Properties

Label 1840.3.k.c
Level $1840$
Weight $3$
Character orbit 1840.k
Analytic conductor $50.136$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1840.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(50.1363686423\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + 38013345 x^{4} - 8913096 x^{3} + 327235800 x^{2} + 72566336 x + 1535848276\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 460)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} -\beta_{4} q^{5} -\beta_{7} q^{7} + ( 4 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} -\beta_{4} q^{5} -\beta_{7} q^{7} + ( 4 + \beta_{2} ) q^{9} + ( 2 \beta_{4} - \beta_{9} ) q^{11} + ( -1 - \beta_{11} ) q^{13} + \beta_{1} q^{15} + ( \beta_{1} - \beta_{10} ) q^{17} + ( -\beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{6} + \beta_{7} + \beta_{9} ) q^{21} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{15} ) q^{23} -5 q^{25} + ( -3 - \beta_{2} - 6 \beta_{3} + \beta_{5} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{27} + ( 5 + 4 \beta_{3} + 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{29} + ( -1 + 2 \beta_{3} + \beta_{5} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{31} + ( -4 \beta_{1} - 6 \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{15} ) q^{33} + ( -2 + \beta_{13} ) q^{35} + ( 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{15} ) q^{37} + ( -2 + 2 \beta_{3} - \beta_{5} - \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{39} + ( 12 + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{41} + ( -2 \beta_{1} - 4 \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{15} ) q^{43} + ( -5 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{15} ) q^{45} + ( 20 + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{47} + ( 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{49} + ( -12 \beta_{4} + \beta_{6} + \beta_{7} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{15} ) q^{51} + ( 2 \beta_{1} - 8 \beta_{4} + \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{15} ) q^{53} + ( 8 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{11} - \beta_{13} ) q^{55} + ( \beta_{1} - 8 \beta_{4} + 5 \beta_{7} + 6 \beta_{9} + \beta_{10} - 2 \beta_{15} ) q^{57} + ( 6 + \beta_{2} + 6 \beta_{3} - \beta_{5} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{59} + ( 2 \beta_{1} + 6 \beta_{4} - 2 \beta_{6} - 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{15} ) q^{61} + ( -3 \beta_{1} + 8 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{63} + ( 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{65} + ( 4 \beta_{1} - 6 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{9} - \beta_{10} - 3 \beta_{15} ) q^{67} + ( -15 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 10 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{69} + ( 16 + \beta_{2} - 12 \beta_{3} - \beta_{5} + \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{71} + ( -16 - \beta_{2} + \beta_{3} - 3 \beta_{5} - 5 \beta_{13} + \beta_{14} ) q^{73} + 5 \beta_{3} q^{75} + ( 21 - 9 \beta_{3} - 2 \beta_{5} + \beta_{11} - 2 \beta_{13} ) q^{77} + ( -2 \beta_{4} - \beta_{6} + 6 \beta_{7} + \beta_{8} - 3 \beta_{9} + 4 \beta_{10} ) q^{79} + ( 44 + 5 \beta_{2} + 14 \beta_{3} - 4 \beta_{5} + 3 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} ) q^{81} + ( 2 \beta_{1} + 8 \beta_{4} - \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{83} + ( -2 + 4 \beta_{3} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{85} + ( -46 - 3 \beta_{2} - 5 \beta_{3} - \beta_{5} + 3 \beta_{13} - \beta_{14} ) q^{87} + ( -12 \beta_{1} + 10 \beta_{4} + \beta_{6} + \beta_{8} - 5 \beta_{9} - 4 \beta_{15} ) q^{89} + ( -8 \beta_{1} - 8 \beta_{7} - 5 \beta_{9} + 2 \beta_{15} ) q^{91} + ( -25 + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 3 \beta_{11} + 2 \beta_{12} + 5 \beta_{13} - \beta_{14} ) q^{93} + ( -5 + \beta_{3} - 3 \beta_{5} - \beta_{13} - \beta_{14} ) q^{95} + ( -3 \beta_{1} + 8 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} + 4 \beta_{10} - \beta_{15} ) q^{97} + ( 12 \beta_{1} + 38 \beta_{4} - \beta_{6} - 5 \beta_{7} - 10 \beta_{9} - 4 \beta_{10} + 8 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 64q^{9} + O(q^{10}) \) \( 16q + 64q^{9} - 12q^{13} + 14q^{23} - 80q^{25} - 48q^{27} + 90q^{29} - 10q^{31} - 30q^{35} - 20q^{39} + 186q^{41} + 320q^{47} + 2q^{49} + 120q^{55} + 90q^{59} - 232q^{69} + 238q^{71} - 280q^{73} + 324q^{77} + 704q^{81} - 30q^{85} - 724q^{87} - 380q^{93} - 80q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + 38013345 x^{4} - 8913096 x^{3} + 327235800 x^{2} + 72566336 x + 1535848276\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(23\!\cdots\!29\)\( \nu^{15} - \)\(42\!\cdots\!34\)\( \nu^{14} - \)\(12\!\cdots\!60\)\( \nu^{13} + \)\(23\!\cdots\!76\)\( \nu^{12} + \)\(11\!\cdots\!72\)\( \nu^{11} - \)\(79\!\cdots\!84\)\( \nu^{10} - \)\(26\!\cdots\!08\)\( \nu^{9} + \)\(74\!\cdots\!49\)\( \nu^{8} - \)\(57\!\cdots\!86\)\( \nu^{7} - \)\(12\!\cdots\!80\)\( \nu^{6} - \)\(12\!\cdots\!44\)\( \nu^{5} - \)\(16\!\cdots\!66\)\( \nu^{4} + \)\(83\!\cdots\!91\)\( \nu^{3} - \)\(56\!\cdots\!87\)\( \nu^{2} + \)\(61\!\cdots\!34\)\( \nu - \)\(21\!\cdots\!52\)\(\)\()/ \)\(97\!\cdots\!90\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(23\!\cdots\!29\)\( \nu^{15} - \)\(42\!\cdots\!34\)\( \nu^{14} - \)\(12\!\cdots\!60\)\( \nu^{13} + \)\(23\!\cdots\!76\)\( \nu^{12} + \)\(11\!\cdots\!72\)\( \nu^{11} - \)\(79\!\cdots\!84\)\( \nu^{10} - \)\(26\!\cdots\!08\)\( \nu^{9} + \)\(74\!\cdots\!49\)\( \nu^{8} - \)\(57\!\cdots\!86\)\( \nu^{7} - \)\(12\!\cdots\!80\)\( \nu^{6} - \)\(12\!\cdots\!44\)\( \nu^{5} - \)\(16\!\cdots\!66\)\( \nu^{4} + \)\(83\!\cdots\!91\)\( \nu^{3} - \)\(76\!\cdots\!42\)\( \nu^{2} + \)\(61\!\cdots\!34\)\( \nu - \)\(60\!\cdots\!12\)\(\)\()/ \)\(48\!\cdots\!45\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(34\!\cdots\!21\)\( \nu^{15} + \)\(46\!\cdots\!58\)\( \nu^{14} + \)\(21\!\cdots\!76\)\( \nu^{13} + \)\(53\!\cdots\!16\)\( \nu^{12} - \)\(73\!\cdots\!40\)\( \nu^{11} - \)\(20\!\cdots\!64\)\( \nu^{10} + \)\(89\!\cdots\!58\)\( \nu^{9} - \)\(48\!\cdots\!76\)\( \nu^{8} - \)\(11\!\cdots\!90\)\( \nu^{7} - \)\(79\!\cdots\!96\)\( \nu^{6} - \)\(69\!\cdots\!16\)\( \nu^{5} + \)\(12\!\cdots\!36\)\( \nu^{4} - \)\(16\!\cdots\!77\)\( \nu^{3} + \)\(47\!\cdots\!98\)\( \nu^{2} - \)\(32\!\cdots\!94\)\( \nu - \)\(12\!\cdots\!88\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(34\!\cdots\!21\)\( \nu^{15} + \)\(46\!\cdots\!58\)\( \nu^{14} + \)\(21\!\cdots\!76\)\( \nu^{13} + \)\(53\!\cdots\!16\)\( \nu^{12} - \)\(73\!\cdots\!40\)\( \nu^{11} - \)\(20\!\cdots\!64\)\( \nu^{10} + \)\(89\!\cdots\!58\)\( \nu^{9} - \)\(48\!\cdots\!76\)\( \nu^{8} - \)\(11\!\cdots\!90\)\( \nu^{7} - \)\(79\!\cdots\!96\)\( \nu^{6} - \)\(69\!\cdots\!16\)\( \nu^{5} + \)\(12\!\cdots\!36\)\( \nu^{4} - \)\(16\!\cdots\!77\)\( \nu^{3} + \)\(47\!\cdots\!98\)\( \nu^{2} - \)\(22\!\cdots\!74\)\( \nu - \)\(12\!\cdots\!88\)\(\)\()/ \)\(19\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(14\!\cdots\!29\)\( \nu^{15} - \)\(59\!\cdots\!16\)\( \nu^{14} + \)\(15\!\cdots\!93\)\( \nu^{13} + \)\(31\!\cdots\!88\)\( \nu^{12} - \)\(79\!\cdots\!61\)\( \nu^{11} - \)\(10\!\cdots\!24\)\( \nu^{10} + \)\(22\!\cdots\!75\)\( \nu^{9} + \)\(76\!\cdots\!88\)\( \nu^{8} - \)\(40\!\cdots\!79\)\( \nu^{7} - \)\(15\!\cdots\!52\)\( \nu^{6} + \)\(33\!\cdots\!83\)\( \nu^{5} - \)\(16\!\cdots\!00\)\( \nu^{4} - \)\(12\!\cdots\!76\)\( \nu^{3} - \)\(53\!\cdots\!48\)\( \nu^{2} + \)\(27\!\cdots\!40\)\( \nu - \)\(33\!\cdots\!12\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(18\!\cdots\!44\)\( \nu^{15} - \)\(21\!\cdots\!33\)\( \nu^{14} + \)\(81\!\cdots\!10\)\( \nu^{13} + \)\(13\!\cdots\!45\)\( \nu^{12} - \)\(20\!\cdots\!14\)\( \nu^{11} - \)\(10\!\cdots\!25\)\( \nu^{10} - \)\(11\!\cdots\!54\)\( \nu^{9} - \)\(14\!\cdots\!25\)\( \nu^{8} - \)\(28\!\cdots\!18\)\( \nu^{7} + \)\(10\!\cdots\!97\)\( \nu^{6} - \)\(67\!\cdots\!66\)\( \nu^{5} - \)\(65\!\cdots\!09\)\( \nu^{4} - \)\(17\!\cdots\!26\)\( \nu^{3} + \)\(23\!\cdots\!64\)\( \nu^{2} - \)\(16\!\cdots\!08\)\( \nu + \)\(93\!\cdots\!56\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(20\!\cdots\!08\)\( \nu^{15} + \)\(98\!\cdots\!41\)\( \nu^{14} + \)\(15\!\cdots\!94\)\( \nu^{13} - \)\(59\!\cdots\!17\)\( \nu^{12} - \)\(60\!\cdots\!50\)\( \nu^{11} + \)\(20\!\cdots\!57\)\( \nu^{10} + \)\(10\!\cdots\!34\)\( \nu^{9} - \)\(32\!\cdots\!63\)\( \nu^{8} - \)\(93\!\cdots\!14\)\( \nu^{7} + \)\(42\!\cdots\!31\)\( \nu^{6} - \)\(10\!\cdots\!22\)\( \nu^{5} + \)\(68\!\cdots\!13\)\( \nu^{4} - \)\(51\!\cdots\!74\)\( \nu^{3} + \)\(36\!\cdots\!04\)\( \nu^{2} - \)\(46\!\cdots\!80\)\( \nu + \)\(72\!\cdots\!04\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(31\!\cdots\!80\)\( \nu^{15} - \)\(32\!\cdots\!25\)\( \nu^{14} - \)\(18\!\cdots\!14\)\( \nu^{13} + \)\(21\!\cdots\!01\)\( \nu^{12} + \)\(63\!\cdots\!66\)\( \nu^{11} - \)\(77\!\cdots\!21\)\( \nu^{10} - \)\(78\!\cdots\!30\)\( \nu^{9} + \)\(12\!\cdots\!75\)\( \nu^{8} + \)\(45\!\cdots\!46\)\( \nu^{7} - \)\(15\!\cdots\!19\)\( \nu^{6} - \)\(20\!\cdots\!30\)\( \nu^{5} + \)\(28\!\cdots\!47\)\( \nu^{4} + \)\(26\!\cdots\!30\)\( \nu^{3} - \)\(17\!\cdots\!96\)\( \nu^{2} + \)\(14\!\cdots\!40\)\( \nu - \)\(54\!\cdots\!00\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(16\!\cdots\!59\)\( \nu^{15} - \)\(29\!\cdots\!08\)\( \nu^{14} - \)\(11\!\cdots\!96\)\( \nu^{13} + \)\(20\!\cdots\!48\)\( \nu^{12} + \)\(42\!\cdots\!46\)\( \nu^{11} - \)\(88\!\cdots\!36\)\( \nu^{10} - \)\(70\!\cdots\!06\)\( \nu^{9} + \)\(22\!\cdots\!48\)\( \nu^{8} + \)\(72\!\cdots\!94\)\( \nu^{7} - \)\(28\!\cdots\!36\)\( \nu^{6} + \)\(21\!\cdots\!26\)\( \nu^{5} + \)\(47\!\cdots\!04\)\( \nu^{4} + \)\(20\!\cdots\!31\)\( \nu^{3} - \)\(38\!\cdots\!40\)\( \nu^{2} - \)\(20\!\cdots\!78\)\( \nu + \)\(32\!\cdots\!44\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(60\!\cdots\!66\)\( \nu^{15} + \)\(51\!\cdots\!07\)\( \nu^{14} - \)\(42\!\cdots\!58\)\( \nu^{13} - \)\(58\!\cdots\!35\)\( \nu^{12} + \)\(15\!\cdots\!46\)\( \nu^{11} + \)\(26\!\cdots\!39\)\( \nu^{10} - \)\(23\!\cdots\!22\)\( \nu^{9} - \)\(48\!\cdots\!61\)\( \nu^{8} + \)\(26\!\cdots\!30\)\( \nu^{7} + \)\(79\!\cdots\!49\)\( \nu^{6} + \)\(59\!\cdots\!86\)\( \nu^{5} - \)\(82\!\cdots\!41\)\( \nu^{4} - \)\(13\!\cdots\!16\)\( \nu^{3} - \)\(62\!\cdots\!00\)\( \nu^{2} + \)\(16\!\cdots\!44\)\( \nu + \)\(41\!\cdots\!16\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(32\!\cdots\!82\)\( \nu^{15} + \)\(27\!\cdots\!58\)\( \nu^{14} + \)\(23\!\cdots\!93\)\( \nu^{13} - \)\(13\!\cdots\!24\)\( \nu^{12} - \)\(88\!\cdots\!67\)\( \nu^{11} + \)\(39\!\cdots\!00\)\( \nu^{10} + \)\(14\!\cdots\!29\)\( \nu^{9} - \)\(10\!\cdots\!72\)\( \nu^{8} - \)\(18\!\cdots\!13\)\( \nu^{7} + \)\(48\!\cdots\!04\)\( \nu^{6} + \)\(71\!\cdots\!09\)\( \nu^{5} + \)\(30\!\cdots\!92\)\( \nu^{4} - \)\(15\!\cdots\!49\)\( \nu^{3} + \)\(17\!\cdots\!02\)\( \nu^{2} - \)\(18\!\cdots\!34\)\( \nu + \)\(87\!\cdots\!64\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(66\!\cdots\!45\)\( \nu^{15} + \)\(54\!\cdots\!24\)\( \nu^{14} + \)\(39\!\cdots\!05\)\( \nu^{13} - \)\(34\!\cdots\!80\)\( \nu^{12} - \)\(14\!\cdots\!53\)\( \nu^{11} + \)\(11\!\cdots\!64\)\( \nu^{10} + \)\(18\!\cdots\!07\)\( \nu^{9} - \)\(16\!\cdots\!08\)\( \nu^{8} - \)\(13\!\cdots\!79\)\( \nu^{7} + \)\(11\!\cdots\!84\)\( \nu^{6} - \)\(92\!\cdots\!13\)\( \nu^{5} + \)\(18\!\cdots\!80\)\( \nu^{4} - \)\(60\!\cdots\!04\)\( \nu^{3} + \)\(69\!\cdots\!84\)\( \nu^{2} - \)\(21\!\cdots\!04\)\( \nu + \)\(10\!\cdots\!68\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(67\!\cdots\!87\)\( \nu^{15} - \)\(14\!\cdots\!24\)\( \nu^{14} + \)\(47\!\cdots\!37\)\( \nu^{13} + \)\(11\!\cdots\!68\)\( \nu^{12} - \)\(17\!\cdots\!85\)\( \nu^{11} - \)\(41\!\cdots\!76\)\( \nu^{10} + \)\(28\!\cdots\!59\)\( \nu^{9} + \)\(44\!\cdots\!48\)\( \nu^{8} - \)\(38\!\cdots\!91\)\( \nu^{7} - \)\(41\!\cdots\!04\)\( \nu^{6} + \)\(11\!\cdots\!35\)\( \nu^{5} - \)\(30\!\cdots\!56\)\( \nu^{4} - \)\(18\!\cdots\!22\)\( \nu^{3} - \)\(15\!\cdots\!60\)\( \nu^{2} + \)\(29\!\cdots\!80\)\( \nu - \)\(20\!\cdots\!12\)\(\)\()/ \)\(34\!\cdots\!80\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(59\!\cdots\!01\)\( \nu^{15} + \)\(11\!\cdots\!38\)\( \nu^{14} + \)\(34\!\cdots\!10\)\( \nu^{13} - \)\(65\!\cdots\!40\)\( \nu^{12} - \)\(10\!\cdots\!32\)\( \nu^{11} + \)\(22\!\cdots\!16\)\( \nu^{10} + \)\(86\!\cdots\!40\)\( \nu^{9} - \)\(43\!\cdots\!00\)\( \nu^{8} - \)\(27\!\cdots\!52\)\( \nu^{7} + \)\(14\!\cdots\!64\)\( \nu^{6} - \)\(24\!\cdots\!00\)\( \nu^{5} + \)\(61\!\cdots\!72\)\( \nu^{4} - \)\(30\!\cdots\!83\)\( \nu^{3} + \)\(43\!\cdots\!66\)\( \nu^{2} - \)\(98\!\cdots\!46\)\( \nu + \)\(15\!\cdots\!28\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(64\!\cdots\!75\)\( \nu^{15} - \)\(21\!\cdots\!42\)\( \nu^{14} - \)\(43\!\cdots\!42\)\( \nu^{13} + \)\(10\!\cdots\!92\)\( \nu^{12} + \)\(15\!\cdots\!04\)\( \nu^{11} - \)\(47\!\cdots\!92\)\( \nu^{10} - \)\(23\!\cdots\!34\)\( \nu^{9} + \)\(29\!\cdots\!16\)\( \nu^{8} + \)\(25\!\cdots\!36\)\( \nu^{7} - \)\(19\!\cdots\!44\)\( \nu^{6} + \)\(91\!\cdots\!40\)\( \nu^{5} - \)\(99\!\cdots\!56\)\( \nu^{4} + \)\(11\!\cdots\!63\)\( \nu^{3} + \)\(14\!\cdots\!62\)\( \nu^{2} + \)\(14\!\cdots\!02\)\( \nu + \)\(16\!\cdots\!20\)\(\)\()/ \)\(17\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{4} + \beta_{3}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2 \beta_{1} + 8\)
\(\nu^{3}\)\(=\)\(-3 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} + 6 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} - \beta_{5} - 37 \beta_{4} + 9 \beta_{3} + \beta_{2} + 3\)
\(\nu^{4}\)\(=\)\(-12 \beta_{15} - 3 \beta_{14} + 2 \beta_{12} + 3 \beta_{11} + 12 \beta_{10} + 16 \beta_{9} + 8 \beta_{8} + 4 \beta_{7} - 4 \beta_{5} - 24 \beta_{4} + 14 \beta_{3} + 2 \beta_{2} - 80 \beta_{1} - 51\)
\(\nu^{5}\)\(=\)\(-105 \beta_{15} + 12 \beta_{14} + 14 \beta_{13} + \beta_{11} + 50 \beta_{10} + 260 \beta_{9} + 25 \beta_{8} + 180 \beta_{7} + 105 \beta_{6} + 8 \beta_{5} - 1125 \beta_{4} - 403 \beta_{3} + 10 \beta_{2} - 70 \beta_{1} + 121\)
\(\nu^{6}\)\(=\)\(-492 \beta_{15} + 76 \beta_{14} + 2 \beta_{13} - 67 \beta_{12} - 88 \beta_{11} + 438 \beta_{10} + 902 \beta_{9} + 202 \beta_{8} + 470 \beta_{7} + 54 \beta_{6} + 105 \beta_{5} - 1992 \beta_{4} - 96 \beta_{3} - 1029 \beta_{2} - 1886 \beta_{1} - 9948\)
\(\nu^{7}\)\(=\)\(-2093 \beta_{15} + 1833 \beta_{14} + 1717 \beta_{13} + 6 \beta_{12} - 2546 \beta_{11} + 1554 \beta_{10} + 5040 \beta_{9} + 1036 \beta_{8} + 4452 \beta_{7} + 1694 \beta_{6} + 2005 \beta_{5} - 17921 \beta_{4} - 30306 \beta_{3} - 2749 \beta_{2} - 4368 \beta_{1} - 10828\)
\(\nu^{8}\)\(=\)\(-7032 \beta_{15} + 9718 \beta_{14} - 476 \beta_{13} - 5252 \beta_{12} - 8558 \beta_{11} + 3808 \beta_{10} + 17352 \beta_{9} + 1272 \beta_{8} + 11040 \beta_{7} + 3464 \beta_{6} + 12708 \beta_{5} - 60064 \beta_{4} - 63452 \beta_{3} - 55267 \beta_{2} - 5576 \beta_{1} - 467398\)
\(\nu^{9}\)\(=\)\(28053 \beta_{15} + 78623 \beta_{14} + 72687 \beta_{13} - 80 \beta_{12} - 130637 \beta_{11} - 1824 \beta_{10} - 75510 \beta_{9} + 1314 \beta_{8} - 47247 \beta_{7} - 39969 \beta_{6} + 100283 \beta_{5} + 425261 \beta_{4} - 1124603 \beta_{3} - 196087 \beta_{2} - 57408 \beta_{1} - 1205919\)
\(\nu^{10}\)\(=\)\(401930 \beta_{15} + 404837 \beta_{14} + 32552 \beta_{13} - 167844 \beta_{12} - 363817 \beta_{11} - 385430 \beta_{10} - 679140 \beta_{9} - 191720 \beta_{8} - 294010 \beta_{7} + 15500 \beta_{6} + 512402 \beta_{5} + 1117380 \beta_{4} - 3723658 \beta_{3} - 1680140 \beta_{2} + 2017660 \beta_{1} - 13138755\)
\(\nu^{11}\)\(=\)\(4719737 \beta_{15} + 1871644 \beta_{14} + 1472966 \beta_{13} - 152960 \beta_{12} - 3126813 \beta_{11} - 2668270 \beta_{10} - 11292204 \beta_{9} - 1695265 \beta_{8} - 8956222 \beta_{7} - 4319887 \beta_{6} + 2526084 \beta_{5} + 46905189 \beta_{4} - 22889857 \beta_{3} - 6671334 \beta_{2} + 6479946 \beta_{1} - 49274749\)
\(\nu^{12}\)\(=\)\(35357024 \beta_{15} + 7306256 \beta_{14} + 3282590 \beta_{13} - 1523959 \beta_{12} - 8191892 \beta_{11} - 27317252 \beta_{10} - 69695660 \beta_{9} - 12986388 \beta_{8} - 37465716 \beta_{7} - 8593372 \beta_{6} + 8891241 \beta_{5} + 192058512 \beta_{4} - 93874956 \beta_{3} - 19001651 \beta_{2} + 119396812 \beta_{1} - 121656184\)
\(\nu^{13}\)\(=\)\(238683133 \beta_{15} - 5803841 \beta_{14} - 19627253 \beta_{13} - 7811938 \beta_{12} + 17346326 \beta_{11} - 155666914 \beta_{10} - 544563500 \beta_{9} - 97260826 \beta_{8} - 422615622 \beta_{7} - 182130754 \beta_{6} - 4851545 \beta_{5} + 2069826045 \beta_{4} + 226480374 \beta_{3} - 47695775 \beta_{2} + 504243844 \beta_{1} - 546243096\)
\(\nu^{14}\)\(=\)\(1439013442 \beta_{15} - 228382132 \beta_{14} + 68953924 \beta_{13} + 144852348 \beta_{12} + 139890736 \beta_{11} - 1015933716 \beta_{10} - 2988873050 \beta_{9} - 496930214 \beta_{8} - 1770634388 \beta_{7} - 573467386 \beta_{6} - 294366192 \beta_{5} + 9491328720 \beta_{4} + 1346596748 \beta_{3} + 1299106089 \beta_{2} + 4014142154 \beta_{1} + 11047082428\)
\(\nu^{15}\)\(=\)\(6968471215 \beta_{15} - 3180242887 \beta_{14} - 2976448991 \beta_{13} + 10488064 \beta_{12} + 5374132165 \beta_{11} - 4918684580 \beta_{10} - 15147940310 \beta_{9} - 2891101520 \beta_{8} - 10933071245 \beta_{7} - 4220391785 \beta_{6} - 4092112867 \beta_{5} + 52749131295 \beta_{4} + 45082668763 \beta_{3} + 8735228059 \beta_{2} + 18359505520 \beta_{1} + 57576702111\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
5.89296 2.23607i
5.89296 + 2.23607i
3.41677 2.23607i
3.41677 + 2.23607i
2.47022 2.23607i
2.47022 + 2.23607i
0.613622 2.23607i
0.613622 + 2.23607i
−0.886481 2.23607i
−0.886481 + 2.23607i
−1.83366 2.23607i
−1.83366 + 2.23607i
−4.53300 2.23607i
−4.53300 + 2.23607i
−5.14043 2.23607i
−5.14043 + 2.23607i
0 −5.89296 0 2.23607i 0 1.21480i 0 25.7269 0
321.2 0 −5.89296 0 2.23607i 0 1.21480i 0 25.7269 0
321.3 0 −3.41677 0 2.23607i 0 11.7428i 0 2.67432 0
321.4 0 −3.41677 0 2.23607i 0 11.7428i 0 2.67432 0
321.5 0 −2.47022 0 2.23607i 0 8.25674i 0 −2.89803 0
321.6 0 −2.47022 0 2.23607i 0 8.25674i 0 −2.89803 0
321.7 0 −0.613622 0 2.23607i 0 2.35348i 0 −8.62347 0
321.8 0 −0.613622 0 2.23607i 0 2.35348i 0 −8.62347 0
321.9 0 0.886481 0 2.23607i 0 9.10808i 0 −8.21415 0
321.10 0 0.886481 0 2.23607i 0 9.10808i 0 −8.21415 0
321.11 0 1.83366 0 2.23607i 0 0.167846i 0 −5.63770 0
321.12 0 1.83366 0 2.23607i 0 0.167846i 0 −5.63770 0
321.13 0 4.53300 0 2.23607i 0 5.73038i 0 11.5481 0
321.14 0 4.53300 0 2.23607i 0 5.73038i 0 11.5481 0
321.15 0 5.14043 0 2.23607i 0 7.88014i 0 17.4240 0
321.16 0 5.14043 0 2.23607i 0 7.88014i 0 17.4240 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.3.k.c 16
4.b odd 2 1 460.3.f.a 16
12.b even 2 1 4140.3.d.a 16
20.d odd 2 1 2300.3.f.e 16
20.e even 4 2 2300.3.d.b 32
23.b odd 2 1 inner 1840.3.k.c 16
92.b even 2 1 460.3.f.a 16
276.h odd 2 1 4140.3.d.a 16
460.g even 2 1 2300.3.f.e 16
460.k odd 4 2 2300.3.d.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.3.f.a 16 4.b odd 2 1
460.3.f.a 16 92.b even 2 1
1840.3.k.c 16 1.a even 1 1 trivial
1840.3.k.c 16 23.b odd 2 1 inner
2300.3.d.b 32 20.e even 4 2
2300.3.d.b 32 460.k odd 4 2
2300.3.f.e 16 20.d odd 2 1
2300.3.f.e 16 460.g even 2 1
4140.3.d.a 16 12.b even 2 1
4140.3.d.a 16 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 52 T_{3}^{6} + 8 T_{3}^{5} + 724 T_{3}^{4} + 12 T_{3}^{3} - 2557 T_{3}^{2} + 472 T_{3} + 1156 \) acting on \(S_{3}^{\mathrm{new}}(1840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 1156 + 472 T - 2557 T^{2} + 12 T^{3} + 724 T^{4} + 8 T^{5} - 52 T^{6} + T^{8} )^{2} \)
$5$ \( ( 5 + T^{2} )^{8} \)
$7$ \( 366186496 + 13341794000 T^{2} + 12272252524 T^{4} + 2524614569 T^{6} + 155886747 T^{8} + 4281913 T^{10} + 58685 T^{12} + 391 T^{14} + T^{16} \)
$11$ \( 538982656 + 20926554176 T^{2} + 62650963968 T^{4} + 23451968484 T^{6} + 1682709097 T^{8} + 32012450 T^{10} + 249475 T^{12} + 842 T^{14} + T^{16} \)
$13$ \( ( 12532480 + 55269430 T - 8146861 T^{2} - 1210348 T^{3} + 209672 T^{4} + 408 T^{5} - 832 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$17$ \( 645469713961216 + 1095995238952528 T^{2} + 183863288669160 T^{4} + 7139592083589 T^{6} + 103619670159 T^{8} + 612883733 T^{10} + 1669657 T^{12} + 2107 T^{14} + T^{16} \)
$19$ \( 483658046479878400 + 209327202365092160 T^{2} + 19764941764557056 T^{4} + 263607893757396 T^{6} + 1388771053737 T^{8} + 3650498706 T^{10} + 5083907 T^{12} + 3578 T^{14} + T^{16} \)
$23$ \( \)\(61\!\cdots\!61\)\( - \)\(16\!\cdots\!26\)\( T + 18495943020625150924 T^{2} - 1274362337954270538 T^{3} + 79976503315956308 T^{4} - 3955052048886094 T^{5} + 202509260091572 T^{6} - 10158790780410 T^{7} + 461135845846 T^{8} - 19203763290 T^{9} + 723658292 T^{10} - 26716846 T^{11} + 1021268 T^{12} - 30762 T^{13} + 844 T^{14} - 14 T^{15} + T^{16} \)
$29$ \( ( -519358880 + 539930480 T + 343011096 T^{2} - 46866036 T^{3} + 432474 T^{4} + 110777 T^{5} - 2411 T^{6} - 45 T^{7} + T^{8} )^{2} \)
$31$ \( ( -162152849534 - 39818067751 T - 2322982667 T^{2} + 53506228 T^{3} + 5788622 T^{4} - 20974 T^{5} - 4394 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$37$ \( 216591737307136 + 166004797403668480 T^{2} + 22822976416410624 T^{4} + 848256676333312 T^{6} + 5630424572096 T^{8} + 14212671024 T^{10} + 15172360 T^{12} + 6637 T^{14} + T^{16} \)
$41$ \( ( 580968030770 + 39640831075 T - 3870477691 T^{2} - 240145676 T^{3} + 4624142 T^{4} + 285874 T^{5} - 2586 T^{6} - 93 T^{7} + T^{8} )^{2} \)
$43$ \( \)\(17\!\cdots\!56\)\( + \)\(18\!\cdots\!20\)\( T^{2} + \)\(66\!\cdots\!24\)\( T^{4} + 1049280834589435904 T^{6} + 875691541868032 T^{8} + 416545461888 T^{10} + 113608080 T^{12} + 16556 T^{14} + T^{16} \)
$47$ \( ( -66361863793216 + 1164169651008 T + 103994227520 T^{2} - 2253983984 T^{3} - 41346840 T^{4} + 1249104 T^{5} - 997 T^{6} - 160 T^{7} + T^{8} )^{2} \)
$53$ \( \)\(55\!\cdots\!76\)\( + \)\(11\!\cdots\!76\)\( T^{2} + 82687734869125993472 T^{4} + 264266260655688960 T^{6} + 408063006271872 T^{8} + 306592228448 T^{10} + 110793796 T^{12} + 18049 T^{14} + T^{16} \)
$59$ \( ( 199085136256 + 130516990272 T - 1515485024 T^{2} - 516164720 T^{3} + 9479032 T^{4} + 399932 T^{5} - 7890 T^{6} - 45 T^{7} + T^{8} )^{2} \)
$61$ \( \)\(21\!\cdots\!76\)\( + \)\(90\!\cdots\!72\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{4} + 14634885952073347364 T^{6} + 7721448036303089 T^{8} + 2321502596466 T^{10} + 380794547 T^{12} + 31266 T^{14} + T^{16} \)
$67$ \( \)\(11\!\cdots\!96\)\( + \)\(10\!\cdots\!32\)\( T^{2} + \)\(88\!\cdots\!68\)\( T^{4} + 22176717437650435072 T^{6} + 15511162691402304 T^{8} + 4555411323968 T^{10} + 629483568 T^{12} + 40765 T^{14} + T^{16} \)
$71$ \( ( -318523111508570 + 24525587575845 T + 44577347181 T^{2} - 14923409512 T^{3} + 80275002 T^{4} + 2470446 T^{5} - 18498 T^{6} - 119 T^{7} + T^{8} )^{2} \)
$73$ \( ( -71365498673344 + 873793190848 T + 664172759456 T^{2} + 16845279680 T^{3} - 25382248 T^{4} - 3676292 T^{5} - 19845 T^{6} + 140 T^{7} + T^{8} )^{2} \)
$79$ \( \)\(17\!\cdots\!16\)\( + \)\(28\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!24\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{6} + 63613047675516672 T^{8} + 12545839071936 T^{10} + 1215192880 T^{12} + 56440 T^{14} + T^{16} \)
$83$ \( \)\(14\!\cdots\!16\)\( + \)\(10\!\cdots\!52\)\( T^{2} + \)\(78\!\cdots\!00\)\( T^{4} + 87029274096726457344 T^{6} + 38243523522303296 T^{8} + 8309783161072 T^{10} + 936578764 T^{12} + 51025 T^{14} + T^{16} \)
$89$ \( \)\(43\!\cdots\!16\)\( + \)\(67\!\cdots\!16\)\( T^{2} + \)\(32\!\cdots\!36\)\( T^{4} + \)\(74\!\cdots\!64\)\( T^{6} + 986394349960649472 T^{8} + 78193127349184 T^{10} + 3666695728 T^{12} + 93608 T^{14} + T^{16} \)
$97$ \( \)\(31\!\cdots\!56\)\( + \)\(13\!\cdots\!40\)\( T^{2} + \)\(16\!\cdots\!44\)\( T^{4} + 61013760166167497248 T^{6} + 36744767797548921 T^{8} + 8800917147286 T^{10} + 986183835 T^{12} + 51438 T^{14} + T^{16} \)
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