beta_{10} ) q^{46} + ( -\beta_{2} - \beta_{4} - 12 \beta_{6} - \beta_{15} ) q^{47} + ( -2 \beta_{2} - 10 \beta_{3} - 10 \beta_{6} + 2 \beta_{7} + 4 \beta_{9} + 2 \beta_{14} ) q^{48} + ( -10 + 6 \beta_{1} - 13 \beta_{10} - 2 \beta_{11} ) q^{49} + ( 10 \beta_{2} + 15 \beta_{3} + 15 \beta_{6} - 10 \beta_{8} ) q^{50} + ( 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{10} + 3 \beta_{11} + 13 \beta_{12} - 2 \beta_{15} ) q^{51} + ( -37 \beta_{5} + 37 \beta_{7} - \beta_{9} + \beta_{13} ) q^{52} + ( 12 \beta_{3} - 6 \beta_{4} - 6 \beta_{14} - 6 \beta_{15} ) q^{53} + ( -25 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} + 13 \beta_{10} + 13 \beta_{11} + \beta_{12} - 4 \beta_{15} ) q^{54} + ( 5 \beta_{7} + 20 \beta_{9} + 5 \beta_{11} ) q^{55} + ( -\beta_{2} + 3 \beta_{4} + 18 \beta_{8} + 6 \beta_{12} + 2 \beta_{14} - 3 \beta_{15} ) q^{56} + ( -\beta_{2} - 9 \beta_{3} - 9 \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{14} ) q^{57} + ( 31 \beta_{5} + 9 \beta_{13} ) q^{58} + ( -8 \beta_{4} - 31 \beta_{8} - 31 \beta_{12} - 8 \beta_{14} + 8 \beta_{15} ) q^{59} + ( 5 + 5 \beta_{1} - \beta_{3} + 25 \beta_{5} - 25 \beta_{7} - 11 \beta_{8} - 5 \beta_{9} - 5 \beta_{10} - 11 \beta_{12} + 5 \beta_{13} + 4 \beta_{15} ) q^{60} + ( 49 \beta_{1} - 5 \beta_{10} - 5 \beta_{11} ) q^{61} + ( -2 \beta_{2} + 37 \beta_{3} + 37 \beta_{6} + 2 \beta_{14} ) q^{62} + ( -6 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 23 \beta_{5} + 4 \beta_{6} + 20 \beta_{7} + 11 \beta_{9} + 8 \beta_{13} + 2 \beta_{14} - 4 \beta_{15} ) q^{63} + ( -34 - 14 \beta_{11} ) q^{64} + ( -11 \beta_{2} - 29 \beta_{6} + 4 \beta_{12} - 11 \beta_{15} ) q^{65} + ( 15 + 15 \beta_{1} - 5 \beta_{4} + 5 \beta_{10} - 5 \beta_{14} + 5 \beta_{15} ) q^{66} + ( 26 \beta_{5} - 26 \beta_{7} - 13 \beta_{9} + 13 \beta_{13} ) q^{67} + ( -5 \beta_{2} - 5 \beta_{4} + 8 \beta_{6} - 5 \beta_{15} ) q^{68} + ( -15 + 4 \beta_{2} + 21 \beta_{8} - 11 \beta_{11} + 4 \beta_{14} ) q^{69} + ( -50 - 5 \beta_{1} + 5 \beta_{5} - 15 \beta_{7} - 10 \beta_{9} + 5 \beta_{10} - 10 \beta_{11} + 15 \beta_{13} ) q^{70} + ( -2 \beta_{2} - 25 \beta_{8} - 2 \beta_{14} ) q^{71} + ( -5 \beta_{2} - 5 \beta_{4} - 4 \beta_{5} + 12 \beta_{6} - 16 \beta_{13} - 5 \beta_{15} ) q^{72} + ( -37 \beta_{5} + 37 \beta_{7} + 4 \beta_{9} - 4 \beta_{13} ) q^{73} + ( -17 \beta_{4} + 23 \beta_{8} + 23 \beta_{12} - 17 \beta_{14} + 17 \beta_{15} ) q^{74} + ( -25 \beta_{1} + 10 \beta_{2} - 5 \beta_{5} + 15 \beta_{6} - 5 \beta_{10} - 5 \beta_{11} + 10 \beta_{12} - 20 \beta_{13} + 10 \beta_{15} ) q^{75} + ( -21 - 7 \beta_{11} ) q^{76} + ( -6 \beta_{2} - 15 \beta_{3} - 11 \beta_{4} - 3 \beta_{6} - 5 \beta_{14} - 11 \beta_{15} ) q^{77} + ( 3 \beta_{2} + 5 \beta_{3} + 5 \beta_{6} + 69 \beta_{7} + 11 \beta_{9} - 3 \beta_{14} ) q^{78} + ( -76 \beta_{1} - 3 \beta_{10} - 3 \beta_{11} ) q^{79} + ( -4 \beta_{3} + 10 \beta_{4} - 4 \beta_{8} - 4 \beta_{12} + 10 \beta_{14} - 4 \beta_{15} ) q^{80} + ( -41 - 41 \beta_{1} + 8 \beta_{4} + 20 \beta_{8} + 8 \beta_{10} + 20 \beta_{12} + 8 \beta_{14} - 8 \beta_{15} ) q^{81} + ( -27 \beta_{5} + 17 \beta_{13} ) q^{82} + ( -14 \beta_{2} - 29 \beta_{3} - 29 \beta_{6} + 14 \beta_{14} ) q^{83} + ( -9 + 25 \beta_{1} - 2 \beta_{2} + 13 \beta_{4} - 6 \beta_{8} + 3 \beta_{10} + 8 \beta_{11} - 16 \beta_{12} + 11 \beta_{14} - 13 \beta_{15} ) q^{84} + ( -35 - 5 \beta_{7} - 10 \beta_{9} + 10 \beta_{11} ) q^{85} + ( -17 \beta_{2} + 17 \beta_{4} - \beta_{12} - 17 \beta_{15} ) q^{86} + ( \beta_{3} + 5 \beta_{4} + 31 \beta_{5} - 31 \beta_{7} - 9 \beta_{9} + 9 \beta_{13} + 5 \beta_{14} + 5 \beta_{15} ) q^{87} + ( 30 \beta_{9} - 30 \beta_{13} ) q^{88} + ( 11 \beta_{2} - 11 \beta_{4} + 51 \beta_{12} + 11 \beta_{15} ) q^{89} + ( 5 - 10 \beta_{3} - 10 \beta_{6} - 45 \beta_{7} + 10 \beta_{8} - 10 \beta_{9} + 10 \beta_{11} ) q^{90} + ( 41 - 54 \beta_{1} + 19 \beta_{10} + 18 \beta_{11} ) q^{91} + ( 5 \beta_{2} - 36 \beta_{3} - 36 \beta_{6} - 5 \beta_{14} ) q^{92} + ( -2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 37 \beta_{6} + 51 \beta_{13} - 2 \beta_{15} ) q^{93} + ( 60 + 60 \beta_{1} - 14 \beta_{10} ) q^{94} + ( -6 \beta_{3} + 5 \beta_{4} - 6 \beta_{8} - 6 \beta_{12} + 5 \beta_{14} - 6 \beta_{15} ) q^{95} + ( 50 \beta_{1} + 6 \beta_{2} - 6 \beta_{4} - 22 \beta_{10} - 22 \beta_{11} - 26 \beta_{12} + 6 \beta_{15} ) q^{96} + ( -5 \beta_{7} - \beta_{9} ) q^{97} + ( 11 \beta_{2} + 45 \beta_{3} - 2 \beta_{4} + 23 \beta_{6} - 13 \beta_{14} - 2 \beta_{15} ) q^{98} + ( 30 - \beta_{2} + 27 \beta_{8} - 10 \beta_{11} - \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{4} - 80q^{6} - 8q^{9} + O(q^{10}) \) \( 16q - 8q^{4} - 80q^{6} - 8q^{9} - 40q^{10} - 80q^{15} + 32q^{16} + 48q^{19} - 8q^{21} + 40q^{30} + 344q^{31} - 80q^{34} + 496q^{36} - 32q^{39} + 120q^{40} - 80q^{45} - 120q^{46} - 208q^{49} - 40q^{51} + 200q^{54} + 40q^{60} - 392q^{61} - 544q^{64} + 120q^{66} - 240q^{69} - 760q^{70} + 200q^{75} - 336q^{76} + 608q^{79} - 328q^{81} - 344q^{84} - 560q^{85} + 80q^{90} + 1088q^{91} + 480q^{94} - 400q^{96} + 480q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} - 4 x^{14} + 12 x^{13} + 162 x^{12} - 524 x^{11} - 88 x^{10} + 1492 x^{9} + 6266 x^{8} - 41348 x^{7} + 103092 x^{6} - 152724 x^{5} + 148342 x^{4} - 96348 x^{3} + 41016 x^{2} - 10764 x + 1521\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1282138 \nu^{15} + 19217450 \nu^{14} - 57698230 \nu^{13} - 201080763 \nu^{12} + 155658870 \nu^{11} + 3374884060 \nu^{10} - 5440348978 \nu^{9} - 13893531470 \nu^{8} + 20977752170 \nu^{7} + 138561264022 \nu^{6} - 538089884310 \nu^{5} + 898049014875 \nu^{4} - 857050300178 \nu^{3} + 492147526380 \nu^{2} - 166399335570 \nu + 29387553354\)\()/ 3843901530 \)
\(\beta_{2}\)\(=\)\((\)\(-945353839 \nu^{15} + 19734882097 \nu^{14} - 9615662912 \nu^{13} - 208584356535 \nu^{12} - 405935897763 \nu^{11} + 3186101541443 \nu^{10} + 609707566390 \nu^{9} - 16797143566483 \nu^{8} - 15339757529897 \nu^{7} + 178644540686345 \nu^{6} - 345522051469818 \nu^{5} + 247141137866943 \nu^{4} + 51902331774725 \nu^{3} - 196224803262363 \nu^{2} + 117682690173798 \nu - 29454867221481\)\()/ 2648448154170 \)
\(\beta_{3}\)\(=\)\((\)\(6799338422 \nu^{15} + 5815573694 \nu^{14} - 110020669305 \nu^{13} - 206978139649 \nu^{12} + 1133812525284 \nu^{11} + 2148166140275 \nu^{10} - 9264008158129 \nu^{9} - 11285354004386 \nu^{8} + 65976864749420 \nu^{7} - 15627298002614 \nu^{6} - 280505779827521 \nu^{5} + 632861897955195 \nu^{4} - 685350454209814 \nu^{3} + 432096143403529 \nu^{2} - 162853890704445 \nu + 36969480533544\)\()/ 2648448154170 \)
\(\beta_{4}\)\(=\)\((\)\(6939940550 \nu^{15} - 489734378 \nu^{14} - 79452033908 \nu^{13} - 161040415944 \nu^{12} + 982921499997 \nu^{11} + 730190682377 \nu^{10} - 5532154780559 \nu^{9} - 6603544904488 \nu^{8} + 52540277384872 \nu^{7} - 79779869477764 \nu^{6} + 17387330869812 \nu^{5} + 80707484425902 \nu^{4} - 89523726958339 \nu^{3} + 32060282480337 \nu^{2} - 12117424237953 \nu + 10360850614686\)\()/ 2648448154170 \)
\(\beta_{5}\)\(=\)\((\)\(-8517854635 \nu^{15} + 47901289016 \nu^{14} + 10808846529 \nu^{13} - 254360686162 \nu^{12} - 1466326909149 \nu^{11} + 6905401670414 \nu^{10} - 814254572587 \nu^{9} - 25080362663489 \nu^{8} - 51676379354911 \nu^{7} + 473690676980758 \nu^{6} - 1192570494771599 \nu^{5} + 1642118210452914 \nu^{4} - 1379621746034317 \nu^{3} + 705130916855266 \nu^{2} - 200992392789411 \nu + 26162339620443\)\()/ 2648448154170 \)
\(\beta_{6}\)\(=\)\((\)\(-10954917349 \nu^{15} + 11841164258 \nu^{14} + 138130967169 \nu^{13} + 123632263943 \nu^{12} - 1940234326557 \nu^{11} + 129659506964 \nu^{10} + 11668264098683 \nu^{9} + 2213398877473 \nu^{8} - 101554428867421 \nu^{7} + 199436126741848 \nu^{6} - 64996936610357 \nu^{5} - 287786946129621 \nu^{4} + 499088374704863 \nu^{3} - 386333932086932 \nu^{2} + 158881305502329 \nu - 31317616718193\)\()/ 2648448154170 \)
\(\beta_{7}\)\(=\)\((\)\(1683778105 \nu^{15} - 2653967414 \nu^{14} - 17809640997 \nu^{13} - 13603646729 \nu^{12} + 286447821441 \nu^{11} - 172395647507 \nu^{10} - 1376673058709 \nu^{9} - 14364679084 \nu^{8} + 14179299874903 \nu^{7} - 36953946144364 \nu^{6} + 44710115365661 \nu^{5} - 29789710852077 \nu^{4} + 13481470583671 \nu^{3} - 8129675739049 \nu^{2} + 5486210722353 \nu - 1451250345714\)\()/ 378349736310 \)
\(\beta_{8}\)\(=\)\((\)\(58793534 \nu^{15} - 247088261 \nu^{14} - 186701243 \nu^{13} + 925377984 \nu^{12} + 9206342304 \nu^{11} - 34540664458 \nu^{10} - 1337656541 \nu^{9} + 115013085509 \nu^{8} + 345132534322 \nu^{7} - 2630197201531 \nu^{6} + 6496113860289 \nu^{5} - 8909194143438 \nu^{4} + 7427499314864 \nu^{3} - 3732355604436 \nu^{2} + 1074418113927 \nu - 169644585825\)\()/ 11081373030 \)
\(\beta_{9}\)\(=\)\((\)\(2053516175 \nu^{15} - 8267524636 \nu^{14} - 7910711853 \nu^{13} + 29044383254 \nu^{12} + 327126906459 \nu^{11} - 1131332511223 \nu^{10} - 205259572141 \nu^{9} + 3717296483509 \nu^{8} + 12657496029497 \nu^{7} - 88386086837336 \nu^{6} + 212751901268509 \nu^{5} - 288215409944898 \nu^{4} + 239544740552129 \nu^{3} - 121253930827751 \nu^{2} + 35732535181197 \nu - 5795645027601\)\()/ 378349736310 \)
\(\beta_{10}\)\(=\)\((\)\(-15731411194 \nu^{15} + 64536858778 \nu^{14} + 74366831386 \nu^{13} - 224596420089 \nu^{12} - 2680722251562 \nu^{11} + 8348914959986 \nu^{10} + 3165045634846 \nu^{9} - 26385199034992 \nu^{8} - 105149224552934 \nu^{7} + 663696947741006 \nu^{6} - 1566177154674402 \nu^{5} + 2135855292432891 \nu^{4} - 1800341639918254 \nu^{3} + 910012532192418 \nu^{2} - 247428058699014 \nu + 33091548064380\)\()/ 2648448154170 \)
\(\beta_{11}\)\(=\)\((\)\(-394610 \nu^{15} + 1397252 \nu^{14} + 2812046 \nu^{13} - 4855143 \nu^{12} - 71847798 \nu^{11} + 173697436 \nu^{10} + 220205102 \nu^{9} - 621978803 \nu^{8} - 3201338854 \nu^{7} + 15191883352 \nu^{6} - 28706014158 \nu^{5} + 30360602271 \nu^{4} - 18935670938 \nu^{3} + 6793325772 \nu^{2} - 1466540094 \nu + 209664117\)\()/32307210\)
\(\beta_{12}\)\(=\)\((\)\(5555945704 \nu^{15} - 12683893446 \nu^{14} - 51058434102 \nu^{13} - 5680074782 \nu^{12} + 954757786289 \nu^{11} - 1256911149637 \nu^{10} - 3836441442682 \nu^{9} + 3210138065694 \nu^{8} + 45084588651418 \nu^{7} - 155963493715472 \nu^{6} + 248666737552669 \nu^{5} - 228244343991577 \nu^{4} + 123536409068243 \nu^{3} - 38369553659586 \nu^{2} + 7264577187093 \nu - 550491674226\)\()/ 441408025695 \)
\(\beta_{13}\)\(=\)\((\)\(-34807975271 \nu^{15} + 87110547559 \nu^{14} + 303155003931 \nu^{13} - 35455767722 \nu^{12} - 5998145503491 \nu^{11} + 9176128816621 \nu^{10} + 22425615715333 \nu^{9} - 25378266515371 \nu^{8} - 278639362190279 \nu^{7} + 1038716795984663 \nu^{6} - 1766588211189121 \nu^{5} + 1759996937587836 \nu^{4} - 1077064039952387 \nu^{3} + 404857061994299 \nu^{2} - 93916582841499 \nu + 9989689919079\)\()/ 2648448154170 \)
\(\beta_{14}\)\(=\)\((\)\(37024534763 \nu^{15} - 117279863141 \nu^{14} - 278878852346 \nu^{13} + 286708401207 \nu^{12} + 6550796205339 \nu^{11} - 13918910002351 \nu^{10} - 20653048094978 \nu^{9} + 45252236888369 \nu^{8} + 294321980014789 \nu^{7} - 1303927051609243 \nu^{6} + 2454306807909024 \nu^{5} - 2696309558207121 \nu^{4} + 1862429268634547 \nu^{3} - 827380101175731 \nu^{2} + 248779837958034 \nu - 41746013359011\)\()/ 2648448154170 \)
\(\beta_{15}\)\(=\)\((\)\(-62968587943 \nu^{15} + 235645881604 \nu^{14} + 369876667756 \nu^{13} - 753064240620 \nu^{12} - 10928412057036 \nu^{11} + 29737319342261 \nu^{10} + 22220875870225 \nu^{9} - 96167078872606 \nu^{8} - 456657473810939 \nu^{7} + 2494464102157640 \nu^{6} - 5413835800346586 \nu^{5} + 6874644775014066 \nu^{4} - 5513797579347700 \nu^{3} + 2778386358592989 \nu^{2} - 828557161276569 \nu + 128382868343808\)\()/ 2648448154170 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2} + 3 \beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{2} - 9 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-17 \beta_{15} - 7 \beta_{14} + 29 \beta_{13} + 18 \beta_{12} + 2 \beta_{11} + 13 \beta_{10} - 19 \beta_{9} + 18 \beta_{8} - \beta_{7} - 6 \beta_{6} + 11 \beta_{5} - 5 \beta_{4} - 12 \beta_{3} - 4 \beta_{2} + 45 \beta_{1} + 51\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(6 \beta_{15} + 12 \beta_{14} - 30 \beta_{13} - 30 \beta_{12} + 14 \beta_{11} - 5 \beta_{10} + 60 \beta_{9} - 48 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} + 6 \beta_{4} + 12 \beta_{3} - 18 \beta_{2} - 21 \beta_{1} - 66\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(35 \beta_{15} + 5 \beta_{14} - 51 \beta_{13} - 60 \beta_{12} - 90 \beta_{11} - 45 \beta_{10} - 363 \beta_{9} + 300 \beta_{8} - 63 \beta_{7} - 90 \beta_{6} + 39 \beta_{5} + 55 \beta_{4} - 60 \beta_{3} + 130 \beta_{2} - 153 \beta_{1} + 225\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(-188 \beta_{15} - 49 \beta_{14} + 620 \beta_{13} + 495 \beta_{12} + 74 \beta_{11} + 103 \beta_{10} + 188 \beta_{9} - 156 \beta_{8} + 2 \beta_{7} + 45 \beta_{6} + 125 \beta_{5} - 8 \beta_{4} - 60 \beta_{3} - 169 \beta_{2} + 450 \beta_{1} + 198\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(851 \beta_{15} + 185 \beta_{14} - 5865 \beta_{13} - 4872 \beta_{12} + 324 \beta_{11} - 237 \beta_{10} + 2745 \beta_{9} - 2268 \beta_{8} + 267 \beta_{7} + 180 \beta_{6} - 573 \beta_{5} - 437 \beta_{4} + 258 \beta_{3} + 64 \beta_{2} - 627 \beta_{1} - 1677\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-140 \beta_{15} + 1192 \beta_{14} + 7656 \beta_{13} + 6252 \beta_{12} - 770 \beta_{11} - 904 \beta_{10} - 9204 \beta_{9} + 7560 \beta_{8} - 1224 \beta_{7} - 756 \beta_{6} + 1128 \beta_{5} + 1796 \beta_{4} - 144 \beta_{3} + 908 \beta_{2} - 3174 \beta_{1} + 105\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-6493 \beta_{15} - 15521 \beta_{14} - 9587 \beta_{13} - 8064 \beta_{12} - 952 \beta_{11} + 7789 \beta_{10} + 59095 \beta_{9} - 48600 \beta_{8} + 7435 \beta_{7} - 1416 \beta_{6} - 623 \beta_{5} - 8389 \beta_{4} - 8340 \beta_{3} - 5060 \beta_{2} + 31455 \beta_{1} + 36681\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(7580 \beta_{15} + 24505 \beta_{14} - 55682 \beta_{13} - 46005 \beta_{12} + 19685 \beta_{11} - 1310 \beta_{10} - 41807 \beta_{9} + 34410 \beta_{8} - 5237 \beta_{7} + 15465 \beta_{6} - 6437 \beta_{5} - 5320 \beta_{4} + 16050 \beta_{3} - 6965 \beta_{2} - 3861 \beta_{1} - 78336\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(56503 \beta_{15} - 30049 \beta_{14} + 570965 \beta_{13} + 468402 \beta_{12} - 189892 \beta_{11} - 134003 \beta_{10} - 199153 \beta_{9} + 164010 \beta_{8} - 24583 \beta_{7} - 159090 \beta_{6} + 75503 \beta_{5} + 195103 \beta_{4} - 50880 \beta_{3} + 133742 \beta_{2} - 514143 \beta_{1} + 228525\)\()/6\)
\(\nu^{12}\)\(=\)\((\)\(-286602 \beta_{15} - 255672 \beta_{14} - 640926 \beta_{13} - 528462 \beta_{12} + 211166 \beta_{11} + 383911 \beta_{10} + 766044 \beta_{9} - 629628 \beta_{8} + 98226 \beta_{7} + 170946 \beta_{6} - 77814 \beta_{5} - 437610 \beta_{4} - 147024 \beta_{3} - 214854 \beta_{2} + 1492425 \beta_{1} + 681258\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(2083883 \beta_{15} + 3003875 \beta_{14} - 202827 \beta_{13} - 173628 \beta_{12} + 512796 \beta_{11} - 2194869 \beta_{10} - 3834117 \beta_{9} + 3154164 \beta_{8} - 483081 \beta_{7} + 412518 \beta_{6} - 8841 \beta_{5} + 2046391 \beta_{4} + 2205828 \beta_{3} - 5156 \beta_{2} - 8472357 \beta_{1} - 10435617\)\()/6\)
\(\nu^{14}\)\(=\)\((\)\(-1254626 \beta_{15} - 4365001 \beta_{14} + 5874734 \beta_{13} + 4823145 \beta_{12} - 4106908 \beta_{11} + 575965 \beta_{10} - 433264 \beta_{9} + 357630 \beta_{8} - 51064 \beta_{7} - 3384651 \beta_{6} + 765185 \beta_{5} + 163498 \beta_{4} - 3873486 \beta_{3} + 3211385 \beta_{2} + 2268096 \beta_{1} + 18192720\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(-15892705 \beta_{15} + 7986815 \beta_{14} - 33583785 \beta_{13} - 27649920 \beta_{12} + 34806450 \beta_{11} + 21093315 \beta_{10} + 40993695 \beta_{9} - 33703560 \beta_{8} + 5224605 \beta_{7} + 28597260 \beta_{6} - 4184565 \beta_{5} - 23876285 \beta_{4} + 11173710 \beta_{3} - 33700730 \beta_{2} + 81886725 \beta_{1} - 52856019\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
2.22190 + 0.111032i
1.51479 1.11371i
−2.89591 + 1.43281i
−2.18880 + 2.65755i
0.752308 0.673492i
1.45942 + 0.551253i
0.921698 + 0.861704i
0.214591 0.363041i
2.22190 0.111032i
1.51479 + 1.11371i
−2.89591 1.43281i
−2.18880 2.65755i
0.752308 + 0.673492i
1.45942 0.551253i
0.921698 0.861704i
0.214591 + 0.363041i
−1.48938 + 2.57968i 1.18073 + 2.75787i −2.43649 4.22013i 3.23955 + 3.80858i −8.87298 1.06161i 5.16858 + 4.72078i 2.60040 −6.21175 + 6.51262i −14.6498 + 2.68458i
44.2 −1.48938 + 2.57968i 1.79802 + 2.40148i −2.43649 4.22013i −4.91811 0.901243i −8.87298 + 1.06161i −5.16858 4.72078i 2.60040 −2.53422 + 8.63584i 9.64983 11.3448i
44.3 −0.530805 + 0.919382i −1.89916 + 2.32232i 1.43649 + 2.48808i −0.901243 + 4.91811i −1.12702 2.97876i 3.04726 6.30192i −7.29643 −1.78637 8.82093i −4.04323 3.43914i
44.4 −0.530805 + 0.919382i 2.96077 0.483560i 1.43649 + 2.48808i −3.80858 + 3.23955i −1.12702 + 2.97876i −3.04726 + 6.30192i −7.29643 8.53234 2.86342i −0.956769 5.22111i
44.5 0.530805 0.919382i −2.96077 + 0.483560i 1.43649 + 2.48808i 0.901243 4.91811i −1.12702 + 2.97876i 3.04726 6.30192i 7.29643 8.53234 2.86342i −4.04323 3.43914i
44.6 0.530805 0.919382i 1.89916 2.32232i 1.43649 + 2.48808i 3.80858 3.23955i −1.12702 2.97876i −3.04726 + 6.30192i 7.29643 −1.78637 8.82093i −0.956769 5.22111i
44.7 1.48938 2.57968i −1.79802 2.40148i −2.43649 4.22013i −3.23955 3.80858i −8.87298 + 1.06161i 5.16858 + 4.72078i −2.60040 −2.53422 + 8.63584i −14.6498 + 2.68458i
44.8 1.48938 2.57968i −1.18073 2.75787i −2.43649 4.22013i 4.91811 + 0.901243i −8.87298 1.06161i −5.16858 4.72078i −2.60040 −6.21175 + 6.51262i 9.64983 11.3448i
74.1 −1.48938 2.57968i 1.18073 2.75787i −2.43649 + 4.22013i 3.23955 3.80858i −8.87298 + 1.06161i 5.16858 4.72078i 2.60040 −6.21175 6.51262i −14.6498 2.68458i
74.2 −1.48938 2.57968i 1.79802 2.40148i −2.43649 + 4.22013i −4.91811 + 0.901243i −8.87298 1.06161i −5.16858 + 4.72078i 2.60040 −2.53422 8.63584i 9.64983 + 11.3448i
74.3 −0.530805 0.919382i −1.89916 2.32232i 1.43649 2.48808i −0.901243 4.91811i −1.12702 + 2.97876i 3.04726 + 6.30192i −7.29643 −1.78637 + 8.82093i −4.04323 + 3.43914i
74.4 −0.530805 0.919382i 2.96077 + 0.483560i 1.43649 2.48808i −3.80858 3.23955i −1.12702 2.97876i −3.04726 6.30192i −7.29643 8.53234 + 2.86342i −0.956769 + 5.22111i
74.5 0.530805 + 0.919382i −2.96077 0.483560i 1.43649 2.48808i 0.901243 + 4.91811i −1.12702 2.97876i 3.04726 + 6.30192i 7.29643 8.53234 + 2.86342i −4.04323 + 3.43914i
74.6 0.530805 + 0.919382i 1.89916 + 2.32232i 1.43649 2.48808i 3.80858 + 3.23955i −1.12702 + 2.97876i −3.04726 6.30192i 7.29643 −1.78637 + 8.82093i −0.956769 + 5.22111i
74.7 1.48938 + 2.57968i −1.79802 + 2.40148i −2.43649 + 4.22013i −3.23955 + 3.80858i −8.87298 1.06161i 5.16858 4.72078i −2.60040 −2.53422 8.63584i −14.6498 2.68458i
74.8 1.48938 + 2.57968i −1.18073 + 2.75787i −2.43649 + 4.22013i 4.91811 0.901243i −8.87298 + 1.06161i −5.16858 + 4.72078i −2.60040 −6.21175 6.51262i 9.64983 + 11.3448i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 74.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.o.a 16
3.b odd 2 1 inner 105.3.o.a 16
5.b even 2 1 inner 105.3.o.a 16
7.c even 3 1 inner 105.3.o.a 16
15.d odd 2 1 inner 105.3.o.a 16
21.h odd 6 1 inner 105.3.o.a 16
35.j even 6 1 inner 105.3.o.a 16
105.o odd 6 1 inner 105.3.o.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.o.a 16 1.a even 1 1 trivial
105.3.o.a 16 3.b odd 2 1 inner
105.3.o.a 16 5.b even 2 1 inner
105.3.o.a 16 7.c even 3 1 inner
105.3.o.a 16 15.d odd 2 1 inner
105.3.o.a 16 21.h odd 6 1 inner
105.3.o.a 16 35.j even 6 1 inner
105.3.o.a 16 105.o odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10 T_{2}^{6} + 90 T_{2}^{4} + 100 T_{2}^{2} + 100 \) acting on \(S_{3}^{\mathrm{new}}(105, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 6 T^{2} + 10 T^{4} + 36 T^{6} - 156 T^{8} + 576 T^{10} + 2560 T^{12} - 24576 T^{14} + 65536 T^{16} )^{2} \)
$3$ \( 1 + 4 T^{2} + 90 T^{4} - 944 T^{6} - 2381 T^{8} - 76464 T^{10} + 590490 T^{12} + 2125764 T^{14} + 43046721 T^{16} \)
$5$ \( 1 + 250 T^{4} - 328125 T^{8} + 97656250 T^{12} + 152587890625 T^{16} \)
$7$ \( ( 1 + 52 T^{2} + 4263 T^{4} + 124852 T^{6} + 5764801 T^{8} )^{2} \)
$11$ \( ( 1 + 334 T^{2} + 57760 T^{4} + 8187676 T^{6} + 1046359339 T^{8} + 119875764316 T^{10} + 12381368966560 T^{12} + 1048235077824814 T^{14} + 45949729863572161 T^{16} )^{2} \)
$13$ \( ( 1 - 188 T^{2} + 20583 T^{4} - 5369468 T^{6} + 815730721 T^{8} )^{4} \)
$17$ \( ( 1 - 906 T^{2} + 463000 T^{4} - 172859364 T^{6} + 52513801899 T^{8} - 14437386940644 T^{10} + 3229775695183000 T^{12} - 527855746930163466 T^{14} + 48661191875666868481 T^{16} )^{2} \)
$19$ \( ( 1 - 12 T - 599 T^{2} - 252 T^{3} + 369744 T^{4} - 90972 T^{5} - 78062279 T^{6} - 564550572 T^{7} + 16983563041 T^{8} )^{4} \)
$23$ \( ( 1 - 1066 T^{2} + 315400 T^{4} - 278518084 T^{6} + 277658769259 T^{8} - 77940779144644 T^{10} + 24699284757627400 T^{12} - 23360989644533662186 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \)
$29$ \( ( 1 - 2394 T^{2} + 2753756 T^{4} - 1693230714 T^{6} + 500246412961 T^{8} )^{4} \)
$31$ \( ( 1 - 86 T + 3685 T^{2} - 153854 T^{3} + 5740444 T^{4} - 147853694 T^{5} + 3403174885 T^{6} - 76325316566 T^{7} + 852891037441 T^{8} )^{4} \)
$37$ \( ( 1 + 2468 T^{2} + 1780081 T^{4} + 1388548628 T^{6} + 3656190464464 T^{8} + 2602363685201108 T^{10} + 6252497938815147601 T^{12} + \)\(16\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!41\)\( T^{16} )^{2} \)
$41$ \( ( 1 - 3054 T^{2} + 6815636 T^{4} - 8629874094 T^{6} + 7984925229121 T^{8} )^{4} \)
$43$ \( ( 1 - 4124 T^{2} + 11073111 T^{4} - 14099135324 T^{6} + 11688200277601 T^{8} )^{4} \)
$47$ \( ( 1 - 7336 T^{2} + 31122250 T^{4} - 94893243424 T^{6} + 228626385513379 T^{8} - 463048756964467744 T^{10} + \)\(74\!\cdots\!50\)\( T^{12} - \)\(85\!\cdots\!76\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} )^{2} \)
$53$ \( ( 1 - 7636 T^{2} + 28650250 T^{4} - 105966940624 T^{6} + 357269785558579 T^{8} - 836130131621800144 T^{10} + \)\(17\!\cdots\!50\)\( T^{12} - \)\(37\!\cdots\!76\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} )^{2} \)
$59$ \( ( 1 + 474 T^{2} + 12860200 T^{4} - 17476496604 T^{6} + 10103548950219 T^{8} - 211769018365942044 T^{10} + \)\(18\!\cdots\!00\)\( T^{12} + \)\(84\!\cdots\!94\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \)
$61$ \( ( 1 + 98 T + 136 T^{2} + 198548 T^{3} + 40060699 T^{4} + 738797108 T^{5} + 1883034376 T^{6} + 5048996687378 T^{7} + 191707312997281 T^{8} )^{4} \)
$67$ \( ( 1 + 12548 T^{2} + 81642721 T^{4} + 445546114868 T^{6} + 2168184767444464 T^{8} + 8978253671784967028 T^{10} + \)\(33\!\cdots\!61\)\( T^{12} + \)\(10\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} )^{2} \)
$71$ \( ( 1 - 13674 T^{2} + 87943916 T^{4} - 347479325994 T^{6} + 645753531245761 T^{8} )^{4} \)
$73$ \( ( 1 + 10828 T^{2} + 58596841 T^{4} + 20056282108 T^{6} - 696353387031776 T^{8} + 569563132866972028 T^{10} + \)\(47\!\cdots\!21\)\( T^{12} + \)\(24\!\cdots\!88\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} )^{2} \)
$79$ \( ( 1 - 152 T + 4981 T^{2} - 857432 T^{3} + 145300984 T^{4} - 5351233112 T^{5} + 194010353461 T^{6} - 36949293239192 T^{7} + 1517108809906561 T^{8} )^{4} \)
$83$ \( ( 1 + 7386 T^{2} + 83632076 T^{4} + 350527158906 T^{6} + 2252292232139041 T^{8} )^{4} \)
$89$ \( ( 1 - 1586 T^{2} - 121555520 T^{4} + 2241915676 T^{6} + 11299181809334539 T^{8} + 140662813645269916 T^{10} - \)\(47\!\cdots\!20\)\( T^{12} - \)\(39\!\cdots\!06\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \)
$97$ \( ( 1 - 37448 T^{2} + 527638098 T^{4} - 3315244514888 T^{6} + 7837433594376961 T^{8} )^{4} \)
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