Properties

Label 1.44.a.a
Level 1
Weight 44
Character orbit 1.a
Self dual Yes
Analytic conductor 11.711
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 44 \)
Character orbit: \([\chi]\) = 1.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(11.7110395346\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4}\cdot 5\cdot 7 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -736648 - \beta_{1} ) q^{2} \) \( + ( 8133812604 + 2135 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 3274206140608 + 810816 \beta_{1} - 408 \beta_{2} ) q^{4} \) \( + ( 178401793591390 - 23658932 \beta_{1} - 29844 \beta_{2} ) q^{5} \) \( + ( -30597991586147808 - 21343496316 \beta_{1} + 2270208 \beta_{2} ) q^{6} \) \( + ( 100657141888492952 - 411863382714 \beta_{1} - 60797226 \beta_{2} ) q^{7} \) \( + ( -5276954637929116160 + 136805587456 \beta_{1} + 901657152 \beta_{2} ) q^{8} \) \( + ( 159160708390355581077 + 108619653530952 \beta_{1} - 7904835576 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(-736648 - \beta_{1}) q^{2}\) \(+(8133812604 + 2135 \beta_{1} - \beta_{2}) q^{3}\) \(+(3274206140608 + 810816 \beta_{1} - 408 \beta_{2}) q^{4}\) \(+(178401793591390 - 23658932 \beta_{1} - 29844 \beta_{2}) q^{5}\) \(+(-30597991586147808 - 21343496316 \beta_{1} + 2270208 \beta_{2}) q^{6}\) \(+(100657141888492952 - 411863382714 \beta_{1} - 60797226 \beta_{2}) q^{7}\) \(+(-5276954637929116160 + 136805587456 \beta_{1} + 901657152 \beta_{2}) q^{8}\) \(+(\)\(15\!\cdots\!77\)\( + 108619653530952 \beta_{1} - 7904835576 \beta_{2}) q^{9}\) \(+(\)\(14\!\cdots\!40\)\( - 566151100617822 \beta_{1} + 32102731776 \beta_{2}) q^{10}\) \(+(\)\(88\!\cdots\!32\)\( - 4605494939944195 \beta_{1} + 137241701685 \beta_{2}) q^{11}\) \(+(\)\(19\!\cdots\!12\)\( + 43030582618825472 \beta_{1} - 3088365053344 \beta_{2}) q^{12}\) \(+(\)\(89\!\cdots\!94\)\( - 6729074743426932 \beta_{1} + 22807920190380 \beta_{2}) q^{13}\) \(+(\)\(46\!\cdots\!76\)\( - 863595024061582232 \beta_{1} - 82977158928384 \beta_{2}) q^{14}\) \(+(\)\(11\!\cdots\!80\)\( + 1376784350493064386 \beta_{1} - 37572060903438 \beta_{2}) q^{15}\) \(+(-\)\(26\!\cdots\!64\)\( + 9902624655975026688 \beta_{1} + 2383088864979456 \beta_{2}) q^{16}\) \(+(-\)\(13\!\cdots\!98\)\( - 4605229780369521720 \beta_{1} - 15383154249178488 \beta_{2}) q^{17}\) \(+(-\)\(13\!\cdots\!16\)\( - \)\(27\!\cdots\!45\)\( \beta_{1} + 55376695430406144 \beta_{2}) q^{18}\) \(+(\)\(52\!\cdots\!00\)\( + \)\(85\!\cdots\!19\)\( \beta_{1} - 107308048565285397 \beta_{2}) q^{19}\) \(+(\)\(48\!\cdots\!20\)\( + \)\(52\!\cdots\!44\)\( \beta_{1} - 13394879801587152 \beta_{2}) q^{20}\) \(+(\)\(13\!\cdots\!12\)\( - \)\(49\!\cdots\!32\)\( \beta_{1} + 743940509412748816 \beta_{2}) q^{21}\) \(+(\)\(46\!\cdots\!64\)\( - \)\(67\!\cdots\!52\)\( \beta_{1} - 2071060643092362240 \beta_{2}) q^{22}\) \(+(-\)\(40\!\cdots\!16\)\( + \)\(54\!\cdots\!86\)\( \beta_{1} + 1876113512966847906 \beta_{2}) q^{23}\) \(+(-\)\(37\!\cdots\!00\)\( - \)\(52\!\cdots\!88\)\( \beta_{1} + 1908534981075097344 \beta_{2}) q^{24}\) \(+(-\)\(77\!\cdots\!25\)\( - \)\(83\!\cdots\!40\)\( \beta_{1} + 1683060411116573520 \beta_{2}) q^{25}\) \(+(-\)\(57\!\cdots\!48\)\( - \)\(59\!\cdots\!58\)\( \beta_{1} - 34656662255444176896 \beta_{2}) q^{26}\) \(+(\)\(42\!\cdots\!60\)\( + \)\(21\!\cdots\!74\)\( \beta_{1} + 30213151036337641158 \beta_{2}) q^{27}\) \(+(\)\(56\!\cdots\!56\)\( - \)\(20\!\cdots\!76\)\( \beta_{1} + \)\(29\!\cdots\!04\)\( \beta_{2}) q^{28}\) \(+(\)\(19\!\cdots\!50\)\( + \)\(17\!\cdots\!76\)\( \beta_{1} - \)\(10\!\cdots\!88\)\( \beta_{2}) q^{29}\) \(+(-\)\(24\!\cdots\!20\)\( - \)\(12\!\cdots\!44\)\( \beta_{1} + \)\(61\!\cdots\!52\)\( \beta_{2}) q^{30}\) \(+(-\)\(85\!\cdots\!08\)\( + \)\(11\!\cdots\!40\)\( \beta_{1} + \)\(33\!\cdots\!80\)\( \beta_{2}) q^{31}\) \(+(-\)\(48\!\cdots\!28\)\( + \)\(55\!\cdots\!24\)\( \beta_{1} - \)\(72\!\cdots\!80\)\( \beta_{2}) q^{32}\) \(+(-\)\(91\!\cdots\!72\)\( - \)\(78\!\cdots\!20\)\( \beta_{1} - \)\(23\!\cdots\!12\)\( \beta_{2}) q^{33}\) \(+(\)\(15\!\cdots\!96\)\( - \)\(64\!\cdots\!58\)\( \beta_{1} + \)\(19\!\cdots\!04\)\( \beta_{2}) q^{34}\) \(+(\)\(79\!\cdots\!40\)\( - \)\(22\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!36\)\( \beta_{2}) q^{35}\) \(+(\)\(27\!\cdots\!16\)\( + \)\(11\!\cdots\!68\)\( \beta_{1} - \)\(11\!\cdots\!84\)\( \beta_{2}) q^{36}\) \(+(-\)\(77\!\cdots\!98\)\( - \)\(43\!\cdots\!72\)\( \beta_{1} + \)\(46\!\cdots\!08\)\( \beta_{2}) q^{37}\) \(+(-\)\(10\!\cdots\!40\)\( - \)\(19\!\cdots\!96\)\( \beta_{1} + \)\(49\!\cdots\!68\)\( \beta_{2}) q^{38}\) \(+(-\)\(13\!\cdots\!76\)\( + \)\(65\!\cdots\!78\)\( \beta_{1} - \)\(96\!\cdots\!14\)\( \beta_{2}) q^{39}\) \(+(-\)\(10\!\cdots\!00\)\( - \)\(86\!\cdots\!00\)\( \beta_{1} - \)\(48\!\cdots\!00\)\( \beta_{2}) q^{40}\) \(+(\)\(85\!\cdots\!22\)\( + \)\(87\!\cdots\!40\)\( \beta_{1} + \)\(23\!\cdots\!80\)\( \beta_{2}) q^{41}\) \(+(\)\(47\!\cdots\!44\)\( - \)\(30\!\cdots\!24\)\( \beta_{1} - \)\(30\!\cdots\!04\)\( \beta_{2}) q^{42}\) \(+(\)\(80\!\cdots\!64\)\( - \)\(12\!\cdots\!83\)\( \beta_{1} + \)\(17\!\cdots\!73\)\( \beta_{2}) q^{43}\) \(+(-\)\(34\!\cdots\!44\)\( - \)\(32\!\cdots\!48\)\( \beta_{1} - \)\(10\!\cdots\!76\)\( \beta_{2}) q^{44}\) \(+(\)\(85\!\cdots\!30\)\( + \)\(62\!\cdots\!36\)\( \beta_{1} - \)\(40\!\cdots\!88\)\( \beta_{2}) q^{45}\) \(+(-\)\(63\!\cdots\!88\)\( + \)\(20\!\cdots\!60\)\( \beta_{1} + \)\(19\!\cdots\!20\)\( \beta_{2}) q^{46}\) \(+(\)\(10\!\cdots\!52\)\( + \)\(31\!\cdots\!32\)\( \beta_{1} - \)\(25\!\cdots\!84\)\( \beta_{2}) q^{47}\) \(+(-\)\(85\!\cdots\!16\)\( + \)\(22\!\cdots\!16\)\( \beta_{1} + \)\(29\!\cdots\!16\)\( \beta_{2}) q^{48}\) \(+(\)\(11\!\cdots\!93\)\( - \)\(77\!\cdots\!00\)\( \beta_{1} - \)\(30\!\cdots\!00\)\( \beta_{2}) q^{49}\) \(+(\)\(66\!\cdots\!00\)\( + \)\(79\!\cdots\!85\)\( \beta_{1} - \)\(57\!\cdots\!80\)\( \beta_{2}) q^{50}\) \(+(\)\(44\!\cdots\!52\)\( + \)\(37\!\cdots\!26\)\( \beta_{1} + \)\(19\!\cdots\!62\)\( \beta_{2}) q^{51}\) \(+(-\)\(57\!\cdots\!68\)\( + \)\(23\!\cdots\!76\)\( \beta_{1} - \)\(39\!\cdots\!16\)\( \beta_{2}) q^{52}\) \(+(\)\(50\!\cdots\!54\)\( + \)\(47\!\cdots\!80\)\( \beta_{1} - \)\(90\!\cdots\!36\)\( \beta_{2}) q^{53}\) \(+(-\)\(28\!\cdots\!00\)\( - \)\(39\!\cdots\!36\)\( \beta_{1} + \)\(84\!\cdots\!68\)\( \beta_{2}) q^{54}\) \(+(\)\(13\!\cdots\!80\)\( - \)\(29\!\cdots\!74\)\( \beta_{1} - \)\(24\!\cdots\!58\)\( \beta_{2}) q^{55}\) \(+(-\)\(21\!\cdots\!00\)\( + \)\(60\!\cdots\!96\)\( \beta_{1} - \)\(53\!\cdots\!48\)\( \beta_{2}) q^{56}\) \(+(\)\(64\!\cdots\!20\)\( + \)\(23\!\cdots\!28\)\( \beta_{1} - \)\(14\!\cdots\!24\)\( \beta_{2}) q^{57}\) \(+(-\)\(34\!\cdots\!60\)\( - \)\(32\!\cdots\!34\)\( \beta_{1} + \)\(21\!\cdots\!72\)\( \beta_{2}) q^{58}\) \(+(\)\(75\!\cdots\!00\)\( - \)\(62\!\cdots\!43\)\( \beta_{1} + \)\(37\!\cdots\!09\)\( \beta_{2}) q^{59}\) \(+(\)\(57\!\cdots\!40\)\( + \)\(21\!\cdots\!88\)\( \beta_{1} - \)\(56\!\cdots\!04\)\( \beta_{2}) q^{60}\) \(+(-\)\(31\!\cdots\!18\)\( - \)\(68\!\cdots\!00\)\( \beta_{1} - \)\(30\!\cdots\!00\)\( \beta_{2}) q^{61}\) \(+(-\)\(69\!\cdots\!16\)\( + \)\(12\!\cdots\!48\)\( \beta_{1} - \)\(20\!\cdots\!20\)\( \beta_{2}) q^{62}\) \(+(-\)\(32\!\cdots\!16\)\( + \)\(28\!\cdots\!14\)\( \beta_{1} + \)\(11\!\cdots\!62\)\( \beta_{2}) q^{63}\) \(+(-\)\(37\!\cdots\!12\)\( - \)\(13\!\cdots\!68\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2}) q^{64}\) \(+(-\)\(90\!\cdots\!20\)\( - \)\(32\!\cdots\!84\)\( \beta_{1} - \)\(31\!\cdots\!28\)\( \beta_{2}) q^{65}\) \(+(\)\(97\!\cdots\!44\)\( + \)\(67\!\cdots\!48\)\( \beta_{1} - \)\(28\!\cdots\!24\)\( \beta_{2}) q^{66}\) \(+(-\)\(24\!\cdots\!48\)\( - \)\(23\!\cdots\!25\)\( \beta_{1} + \)\(46\!\cdots\!27\)\( \beta_{2}) q^{67}\) \(+(\)\(18\!\cdots\!56\)\( + \)\(14\!\cdots\!36\)\( \beta_{1} + \)\(81\!\cdots\!28\)\( \beta_{2}) q^{68}\) \(+(\)\(65\!\cdots\!44\)\( + \)\(99\!\cdots\!24\)\( \beta_{1} - \)\(90\!\cdots\!12\)\( \beta_{2}) q^{69}\) \(+(\)\(19\!\cdots\!40\)\( - \)\(58\!\cdots\!32\)\( \beta_{1} - \)\(11\!\cdots\!44\)\( \beta_{2}) q^{70}\) \(+(-\)\(61\!\cdots\!88\)\( - \)\(47\!\cdots\!50\)\( \beta_{1} + \)\(97\!\cdots\!50\)\( \beta_{2}) q^{71}\) \(+(-\)\(32\!\cdots\!20\)\( - \)\(19\!\cdots\!68\)\( \beta_{1} + \)\(15\!\cdots\!44\)\( \beta_{2}) q^{72}\) \(+(-\)\(67\!\cdots\!66\)\( + \)\(18\!\cdots\!36\)\( \beta_{1} - \)\(13\!\cdots\!44\)\( \beta_{2}) q^{73}\) \(+(\)\(55\!\cdots\!36\)\( + \)\(14\!\cdots\!50\)\( \beta_{1} - \)\(24\!\cdots\!00\)\( \beta_{2}) q^{74}\) \(+(-\)\(70\!\cdots\!00\)\( - \)\(18\!\cdots\!55\)\( \beta_{1} + \)\(77\!\cdots\!65\)\( \beta_{2}) q^{75}\) \(+(\)\(25\!\cdots\!00\)\( + \)\(93\!\cdots\!92\)\( \beta_{1} - \)\(56\!\cdots\!96\)\( \beta_{2}) q^{76}\) \(+(\)\(19\!\cdots\!64\)\( - \)\(84\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!72\)\( \beta_{2}) q^{77}\) \(+(-\)\(65\!\cdots\!32\)\( - \)\(11\!\cdots\!76\)\( \beta_{1} + \)\(16\!\cdots\!16\)\( \beta_{2}) q^{78}\) \(+(\)\(50\!\cdots\!00\)\( + \)\(26\!\cdots\!96\)\( \beta_{1} + \)\(11\!\cdots\!52\)\( \beta_{2}) q^{79}\) \(+(-\)\(33\!\cdots\!60\)\( + \)\(56\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!16\)\( \beta_{2}) q^{80}\) \(+(\)\(24\!\cdots\!21\)\( + \)\(14\!\cdots\!44\)\( \beta_{1} - \)\(50\!\cdots\!72\)\( \beta_{2}) q^{81}\) \(+(-\)\(10\!\cdots\!56\)\( + \)\(20\!\cdots\!18\)\( \beta_{1} + \)\(32\!\cdots\!80\)\( \beta_{2}) q^{82}\) \(+(-\)\(29\!\cdots\!76\)\( - \)\(22\!\cdots\!81\)\( \beta_{1} + \)\(66\!\cdots\!87\)\( \beta_{2}) q^{83}\) \(+(-\)\(11\!\cdots\!04\)\( - \)\(43\!\cdots\!84\)\( \beta_{1} - \)\(34\!\cdots\!08\)\( \beta_{2}) q^{84}\) \(+(\)\(14\!\cdots\!40\)\( + \)\(55\!\cdots\!68\)\( \beta_{1} + \)\(76\!\cdots\!56\)\( \beta_{2}) q^{85}\) \(+(\)\(83\!\cdots\!32\)\( - \)\(55\!\cdots\!84\)\( \beta_{1} - \)\(75\!\cdots\!08\)\( \beta_{2}) q^{86}\) \(+(\)\(57\!\cdots\!80\)\( + \)\(12\!\cdots\!62\)\( \beta_{1} - \)\(18\!\cdots\!46\)\( \beta_{2}) q^{87}\) \(+(-\)\(85\!\cdots\!20\)\( + \)\(82\!\cdots\!92\)\( \beta_{1} + \)\(64\!\cdots\!64\)\( \beta_{2}) q^{88}\) \(+(\)\(67\!\cdots\!50\)\( - \)\(45\!\cdots\!52\)\( \beta_{1} + \)\(31\!\cdots\!76\)\( \beta_{2}) q^{89}\) \(+(-\)\(77\!\cdots\!20\)\( - \)\(14\!\cdots\!94\)\( \beta_{1} + \)\(31\!\cdots\!52\)\( \beta_{2}) q^{90}\) \(+(-\)\(38\!\cdots\!28\)\( - \)\(28\!\cdots\!64\)\( \beta_{1} - \)\(77\!\cdots\!68\)\( \beta_{2}) q^{91}\) \(+(\)\(22\!\cdots\!52\)\( + \)\(40\!\cdots\!60\)\( \beta_{1} - \)\(35\!\cdots\!28\)\( \beta_{2}) q^{92}\) \(+(-\)\(16\!\cdots\!32\)\( - \)\(11\!\cdots\!00\)\( \beta_{1} + \)\(55\!\cdots\!68\)\( \beta_{2}) q^{93}\) \(+(-\)\(44\!\cdots\!44\)\( - \)\(43\!\cdots\!16\)\( \beta_{1} + \)\(48\!\cdots\!08\)\( \beta_{2}) q^{94}\) \(+(\)\(10\!\cdots\!00\)\( + \)\(52\!\cdots\!50\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2}) q^{95}\) \(+(\)\(36\!\cdots\!12\)\( + \)\(13\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!72\)\( \beta_{2}) q^{96}\) \(+(-\)\(12\!\cdots\!98\)\( - \)\(11\!\cdots\!28\)\( \beta_{1} + \)\(21\!\cdots\!96\)\( \beta_{2}) q^{97}\) \(+(\)\(80\!\cdots\!36\)\( - \)\(10\!\cdots\!93\)\( \beta_{1} - \)\(31\!\cdots\!00\)\( \beta_{2}) q^{98}\) \(+(-\)\(47\!\cdots\!36\)\( - \)\(12\!\cdots\!51\)\( \beta_{1} + \)\(17\!\cdots\!13\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 2209944q^{2} \) \(\mathstrut +\mathstrut 24401437812q^{3} \) \(\mathstrut +\mathstrut 9822618421824q^{4} \) \(\mathstrut +\mathstrut 535205380774170q^{5} \) \(\mathstrut -\mathstrut 91793974758443424q^{6} \) \(\mathstrut +\mathstrut 301971425665478856q^{7} \) \(\mathstrut -\mathstrut 15830863913787348480q^{8} \) \(\mathstrut +\mathstrut 477482125171066743231q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 2209944q^{2} \) \(\mathstrut +\mathstrut 24401437812q^{3} \) \(\mathstrut +\mathstrut 9822618421824q^{4} \) \(\mathstrut +\mathstrut 535205380774170q^{5} \) \(\mathstrut -\mathstrut 91793974758443424q^{6} \) \(\mathstrut +\mathstrut 301971425665478856q^{7} \) \(\mathstrut -\mathstrut 15830863913787348480q^{8} \) \(\mathstrut +\mathstrut \)\(47\!\cdots\!31\)\(q^{9} \) \(\mathstrut +\mathstrut \)\(42\!\cdots\!20\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!96\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!36\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!82\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!28\)\(q^{14} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!40\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!92\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!94\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(41\!\cdots\!48\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!60\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!36\)\(q^{21} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!92\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!48\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!75\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!44\)\(q^{26} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!80\)\(q^{27} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!68\)\(q^{28} \) \(\mathstrut +\mathstrut \)\(57\!\cdots\!50\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!60\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!24\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!84\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!16\)\(q^{33} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!88\)\(q^{34} \) \(\mathstrut +\mathstrut \)\(23\!\cdots\!20\)\(q^{35} \) \(\mathstrut +\mathstrut \)\(81\!\cdots\!48\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!94\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!20\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(39\!\cdots\!28\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!66\)\(q^{41} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!32\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!92\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!32\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!90\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!64\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!56\)\(q^{47} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!48\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!79\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!56\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!04\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!62\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!00\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!40\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!00\)\(q^{56} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!60\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{58} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{59} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!20\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(93\!\cdots\!54\)\(q^{61} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!48\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(96\!\cdots\!48\)\(q^{63} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!36\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!60\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(29\!\cdots\!32\)\(q^{66} \) \(\mathstrut -\mathstrut \)\(73\!\cdots\!44\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(54\!\cdots\!68\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!32\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!20\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!64\)\(q^{71} \) \(\mathstrut -\mathstrut \)\(98\!\cdots\!60\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!98\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!08\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!00\)\(q^{75} \) \(\mathstrut +\mathstrut \)\(77\!\cdots\!00\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(59\!\cdots\!92\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!96\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!80\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(73\!\cdots\!63\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(32\!\cdots\!68\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(89\!\cdots\!28\)\(q^{83} \) \(\mathstrut -\mathstrut \)\(34\!\cdots\!12\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!20\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(25\!\cdots\!96\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!40\)\(q^{87} \) \(\mathstrut -\mathstrut \)\(25\!\cdots\!60\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!50\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!60\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!84\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(68\!\cdots\!56\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(49\!\cdots\!96\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!32\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!00\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!36\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(38\!\cdots\!94\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(24\!\cdots\!08\)\(q^{98} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!08\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(11258260111\) \(x\mathstrut -\mathstrut \) \(264759545317170\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -6 \nu^{2} + 343074 \nu + 45033040444 \)\()/8369\)
\(\beta_{2}\)\(=\)\((\)\( 14838 \nu^{2} + 1176206478 \nu - 111366709018012 \)\()/8369\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(2473\) \(\beta_{1}\)\()/241920\)
\(\nu^{2}\)\(=\)\((\)\(57179\) \(\beta_{2}\mathstrut -\mathstrut \) \(196034413\) \(\beta_{1}\mathstrut +\mathstrut \) \(1815732190702080\)\()/241920\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−24885.9
116336.
−91450.2
−4.65343e6 3.22027e10 1.28583e13 5.54482e14 −1.49853e17 −5.57604e17 −1.89031e19 7.08758e20 −2.58024e21
1.2 −1.18359e6 −1.79507e10 −7.39521e12 −6.39116e14 2.12463e16 −1.72730e18 1.91639e19 −6.02939e18 7.56451e20
1.3 3.62707e6 1.01494e10 4.35955e12 6.19839e14 3.68127e16 2.58688e18 −1.60917e19 −2.25246e20 2.24820e21
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Hecke kernels

There are no other newforms in \(S_{44}^{\mathrm{new}}(\Gamma_0(1))\).