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

Related objects

Downloads

Learn more about

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: \( 2^{22}\cdot 3^{6}\cdot 5^{3}\cdot 7 \)
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.

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

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

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

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.

Atkin-Lehner signs

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

Hecke kernels

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