Properties

Label 2736.2.dc.e
Level $2736$
Weight $2$
Character orbit 2736.dc
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + 827916 x^{8} + 179664 x^{7} - 1551960 x^{6} - 1894880 x^{5} + 305564 x^{4} + 2772832 x^{3} + 3072904 x^{2} + 1645728 x + 414864\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{5} + ( -\beta_{7} - \beta_{10} ) q^{7} +O(q^{10})\) \( q + \beta_{14} q^{5} + ( -\beta_{7} - \beta_{10} ) q^{7} + ( \beta_{6} + \beta_{18} ) q^{11} + ( 1 - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{18} ) q^{13} + ( -\beta_{1} + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{9} + \beta_{13} - 2 \beta_{14} + \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{17} + ( 2 \beta_{1} - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{17} - \beta_{19} ) q^{19} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{25} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{29} + ( -1 - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{18} ) q^{31} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{15} - \beta_{18} + \beta_{19} ) q^{35} + ( \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{17} - \beta_{18} ) q^{37} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{18} - \beta_{19} ) q^{41} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{11} - 3 \beta_{13} + 3 \beta_{14} - 2 \beta_{16} + 2 \beta_{18} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{14} - \beta_{16} - \beta_{17} - \beta_{19} ) q^{47} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{49} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} - \beta_{16} + 3 \beta_{18} - 3 \beta_{19} ) q^{53} + ( \beta_{2} + 3 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{55} + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{14} - \beta_{16} + 2 \beta_{18} ) q^{59} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{12} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{19} ) q^{61} + ( \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - 2 \beta_{16} + \beta_{18} - \beta_{19} ) q^{65} + ( -1 + 3 \beta_{2} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} - \beta_{19} ) q^{67} + ( -1 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{14} + 3 \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{71} + ( 2 + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{16} - 2 \beta_{18} - 2 \beta_{19} ) q^{73} + ( 3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} - 3 \beta_{16} + \beta_{17} ) q^{77} + ( -4 - 2 \beta_{1} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - \beta_{11} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{79} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{16} + \beta_{17} ) q^{83} + ( -6 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{85} + ( 4 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 6 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{89} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{91} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - 4 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{95} + ( 2 + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} + \beta_{13} - 4 \beta_{14} + \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{7} + O(q^{10}) \) \( 20q + 4q^{7} + 6q^{13} - 12q^{17} - 4q^{19} + 14q^{25} - 12q^{35} - 8q^{41} + 2q^{43} - 36q^{47} + 32q^{49} + 8q^{53} - 12q^{55} + 8q^{59} - 2q^{61} - 8q^{65} - 30q^{67} - 4q^{71} - 22q^{73} - 54q^{79} - 4q^{85} + 32q^{89} - 18q^{91} - 32q^{95} + 12q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + 30728 x^{12} - 110512 x^{11} - 211368 x^{10} + 231024 x^{9} + 827916 x^{8} + 179664 x^{7} - 1551960 x^{6} - 1894880 x^{5} + 305564 x^{4} + 2772832 x^{3} + 3072904 x^{2} + 1645728 x + 414864\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(16\!\cdots\!17\)\( \nu^{19} - \)\(14\!\cdots\!14\)\( \nu^{18} - \)\(16\!\cdots\!50\)\( \nu^{17} + \)\(39\!\cdots\!54\)\( \nu^{16} - \)\(16\!\cdots\!04\)\( \nu^{15} - \)\(51\!\cdots\!44\)\( \nu^{14} + \)\(77\!\cdots\!76\)\( \nu^{13} + \)\(39\!\cdots\!68\)\( \nu^{12} + \)\(15\!\cdots\!36\)\( \nu^{11} - \)\(19\!\cdots\!32\)\( \nu^{10} - \)\(17\!\cdots\!72\)\( \nu^{9} + \)\(51\!\cdots\!16\)\( \nu^{8} + \)\(89\!\cdots\!16\)\( \nu^{7} - \)\(41\!\cdots\!00\)\( \nu^{6} - \)\(20\!\cdots\!96\)\( \nu^{5} - \)\(13\!\cdots\!04\)\( \nu^{4} + \)\(14\!\cdots\!16\)\( \nu^{3} + \)\(31\!\cdots\!48\)\( \nu^{2} + \)\(25\!\cdots\!16\)\( \nu + \)\(88\!\cdots\!20\)\(\)\()/ \)\(13\!\cdots\!68\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(13\!\cdots\!15\)\( \nu^{19} + \)\(26\!\cdots\!19\)\( \nu^{18} - \)\(16\!\cdots\!28\)\( \nu^{17} + \)\(46\!\cdots\!10\)\( \nu^{16} + \)\(33\!\cdots\!05\)\( \nu^{15} - \)\(10\!\cdots\!26\)\( \nu^{14} - \)\(34\!\cdots\!49\)\( \nu^{13} + \)\(10\!\cdots\!22\)\( \nu^{12} + \)\(24\!\cdots\!49\)\( \nu^{11} - \)\(54\!\cdots\!14\)\( \nu^{10} - \)\(12\!\cdots\!20\)\( \nu^{9} + \)\(14\!\cdots\!16\)\( \nu^{8} + \)\(43\!\cdots\!76\)\( \nu^{7} - \)\(55\!\cdots\!08\)\( \nu^{6} - \)\(85\!\cdots\!72\)\( \nu^{5} - \)\(58\!\cdots\!40\)\( \nu^{4} + \)\(58\!\cdots\!02\)\( \nu^{3} + \)\(11\!\cdots\!60\)\( \nu^{2} + \)\(73\!\cdots\!16\)\( \nu + \)\(15\!\cdots\!44\)\(\)\()/ \)\(73\!\cdots\!72\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(10\!\cdots\!53\)\( \nu^{19} - \)\(12\!\cdots\!97\)\( \nu^{18} + \)\(11\!\cdots\!58\)\( \nu^{17} - \)\(34\!\cdots\!58\)\( \nu^{16} - \)\(23\!\cdots\!74\)\( \nu^{15} + \)\(78\!\cdots\!86\)\( \nu^{14} + \)\(24\!\cdots\!84\)\( \nu^{13} - \)\(76\!\cdots\!40\)\( \nu^{12} - \)\(17\!\cdots\!54\)\( \nu^{11} + \)\(38\!\cdots\!18\)\( \nu^{10} + \)\(91\!\cdots\!80\)\( \nu^{9} - \)\(96\!\cdots\!88\)\( \nu^{8} - \)\(31\!\cdots\!92\)\( \nu^{7} + \)\(20\!\cdots\!20\)\( \nu^{6} + \)\(62\!\cdots\!40\)\( \nu^{5} + \)\(46\!\cdots\!44\)\( \nu^{4} - \)\(38\!\cdots\!32\)\( \nu^{3} - \)\(86\!\cdots\!60\)\( \nu^{2} - \)\(58\!\cdots\!52\)\( \nu - \)\(15\!\cdots\!36\)\(\)\()/ \)\(49\!\cdots\!48\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(43\!\cdots\!24\)\( \nu^{19} - \)\(48\!\cdots\!49\)\( \nu^{18} + \)\(52\!\cdots\!88\)\( \nu^{17} + \)\(10\!\cdots\!12\)\( \nu^{16} - \)\(22\!\cdots\!00\)\( \nu^{15} - \)\(11\!\cdots\!32\)\( \nu^{14} + \)\(26\!\cdots\!88\)\( \nu^{13} + \)\(84\!\cdots\!10\)\( \nu^{12} - \)\(15\!\cdots\!84\)\( \nu^{11} - \)\(46\!\cdots\!22\)\( \nu^{10} + \)\(47\!\cdots\!72\)\( \nu^{9} + \)\(18\!\cdots\!68\)\( \nu^{8} - \)\(45\!\cdots\!20\)\( \nu^{7} - \)\(46\!\cdots\!64\)\( \nu^{6} - \)\(17\!\cdots\!88\)\( \nu^{5} + \)\(65\!\cdots\!76\)\( \nu^{4} + \)\(64\!\cdots\!44\)\( \nu^{3} - \)\(22\!\cdots\!36\)\( \nu^{2} - \)\(60\!\cdots\!36\)\( \nu - \)\(29\!\cdots\!00\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(45\!\cdots\!15\)\( \nu^{19} + \)\(45\!\cdots\!37\)\( \nu^{18} + \)\(27\!\cdots\!78\)\( \nu^{17} - \)\(11\!\cdots\!67\)\( \nu^{16} + \)\(14\!\cdots\!02\)\( \nu^{15} + \)\(13\!\cdots\!32\)\( \nu^{14} - \)\(19\!\cdots\!90\)\( \nu^{13} - \)\(10\!\cdots\!72\)\( \nu^{12} + \)\(10\!\cdots\!62\)\( \nu^{11} + \)\(47\!\cdots\!02\)\( \nu^{10} - \)\(21\!\cdots\!12\)\( \nu^{9} - \)\(14\!\cdots\!82\)\( \nu^{8} - \)\(35\!\cdots\!80\)\( \nu^{7} + \)\(25\!\cdots\!04\)\( \nu^{6} + \)\(25\!\cdots\!24\)\( \nu^{5} - \)\(96\!\cdots\!60\)\( \nu^{4} - \)\(37\!\cdots\!44\)\( \nu^{3} - \)\(33\!\cdots\!12\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu - \)\(35\!\cdots\!84\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{6}\)\(=\)\((\)\(155949130665898008 \nu^{19} - 1426701888785846434 \nu^{18} - 1021833237456926570 \nu^{17} + 35737623454942883911 \nu^{16} - 6909492216299799072 \nu^{15} - 448971481003687507060 \nu^{14} + 11658639337487252752 \nu^{13} + 3458587877297465561562 \nu^{12} + 1726734584144648759712 \nu^{11} - 16916575613597789414324 \nu^{10} - 18666595158931711807576 \nu^{9} + 46715454745291975296686 \nu^{8} + 91165988185656377603072 \nu^{7} - 41057110277847312776644 \nu^{6} - 215042516118489255052816 \nu^{5} - 122804429110328283320416 \nu^{4} + 174384646298544210182272 \nu^{3} + 305343803120447027306372 \nu^{2} + 194224008953817686819344 \nu + 44015430934707192303408\)\()/ \)\(34\!\cdots\!72\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(66\!\cdots\!42\)\( \nu^{19} + \)\(64\!\cdots\!99\)\( \nu^{18} + \)\(43\!\cdots\!10\)\( \nu^{17} - \)\(18\!\cdots\!31\)\( \nu^{16} + \)\(15\!\cdots\!74\)\( \nu^{15} + \)\(25\!\cdots\!88\)\( \nu^{14} - \)\(27\!\cdots\!52\)\( \nu^{13} - \)\(20\!\cdots\!98\)\( \nu^{12} + \)\(18\!\cdots\!12\)\( \nu^{11} + \)\(10\!\cdots\!86\)\( \nu^{10} - \)\(55\!\cdots\!16\)\( \nu^{9} - \)\(38\!\cdots\!50\)\( \nu^{8} + \)\(69\!\cdots\!16\)\( \nu^{7} + \)\(85\!\cdots\!40\)\( \nu^{6} + \)\(43\!\cdots\!52\)\( \nu^{5} - \)\(10\!\cdots\!92\)\( \nu^{4} - \)\(11\!\cdots\!48\)\( \nu^{3} + \)\(10\!\cdots\!48\)\( \nu^{2} + \)\(84\!\cdots\!48\)\( \nu + \)\(48\!\cdots\!00\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(35\!\cdots\!12\)\( \nu^{19} - \)\(32\!\cdots\!98\)\( \nu^{18} - \)\(19\!\cdots\!88\)\( \nu^{17} + \)\(82\!\cdots\!41\)\( \nu^{16} - \)\(37\!\cdots\!05\)\( \nu^{15} - \)\(99\!\cdots\!89\)\( \nu^{14} + \)\(34\!\cdots\!63\)\( \nu^{13} + \)\(71\!\cdots\!83\)\( \nu^{12} + \)\(15\!\cdots\!04\)\( \nu^{11} - \)\(31\!\cdots\!00\)\( \nu^{10} - \)\(32\!\cdots\!44\)\( \nu^{9} + \)\(76\!\cdots\!88\)\( \nu^{8} + \)\(17\!\cdots\!72\)\( \nu^{7} - \)\(12\!\cdots\!40\)\( \nu^{6} - \)\(39\!\cdots\!54\)\( \nu^{5} - \)\(38\!\cdots\!14\)\( \nu^{4} + \)\(21\!\cdots\!48\)\( \nu^{3} + \)\(72\!\cdots\!00\)\( \nu^{2} + \)\(60\!\cdots\!88\)\( \nu + \)\(22\!\cdots\!60\)\(\)\()/ \)\(73\!\cdots\!72\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(24\!\cdots\!12\)\( \nu^{19} - \)\(30\!\cdots\!39\)\( \nu^{18} + \)\(67\!\cdots\!58\)\( \nu^{17} + \)\(52\!\cdots\!86\)\( \nu^{16} - \)\(20\!\cdots\!12\)\( \nu^{15} - \)\(40\!\cdots\!68\)\( \nu^{14} + \)\(22\!\cdots\!90\)\( \nu^{13} + \)\(18\!\cdots\!02\)\( \nu^{12} - \)\(13\!\cdots\!12\)\( \nu^{11} - \)\(80\!\cdots\!78\)\( \nu^{10} + \)\(49\!\cdots\!92\)\( \nu^{9} + \)\(33\!\cdots\!12\)\( \nu^{8} - \)\(11\!\cdots\!68\)\( \nu^{7} - \)\(10\!\cdots\!84\)\( \nu^{6} + \)\(13\!\cdots\!64\)\( \nu^{5} + \)\(20\!\cdots\!44\)\( \nu^{4} - \)\(12\!\cdots\!16\)\( \nu^{3} - \)\(15\!\cdots\!36\)\( \nu^{2} - \)\(92\!\cdots\!92\)\( \nu - \)\(96\!\cdots\!60\)\(\)\()/ \)\(49\!\cdots\!48\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(74\!\cdots\!97\)\( \nu^{19} + \)\(80\!\cdots\!05\)\( \nu^{18} - \)\(65\!\cdots\!50\)\( \nu^{17} - \)\(17\!\cdots\!95\)\( \nu^{16} + \)\(32\!\cdots\!98\)\( \nu^{15} + \)\(19\!\cdots\!76\)\( \nu^{14} - \)\(37\!\cdots\!48\)\( \nu^{13} - \)\(14\!\cdots\!22\)\( \nu^{12} + \)\(20\!\cdots\!40\)\( \nu^{11} + \)\(78\!\cdots\!58\)\( \nu^{10} - \)\(57\!\cdots\!36\)\( \nu^{9} - \)\(28\!\cdots\!58\)\( \nu^{8} + \)\(22\!\cdots\!00\)\( \nu^{7} + \)\(66\!\cdots\!48\)\( \nu^{6} + \)\(31\!\cdots\!52\)\( \nu^{5} - \)\(77\!\cdots\!16\)\( \nu^{4} - \)\(84\!\cdots\!12\)\( \nu^{3} + \)\(95\!\cdots\!28\)\( \nu^{2} + \)\(50\!\cdots\!96\)\( \nu + \)\(27\!\cdots\!20\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(41\!\cdots\!81\)\( \nu^{19} + \)\(29\!\cdots\!40\)\( \nu^{18} + \)\(12\!\cdots\!63\)\( \nu^{17} - \)\(10\!\cdots\!63\)\( \nu^{16} - \)\(16\!\cdots\!09\)\( \nu^{15} + \)\(16\!\cdots\!35\)\( \nu^{14} + \)\(17\!\cdots\!40\)\( \nu^{13} - \)\(13\!\cdots\!25\)\( \nu^{12} - \)\(15\!\cdots\!73\)\( \nu^{11} + \)\(62\!\cdots\!82\)\( \nu^{10} + \)\(10\!\cdots\!60\)\( \nu^{9} - \)\(15\!\cdots\!52\)\( \nu^{8} - \)\(41\!\cdots\!98\)\( \nu^{7} + \)\(38\!\cdots\!08\)\( \nu^{6} + \)\(86\!\cdots\!10\)\( \nu^{5} + \)\(71\!\cdots\!82\)\( \nu^{4} - \)\(51\!\cdots\!74\)\( \nu^{3} - \)\(13\!\cdots\!56\)\( \nu^{2} - \)\(10\!\cdots\!92\)\( \nu - \)\(33\!\cdots\!28\)\(\)\()/ \)\(73\!\cdots\!72\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(99\!\cdots\!07\)\( \nu^{19} + \)\(10\!\cdots\!86\)\( \nu^{18} - \)\(49\!\cdots\!34\)\( \nu^{17} - \)\(26\!\cdots\!41\)\( \nu^{16} + \)\(45\!\cdots\!96\)\( \nu^{15} + \)\(33\!\cdots\!72\)\( \nu^{14} - \)\(63\!\cdots\!58\)\( \nu^{13} - \)\(25\!\cdots\!50\)\( \nu^{12} + \)\(43\!\cdots\!68\)\( \nu^{11} + \)\(14\!\cdots\!88\)\( \nu^{10} - \)\(16\!\cdots\!88\)\( \nu^{9} - \)\(54\!\cdots\!30\)\( \nu^{8} + \)\(26\!\cdots\!56\)\( \nu^{7} + \)\(14\!\cdots\!16\)\( \nu^{6} + \)\(18\!\cdots\!40\)\( \nu^{5} - \)\(20\!\cdots\!52\)\( \nu^{4} - \)\(15\!\cdots\!96\)\( \nu^{3} + \)\(98\!\cdots\!12\)\( \nu^{2} + \)\(18\!\cdots\!32\)\( \nu + \)\(91\!\cdots\!84\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(10\!\cdots\!67\)\( \nu^{19} - \)\(82\!\cdots\!39\)\( \nu^{18} - \)\(21\!\cdots\!72\)\( \nu^{17} + \)\(26\!\cdots\!20\)\( \nu^{16} + \)\(22\!\cdots\!36\)\( \nu^{15} - \)\(36\!\cdots\!64\)\( \nu^{14} - \)\(24\!\cdots\!32\)\( \nu^{13} + \)\(29\!\cdots\!48\)\( \nu^{12} + \)\(27\!\cdots\!38\)\( \nu^{11} - \)\(13\!\cdots\!66\)\( \nu^{10} - \)\(20\!\cdots\!76\)\( \nu^{9} + \)\(33\!\cdots\!60\)\( \nu^{8} + \)\(86\!\cdots\!96\)\( \nu^{7} - \)\(93\!\cdots\!40\)\( \nu^{6} - \)\(18\!\cdots\!52\)\( \nu^{5} - \)\(15\!\cdots\!64\)\( \nu^{4} + \)\(11\!\cdots\!04\)\( \nu^{3} + \)\(30\!\cdots\!20\)\( \nu^{2} + \)\(22\!\cdots\!48\)\( \nu + \)\(68\!\cdots\!36\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(10\!\cdots\!40\)\( \nu^{19} + \)\(82\!\cdots\!79\)\( \nu^{18} + \)\(25\!\cdots\!44\)\( \nu^{17} - \)\(28\!\cdots\!92\)\( \nu^{16} - \)\(26\!\cdots\!50\)\( \nu^{15} + \)\(42\!\cdots\!42\)\( \nu^{14} + \)\(22\!\cdots\!08\)\( \nu^{13} - \)\(35\!\cdots\!08\)\( \nu^{12} - \)\(21\!\cdots\!74\)\( \nu^{11} + \)\(18\!\cdots\!58\)\( \nu^{10} + \)\(16\!\cdots\!84\)\( \nu^{9} - \)\(53\!\cdots\!08\)\( \nu^{8} - \)\(77\!\cdots\!44\)\( \nu^{7} + \)\(65\!\cdots\!44\)\( \nu^{6} + \)\(18\!\cdots\!00\)\( \nu^{5} + \)\(57\!\cdots\!68\)\( \nu^{4} - \)\(17\!\cdots\!40\)\( \nu^{3} - \)\(23\!\cdots\!96\)\( \nu^{2} - \)\(11\!\cdots\!64\)\( \nu - \)\(22\!\cdots\!12\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(17\!\cdots\!81\)\( \nu^{19} - \)\(14\!\cdots\!16\)\( \nu^{18} - \)\(22\!\cdots\!12\)\( \nu^{17} + \)\(41\!\cdots\!75\)\( \nu^{16} + \)\(12\!\cdots\!22\)\( \nu^{15} - \)\(55\!\cdots\!76\)\( \nu^{14} - \)\(15\!\cdots\!58\)\( \nu^{13} + \)\(43\!\cdots\!26\)\( \nu^{12} + \)\(28\!\cdots\!66\)\( \nu^{11} - \)\(21\!\cdots\!36\)\( \nu^{10} - \)\(24\!\cdots\!96\)\( \nu^{9} + \)\(58\!\cdots\!38\)\( \nu^{8} + \)\(11\!\cdots\!20\)\( \nu^{7} - \)\(51\!\cdots\!76\)\( \nu^{6} - \)\(26\!\cdots\!88\)\( \nu^{5} - \)\(15\!\cdots\!88\)\( \nu^{4} + \)\(20\!\cdots\!04\)\( \nu^{3} + \)\(37\!\cdots\!56\)\( \nu^{2} + \)\(23\!\cdots\!84\)\( \nu + \)\(63\!\cdots\!16\)\(\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(89\!\cdots\!29\)\( \nu^{19} + \)\(64\!\cdots\!55\)\( \nu^{18} + \)\(24\!\cdots\!41\)\( \nu^{17} - \)\(22\!\cdots\!06\)\( \nu^{16} - \)\(34\!\cdots\!93\)\( \nu^{15} + \)\(34\!\cdots\!71\)\( \nu^{14} + \)\(37\!\cdots\!84\)\( \nu^{13} - \)\(28\!\cdots\!46\)\( \nu^{12} - \)\(34\!\cdots\!02\)\( \nu^{11} + \)\(13\!\cdots\!66\)\( \nu^{10} + \)\(22\!\cdots\!00\)\( \nu^{9} - \)\(36\!\cdots\!98\)\( \nu^{8} - \)\(91\!\cdots\!58\)\( \nu^{7} + \)\(18\!\cdots\!94\)\( \nu^{6} + \)\(19\!\cdots\!16\)\( \nu^{5} + \)\(13\!\cdots\!72\)\( \nu^{4} - \)\(13\!\cdots\!20\)\( \nu^{3} - \)\(28\!\cdots\!20\)\( \nu^{2} - \)\(19\!\cdots\!24\)\( \nu - \)\(56\!\cdots\!72\)\(\)\()/ \)\(73\!\cdots\!72\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(70\!\cdots\!85\)\( \nu^{19} + \)\(73\!\cdots\!21\)\( \nu^{18} - \)\(39\!\cdots\!06\)\( \nu^{17} - \)\(16\!\cdots\!04\)\( \nu^{16} + \)\(27\!\cdots\!52\)\( \nu^{15} + \)\(19\!\cdots\!86\)\( \nu^{14} - \)\(32\!\cdots\!04\)\( \nu^{13} - \)\(13\!\cdots\!90\)\( \nu^{12} + \)\(16\!\cdots\!08\)\( \nu^{11} + \)\(67\!\cdots\!78\)\( \nu^{10} - \)\(38\!\cdots\!00\)\( \nu^{9} - \)\(22\!\cdots\!92\)\( \nu^{8} - \)\(27\!\cdots\!68\)\( \nu^{7} + \)\(42\!\cdots\!44\)\( \nu^{6} + \)\(33\!\cdots\!64\)\( \nu^{5} - \)\(28\!\cdots\!80\)\( \nu^{4} - \)\(58\!\cdots\!28\)\( \nu^{3} - \)\(30\!\cdots\!32\)\( \nu^{2} + \)\(89\!\cdots\!04\)\( \nu + \)\(43\!\cdots\!72\)\(\)\()/ \)\(49\!\cdots\!48\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(75\!\cdots\!00\)\( \nu^{19} + \)\(57\!\cdots\!60\)\( \nu^{18} + \)\(17\!\cdots\!14\)\( \nu^{17} - \)\(18\!\cdots\!63\)\( \nu^{16} - \)\(22\!\cdots\!54\)\( \nu^{15} + \)\(27\!\cdots\!34\)\( \nu^{14} + \)\(25\!\cdots\!64\)\( \nu^{13} - \)\(22\!\cdots\!82\)\( \nu^{12} - \)\(25\!\cdots\!68\)\( \nu^{11} + \)\(11\!\cdots\!36\)\( \nu^{10} + \)\(17\!\cdots\!96\)\( \nu^{9} - \)\(28\!\cdots\!90\)\( \nu^{8} - \)\(71\!\cdots\!52\)\( \nu^{7} + \)\(16\!\cdots\!28\)\( \nu^{6} + \)\(15\!\cdots\!72\)\( \nu^{5} + \)\(10\!\cdots\!84\)\( \nu^{4} - \)\(10\!\cdots\!72\)\( \nu^{3} - \)\(22\!\cdots\!56\)\( \nu^{2} - \)\(15\!\cdots\!96\)\( \nu - \)\(41\!\cdots\!00\)\(\)\()/ \)\(49\!\cdots\!48\)\( \)
\(\beta_{19}\)\(=\)\((\)\(-\)\(82\!\cdots\!92\)\( \nu^{19} + \)\(70\!\cdots\!23\)\( \nu^{18} + \)\(12\!\cdots\!16\)\( \nu^{17} - \)\(20\!\cdots\!98\)\( \nu^{16} - \)\(63\!\cdots\!28\)\( \nu^{15} + \)\(28\!\cdots\!84\)\( \nu^{14} + \)\(56\!\cdots\!98\)\( \nu^{13} - \)\(22\!\cdots\!86\)\( \nu^{12} - \)\(11\!\cdots\!72\)\( \nu^{11} + \)\(10\!\cdots\!78\)\( \nu^{10} + \)\(10\!\cdots\!52\)\( \nu^{9} - \)\(29\!\cdots\!60\)\( \nu^{8} - \)\(51\!\cdots\!64\)\( \nu^{7} + \)\(27\!\cdots\!00\)\( \nu^{6} + \)\(12\!\cdots\!20\)\( \nu^{5} + \)\(68\!\cdots\!36\)\( \nu^{4} - \)\(96\!\cdots\!72\)\( \nu^{3} - \)\(17\!\cdots\!12\)\( \nu^{2} - \)\(11\!\cdots\!16\)\( \nu - \)\(30\!\cdots\!20\)\(\)\()/ \)\(49\!\cdots\!48\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{18} + \beta_{17} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2\)\()/2\)
\(\nu^{2}\)\(=\)\(-2 \beta_{18} + \beta_{16} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{19} - 9 \beta_{18} + 3 \beta_{17} + 2 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + 8 \beta_{13} + \beta_{12} + 8 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} - 9 \beta_{8} - 7 \beta_{7} - 15 \beta_{6} + 10 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 2 \beta_{1} + 16\)
\(\nu^{4}\)\(=\)\(3 \beta_{19} - 45 \beta_{18} + \beta_{17} + 24 \beta_{16} + 8 \beta_{15} - 18 \beta_{14} + 34 \beta_{13} + 14 \beta_{12} + 32 \beta_{11} + \beta_{10} + 15 \beta_{9} - 37 \beta_{8} - 27 \beta_{7} - 69 \beta_{6} + 34 \beta_{5} - 20 \beta_{4} + 17 \beta_{3} - 19 \beta_{2} - \beta_{1} + 64\)
\(\nu^{5}\)\(=\)\(10 \beta_{19} - 183 \beta_{18} + 19 \beta_{17} + 108 \beta_{16} + 85 \beta_{15} - 109 \beta_{14} + 197 \beta_{13} + 28 \beta_{12} + 168 \beta_{11} - \beta_{10} + 53 \beta_{9} - 177 \beta_{8} - 56 \beta_{7} - 378 \beta_{6} + 163 \beta_{5} - 88 \beta_{4} + 124 \beta_{3} - 66 \beta_{2} + 39 \beta_{1} + 201\)
\(\nu^{6}\)\(=\)\(-\beta_{19} - 859 \beta_{18} + 11 \beta_{17} + 664 \beta_{16} + 388 \beta_{15} - 545 \beta_{14} + 931 \beta_{13} + 116 \beta_{12} + 767 \beta_{11} - 159 \beta_{10} + 64 \beta_{9} - 777 \beta_{8} - 20 \beta_{7} - 1795 \beta_{6} + 593 \beta_{5} - 425 \beta_{4} + 595 \beta_{3} - 179 \beta_{2} + 143 \beta_{1} + 687\)
\(\nu^{7}\)\(=\)\(-288 \beta_{19} - 3480 \beta_{18} - 4 \beta_{17} + 3375 \beta_{16} + 2366 \beta_{15} - 2696 \beta_{14} + 4670 \beta_{13} + 66 \beta_{12} + 3591 \beta_{11} - 1184 \beta_{10} - 375 \beta_{9} - 3754 \beta_{8} + 1445 \beta_{7} - 8772 \beta_{6} + 2053 \beta_{5} - 1855 \beta_{4} + 3132 \beta_{3} - 200 \beta_{2} + 1033 \beta_{1} + 1742\)
\(\nu^{8}\)\(=\)\(-2780 \beta_{19} - 15022 \beta_{18} - 705 \beta_{17} + 17713 \beta_{16} + 11856 \beta_{15} - 13184 \beta_{14} + 21833 \beta_{13} - 1565 \beta_{12} + 16282 \beta_{11} - 7451 \beta_{10} - 4970 \beta_{9} - 17187 \beta_{8} + 13298 \beta_{7} - 41334 \beta_{6} + 5914 \beta_{5} - 8418 \beta_{4} + 15647 \beta_{3} + 1808 \beta_{2} + 5247 \beta_{1} + 2001\)
\(\nu^{9}\)\(=\)\(-21092 \beta_{19} - 58465 \beta_{18} - 6619 \beta_{17} + 88838 \beta_{16} + 61746 \beta_{15} - 61647 \beta_{14} + 101421 \beta_{13} - 16732 \beta_{12} + 72535 \beta_{11} - 42924 \beta_{10} - 38745 \beta_{9} - 80705 \beta_{8} + 95533 \beta_{7} - 192103 \beta_{6} + 7315 \beta_{5} - 34939 \beta_{4} + 76043 \beta_{3} + 22833 \beta_{2} + 29814 \beta_{1} - 21606\)
\(\nu^{10}\)\(=\)\(-135802 \beta_{19} - 223950 \beta_{18} - 43510 \beta_{17} + 438744 \beta_{16} + 304234 \beta_{15} - 286578 \beta_{14} + 455088 \beta_{13} - 125858 \beta_{12} + 315842 \beta_{11} - 227586 \beta_{10} - 246372 \beta_{9} - 365298 \beta_{8} + 595106 \beta_{7} - 876488 \beta_{6} - 56174 \beta_{5} - 140686 \beta_{4} + 365442 \beta_{3} + 169258 \beta_{2} + 158012 \beta_{1} - 247596\)
\(\nu^{11}\)\(=\)\(-820442 \beta_{19} - 752876 \beta_{18} - 278560 \beta_{17} + 2121960 \beta_{16} + 1478444 \beta_{15} - 1280848 \beta_{14} + 1984514 \beta_{13} - 794702 \beta_{12} + 1330152 \beta_{11} - 1179368 \beta_{10} - 1464042 \beta_{9} - 1632980 \beta_{8} + 3465722 \beta_{7} - 3897358 \beta_{6} - 729414 \beta_{5} - 510102 \beta_{4} + 1697612 \beta_{3} + 1097516 \beta_{2} + 841830 \beta_{1} - 1854886\)
\(\nu^{12}\)\(=\)\(-4655336 \beta_{19} - 2085544 \beta_{18} - 1584368 \beta_{17} + 10040992 \beta_{16} + 6999828 \beta_{15} - 5611710 \beta_{14} + 8347666 \beta_{13} - 4705280 \beta_{12} + 5388026 \beta_{11} - 5860036 \beta_{10} - 8144282 \beta_{9} - 7041224 \beta_{8} + 19173438 \beta_{7} - 16905084 \beta_{6} - 5506858 \beta_{5} - 1595482 \beta_{4} + 7758840 \beta_{3} + 6473996 \beta_{2} + 4376316 \beta_{1} - 11948222\)
\(\nu^{13}\)\(=\)\(-25452540 \beta_{19} - 2320366 \beta_{18} - 8851794 \beta_{17} + 46569422 \beta_{16} + 32359158 \beta_{15} - 23578942 \beta_{14} + 33425950 \beta_{13} - 26180568 \beta_{12} + 20507522 \beta_{11} - 28577454 \beta_{10} - 43824724 \beta_{9} - 29269134 \beta_{8} + 102297558 \beta_{7} - 70646780 \beta_{6} - 35748548 \beta_{5} - 3222566 \beta_{4} + 34318320 \beta_{3} + 36360760 \beta_{2} + 22391748 \beta_{1} - 70792962\)
\(\nu^{14}\)\(=\)\(-134388246 \beta_{19} + 25660522 \beta_{18} - 47035156 \beta_{17} + 210719766 \beta_{16} + 146020760 \beta_{15} - 94912194 \beta_{14} + 125040980 \beta_{13} - 140750702 \beta_{12} + 70843422 \beta_{11} - 135615828 \beta_{10} - 227950052 \beta_{9} - 115110028 \beta_{8} + 529651108 \beta_{7} - 282098330 \beta_{6} - 211670802 \beta_{5} + 6513782 \beta_{4} + 147529928 \beta_{3} + 195915542 \beta_{2} + 112472160 \beta_{1} - 397964712\)
\(\nu^{15}\)\(=\)\(-690653692 \beta_{19} + 297922790 \beta_{18} - 244628990 \beta_{17} + 929045294 \beta_{16} + 639608512 \beta_{15} - 356188698 \beta_{14} + 415162054 \beta_{13} - 731858536 \beta_{12} + 199592544 \beta_{11} - 630221232 \beta_{10} - 1156913516 \beta_{9} - 416869170 \beta_{8} + 2672840408 \beta_{7} - 1049166142 \beta_{6} - 1188490904 \beta_{5} + 137698300 \beta_{4} + 607842998 \beta_{3} + 1024249042 \beta_{2} + 554078218 \beta_{1} - 2149996356\)
\(\nu^{16}\)\(=\)\(-3461541484 \beta_{19} + 2240581900 \beta_{18} - 1236318086 \beta_{17} + 3963909834 \beta_{16} + 2709579968 \beta_{15} - 1194745804 \beta_{14} + 1058352742 \beta_{13} - 3710565346 \beta_{12} + 262169316 \beta_{11} - 2855594598 \beta_{10} - 5732092912 \beta_{9} - 1297261218 \beta_{8} + 13178875340 \beta_{7} - 3462649844 \beta_{6} - 6405708988 \beta_{5} + 1163304956 \beta_{4} + 2372837210 \beta_{3} + 5213381052 \beta_{2} + 2678546914 \beta_{1} - 11261926130\)
\(\nu^{17}\)\(=\)\(-16959549872 \beta_{19} + 14425981062 \beta_{18} - 6116742626 \beta_{17} + 16233773096 \beta_{16} + 10971195820 \beta_{15} - 3117205526 \beta_{14} + 450708166 \beta_{13} - 18364297812 \beta_{12} - 2127524046 \beta_{11} - 12599313768 \beta_{10} - 27791183714 \beta_{9} - 2672139126 \beta_{8} + 63557904558 \beta_{7} - 8663409282 \beta_{6} - 33464412710 \beta_{5} + 7892332914 \beta_{4} + 8500322086 \beta_{3} + 25921089806 \beta_{2} + 12684019076 \beta_{1} - 57441958996\)
\(\nu^{18}\)\(=\)\(-81274368308 \beta_{19} + 85464416684 \beta_{18} - 29589942532 \beta_{17} + 62744460160 \beta_{16} + 41717395868 \beta_{15} - 2005906720 \beta_{14} - 20209488412 \beta_{13} - 88907191572 \beta_{12} - 26319786080 \beta_{11} - 53818234652 \beta_{10} - 131834847452 \beta_{9} + 4672078244 \beta_{8} + 299871371344 \beta_{7} - 2313837800 \beta_{6} - 170216172552 \beta_{5} + 48225023272 \beta_{4} + 25975905700 \beta_{3} + 126054765484 \beta_{2} + 58799282976 \beta_{1} - 286157119204\)
\(\nu^{19}\)\(=\)\(-380917114036 \beta_{19} + 479521118100 \beta_{18} - 140105517716 \beta_{17} + 221449454948 \beta_{16} + 142983995892 \beta_{15} + 53142131180 \beta_{14} - 202922291944 \beta_{13} - 421057486916 \beta_{12} - 201348890452 \beta_{11} - 220663144468 \beta_{10} - 611583552244 \beta_{9} + 106670815220 \beta_{8} + 1382824831504 \beta_{7} + 178701519436 \beta_{6} - 845516037228 \beta_{5} + 276364096816 \beta_{4} + 50361423168 \beta_{3} + 599837850392 \beta_{2} + 266138655220 \beta_{1} - 1394237620712\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.12095 0.707107i
−0.476437 + 0.707107i
3.16384 + 0.707107i
−0.561107 0.707107i
1.85589 0.707107i
−2.54394 + 0.707107i
2.25091 0.707107i
4.76099 + 0.707107i
−1.67971 + 0.707107i
−1.64948 0.707107i
−1.12095 + 0.707107i
−0.476437 0.707107i
3.16384 0.707107i
−0.561107 + 0.707107i
1.85589 + 0.707107i
−2.54394 0.707107i
2.25091 + 0.707107i
4.76099 0.707107i
−1.67971 0.707107i
−1.64948 + 0.707107i
0 0 0 −3.57821 + 2.06588i 0 0.233685 0 0 0
449.2 0 0 0 −3.20539 + 1.85063i 0 4.57072 0 0 0
449.3 0 0 0 −1.76516 + 1.01912i 0 −3.69902 0 0 0
449.4 0 0 0 −0.524572 + 0.302862i 0 −4.37361 0 0 0
449.5 0 0 0 −0.0242659 + 0.0140099i 0 1.95697 0 0 0
449.6 0 0 0 0.501439 0.289506i 0 0.198579 0 0 0
449.7 0 0 0 0.967070 0.558338i 0 1.95123 0 0 0
449.8 0 0 0 1.80341 1.04120i 0 4.37182 0 0 0
449.9 0 0 0 2.66570 1.53904i 0 −1.99260 0 0 0
449.10 0 0 0 3.15998 1.82441i 0 −1.21776 0 0 0
1889.1 0 0 0 −3.57821 2.06588i 0 0.233685 0 0 0
1889.2 0 0 0 −3.20539 1.85063i 0 4.57072 0 0 0
1889.3 0 0 0 −1.76516 1.01912i 0 −3.69902 0 0 0
1889.4 0 0 0 −0.524572 0.302862i 0 −4.37361 0 0 0
1889.5 0 0 0 −0.0242659 0.0140099i 0 1.95697 0 0 0
1889.6 0 0 0 0.501439 + 0.289506i 0 0.198579 0 0 0
1889.7 0 0 0 0.967070 + 0.558338i 0 1.95123 0 0 0
1889.8 0 0 0 1.80341 + 1.04120i 0 4.37182 0 0 0
1889.9 0 0 0 2.66570 + 1.53904i 0 −1.99260 0 0 0
1889.10 0 0 0 3.15998 + 1.82441i 0 −1.21776 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.dc.e 20
3.b odd 2 1 2736.2.dc.f 20
4.b odd 2 1 1368.2.cu.a 20
12.b even 2 1 1368.2.cu.b yes 20
19.d odd 6 1 2736.2.dc.f 20
57.f even 6 1 inner 2736.2.dc.e 20
76.f even 6 1 1368.2.cu.b yes 20
228.n odd 6 1 1368.2.cu.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.cu.a 20 4.b odd 2 1
1368.2.cu.a 20 228.n odd 6 1
1368.2.cu.b yes 20 12.b even 2 1
1368.2.cu.b yes 20 76.f even 6 1
2736.2.dc.e 20 1.a even 1 1 trivial
2736.2.dc.e 20 57.f even 6 1 inner
2736.2.dc.f 20 3.b odd 2 1
2736.2.dc.f 20 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\(T_{5}^{20} - \cdots\)
\(T_{17}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( 64 + 3840 T + 74208 T^{2} - 155520 T^{3} - 140400 T^{4} + 478656 T^{5} + 378496 T^{6} - 1205184 T^{7} + 522512 T^{8} + 370848 T^{9} - 217160 T^{10} - 88896 T^{11} + 66892 T^{12} + 6096 T^{13} - 8096 T^{14} - 336 T^{15} + 708 T^{16} - 32 T^{18} + T^{20} \)
$7$ \( ( 139 - 1246 T + 2433 T^{2} + 2036 T^{3} - 2110 T^{4} - 672 T^{5} + 522 T^{6} + 68 T^{7} - 41 T^{8} - 2 T^{9} + T^{10} )^{2} \)
$11$ \( 4260096 + 35926656 T^{2} + 90664720 T^{4} + 77271232 T^{6} + 32073024 T^{8} + 7468208 T^{10} + 1030184 T^{12} + 84752 T^{14} + 4020 T^{16} + 100 T^{18} + T^{20} \)
$13$ \( 1176147025 + 1559942370 T - 65551463 T^{2} - 1001647206 T^{3} - 95879073 T^{4} + 387804972 T^{5} + 56338054 T^{6} - 100961940 T^{7} - 9954515 T^{8} + 18205086 T^{9} + 907231 T^{10} - 2379882 T^{11} + 78557 T^{12} + 185868 T^{13} - 12602 T^{14} - 10308 T^{15} + 1167 T^{16} + 306 T^{17} - 39 T^{18} - 6 T^{19} + T^{20} \)
$17$ \( 2064793600 + 11690803200 T + 29369630720 T^{2} + 41362391040 T^{3} + 34264899584 T^{4} + 15318417408 T^{5} + 1866555392 T^{6} - 1310423040 T^{7} - 401001728 T^{8} + 138484224 T^{9} + 85142144 T^{10} + 6802944 T^{11} - 3689088 T^{12} - 594432 T^{13} + 135360 T^{14} + 37344 T^{15} - 348 T^{16} - 744 T^{17} - 14 T^{18} + 12 T^{19} + T^{20} \)
$19$ \( 6131066257801 + 1290750791116 T + 169835630410 T^{2} + 210953730404 T^{3} + 40506503541 T^{4} + 11013688352 T^{5} + 4492425512 T^{6} + 677120480 T^{7} + 249278442 T^{8} + 66628744 T^{9} + 9079356 T^{10} + 3506776 T^{11} + 690522 T^{12} + 98720 T^{13} + 34472 T^{14} + 4448 T^{15} + 861 T^{16} + 236 T^{17} + 10 T^{18} + 4 T^{19} + T^{20} \)
$23$ \( 31021072384 - 98609135616 T + 123620802560 T^{2} - 60826733568 T^{3} - 7020013808 T^{4} + 16813912896 T^{5} - 1081550432 T^{6} - 3548552064 T^{7} + 851118352 T^{8} + 309060672 T^{9} - 97887248 T^{10} - 20424288 T^{11} + 7699580 T^{12} + 732144 T^{13} - 314472 T^{14} - 17376 T^{15} + 9368 T^{16} - 114 T^{18} + T^{20} \)
$29$ \( 34924134400 - 130511011840 T + 411650228224 T^{2} - 317369876480 T^{3} + 244009566208 T^{4} - 87252860928 T^{5} + 45005864960 T^{6} - 10785046528 T^{7} + 5658897408 T^{8} - 809971712 T^{9} + 388404736 T^{10} - 22563840 T^{11} + 19106944 T^{12} - 399360 T^{13} + 575168 T^{14} + 3136 T^{15} + 12356 T^{16} + 64 T^{17} + 134 T^{18} + T^{20} \)
$31$ \( 18420189841 + 293093517986 T^{2} + 294064269917 T^{4} + 106359489920 T^{6} + 18038602546 T^{8} + 1588306820 T^{10} + 78144018 T^{12} + 2214096 T^{14} + 35565 T^{16} + 298 T^{18} + T^{20} \)
$37$ \( 266049536550625 + 200286400021250 T^{2} + 51632099933325 T^{4} + 6505422136600 T^{6} + 459903319986 T^{8} + 19574454924 T^{10} + 517550450 T^{12} + 8503192 T^{14} + 83821 T^{16} + 450 T^{18} + T^{20} \)
$41$ \( 4381296138649600 - 1754145174323200 T + 1318297807093760 T^{2} - 180016824975360 T^{3} + 132943508934656 T^{4} - 6904890998784 T^{5} + 10250434471936 T^{6} + 185689128960 T^{7} + 472176530432 T^{8} + 19405842432 T^{9} + 15561008128 T^{10} + 849148416 T^{11} + 338396864 T^{12} + 17748480 T^{13} + 5152960 T^{14} + 255936 T^{15} + 50432 T^{16} + 1728 T^{17} + 296 T^{18} + 8 T^{19} + T^{20} \)
$43$ \( 11964640689 + 307589152554 T + 7783867963959 T^{2} + 3464373445386 T^{3} + 4934993448439 T^{4} - 1516261700852 T^{5} + 1336905163418 T^{6} - 334389996096 T^{7} + 169681349157 T^{8} - 43619308390 T^{9} + 13787005513 T^{10} - 2352874754 T^{11} + 423430693 T^{12} - 40387200 T^{13} + 5433366 T^{14} - 349364 T^{15} + 50331 T^{16} - 1818 T^{17} + 259 T^{18} - 2 T^{19} + T^{20} \)
$47$ \( 2106445849600 + 3378766080000 T + 1097869550080 T^{2} - 1136697216000 T^{3} - 550279700736 T^{4} + 602903485440 T^{5} + 647107269376 T^{6} + 228835729920 T^{7} + 18802184512 T^{8} - 8566564992 T^{9} - 1004140832 T^{10} + 974238912 T^{11} + 404835536 T^{12} + 70734144 T^{13} + 5567152 T^{14} - 21408 T^{15} - 29988 T^{16} + 648 T^{17} + 450 T^{18} + 36 T^{19} + T^{20} \)
$53$ \( 831865578726400 + 258037399244800 T + 446269937996160 T^{2} + 47771570081280 T^{3} + 144363188647184 T^{4} + 13940977326400 T^{5} + 22870802146976 T^{6} - 554693978624 T^{7} + 2044321922560 T^{8} + 53778366176 T^{9} + 60555030224 T^{10} - 353239872 T^{11} + 1032983084 T^{12} - 11076528 T^{13} + 11509040 T^{14} - 256416 T^{15} + 85028 T^{16} - 1568 T^{17} + 386 T^{18} - 8 T^{19} + T^{20} \)
$59$ \( 2030625000000 + 4770900000000 T + 8850871500000 T^{2} + 5985752400000 T^{3} + 3596294626000 T^{4} + 1286366035200 T^{5} + 502841191264 T^{6} + 131168997952 T^{7} + 43453060912 T^{8} + 8146212384 T^{9} + 2080447288 T^{10} + 214079328 T^{11} + 57200668 T^{12} + 3086144 T^{13} + 1269808 T^{14} + 2224 T^{15} + 18572 T^{16} - 464 T^{17} + 208 T^{18} - 8 T^{19} + T^{20} \)
$61$ \( 175961790273481 + 410320455903134 T + 911663940818813 T^{2} + 226375438341026 T^{3} + 164355227152871 T^{4} + 39988682888868 T^{5} + 21052115787590 T^{6} + 4034576417196 T^{7} + 1131173090789 T^{8} + 132618965202 T^{9} + 29571242435 T^{10} + 2624977150 T^{11} + 535991389 T^{12} + 32734532 T^{13} + 6282662 T^{14} + 257468 T^{15} + 54239 T^{16} + 1230 T^{17} + 285 T^{18} + 2 T^{19} + T^{20} \)
$67$ \( 3205873659589225 + 15366800214108750 T + 25074888058992855 T^{2} + 2503024901858250 T^{3} - 2198376233671581 T^{4} - 279123726657924 T^{5} + 144088725611574 T^{6} + 14464486627224 T^{7} - 4464776173707 T^{8} - 427074480810 T^{9} + 98347666165 T^{10} + 9119220534 T^{11} - 1234220451 T^{12} - 124664808 T^{13} + 10822722 T^{14} + 1282932 T^{15} - 40593 T^{16} - 7470 T^{17} + 51 T^{18} + 30 T^{19} + T^{20} \)
$71$ \( 2366192037990400 - 1523963345080320 T + 1404944883403776 T^{2} - 333831840072704 T^{3} + 227840316702976 T^{4} - 16627973577728 T^{5} + 29384346289152 T^{6} + 989111864832 T^{7} + 2435577714048 T^{8} + 350284225024 T^{9} + 166028245056 T^{10} + 19979073984 T^{11} + 3705882032 T^{12} + 201169600 T^{13} + 29281888 T^{14} + 1028096 T^{15} + 171160 T^{16} + 2896 T^{17} + 484 T^{18} + 4 T^{19} + T^{20} \)
$73$ \( 796807954881 + 3075503516118 T + 7946763067095 T^{2} + 11087505132558 T^{3} + 11190687690439 T^{4} + 7564651080692 T^{5} + 3983025568386 T^{6} + 1583645003292 T^{7} + 537316032653 T^{8} + 151677742434 T^{9} + 39989145689 T^{10} + 8848020194 T^{11} + 1724191821 T^{12} + 248086492 T^{13} + 29086562 T^{14} + 2410740 T^{15} + 174119 T^{16} + 9102 T^{17} + 567 T^{18} + 22 T^{19} + T^{20} \)
$79$ \( 18936116060490721 - 1630820798563638 T - 7591884524370655 T^{2} + 657862082550354 T^{3} + 2751364458663063 T^{4} + 647138064681348 T^{5} - 81699104243258 T^{6} - 36287659923000 T^{7} + 2461681914117 T^{8} + 1374866197890 T^{9} + 25617233247 T^{10} - 22203592962 T^{11} - 948778315 T^{12} + 263577000 T^{13} + 22130242 T^{14} - 1086396 T^{15} - 164669 T^{16} + 1782 T^{17} + 1005 T^{18} + 54 T^{19} + T^{20} \)
$83$ \( 98425257984 + 6830596423680 T^{2} + 21108233076736 T^{4} + 7308875194368 T^{6} + 973344635392 T^{8} + 59318019840 T^{10} + 1692584448 T^{12} + 24410496 T^{14} + 183364 T^{16} + 684 T^{18} + T^{20} \)
$89$ \( 196214164026062400 - 283950538972842240 T + 256946419883282784 T^{2} - 143368917921723264 T^{3} + 58749419686080592 T^{4} - 17913048478700352 T^{5} + 4325621229190176 T^{6} - 832018155169600 T^{7} + 132459240752192 T^{8} - 17506268778912 T^{9} + 2002201539768 T^{10} - 198634614528 T^{11} + 17986891548 T^{12} - 1444783984 T^{13} + 107568144 T^{14} - 6820464 T^{15} + 405720 T^{16} - 19760 T^{17} + 972 T^{18} - 32 T^{19} + T^{20} \)
$97$ \( 1172872899661824 + 2377012152250368 T + 1971768236147712 T^{2} + 741698838051840 T^{3} + 47799865144576 T^{4} - 51472278759936 T^{5} - 8133589602560 T^{6} + 4635119740416 T^{7} + 1642598577024 T^{8} + 66234946944 T^{9} - 36665338816 T^{10} - 3012024768 T^{11} + 616782128 T^{12} + 60006816 T^{13} - 6223552 T^{14} - 610176 T^{15} + 52584 T^{16} + 3696 T^{17} - 260 T^{18} - 12 T^{19} + T^{20} \)
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