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: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 14982256920 x + 433388802120300\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 7 \)
Twist minimal: yes
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.

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

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.42.a.a 3
3.b odd 2 1 9.42.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.42.a.a 3 1.a even 1 1 trivial
9.42.a.a 3 3.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 289380 T + 4467538364928 T^{2} + 316609849852428288 T^{3} + \)\(98\!\cdots\!56\)\( T^{4} + \)\(13\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} \)
$3$ \( ( 1 + 3486784401 T )^{3} \)
$5$ \( 1 - 38650546192026 T + \)\(71\!\cdots\!75\)\( T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!75\)\( T^{4} - \)\(79\!\cdots\!50\)\( T^{5} + \)\(94\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 44602211301408768 T - \)\(17\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} - \)\(78\!\cdots\!73\)\( T^{4} + \)\(88\!\cdots\!32\)\( T^{5} + \)\(88\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 - \)\(21\!\cdots\!04\)\( T + \)\(13\!\cdots\!93\)\( T^{2} - \)\(21\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!23\)\( T^{4} - \)\(53\!\cdots\!84\)\( T^{5} + \)\(12\!\cdots\!31\)\( T^{6} \)
$13$ \( 1 - \)\(44\!\cdots\!42\)\( T + \)\(12\!\cdots\!59\)\( T^{2} - \)\(38\!\cdots\!76\)\( T^{3} + \)\(59\!\cdots\!67\)\( T^{4} - \)\(98\!\cdots\!98\)\( T^{5} + \)\(10\!\cdots\!97\)\( T^{6} \)
$17$ \( 1 + \)\(10\!\cdots\!14\)\( T + \)\(12\!\cdots\!83\)\( T^{2} - \)\(39\!\cdots\!52\)\( T^{3} + \)\(34\!\cdots\!11\)\( T^{4} + \)\(86\!\cdots\!46\)\( T^{5} + \)\(22\!\cdots\!13\)\( T^{6} \)
$19$ \( 1 - \)\(19\!\cdots\!28\)\( T + \)\(12\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!16\)\( T^{3} + \)\(34\!\cdots\!27\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{5} + \)\(19\!\cdots\!59\)\( T^{6} \)
$23$ \( 1 + \)\(46\!\cdots\!84\)\( T + \)\(19\!\cdots\!89\)\( T^{2} + \)\(61\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!47\)\( T^{4} + \)\(21\!\cdots\!36\)\( T^{5} + \)\(31\!\cdots\!67\)\( T^{6} \)
$29$ \( 1 + \)\(94\!\cdots\!14\)\( T - \)\(50\!\cdots\!89\)\( T^{2} - \)\(18\!\cdots\!88\)\( T^{3} - \)\(45\!\cdots\!81\)\( T^{4} + \)\(77\!\cdots\!74\)\( T^{5} + \)\(74\!\cdots\!89\)\( T^{6} \)
$31$ \( 1 + \)\(64\!\cdots\!08\)\( T + \)\(51\!\cdots\!93\)\( T^{2} + \)\(18\!\cdots\!96\)\( T^{3} + \)\(72\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!91\)\( T^{6} \)
$37$ \( 1 + \)\(44\!\cdots\!98\)\( T + \)\(12\!\cdots\!71\)\( T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(24\!\cdots\!27\)\( T^{4} + \)\(17\!\cdots\!62\)\( T^{5} + \)\(77\!\cdots\!53\)\( T^{6} \)
$41$ \( 1 + \)\(43\!\cdots\!98\)\( T + \)\(40\!\cdots\!43\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{3} + \)\(53\!\cdots\!63\)\( T^{4} + \)\(77\!\cdots\!38\)\( T^{5} + \)\(23\!\cdots\!21\)\( T^{6} \)
$43$ \( 1 + \)\(13\!\cdots\!76\)\( T + \)\(27\!\cdots\!33\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} + \)\(11\!\cdots\!24\)\( T^{5} + \)\(82\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(64\!\cdots\!68\)\( T + \)\(22\!\cdots\!37\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!39\)\( T^{4} - \)\(83\!\cdots\!12\)\( T^{5} + \)\(46\!\cdots\!23\)\( T^{6} \)
$53$ \( 1 - \)\(34\!\cdots\!46\)\( T + \)\(10\!\cdots\!59\)\( T^{2} - \)\(28\!\cdots\!76\)\( T^{3} + \)\(54\!\cdots\!27\)\( T^{4} - \)\(85\!\cdots\!14\)\( T^{5} + \)\(12\!\cdots\!77\)\( T^{6} \)
$59$ \( 1 + \)\(45\!\cdots\!48\)\( T + \)\(15\!\cdots\!33\)\( T^{2} + \)\(37\!\cdots\!84\)\( T^{3} + \)\(63\!\cdots\!47\)\( T^{4} + \)\(74\!\cdots\!88\)\( T^{5} + \)\(65\!\cdots\!79\)\( T^{6} \)
$61$ \( 1 + \)\(86\!\cdots\!62\)\( T + \)\(51\!\cdots\!19\)\( T^{2} + \)\(21\!\cdots\!96\)\( T^{3} + \)\(81\!\cdots\!59\)\( T^{4} + \)\(21\!\cdots\!02\)\( T^{5} + \)\(39\!\cdots\!81\)\( T^{6} \)
$67$ \( 1 - \)\(32\!\cdots\!16\)\( T + \)\(19\!\cdots\!53\)\( T^{2} - \)\(47\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!51\)\( T^{4} - \)\(17\!\cdots\!24\)\( T^{5} + \)\(40\!\cdots\!63\)\( T^{6} \)
$71$ \( 1 - \)\(14\!\cdots\!96\)\( T + \)\(21\!\cdots\!85\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(95\!\cdots\!36\)\( T^{5} + \)\(50\!\cdots\!11\)\( T^{6} \)
$73$ \( 1 - \)\(66\!\cdots\!46\)\( T + \)\(29\!\cdots\!99\)\( T^{2} + \)\(27\!\cdots\!84\)\( T^{3} + \)\(74\!\cdots\!27\)\( T^{4} - \)\(41\!\cdots\!34\)\( T^{5} + \)\(15\!\cdots\!17\)\( T^{6} \)
$79$ \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(20\!\cdots\!37\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} + \)\(44\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!39\)\( T^{6} \)
$83$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(12\!\cdots\!37\)\( T^{2} + \)\(10\!\cdots\!36\)\( T^{3} + \)\(60\!\cdots\!71\)\( T^{4} + \)\(27\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!87\)\( T^{6} \)
$89$ \( 1 + \)\(29\!\cdots\!02\)\( T + \)\(77\!\cdots\!63\)\( T^{2} + \)\(10\!\cdots\!76\)\( T^{3} + \)\(65\!\cdots\!07\)\( T^{4} + \)\(20\!\cdots\!42\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \)
$97$ \( 1 + \)\(52\!\cdots\!14\)\( T + \)\(46\!\cdots\!23\)\( T^{2} + \)\(30\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!31\)\( T^{4} + \)\(43\!\cdots\!26\)\( T^{5} + \)\(23\!\cdots\!73\)\( T^{6} \)
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