Properties

Label 3.42.a.a
Level 3
Weight 42
Character orbit 3.a
Self dual Yes
Analytic conductor 31.942
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 42 \)
Character orbit: \([\chi]\) = 3.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9415011369\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -96460 - \beta_{1} ) q^{2} \) \( -3486784401 q^{3} \) \( + ( -751422059600 - 327756 \beta_{1} + 10 \beta_{2} ) q^{4} \) \( + ( 12883515397342 - 71244584 \beta_{1} + 2379 \beta_{2} ) q^{5} \) \( + ( 336335223320460 + 3486784401 \beta_{1} ) q^{6} \) \( + ( -14867403767136256 - 73320502584 \beta_{1} - 3812375 \beta_{2} ) q^{7} \) \( + ( 756010982896666432 + 2363361140656 \beta_{1} - 2893800 \beta_{2} ) q^{8} \) \( + 12157665459056928801 q^{9} \) \(+O(q^{10})\) \( q\) \(+(-96460 - \beta_{1}) q^{2}\) \(-3486784401 q^{3}\) \(+(-751422059600 - 327756 \beta_{1} + 10 \beta_{2}) q^{4}\) \(+(12883515397342 - 71244584 \beta_{1} + 2379 \beta_{2}) q^{5}\) \(+(336335223320460 + 3486784401 \beta_{1}) q^{6}\) \(+(-14867403767136256 - 73320502584 \beta_{1} - 3812375 \beta_{2}) q^{7}\) \(+(756010982896666432 + 2363361140656 \beta_{1} - 2893800 \beta_{2}) q^{8}\) \(+12157665459056928801 q^{9}\) \(+(\)\(10\!\cdots\!08\)\( - 149696474137566 \beta_{1} - 755720704 \beta_{2}) q^{10}\) \(+(\)\(72\!\cdots\!68\)\( + 997689368410448 \beta_{1} + 8237865930 \beta_{2}) q^{11}\) \(+(\)\(26\!\cdots\!00\)\( + 1142814508134156 \beta_{1} - 34867844010 \beta_{2}) q^{12}\) \(+(\)\(14\!\cdots\!14\)\( - 29481065101493616 \beta_{1} - 150407596430 \beta_{2}) q^{13}\) \(+(\)\(10\!\cdots\!28\)\( + 154575166302522112 \beta_{1} + 3085958883840 \beta_{2}) q^{14}\) \(+(-\)\(44\!\cdots\!42\)\( + 248414504146934184 \beta_{1} - 8295060089979 \beta_{2}) q^{15}\) \(+(-\)\(18\!\cdots\!32\)\( + 1096962907304736576 \beta_{1} - 43837975805280 \beta_{2}) q^{16}\) \(+(-\)\(36\!\cdots\!38\)\( + 11928719000396751600 \beta_{1} + 216595399364670 \beta_{2}) q^{17}\) \(+(-\)\(11\!\cdots\!60\)\( - 12157665459056928801 \beta_{1}) q^{18}\) \(+(\)\(65\!\cdots\!76\)\( + \)\(18\!\cdots\!28\)\( \beta_{1} - 634756701247710 \beta_{2}) q^{19}\) \(+(\)\(17\!\cdots\!08\)\( + 25796271271357749784 \beta_{1} - 3268129131198804 \beta_{2}) q^{20}\) \(+(\)\(51\!\cdots\!56\)\( + \)\(25\!\cdots\!84\)\( \beta_{1} + 13292929680762375 \beta_{2}) q^{21}\) \(+(-\)\(15\!\cdots\!76\)\( - \)\(66\!\cdots\!00\)\( \beta_{1} - 15060777312680960 \beta_{2}) q^{22}\) \(+(-\)\(15\!\cdots\!28\)\( - \)\(17\!\cdots\!12\)\( \beta_{1} + 22177359062364210 \beta_{2}) q^{23}\) \(+(-\)\(26\!\cdots\!32\)\( - \)\(82\!\cdots\!56\)\( \beta_{1} + 10090056699613800 \beta_{2}) q^{24}\) \(+(-\)\(15\!\cdots\!33\)\( - \)\(64\!\cdots\!84\)\( \beta_{1} - 318019269759106796 \beta_{2}) q^{25}\) \(+(\)\(40\!\cdots\!92\)\( - \)\(20\!\cdots\!70\)\( \beta_{1} + 387632593445360640 \beta_{2}) q^{26}\) \(-\)\(42\!\cdots\!01\)\( q^{27}\) \(+(-\)\(19\!\cdots\!92\)\( - \)\(18\!\cdots\!68\)\( \beta_{1} + 4933293299122352640 \beta_{2}) q^{28}\) \(+(-\)\(31\!\cdots\!38\)\( + \)\(65\!\cdots\!56\)\( \beta_{1} - 14886455450481587655 \beta_{2}) q^{29}\) \(+(-\)\(35\!\cdots\!08\)\( + \)\(52\!\cdots\!66\)\( \beta_{1} + 2635035162219938304 \beta_{2}) q^{30}\) \(+(-\)\(21\!\cdots\!36\)\( + \)\(83\!\cdots\!36\)\( \beta_{1} + 21546178470509132605 \beta_{2}) q^{31}\) \(+(-\)\(30\!\cdots\!96\)\( - \)\(94\!\cdots\!64\)\( \beta_{1} + 22447897460436289920 \beta_{2}) q^{32}\) \(+(-\)\(25\!\cdots\!68\)\( - \)\(34\!\cdots\!48\)\( \beta_{1} - 28723662422253357930 \beta_{2}) q^{33}\) \(+(-\)\(16\!\cdots\!20\)\( - \)\(98\!\cdots\!62\)\( \beta_{1} - \)\(25\!\cdots\!20\)\( \beta_{2}) q^{34}\) \(+(-\)\(51\!\cdots\!00\)\( - \)\(19\!\cdots\!00\)\( \beta_{1} + \)\(42\!\cdots\!50\)\( \beta_{2}) q^{35}\) \(+(-\)\(91\!\cdots\!00\)\( - \)\(39\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!10\)\( \beta_{2}) q^{36}\) \(+(-\)\(14\!\cdots\!66\)\( + \)\(65\!\cdots\!56\)\( \beta_{1} - \)\(11\!\cdots\!20\)\( \beta_{2}) q^{37}\) \(+(-\)\(27\!\cdots\!16\)\( + \)\(42\!\cdots\!72\)\( \beta_{1} - \)\(14\!\cdots\!20\)\( \beta_{2}) q^{38}\) \(+(-\)\(51\!\cdots\!14\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} + \)\(52\!\cdots\!30\)\( \beta_{2}) q^{39}\) \(+(-\)\(27\!\cdots\!24\)\( + \)\(30\!\cdots\!48\)\( \beta_{1} + \)\(34\!\cdots\!12\)\( \beta_{2}) q^{40}\) \(+(-\)\(14\!\cdots\!66\)\( + \)\(34\!\cdots\!96\)\( \beta_{1} + \)\(21\!\cdots\!70\)\( \beta_{2}) q^{41}\) \(+(-\)\(37\!\cdots\!28\)\( - \)\(53\!\cdots\!12\)\( \beta_{1} - \)\(10\!\cdots\!40\)\( \beta_{2}) q^{42}\) \(+(-\)\(43\!\cdots\!92\)\( - \)\(64\!\cdots\!64\)\( \beta_{1} - \)\(25\!\cdots\!90\)\( \beta_{2}) q^{43}\) \(+(-\)\(48\!\cdots\!76\)\( - \)\(29\!\cdots\!20\)\( \beta_{1} - \)\(21\!\cdots\!00\)\( \beta_{2}) q^{44}\) \(+(\)\(15\!\cdots\!42\)\( - \)\(86\!\cdots\!84\)\( \beta_{1} + \)\(28\!\cdots\!79\)\( \beta_{2}) q^{45}\) \(+(\)\(26\!\cdots\!04\)\( - \)\(18\!\cdots\!64\)\( \beta_{1} + \)\(38\!\cdots\!60\)\( \beta_{2}) q^{46}\) \(+(\)\(21\!\cdots\!56\)\( - \)\(94\!\cdots\!48\)\( \beta_{1} - \)\(79\!\cdots\!90\)\( \beta_{2}) q^{47}\) \(+(\)\(63\!\cdots\!32\)\( - \)\(38\!\cdots\!76\)\( \beta_{1} + \)\(15\!\cdots\!80\)\( \beta_{2}) q^{48}\) \(+(\)\(57\!\cdots\!41\)\( + \)\(58\!\cdots\!52\)\( \beta_{1} - \)\(19\!\cdots\!40\)\( \beta_{2}) q^{49}\) \(+(\)\(94\!\cdots\!08\)\( + \)\(13\!\cdots\!09\)\( \beta_{1} + \)\(26\!\cdots\!96\)\( \beta_{2}) q^{50}\) \(+(\)\(12\!\cdots\!38\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} - \)\(75\!\cdots\!70\)\( \beta_{2}) q^{51}\) \(+(-\)\(69\!\cdots\!08\)\( - \)\(22\!\cdots\!80\)\( \beta_{1} + \)\(29\!\cdots\!20\)\( \beta_{2}) q^{52}\) \(+(\)\(11\!\cdots\!82\)\( - \)\(90\!\cdots\!32\)\( \beta_{1} + \)\(15\!\cdots\!85\)\( \beta_{2}) q^{53}\) \(+(\)\(40\!\cdots\!60\)\( + \)\(42\!\cdots\!01\)\( \beta_{1}) q^{54}\) \(+(\)\(33\!\cdots\!32\)\( + \)\(12\!\cdots\!36\)\( \beta_{1} + \)\(11\!\cdots\!84\)\( \beta_{2}) q^{55}\) \(+(-\)\(18\!\cdots\!00\)\( - \)\(36\!\cdots\!20\)\( \beta_{1} - \)\(96\!\cdots\!40\)\( \beta_{2}) q^{56}\) \(+(-\)\(22\!\cdots\!76\)\( - \)\(65\!\cdots\!28\)\( \beta_{1} + \)\(22\!\cdots\!10\)\( \beta_{2}) q^{57}\) \(+(-\)\(91\!\cdots\!32\)\( + \)\(12\!\cdots\!34\)\( \beta_{1} + \)\(26\!\cdots\!20\)\( \beta_{2}) q^{58}\) \(+(-\)\(15\!\cdots\!16\)\( - \)\(49\!\cdots\!48\)\( \beta_{1} + \)\(18\!\cdots\!40\)\( \beta_{2}) q^{59}\) \(+(-\)\(61\!\cdots\!08\)\( - \)\(89\!\cdots\!84\)\( \beta_{1} + \)\(11\!\cdots\!04\)\( \beta_{2}) q^{60}\) \(+(-\)\(28\!\cdots\!54\)\( + \)\(15\!\cdots\!52\)\( \beta_{1} - \)\(40\!\cdots\!40\)\( \beta_{2}) q^{61}\) \(+(\)\(88\!\cdots\!88\)\( + \)\(12\!\cdots\!12\)\( \beta_{1} - \)\(14\!\cdots\!40\)\( \beta_{2}) q^{62}\) \(+(-\)\(18\!\cdots\!56\)\( - \)\(89\!\cdots\!84\)\( \beta_{1} - \)\(46\!\cdots\!75\)\( \beta_{2}) q^{63}\) \(+(\)\(56\!\cdots\!52\)\( - \)\(75\!\cdots\!80\)\( \beta_{1} + \)\(92\!\cdots\!80\)\( \beta_{2}) q^{64}\) \(+(\)\(90\!\cdots\!16\)\( - \)\(60\!\cdots\!32\)\( \beta_{1} + \)\(39\!\cdots\!42\)\( \beta_{2}) q^{65}\) \(+(\)\(52\!\cdots\!76\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} + \)\(52\!\cdots\!60\)\( \beta_{2}) q^{66}\) \(+(\)\(10\!\cdots\!72\)\( - \)\(16\!\cdots\!60\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2}) q^{67}\) \(+(\)\(11\!\cdots\!00\)\( + \)\(14\!\cdots\!28\)\( \beta_{1} - \)\(31\!\cdots\!00\)\( \beta_{2}) q^{68}\) \(+(\)\(54\!\cdots\!28\)\( + \)\(60\!\cdots\!12\)\( \beta_{1} - \)\(77\!\cdots\!10\)\( \beta_{2}) q^{69}\) \(+(\)\(32\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(68\!\cdots\!00\)\( \beta_{2}) q^{70}\) \(+(\)\(49\!\cdots\!32\)\( + \)\(27\!\cdots\!40\)\( \beta_{1} + \)\(90\!\cdots\!10\)\( \beta_{2}) q^{71}\) \(+(\)\(91\!\cdots\!32\)\( + \)\(28\!\cdots\!56\)\( \beta_{1} - \)\(35\!\cdots\!00\)\( \beta_{2}) q^{72}\) \(+(\)\(22\!\cdots\!82\)\( - \)\(12\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2}) q^{73}\) \(+(\)\(48\!\cdots\!48\)\( + \)\(15\!\cdots\!62\)\( \beta_{1} + \)\(79\!\cdots\!60\)\( \beta_{2}) q^{74}\) \(+(\)\(53\!\cdots\!33\)\( + \)\(22\!\cdots\!84\)\( \beta_{1} + \)\(11\!\cdots\!96\)\( \beta_{2}) q^{75}\) \(+(-\)\(17\!\cdots\!36\)\( - \)\(51\!\cdots\!88\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2}) q^{76}\) \(+(-\)\(31\!\cdots\!72\)\( - \)\(30\!\cdots\!68\)\( \beta_{1} - \)\(44\!\cdots\!20\)\( \beta_{2}) q^{77}\) \(+(-\)\(14\!\cdots\!92\)\( + \)\(72\!\cdots\!70\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2}) q^{78}\) \(+(-\)\(37\!\cdots\!00\)\( - \)\(27\!\cdots\!80\)\( \beta_{1} + \)\(32\!\cdots\!65\)\( \beta_{2}) q^{79}\) \(+(-\)\(80\!\cdots\!92\)\( + \)\(19\!\cdots\!84\)\( \beta_{1} + \)\(19\!\cdots\!96\)\( \beta_{2}) q^{80}\) \(+\)\(14\!\cdots\!01\)\( q^{81}\) \(+(-\)\(35\!\cdots\!32\)\( + \)\(66\!\cdots\!02\)\( \beta_{1} - \)\(16\!\cdots\!80\)\( \beta_{2}) q^{82}\) \(+(-\)\(39\!\cdots\!20\)\( + \)\(26\!\cdots\!52\)\( \beta_{1} + \)\(15\!\cdots\!50\)\( \beta_{2}) q^{83}\) \(+(\)\(69\!\cdots\!92\)\( + \)\(63\!\cdots\!68\)\( \beta_{1} - \)\(17\!\cdots\!40\)\( \beta_{2}) q^{84}\) \(+(\)\(20\!\cdots\!84\)\( + \)\(26\!\cdots\!32\)\( \beta_{1} - \)\(30\!\cdots\!42\)\( \beta_{2}) q^{85}\) \(+(\)\(96\!\cdots\!48\)\( + \)\(17\!\cdots\!68\)\( \beta_{1} + \)\(65\!\cdots\!80\)\( \beta_{2}) q^{86}\) \(+(\)\(10\!\cdots\!38\)\( - \)\(22\!\cdots\!56\)\( \beta_{1} + \)\(51\!\cdots\!55\)\( \beta_{2}) q^{87}\) \(+(\)\(37\!\cdots\!52\)\( + \)\(19\!\cdots\!56\)\( \beta_{1} + \)\(37\!\cdots\!20\)\( \beta_{2}) q^{88}\) \(+(-\)\(96\!\cdots\!34\)\( - \)\(78\!\cdots\!32\)\( \beta_{1} - \)\(68\!\cdots\!80\)\( \beta_{2}) q^{89}\) \(+(\)\(12\!\cdots\!08\)\( - \)\(18\!\cdots\!66\)\( \beta_{1} - \)\(91\!\cdots\!04\)\( \beta_{2}) q^{90}\) \(+(\)\(65\!\cdots\!04\)\( + \)\(50\!\cdots\!76\)\( \beta_{1} + \)\(55\!\cdots\!90\)\( \beta_{2}) q^{91}\) \(+(\)\(34\!\cdots\!44\)\( + \)\(93\!\cdots\!96\)\( \beta_{1} - \)\(49\!\cdots\!40\)\( \beta_{2}) q^{92}\) \(+(\)\(75\!\cdots\!36\)\( - \)\(29\!\cdots\!36\)\( \beta_{1} - \)\(75\!\cdots\!05\)\( \beta_{2}) q^{93}\) \(+(\)\(11\!\cdots\!36\)\( - \)\(21\!\cdots\!24\)\( \beta_{1} + \)\(14\!\cdots\!20\)\( \beta_{2}) q^{94}\) \(+(-\)\(28\!\cdots\!32\)\( + \)\(23\!\cdots\!64\)\( \beta_{1} + \)\(33\!\cdots\!16\)\( \beta_{2}) q^{95}\) \(+(\)\(10\!\cdots\!96\)\( + \)\(33\!\cdots\!64\)\( \beta_{1} - \)\(78\!\cdots\!20\)\( \beta_{2}) q^{96}\) \(+(-\)\(17\!\cdots\!38\)\( + \)\(39\!\cdots\!00\)\( \beta_{1} - \)\(40\!\cdots\!80\)\( \beta_{2}) q^{97}\) \(+(-\)\(89\!\cdots\!64\)\( - \)\(23\!\cdots\!09\)\( \beta_{1} - \)\(46\!\cdots\!80\)\( \beta_{2}) q^{98}\) \(+(\)\(87\!\cdots\!68\)\( + \)\(12\!\cdots\!48\)\( \beta_{1} + \)\(10\!\cdots\!30\)\( \beta_{2}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 289380q^{2} \) \(\mathstrut -\mathstrut 10460353203q^{3} \) \(\mathstrut -\mathstrut 2254266178800q^{4} \) \(\mathstrut +\mathstrut 38650546192026q^{5} \) \(\mathstrut +\mathstrut 1009005669961380q^{6} \) \(\mathstrut -\mathstrut 44602211301408768q^{7} \) \(\mathstrut +\mathstrut 2268032948689999296q^{8} \) \(\mathstrut +\mathstrut 36472996377170786403q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 289380q^{2} \) \(\mathstrut -\mathstrut 10460353203q^{3} \) \(\mathstrut -\mathstrut 2254266178800q^{4} \) \(\mathstrut +\mathstrut 38650546192026q^{5} \) \(\mathstrut +\mathstrut 1009005669961380q^{6} \) \(\mathstrut -\mathstrut 44602211301408768q^{7} \) \(\mathstrut +\mathstrut 2268032948689999296q^{8} \) \(\mathstrut +\mathstrut 36472996377170786403q^{9} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!24\)\(q^{10} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!04\)\(q^{11} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!00\)\(q^{12} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!42\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!84\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!26\)\(q^{15} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!96\)\(q^{16} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!14\)\(q^{17} \) \(\mathstrut -\mathstrut \)\(35\!\cdots\!80\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!28\)\(q^{19} \) \(\mathstrut +\mathstrut \)\(53\!\cdots\!24\)\(q^{20} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!68\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!28\)\(q^{22} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!84\)\(q^{23} \) \(\mathstrut -\mathstrut \)\(79\!\cdots\!96\)\(q^{24} \) \(\mathstrut -\mathstrut \)\(46\!\cdots\!99\)\(q^{25} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!76\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(12\!\cdots\!03\)\(q^{27} \) \(\mathstrut -\mathstrut \)\(59\!\cdots\!76\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(94\!\cdots\!14\)\(q^{29} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!24\)\(q^{30} \) \(\mathstrut -\mathstrut \)\(64\!\cdots\!08\)\(q^{31} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!88\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(75\!\cdots\!04\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(50\!\cdots\!60\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!00\)\(q^{35} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!00\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(44\!\cdots\!98\)\(q^{37} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!48\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!42\)\(q^{39} \) \(\mathstrut -\mathstrut \)\(83\!\cdots\!72\)\(q^{40} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!98\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!84\)\(q^{42} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!76\)\(q^{43} \) \(\mathstrut -\mathstrut \)\(14\!\cdots\!28\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!26\)\(q^{45} \) \(\mathstrut +\mathstrut \)\(79\!\cdots\!12\)\(q^{46} \) \(\mathstrut +\mathstrut \)\(64\!\cdots\!68\)\(q^{47} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!96\)\(q^{48} \) \(\mathstrut +\mathstrut \)\(17\!\cdots\!23\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!24\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(38\!\cdots\!14\)\(q^{51} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!24\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!46\)\(q^{53} \) \(\mathstrut +\mathstrut \)\(12\!\cdots\!80\)\(q^{54} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!96\)\(q^{55} \) \(\mathstrut -\mathstrut \)\(56\!\cdots\!00\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(68\!\cdots\!28\)\(q^{57} \) \(\mathstrut -\mathstrut \)\(27\!\cdots\!96\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(45\!\cdots\!48\)\(q^{59} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!24\)\(q^{60} \) \(\mathstrut -\mathstrut \)\(86\!\cdots\!62\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!64\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(54\!\cdots\!68\)\(q^{63} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!56\)\(q^{64} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!48\)\(q^{65} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!28\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!16\)\(q^{67} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!00\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!84\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(98\!\cdots\!00\)\(q^{70} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!96\)\(q^{71} \) \(\mathstrut +\mathstrut \)\(27\!\cdots\!96\)\(q^{72} \) \(\mathstrut +\mathstrut \)\(66\!\cdots\!46\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(14\!\cdots\!44\)\(q^{74} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!99\)\(q^{75} \) \(\mathstrut -\mathstrut \)\(53\!\cdots\!08\)\(q^{76} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!16\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!76\)\(q^{78} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!00\)\(q^{79} \) \(\mathstrut -\mathstrut \)\(24\!\cdots\!76\)\(q^{80} \) \(\mathstrut +\mathstrut \)\(44\!\cdots\!03\)\(q^{81} \) \(\mathstrut -\mathstrut \)\(10\!\cdots\!96\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!60\)\(q^{83} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!76\)\(q^{84} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!52\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(28\!\cdots\!44\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!14\)\(q^{87} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!56\)\(q^{88} \) \(\mathstrut -\mathstrut \)\(29\!\cdots\!02\)\(q^{89} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!24\)\(q^{90} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!12\)\(q^{91} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!32\)\(q^{92} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!08\)\(q^{93} \) \(\mathstrut +\mathstrut \)\(34\!\cdots\!08\)\(q^{94} \) \(\mathstrut -\mathstrut \)\(84\!\cdots\!96\)\(q^{95} \) \(\mathstrut +\mathstrut \)\(32\!\cdots\!88\)\(q^{96} \) \(\mathstrut -\mathstrut \)\(52\!\cdots\!14\)\(q^{97} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!92\)\(q^{98} \) \(\mathstrut +\mathstrut \)\(26\!\cdots\!04\)\(q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(14982256920\) \(x\mathstrut +\mathstrut \) \(433388802120300\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 12 \nu - 4 \)
\(\beta_{2}\)\(=\)\((\)\( 72 \nu^{2} + 3124008 \nu - 719149373520 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(5\) \(\beta_{2}\mathstrut -\mathstrut \) \(260334\) \(\beta_{1}\mathstrut +\mathstrut \) \(719148332184\)\()/72\)

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
103995.
30895.0
−134889.
−1.34440e6 −3.48678e9 −3.91623e11 1.06877e14 4.68762e15 −3.99469e17 3.48285e18 1.21577e19 −1.43685e20
1.2 −467196. −3.48678e9 −1.98075e12 −2.77079e14 1.62901e15 3.80292e17 1.95278e18 1.21577e19 1.29450e20
1.3 1.52221e6 −3.48678e9 1.18107e11 2.08853e14 −5.30763e15 −2.54249e16 −3.16760e18 1.21577e19 3.17919e20
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{3} \) \(\mathstrut +\mathstrut 289380 T_{2}^{2} \) \(\mathstrut -\mathstrut \)\(21\!\cdots\!28\)\( T_{2} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!32\)\( \) acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(3))\).