Properties

Label 2.72.a.b
Level 2
Weight 72
Character orbit 2.a
Self dual yes
Analytic conductor 63.849
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8492321122\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6}\cdot 5^{3}\cdot 7 \)
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.

\(f(q)\) \(=\) \( q -34359738368 q^{2} +(787523430902412 + \beta_{1}) q^{3} +\)\(11\!\cdots\!24\)\( q^{4} +(\)\(15\!\cdots\!50\)\( + 22397358 \beta_{1} - 3071 \beta_{2}) q^{5} +(-\)\(27\!\cdots\!16\)\( - 34359738368 \beta_{1}) q^{6} +(-\)\(22\!\cdots\!24\)\( - 8176070633262 \beta_{1} + 180943420 \beta_{2}) q^{7} -\)\(40\!\cdots\!32\)\( q^{8} +(-\)\(15\!\cdots\!03\)\( - 51420829207969836 \beta_{1} - 428616082170 \beta_{2}) q^{9} +O(q^{10})\) \( q -34359738368 q^{2} +(787523430902412 + \beta_{1}) q^{3} +\)\(11\!\cdots\!24\)\( q^{4} +(\)\(15\!\cdots\!50\)\( + 22397358 \beta_{1} - 3071 \beta_{2}) q^{5} +(-\)\(27\!\cdots\!16\)\( - 34359738368 \beta_{1}) q^{6} +(-\)\(22\!\cdots\!24\)\( - 8176070633262 \beta_{1} + 180943420 \beta_{2}) q^{7} -\)\(40\!\cdots\!32\)\( q^{8} +(-\)\(15\!\cdots\!03\)\( - 51420829207969836 \beta_{1} - 428616082170 \beta_{2}) q^{9} +(-\)\(54\!\cdots\!00\)\( - 769567361014431744 \beta_{1} + 105518756528128 \beta_{2}) q^{10} +(\)\(13\!\cdots\!52\)\( - \)\(12\!\cdots\!69\)\( \beta_{1} - 2154477364029320 \beta_{2}) q^{11} +(\)\(92\!\cdots\!88\)\( + \)\(11\!\cdots\!24\)\( \beta_{1}) q^{12} +(\)\(84\!\cdots\!82\)\( - \)\(10\!\cdots\!58\)\( \beta_{1} - 1606434955399360255 \beta_{2}) q^{13} +(\)\(78\!\cdots\!32\)\( + \)\(28\!\cdots\!16\)\( \beta_{1} - 6217168570611138560 \beta_{2}) q^{14} +(\)\(13\!\cdots\!00\)\( + \)\(18\!\cdots\!06\)\( \beta_{1} - \)\(17\!\cdots\!72\)\( \beta_{2}) q^{15} +\)\(13\!\cdots\!76\)\( q^{16} +(\)\(11\!\cdots\!06\)\( - \)\(37\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!10\)\( \beta_{2}) q^{17} +(\)\(51\!\cdots\!04\)\( + \)\(17\!\cdots\!48\)\( \beta_{1} + \)\(14\!\cdots\!60\)\( \beta_{2}) q^{18} +(\)\(94\!\cdots\!40\)\( + \)\(11\!\cdots\!89\)\( \beta_{1} - \)\(35\!\cdots\!20\)\( \beta_{2}) q^{19} +(\)\(18\!\cdots\!00\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} - \)\(36\!\cdots\!04\)\( \beta_{2}) q^{20} +(-\)\(49\!\cdots\!88\)\( + \)\(11\!\cdots\!52\)\( \beta_{1} + \)\(13\!\cdots\!80\)\( \beta_{2}) q^{21} +(-\)\(45\!\cdots\!36\)\( + \)\(42\!\cdots\!92\)\( \beta_{1} + \)\(74\!\cdots\!60\)\( \beta_{2}) q^{22} +(\)\(12\!\cdots\!72\)\( + \)\(29\!\cdots\!66\)\( \beta_{1} - \)\(13\!\cdots\!20\)\( \beta_{2}) q^{23} +(-\)\(31\!\cdots\!84\)\( - \)\(40\!\cdots\!32\)\( \beta_{1}) q^{24} +(\)\(23\!\cdots\!75\)\( + \)\(63\!\cdots\!00\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2}) q^{25} +(-\)\(28\!\cdots\!76\)\( + \)\(34\!\cdots\!44\)\( \beta_{1} + \)\(55\!\cdots\!40\)\( \beta_{2}) q^{26} +(-\)\(31\!\cdots\!00\)\( - \)\(61\!\cdots\!22\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2}) q^{27} +(-\)\(26\!\cdots\!76\)\( - \)\(96\!\cdots\!88\)\( \beta_{1} + \)\(21\!\cdots\!80\)\( \beta_{2}) q^{28} +(\)\(94\!\cdots\!90\)\( - \)\(29\!\cdots\!82\)\( \beta_{1} - \)\(33\!\cdots\!95\)\( \beta_{2}) q^{29} +(-\)\(46\!\cdots\!00\)\( - \)\(62\!\cdots\!08\)\( \beta_{1} + \)\(60\!\cdots\!96\)\( \beta_{2}) q^{30} +(-\)\(20\!\cdots\!88\)\( + \)\(52\!\cdots\!52\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2}) q^{31} -\)\(47\!\cdots\!68\)\( q^{32} +(-\)\(74\!\cdots\!76\)\( + \)\(87\!\cdots\!64\)\( \beta_{1} - \)\(62\!\cdots\!10\)\( \beta_{2}) q^{33} +(-\)\(40\!\cdots\!08\)\( + \)\(12\!\cdots\!44\)\( \beta_{1} - \)\(62\!\cdots\!80\)\( \beta_{2}) q^{34} +(-\)\(50\!\cdots\!00\)\( - \)\(52\!\cdots\!12\)\( \beta_{1} + \)\(23\!\cdots\!44\)\( \beta_{2}) q^{35} +(-\)\(17\!\cdots\!72\)\( - \)\(60\!\cdots\!64\)\( \beta_{1} - \)\(50\!\cdots\!80\)\( \beta_{2}) q^{36} +(-\)\(22\!\cdots\!54\)\( - \)\(15\!\cdots\!38\)\( \beta_{1} - \)\(49\!\cdots\!15\)\( \beta_{2}) q^{37} +(-\)\(32\!\cdots\!20\)\( - \)\(40\!\cdots\!52\)\( \beta_{1} + \)\(12\!\cdots\!60\)\( \beta_{2}) q^{38} +(-\)\(60\!\cdots\!16\)\( + \)\(21\!\cdots\!66\)\( \beta_{1} - \)\(82\!\cdots\!00\)\( \beta_{2}) q^{39} +(-\)\(64\!\cdots\!00\)\( - \)\(90\!\cdots\!56\)\( \beta_{1} + \)\(12\!\cdots\!72\)\( \beta_{2}) q^{40} +(\)\(85\!\cdots\!62\)\( + \)\(12\!\cdots\!08\)\( \beta_{1} + \)\(34\!\cdots\!60\)\( \beta_{2}) q^{41} +(\)\(16\!\cdots\!84\)\( - \)\(39\!\cdots\!36\)\( \beta_{1} - \)\(45\!\cdots\!40\)\( \beta_{2}) q^{42} +(\)\(63\!\cdots\!12\)\( + \)\(54\!\cdots\!43\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2}) q^{43} +(\)\(15\!\cdots\!48\)\( - \)\(14\!\cdots\!56\)\( \beta_{1} - \)\(25\!\cdots\!80\)\( \beta_{2}) q^{44} +(-\)\(90\!\cdots\!50\)\( - \)\(47\!\cdots\!34\)\( \beta_{1} + \)\(12\!\cdots\!33\)\( \beta_{2}) q^{45} +(-\)\(42\!\cdots\!96\)\( - \)\(10\!\cdots\!88\)\( \beta_{1} + \)\(45\!\cdots\!60\)\( \beta_{2}) q^{46} +(-\)\(14\!\cdots\!44\)\( + \)\(31\!\cdots\!68\)\( \beta_{1} - \)\(23\!\cdots\!60\)\( \beta_{2}) q^{47} +(\)\(10\!\cdots\!12\)\( + \)\(13\!\cdots\!76\)\( \beta_{1}) q^{48} +(-\)\(34\!\cdots\!67\)\( + \)\(33\!\cdots\!36\)\( \beta_{1} - \)\(25\!\cdots\!40\)\( \beta_{2}) q^{49} +(-\)\(80\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} + \)\(44\!\cdots\!00\)\( \beta_{2}) q^{50} +(-\)\(22\!\cdots\!28\)\( + \)\(22\!\cdots\!90\)\( \beta_{1} + \)\(11\!\cdots\!80\)\( \beta_{2}) q^{51} +(\)\(99\!\cdots\!68\)\( - \)\(11\!\cdots\!92\)\( \beta_{1} - \)\(18\!\cdots\!20\)\( \beta_{2}) q^{52} +(\)\(10\!\cdots\!02\)\( - \)\(18\!\cdots\!58\)\( \beta_{1} - \)\(26\!\cdots\!35\)\( \beta_{2}) q^{53} +(\)\(10\!\cdots\!00\)\( + \)\(21\!\cdots\!96\)\( \beta_{1} + \)\(34\!\cdots\!60\)\( \beta_{2}) q^{54} +(\)\(27\!\cdots\!00\)\( + \)\(20\!\cdots\!26\)\( \beta_{1} + \)\(14\!\cdots\!88\)\( \beta_{2}) q^{55} +(\)\(92\!\cdots\!68\)\( + \)\(33\!\cdots\!84\)\( \beta_{1} - \)\(73\!\cdots\!40\)\( \beta_{2}) q^{56} +(\)\(70\!\cdots\!80\)\( + \)\(50\!\cdots\!68\)\( \beta_{1} - \)\(24\!\cdots\!70\)\( \beta_{2}) q^{57} +(-\)\(32\!\cdots\!20\)\( + \)\(10\!\cdots\!76\)\( \beta_{1} + \)\(11\!\cdots\!60\)\( \beta_{2}) q^{58} +(\)\(15\!\cdots\!80\)\( - \)\(10\!\cdots\!89\)\( \beta_{1} - \)\(25\!\cdots\!40\)\( \beta_{2}) q^{59} +(\)\(16\!\cdots\!00\)\( + \)\(21\!\cdots\!44\)\( \beta_{1} - \)\(20\!\cdots\!28\)\( \beta_{2}) q^{60} +(\)\(55\!\cdots\!22\)\( + \)\(79\!\cdots\!54\)\( \beta_{1} + \)\(11\!\cdots\!85\)\( \beta_{2}) q^{61} +(\)\(69\!\cdots\!84\)\( - \)\(17\!\cdots\!36\)\( \beta_{1} - \)\(38\!\cdots\!40\)\( \beta_{2}) q^{62} +(\)\(23\!\cdots\!72\)\( + \)\(29\!\cdots\!30\)\( \beta_{1} - \)\(69\!\cdots\!20\)\( \beta_{2}) q^{63} +\)\(16\!\cdots\!24\)\( q^{64} +(\)\(31\!\cdots\!00\)\( + \)\(29\!\cdots\!76\)\( \beta_{1} - \)\(30\!\cdots\!12\)\( \beta_{2}) q^{65} +(\)\(25\!\cdots\!68\)\( - \)\(30\!\cdots\!52\)\( \beta_{1} + \)\(21\!\cdots\!80\)\( \beta_{2}) q^{66} +(\)\(98\!\cdots\!36\)\( - \)\(21\!\cdots\!31\)\( \beta_{1} + \)\(51\!\cdots\!40\)\( \beta_{2}) q^{67} +(\)\(13\!\cdots\!44\)\( - \)\(43\!\cdots\!92\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2}) q^{68} +(\)\(17\!\cdots\!64\)\( - \)\(13\!\cdots\!96\)\( \beta_{1} - \)\(19\!\cdots\!60\)\( \beta_{2}) q^{69} +(\)\(17\!\cdots\!00\)\( + \)\(18\!\cdots\!16\)\( \beta_{1} - \)\(81\!\cdots\!92\)\( \beta_{2}) q^{70} +(\)\(30\!\cdots\!32\)\( + \)\(56\!\cdots\!94\)\( \beta_{1} + \)\(28\!\cdots\!80\)\( \beta_{2}) q^{71} +(\)\(60\!\cdots\!96\)\( + \)\(20\!\cdots\!52\)\( \beta_{1} + \)\(17\!\cdots\!40\)\( \beta_{2}) q^{72} +(\)\(93\!\cdots\!62\)\( - \)\(13\!\cdots\!28\)\( \beta_{1} + \)\(31\!\cdots\!90\)\( \beta_{2}) q^{73} +(\)\(78\!\cdots\!72\)\( + \)\(53\!\cdots\!84\)\( \beta_{1} + \)\(16\!\cdots\!20\)\( \beta_{2}) q^{74} +(\)\(38\!\cdots\!00\)\( - \)\(40\!\cdots\!25\)\( \beta_{1} - \)\(96\!\cdots\!00\)\( \beta_{2}) q^{75} +(\)\(11\!\cdots\!60\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(42\!\cdots\!80\)\( \beta_{2}) q^{76} +(\)\(32\!\cdots\!52\)\( - \)\(55\!\cdots\!48\)\( \beta_{1} - \)\(13\!\cdots\!40\)\( \beta_{2}) q^{77} +(\)\(20\!\cdots\!88\)\( - \)\(72\!\cdots\!88\)\( \beta_{1} + \)\(28\!\cdots\!00\)\( \beta_{2}) q^{78} +(\)\(11\!\cdots\!80\)\( + \)\(18\!\cdots\!44\)\( \beta_{1} + \)\(14\!\cdots\!60\)\( \beta_{2}) q^{79} +(\)\(22\!\cdots\!00\)\( + \)\(31\!\cdots\!08\)\( \beta_{1} - \)\(42\!\cdots\!96\)\( \beta_{2}) q^{80} +(-\)\(25\!\cdots\!59\)\( + \)\(39\!\cdots\!48\)\( \beta_{1} + \)\(57\!\cdots\!90\)\( \beta_{2}) q^{81} +(-\)\(29\!\cdots\!16\)\( - \)\(41\!\cdots\!44\)\( \beta_{1} - \)\(11\!\cdots\!80\)\( \beta_{2}) q^{82} +(-\)\(16\!\cdots\!88\)\( + \)\(80\!\cdots\!45\)\( \beta_{1} - \)\(16\!\cdots\!60\)\( \beta_{2}) q^{83} +(-\)\(58\!\cdots\!12\)\( + \)\(13\!\cdots\!48\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2}) q^{84} +(-\)\(38\!\cdots\!00\)\( - \)\(37\!\cdots\!32\)\( \beta_{1} + \)\(52\!\cdots\!34\)\( \beta_{2}) q^{85} +(-\)\(21\!\cdots\!16\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} + \)\(43\!\cdots\!60\)\( \beta_{2}) q^{86} +(-\)\(17\!\cdots\!20\)\( + \)\(40\!\cdots\!26\)\( \beta_{1} - \)\(16\!\cdots\!00\)\( \beta_{2}) q^{87} +(-\)\(53\!\cdots\!64\)\( + \)\(50\!\cdots\!08\)\( \beta_{1} + \)\(87\!\cdots\!40\)\( \beta_{2}) q^{88} +(-\)\(59\!\cdots\!90\)\( + \)\(62\!\cdots\!20\)\( \beta_{1} - \)\(72\!\cdots\!50\)\( \beta_{2}) q^{89} +(\)\(31\!\cdots\!00\)\( + \)\(16\!\cdots\!12\)\( \beta_{1} - \)\(44\!\cdots\!44\)\( \beta_{2}) q^{90} +(-\)\(15\!\cdots\!68\)\( - \)\(30\!\cdots\!52\)\( \beta_{1} + \)\(95\!\cdots\!40\)\( \beta_{2}) q^{91} +(\)\(14\!\cdots\!28\)\( + \)\(34\!\cdots\!84\)\( \beta_{1} - \)\(15\!\cdots\!80\)\( \beta_{2}) q^{92} +(\)\(29\!\cdots\!44\)\( - \)\(28\!\cdots\!84\)\( \beta_{1} + \)\(57\!\cdots\!20\)\( \beta_{2}) q^{93} +(\)\(50\!\cdots\!92\)\( - \)\(10\!\cdots\!24\)\( \beta_{1} + \)\(79\!\cdots\!80\)\( \beta_{2}) q^{94} +(\)\(10\!\cdots\!00\)\( + \)\(10\!\cdots\!10\)\( \beta_{1} - \)\(54\!\cdots\!20\)\( \beta_{2}) q^{95} +(-\)\(37\!\cdots\!16\)\( - \)\(47\!\cdots\!68\)\( \beta_{1}) q^{96} +(\)\(13\!\cdots\!86\)\( + \)\(31\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!90\)\( \beta_{2}) q^{97} +(\)\(11\!\cdots\!56\)\( - \)\(11\!\cdots\!48\)\( \beta_{1} + \)\(87\!\cdots\!20\)\( \beta_{2}) q^{98} +(\)\(42\!\cdots\!44\)\( - \)\(24\!\cdots\!05\)\( \beta_{1} + \)\(90\!\cdots\!40\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 103079215104q^{2} + 2362570292707236q^{3} + \)\(35\!\cdots\!72\)\(q^{4} + \)\(47\!\cdots\!50\)\(q^{5} - \)\(81\!\cdots\!48\)\(q^{6} - \)\(68\!\cdots\!72\)\(q^{7} - \)\(12\!\cdots\!96\)\(q^{8} - \)\(45\!\cdots\!09\)\(q^{9} + O(q^{10}) \) \( 3q - 103079215104q^{2} + 2362570292707236q^{3} + \)\(35\!\cdots\!72\)\(q^{4} + \)\(47\!\cdots\!50\)\(q^{5} - \)\(81\!\cdots\!48\)\(q^{6} - \)\(68\!\cdots\!72\)\(q^{7} - \)\(12\!\cdots\!96\)\(q^{8} - \)\(45\!\cdots\!09\)\(q^{9} - \)\(16\!\cdots\!00\)\(q^{10} + \)\(39\!\cdots\!56\)\(q^{11} + \)\(27\!\cdots\!64\)\(q^{12} + \)\(25\!\cdots\!46\)\(q^{13} + \)\(23\!\cdots\!96\)\(q^{14} + \)\(40\!\cdots\!00\)\(q^{15} + \)\(41\!\cdots\!28\)\(q^{16} + \)\(35\!\cdots\!18\)\(q^{17} + \)\(15\!\cdots\!12\)\(q^{18} + \)\(28\!\cdots\!20\)\(q^{19} + \)\(56\!\cdots\!00\)\(q^{20} - \)\(14\!\cdots\!64\)\(q^{21} - \)\(13\!\cdots\!08\)\(q^{22} + \)\(37\!\cdots\!16\)\(q^{23} - \)\(95\!\cdots\!52\)\(q^{24} + \)\(69\!\cdots\!25\)\(q^{25} - \)\(86\!\cdots\!28\)\(q^{26} - \)\(94\!\cdots\!00\)\(q^{27} - \)\(80\!\cdots\!28\)\(q^{28} + \)\(28\!\cdots\!70\)\(q^{29} - \)\(13\!\cdots\!00\)\(q^{30} - \)\(60\!\cdots\!64\)\(q^{31} - \)\(14\!\cdots\!04\)\(q^{32} - \)\(22\!\cdots\!28\)\(q^{33} - \)\(12\!\cdots\!24\)\(q^{34} - \)\(15\!\cdots\!00\)\(q^{35} - \)\(53\!\cdots\!16\)\(q^{36} - \)\(68\!\cdots\!62\)\(q^{37} - \)\(97\!\cdots\!60\)\(q^{38} - \)\(18\!\cdots\!48\)\(q^{39} - \)\(19\!\cdots\!00\)\(q^{40} + \)\(25\!\cdots\!86\)\(q^{41} + \)\(50\!\cdots\!52\)\(q^{42} + \)\(18\!\cdots\!36\)\(q^{43} + \)\(46\!\cdots\!44\)\(q^{44} - \)\(27\!\cdots\!50\)\(q^{45} - \)\(12\!\cdots\!88\)\(q^{46} - \)\(43\!\cdots\!32\)\(q^{47} + \)\(32\!\cdots\!36\)\(q^{48} - \)\(10\!\cdots\!01\)\(q^{49} - \)\(24\!\cdots\!00\)\(q^{50} - \)\(66\!\cdots\!84\)\(q^{51} + \)\(29\!\cdots\!04\)\(q^{52} + \)\(30\!\cdots\!06\)\(q^{53} + \)\(32\!\cdots\!00\)\(q^{54} + \)\(82\!\cdots\!00\)\(q^{55} + \)\(27\!\cdots\!04\)\(q^{56} + \)\(21\!\cdots\!40\)\(q^{57} - \)\(97\!\cdots\!60\)\(q^{58} + \)\(46\!\cdots\!40\)\(q^{59} + \)\(48\!\cdots\!00\)\(q^{60} + \)\(16\!\cdots\!66\)\(q^{61} + \)\(20\!\cdots\!52\)\(q^{62} + \)\(71\!\cdots\!16\)\(q^{63} + \)\(49\!\cdots\!72\)\(q^{64} + \)\(94\!\cdots\!00\)\(q^{65} + \)\(76\!\cdots\!04\)\(q^{66} + \)\(29\!\cdots\!08\)\(q^{67} + \)\(41\!\cdots\!32\)\(q^{68} + \)\(52\!\cdots\!92\)\(q^{69} + \)\(51\!\cdots\!00\)\(q^{70} + \)\(91\!\cdots\!96\)\(q^{71} + \)\(18\!\cdots\!88\)\(q^{72} + \)\(28\!\cdots\!86\)\(q^{73} + \)\(23\!\cdots\!16\)\(q^{74} + \)\(11\!\cdots\!00\)\(q^{75} + \)\(33\!\cdots\!80\)\(q^{76} + \)\(98\!\cdots\!56\)\(q^{77} + \)\(62\!\cdots\!64\)\(q^{78} + \)\(35\!\cdots\!40\)\(q^{79} + \)\(66\!\cdots\!00\)\(q^{80} - \)\(77\!\cdots\!77\)\(q^{81} - \)\(87\!\cdots\!48\)\(q^{82} - \)\(49\!\cdots\!64\)\(q^{83} - \)\(17\!\cdots\!36\)\(q^{84} - \)\(11\!\cdots\!00\)\(q^{85} - \)\(64\!\cdots\!48\)\(q^{86} - \)\(53\!\cdots\!60\)\(q^{87} - \)\(16\!\cdots\!92\)\(q^{88} - \)\(17\!\cdots\!70\)\(q^{89} + \)\(93\!\cdots\!00\)\(q^{90} - \)\(46\!\cdots\!04\)\(q^{91} + \)\(44\!\cdots\!84\)\(q^{92} + \)\(89\!\cdots\!32\)\(q^{93} + \)\(15\!\cdots\!76\)\(q^{94} + \)\(30\!\cdots\!00\)\(q^{95} - \)\(11\!\cdots\!48\)\(q^{96} + \)\(41\!\cdots\!58\)\(q^{97} + \)\(35\!\cdots\!68\)\(q^{98} + \)\(12\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 71437129084791448795855051 x - 180952663419752575975880178936282470070\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -720 \nu^{2} + 4460299047094320 \nu + 34289821960699895422010424480 \)\()/ 249228641669 \)
\(\beta_{2}\)\(=\)\((\)\( 18347040 \nu^{2} + 2105974227666339360 \nu - 873773243202554735143669636599360 \)\()/ 249228641669 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 25482 \beta_{1}\)\()/ 464486400 \)
\(\nu^{2}\)\(=\)\((\)\(6194859787631 \beta_{2} - 2924964205092138 \beta_{1} + 22121049943286716534647365040537600\)\()/ 464486400 \)

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
−6.65040e12
9.51117e12
−2.86077e12
−3.43597e10 −1.08417e17 1.18059e21 7.90573e22 3.72520e27 6.08855e29 −4.05648e31 4.24488e33 −2.71639e33
1.2 −3.43597e10 4.72491e16 1.18059e21 −7.30618e24 −1.62347e27 −2.33219e28 −4.05648e31 −5.27699e33 2.51038e35
1.3 −3.43597e10 6.35309e16 1.18059e21 1.19804e25 −2.18290e27 −1.27132e30 −4.05648e31 −3.47329e33 −4.11644e35
\(n\): 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 2.72.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.72.a.b 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - \)\(23\!\cdots\!36\)\( T_{3}^{2} - \)\(90\!\cdots\!68\)\( T_{3} + \)\(32\!\cdots\!72\)\( \) acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 34359738368 T )^{3} \)
$3$ \( 1 - 2362570292707236 T + \)\(13\!\cdots\!73\)\( T^{2} + \)\(28\!\cdots\!88\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(13\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!23\)\( T^{6} \)
$5$ \( 1 - \)\(47\!\cdots\!50\)\( T + \)\(39\!\cdots\!75\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!75\)\( T^{4} - \)\(85\!\cdots\!50\)\( T^{5} + \)\(75\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + \)\(68\!\cdots\!72\)\( T + \)\(22\!\cdots\!57\)\( T^{2} + \)\(13\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!51\)\( T^{4} + \)\(69\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 - \)\(39\!\cdots\!56\)\( T + \)\(83\!\cdots\!45\)\( T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!95\)\( T^{4} - \)\(29\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!31\)\( T^{6} \)
$13$ \( 1 - \)\(25\!\cdots\!46\)\( T + \)\(13\!\cdots\!83\)\( T^{2} - \)\(43\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!71\)\( T^{4} - \)\(38\!\cdots\!74\)\( T^{5} + \)\(18\!\cdots\!53\)\( T^{6} \)
$17$ \( 1 - \)\(35\!\cdots\!18\)\( T + \)\(28\!\cdots\!07\)\( T^{2} - \)\(76\!\cdots\!04\)\( T^{3} + \)\(66\!\cdots\!31\)\( T^{4} - \)\(18\!\cdots\!02\)\( T^{5} + \)\(12\!\cdots\!37\)\( T^{6} \)
$19$ \( 1 - \)\(28\!\cdots\!20\)\( T + \)\(18\!\cdots\!57\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!59\)\( T^{6} \)
$23$ \( 1 - \)\(37\!\cdots\!16\)\( T + \)\(65\!\cdots\!33\)\( T^{2} + \)\(39\!\cdots\!88\)\( T^{3} + \)\(31\!\cdots\!91\)\( T^{4} - \)\(86\!\cdots\!64\)\( T^{5} + \)\(11\!\cdots\!83\)\( T^{6} \)
$29$ \( 1 - \)\(28\!\cdots\!70\)\( T + \)\(87\!\cdots\!87\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(59\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!70\)\( T^{5} + \)\(30\!\cdots\!89\)\( T^{6} \)
$31$ \( 1 + \)\(60\!\cdots\!64\)\( T + \)\(23\!\cdots\!25\)\( T^{2} + \)\(91\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!75\)\( T^{4} + \)\(36\!\cdots\!04\)\( T^{5} + \)\(45\!\cdots\!91\)\( T^{6} \)
$37$ \( 1 + \)\(68\!\cdots\!62\)\( T + \)\(77\!\cdots\!87\)\( T^{2} + \)\(30\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} + \)\(32\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!97\)\( T^{6} \)
$41$ \( 1 - \)\(25\!\cdots\!86\)\( T + \)\(93\!\cdots\!55\)\( T^{2} - \)\(15\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!55\)\( T^{4} - \)\(26\!\cdots\!66\)\( T^{5} + \)\(33\!\cdots\!21\)\( T^{6} \)
$43$ \( 1 - \)\(18\!\cdots\!36\)\( T + \)\(38\!\cdots\!53\)\( T^{2} - \)\(37\!\cdots\!32\)\( T^{3} + \)\(36\!\cdots\!71\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(84\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 + \)\(43\!\cdots\!32\)\( T + \)\(60\!\cdots\!17\)\( T^{2} + \)\(99\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!51\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \)
$53$ \( 1 - \)\(30\!\cdots\!06\)\( T + \)\(74\!\cdots\!03\)\( T^{2} - \)\(15\!\cdots\!72\)\( T^{3} + \)\(19\!\cdots\!91\)\( T^{4} - \)\(21\!\cdots\!54\)\( T^{5} + \)\(18\!\cdots\!73\)\( T^{6} \)
$59$ \( 1 - \)\(46\!\cdots\!40\)\( T + \)\(11\!\cdots\!77\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!43\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!79\)\( T^{6} \)
$61$ \( 1 - \)\(16\!\cdots\!66\)\( T + \)\(47\!\cdots\!35\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!35\)\( T^{4} - \)\(55\!\cdots\!86\)\( T^{5} + \)\(18\!\cdots\!81\)\( T^{6} \)
$67$ \( 1 - \)\(29\!\cdots\!08\)\( T + \)\(41\!\cdots\!37\)\( T^{2} - \)\(35\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!71\)\( T^{4} - \)\(59\!\cdots\!12\)\( T^{5} + \)\(89\!\cdots\!87\)\( T^{6} \)
$71$ \( 1 - \)\(91\!\cdots\!96\)\( T + \)\(80\!\cdots\!85\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!35\)\( T^{4} - \)\(69\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!11\)\( T^{6} \)
$73$ \( 1 - \)\(28\!\cdots\!86\)\( T + \)\(59\!\cdots\!63\)\( T^{2} - \)\(84\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(10\!\cdots\!94\)\( T^{5} + \)\(77\!\cdots\!33\)\( T^{6} \)
$79$ \( 1 - \)\(35\!\cdots\!40\)\( T + \)\(17\!\cdots\!37\)\( T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(91\!\cdots\!23\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!39\)\( T^{6} \)
$83$ \( 1 + \)\(49\!\cdots\!64\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(21\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!11\)\( T^{4} + \)\(15\!\cdots\!96\)\( T^{5} + \)\(58\!\cdots\!63\)\( T^{6} \)
$89$ \( 1 + \)\(17\!\cdots\!70\)\( T + \)\(72\!\cdots\!67\)\( T^{2} + \)\(90\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!63\)\( T^{4} + \)\(11\!\cdots\!70\)\( T^{5} + \)\(16\!\cdots\!69\)\( T^{6} \)
$97$ \( 1 - \)\(41\!\cdots\!58\)\( T + \)\(31\!\cdots\!47\)\( T^{2} - \)\(74\!\cdots\!04\)\( T^{3} + \)\(35\!\cdots\!91\)\( T^{4} - \)\(54\!\cdots\!22\)\( T^{5} + \)\(15\!\cdots\!77\)\( T^{6} \)
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