Properties

Label 2.74.a.a
Level 2
Weight 74
Character orbit 2.a
Self dual Yes
Analytic conductor 67.497
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 74 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(67.4967947474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{7}\cdot 5^{3} \)
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.

\(f(q)\) \(=\) \( q -68719476736 q^{2} +(-101346400339911324 + \beta_{1}) q^{3} +\)\(47\!\cdots\!96\)\( q^{4} +(-\)\(10\!\cdots\!50\)\( + 16642465 \beta_{1} - 7 \beta_{2}) q^{5} +(\)\(69\!\cdots\!64\)\( - 68719476736 \beta_{1}) q^{6} +(-\)\(77\!\cdots\!88\)\( + 7903338308822 \beta_{1} - 1662860 \beta_{2}) q^{7} -\)\(32\!\cdots\!56\)\( q^{8} +(\)\(24\!\cdots\!53\)\( - 16762740084667458 \beta_{1} + 4620715470 \beta_{2}) q^{9} +O(q^{10})\) \( q -68719476736 q^{2} +(-101346400339911324 + \beta_{1}) q^{3} +\)\(47\!\cdots\!96\)\( q^{4} +(-\)\(10\!\cdots\!50\)\( + 16642465 \beta_{1} - 7 \beta_{2}) q^{5} +(\)\(69\!\cdots\!64\)\( - 68719476736 \beta_{1}) q^{6} +(-\)\(77\!\cdots\!88\)\( + 7903338308822 \beta_{1} - 1662860 \beta_{2}) q^{7} -\)\(32\!\cdots\!56\)\( q^{8} +(\)\(24\!\cdots\!53\)\( - 16762740084667458 \beta_{1} + 4620715470 \beta_{2}) q^{9} +(\)\(74\!\cdots\!00\)\( - 1143661486397194240 \beta_{1} + 481036337152 \beta_{2}) q^{10} +(-\)\(26\!\cdots\!48\)\( + 38059855453876711147 \beta_{1} + 15290845419080 \beta_{2}) q^{11} +(-\)\(47\!\cdots\!04\)\( + \)\(47\!\cdots\!96\)\( \beta_{1}) q^{12} +(\)\(50\!\cdots\!46\)\( + \)\(87\!\cdots\!57\)\( \beta_{1} - 2079791218114015 \beta_{2}) q^{13} +(\)\(53\!\cdots\!68\)\( - \)\(54\!\cdots\!92\)\( \beta_{1} + 114270869085224960 \beta_{2}) q^{14} +(\)\(24\!\cdots\!00\)\( - \)\(19\!\cdots\!10\)\( \beta_{1} + 2087835322029899148 \beta_{2}) q^{15} +\)\(22\!\cdots\!16\)\( q^{16} +(-\)\(67\!\cdots\!78\)\( + \)\(12\!\cdots\!38\)\( \beta_{1} - 69779524841156108930 \beta_{2}) q^{17} +(-\)\(17\!\cdots\!08\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} - \)\(31\!\cdots\!20\)\( \beta_{2}) q^{18} +(-\)\(13\!\cdots\!60\)\( - \)\(80\!\cdots\!03\)\( \beta_{1} + \)\(32\!\cdots\!20\)\( \beta_{2}) q^{19} +(-\)\(51\!\cdots\!00\)\( + \)\(78\!\cdots\!40\)\( \beta_{1} - \)\(33\!\cdots\!72\)\( \beta_{2}) q^{20} +(\)\(72\!\cdots\!12\)\( - \)\(24\!\cdots\!36\)\( \beta_{1} + \)\(51\!\cdots\!80\)\( \beta_{2}) q^{21} +(\)\(18\!\cdots\!28\)\( - \)\(26\!\cdots\!92\)\( \beta_{1} - \)\(10\!\cdots\!80\)\( \beta_{2}) q^{22} +(\)\(26\!\cdots\!96\)\( - \)\(19\!\cdots\!94\)\( \beta_{1} - \)\(62\!\cdots\!60\)\( \beta_{2}) q^{23} +(\)\(32\!\cdots\!44\)\( - \)\(32\!\cdots\!56\)\( \beta_{1}) q^{24} +(\)\(32\!\cdots\!75\)\( - \)\(34\!\cdots\!00\)\( \beta_{1} + \)\(12\!\cdots\!00\)\( \beta_{2}) q^{25} +(-\)\(34\!\cdots\!56\)\( - \)\(60\!\cdots\!52\)\( \beta_{1} + \)\(14\!\cdots\!40\)\( \beta_{2}) q^{26} +(\)\(29\!\cdots\!80\)\( - \)\(37\!\cdots\!98\)\( \beta_{1} - \)\(14\!\cdots\!40\)\( \beta_{2}) q^{27} +(-\)\(36\!\cdots\!48\)\( + \)\(37\!\cdots\!12\)\( \beta_{1} - \)\(78\!\cdots\!60\)\( \beta_{2}) q^{28} +(\)\(55\!\cdots\!70\)\( + \)\(64\!\cdots\!49\)\( \beta_{1} + \)\(53\!\cdots\!45\)\( \beta_{2}) q^{29} +(-\)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!60\)\( \beta_{1} - \)\(14\!\cdots\!28\)\( \beta_{2}) q^{30} +(\)\(31\!\cdots\!32\)\( + \)\(27\!\cdots\!04\)\( \beta_{1} + \)\(35\!\cdots\!80\)\( \beta_{2}) q^{31} -\)\(15\!\cdots\!76\)\( q^{32} +(\)\(58\!\cdots\!52\)\( - \)\(17\!\cdots\!46\)\( \beta_{1} - \)\(42\!\cdots\!30\)\( \beta_{2}) q^{33} +(\)\(46\!\cdots\!08\)\( - \)\(86\!\cdots\!68\)\( \beta_{1} + \)\(47\!\cdots\!80\)\( \beta_{2}) q^{34} +(\)\(31\!\cdots\!00\)\( - \)\(86\!\cdots\!20\)\( \beta_{1} + \)\(22\!\cdots\!76\)\( \beta_{2}) q^{35} +(\)\(11\!\cdots\!88\)\( - \)\(79\!\cdots\!68\)\( \beta_{1} + \)\(21\!\cdots\!20\)\( \beta_{2}) q^{36} +(\)\(31\!\cdots\!22\)\( + \)\(47\!\cdots\!73\)\( \beta_{1} - \)\(28\!\cdots\!55\)\( \beta_{2}) q^{37} +(\)\(95\!\cdots\!60\)\( + \)\(55\!\cdots\!08\)\( \beta_{1} - \)\(22\!\cdots\!20\)\( \beta_{2}) q^{38} +(\)\(66\!\cdots\!96\)\( + \)\(94\!\cdots\!58\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{39} +(\)\(35\!\cdots\!00\)\( - \)\(54\!\cdots\!40\)\( \beta_{1} + \)\(22\!\cdots\!92\)\( \beta_{2}) q^{40} +(\)\(50\!\cdots\!82\)\( - \)\(72\!\cdots\!24\)\( \beta_{1} - \)\(17\!\cdots\!40\)\( \beta_{2}) q^{41} +(-\)\(50\!\cdots\!32\)\( + \)\(17\!\cdots\!96\)\( \beta_{1} - \)\(35\!\cdots\!80\)\( \beta_{2}) q^{42} +(-\)\(15\!\cdots\!64\)\( - \)\(16\!\cdots\!17\)\( \beta_{1} + \)\(55\!\cdots\!40\)\( \beta_{2}) q^{43} +(-\)\(12\!\cdots\!08\)\( + \)\(17\!\cdots\!12\)\( \beta_{1} + \)\(72\!\cdots\!80\)\( \beta_{2}) q^{44} +(-\)\(11\!\cdots\!50\)\( + \)\(26\!\cdots\!45\)\( \beta_{1} - \)\(21\!\cdots\!11\)\( \beta_{2}) q^{45} +(-\)\(18\!\cdots\!56\)\( + \)\(13\!\cdots\!84\)\( \beta_{1} + \)\(42\!\cdots\!60\)\( \beta_{2}) q^{46} +(-\)\(41\!\cdots\!88\)\( - \)\(34\!\cdots\!68\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2}) q^{47} +(-\)\(22\!\cdots\!84\)\( + \)\(22\!\cdots\!16\)\( \beta_{1}) q^{48} +(\)\(26\!\cdots\!37\)\( - \)\(19\!\cdots\!12\)\( \beta_{1} + \)\(38\!\cdots\!40\)\( \beta_{2}) q^{49} +(-\)\(22\!\cdots\!00\)\( + \)\(23\!\cdots\!00\)\( \beta_{1} - \)\(87\!\cdots\!00\)\( \beta_{2}) q^{50} +(\)\(17\!\cdots\!72\)\( - \)\(66\!\cdots\!70\)\( \beta_{1} + \)\(25\!\cdots\!80\)\( \beta_{2}) q^{51} +(\)\(23\!\cdots\!16\)\( + \)\(41\!\cdots\!72\)\( \beta_{1} - \)\(98\!\cdots\!40\)\( \beta_{2}) q^{52} +(\)\(47\!\cdots\!86\)\( + \)\(41\!\cdots\!17\)\( \beta_{1} - \)\(12\!\cdots\!55\)\( \beta_{2}) q^{53} +(-\)\(20\!\cdots\!80\)\( + \)\(25\!\cdots\!28\)\( \beta_{1} + \)\(96\!\cdots\!40\)\( \beta_{2}) q^{54} +(-\)\(23\!\cdots\!00\)\( + \)\(53\!\cdots\!30\)\( \beta_{1} + \)\(17\!\cdots\!96\)\( \beta_{2}) q^{55} +(\)\(25\!\cdots\!28\)\( - \)\(25\!\cdots\!32\)\( \beta_{1} + \)\(53\!\cdots\!60\)\( \beta_{2}) q^{56} +(-\)\(52\!\cdots\!60\)\( - \)\(20\!\cdots\!58\)\( \beta_{1} - \)\(46\!\cdots\!90\)\( \beta_{2}) q^{57} +(-\)\(38\!\cdots\!20\)\( - \)\(44\!\cdots\!64\)\( \beta_{1} - \)\(36\!\cdots\!20\)\( \beta_{2}) q^{58} +(-\)\(12\!\cdots\!60\)\( + \)\(26\!\cdots\!83\)\( \beta_{1} + \)\(31\!\cdots\!40\)\( \beta_{2}) q^{59} +(\)\(11\!\cdots\!00\)\( - \)\(91\!\cdots\!60\)\( \beta_{1} + \)\(98\!\cdots\!08\)\( \beta_{2}) q^{60} +(-\)\(44\!\cdots\!38\)\( + \)\(45\!\cdots\!73\)\( \beta_{1} + \)\(13\!\cdots\!85\)\( \beta_{2}) q^{61} +(-\)\(21\!\cdots\!52\)\( - \)\(18\!\cdots\!44\)\( \beta_{1} - \)\(24\!\cdots\!80\)\( \beta_{2}) q^{62} +(-\)\(22\!\cdots\!64\)\( + \)\(71\!\cdots\!30\)\( \beta_{1} - \)\(46\!\cdots\!60\)\( \beta_{2}) q^{63} +\)\(10\!\cdots\!36\)\( q^{64} +(\)\(43\!\cdots\!00\)\( - \)\(23\!\cdots\!60\)\( \beta_{1} + \)\(93\!\cdots\!88\)\( \beta_{2}) q^{65} +(-\)\(40\!\cdots\!72\)\( + \)\(12\!\cdots\!56\)\( \beta_{1} + \)\(28\!\cdots\!80\)\( \beta_{2}) q^{66} +(-\)\(33\!\cdots\!48\)\( - \)\(10\!\cdots\!59\)\( \beta_{1} - \)\(45\!\cdots\!20\)\( \beta_{2}) q^{67} +(-\)\(31\!\cdots\!88\)\( + \)\(59\!\cdots\!48\)\( \beta_{1} - \)\(32\!\cdots\!80\)\( \beta_{2}) q^{68} +(-\)\(19\!\cdots\!04\)\( + \)\(55\!\cdots\!92\)\( \beta_{1} + \)\(86\!\cdots\!60\)\( \beta_{2}) q^{69} +(-\)\(21\!\cdots\!00\)\( + \)\(59\!\cdots\!20\)\( \beta_{1} - \)\(15\!\cdots\!36\)\( \beta_{2}) q^{70} +(-\)\(38\!\cdots\!08\)\( + \)\(33\!\cdots\!58\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2}) q^{71} +(-\)\(81\!\cdots\!68\)\( + \)\(54\!\cdots\!48\)\( \beta_{1} - \)\(14\!\cdots\!20\)\( \beta_{2}) q^{72} +(-\)\(51\!\cdots\!94\)\( - \)\(20\!\cdots\!98\)\( \beta_{1} + \)\(21\!\cdots\!70\)\( \beta_{2}) q^{73} +(-\)\(21\!\cdots\!92\)\( - \)\(32\!\cdots\!28\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2}) q^{74} +(-\)\(31\!\cdots\!00\)\( + \)\(21\!\cdots\!75\)\( \beta_{1} - \)\(52\!\cdots\!00\)\( \beta_{2}) q^{75} +(-\)\(65\!\cdots\!60\)\( - \)\(37\!\cdots\!88\)\( \beta_{1} + \)\(15\!\cdots\!20\)\( \beta_{2}) q^{76} +(-\)\(60\!\cdots\!76\)\( + \)\(13\!\cdots\!68\)\( \beta_{1} + \)\(59\!\cdots\!20\)\( \beta_{2}) q^{77} +(-\)\(45\!\cdots\!56\)\( - \)\(64\!\cdots\!88\)\( \beta_{1} - \)\(68\!\cdots\!00\)\( \beta_{2}) q^{78} +(-\)\(13\!\cdots\!00\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} + \)\(22\!\cdots\!40\)\( \beta_{2}) q^{79} +(-\)\(24\!\cdots\!00\)\( + \)\(37\!\cdots\!40\)\( \beta_{1} - \)\(15\!\cdots\!12\)\( \beta_{2}) q^{80} +(-\)\(50\!\cdots\!39\)\( - \)\(10\!\cdots\!54\)\( \beta_{1} - \)\(81\!\cdots\!10\)\( \beta_{2}) q^{81} +(-\)\(34\!\cdots\!52\)\( + \)\(49\!\cdots\!64\)\( \beta_{1} + \)\(11\!\cdots\!40\)\( \beta_{2}) q^{82} +(\)\(16\!\cdots\!96\)\( - \)\(25\!\cdots\!55\)\( \beta_{1} - \)\(23\!\cdots\!80\)\( \beta_{2}) q^{83} +(\)\(34\!\cdots\!52\)\( - \)\(11\!\cdots\!56\)\( \beta_{1} + \)\(24\!\cdots\!80\)\( \beta_{2}) q^{84} +(\)\(21\!\cdots\!00\)\( - \)\(63\!\cdots\!70\)\( \beta_{1} + \)\(67\!\cdots\!86\)\( \beta_{2}) q^{85} +(\)\(10\!\cdots\!04\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(38\!\cdots\!40\)\( \beta_{2}) q^{86} +(\)\(47\!\cdots\!20\)\( + \)\(18\!\cdots\!54\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2}) q^{87} +(\)\(86\!\cdots\!88\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} - \)\(49\!\cdots\!80\)\( \beta_{2}) q^{88} +(\)\(15\!\cdots\!90\)\( - \)\(48\!\cdots\!90\)\( \beta_{1} + \)\(15\!\cdots\!50\)\( \beta_{2}) q^{89} +(\)\(76\!\cdots\!00\)\( - \)\(18\!\cdots\!20\)\( \beta_{1} + \)\(14\!\cdots\!96\)\( \beta_{2}) q^{90} +(\)\(14\!\cdots\!52\)\( - \)\(32\!\cdots\!44\)\( \beta_{1} + \)\(21\!\cdots\!40\)\( \beta_{2}) q^{91} +(\)\(12\!\cdots\!16\)\( - \)\(94\!\cdots\!24\)\( \beta_{1} - \)\(29\!\cdots\!60\)\( \beta_{2}) q^{92} +(\)\(19\!\cdots\!32\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} - \)\(89\!\cdots\!40\)\( \beta_{2}) q^{93} +(\)\(28\!\cdots\!68\)\( + \)\(23\!\cdots\!48\)\( \beta_{1} + \)\(72\!\cdots\!20\)\( \beta_{2}) q^{94} +(-\)\(17\!\cdots\!00\)\( + \)\(13\!\cdots\!50\)\( \beta_{1} - \)\(15\!\cdots\!20\)\( \beta_{2}) q^{95} +(\)\(15\!\cdots\!24\)\( - \)\(15\!\cdots\!76\)\( \beta_{1}) q^{96} +(\)\(25\!\cdots\!42\)\( + \)\(12\!\cdots\!90\)\( \beta_{1} + \)\(22\!\cdots\!30\)\( \beta_{2}) q^{97} +(-\)\(18\!\cdots\!32\)\( + \)\(13\!\cdots\!32\)\( \beta_{1} - \)\(26\!\cdots\!40\)\( \beta_{2}) q^{98} +(\)\(10\!\cdots\!56\)\( - \)\(28\!\cdots\!65\)\( \beta_{1} + \)\(16\!\cdots\!60\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 206158430208q^{2} - 304039201019733972q^{3} + \)\(14\!\cdots\!88\)\(q^{4} - \)\(32\!\cdots\!50\)\(q^{5} + \)\(20\!\cdots\!92\)\(q^{6} - \)\(23\!\cdots\!64\)\(q^{7} - \)\(97\!\cdots\!68\)\(q^{8} + \)\(74\!\cdots\!59\)\(q^{9} + O(q^{10}) \) \( 3q - 206158430208q^{2} - 304039201019733972q^{3} + \)\(14\!\cdots\!88\)\(q^{4} - \)\(32\!\cdots\!50\)\(q^{5} + \)\(20\!\cdots\!92\)\(q^{6} - \)\(23\!\cdots\!64\)\(q^{7} - \)\(97\!\cdots\!68\)\(q^{8} + \)\(74\!\cdots\!59\)\(q^{9} + \)\(22\!\cdots\!00\)\(q^{10} - \)\(80\!\cdots\!44\)\(q^{11} - \)\(14\!\cdots\!12\)\(q^{12} + \)\(15\!\cdots\!38\)\(q^{13} + \)\(15\!\cdots\!04\)\(q^{14} + \)\(74\!\cdots\!00\)\(q^{15} + \)\(66\!\cdots\!48\)\(q^{16} - \)\(20\!\cdots\!34\)\(q^{17} - \)\(51\!\cdots\!24\)\(q^{18} - \)\(41\!\cdots\!80\)\(q^{19} - \)\(15\!\cdots\!00\)\(q^{20} + \)\(21\!\cdots\!36\)\(q^{21} + \)\(55\!\cdots\!84\)\(q^{22} + \)\(78\!\cdots\!88\)\(q^{23} + \)\(98\!\cdots\!32\)\(q^{24} + \)\(98\!\cdots\!25\)\(q^{25} - \)\(10\!\cdots\!68\)\(q^{26} + \)\(88\!\cdots\!40\)\(q^{27} - \)\(10\!\cdots\!44\)\(q^{28} + \)\(16\!\cdots\!10\)\(q^{29} - \)\(51\!\cdots\!00\)\(q^{30} + \)\(95\!\cdots\!96\)\(q^{31} - \)\(45\!\cdots\!28\)\(q^{32} + \)\(17\!\cdots\!56\)\(q^{33} + \)\(13\!\cdots\!24\)\(q^{34} + \)\(94\!\cdots\!00\)\(q^{35} + \)\(35\!\cdots\!64\)\(q^{36} + \)\(95\!\cdots\!66\)\(q^{37} + \)\(28\!\cdots\!80\)\(q^{38} + \)\(20\!\cdots\!88\)\(q^{39} + \)\(10\!\cdots\!00\)\(q^{40} + \)\(15\!\cdots\!46\)\(q^{41} - \)\(15\!\cdots\!96\)\(q^{42} - \)\(47\!\cdots\!92\)\(q^{43} - \)\(37\!\cdots\!24\)\(q^{44} - \)\(33\!\cdots\!50\)\(q^{45} - \)\(54\!\cdots\!68\)\(q^{46} - \)\(12\!\cdots\!64\)\(q^{47} - \)\(67\!\cdots\!52\)\(q^{48} + \)\(80\!\cdots\!11\)\(q^{49} - \)\(67\!\cdots\!00\)\(q^{50} + \)\(51\!\cdots\!16\)\(q^{51} + \)\(70\!\cdots\!48\)\(q^{52} + \)\(14\!\cdots\!58\)\(q^{53} - \)\(60\!\cdots\!40\)\(q^{54} - \)\(71\!\cdots\!00\)\(q^{55} + \)\(75\!\cdots\!84\)\(q^{56} - \)\(15\!\cdots\!80\)\(q^{57} - \)\(11\!\cdots\!60\)\(q^{58} - \)\(38\!\cdots\!80\)\(q^{59} + \)\(35\!\cdots\!00\)\(q^{60} - \)\(13\!\cdots\!14\)\(q^{61} - \)\(65\!\cdots\!56\)\(q^{62} - \)\(67\!\cdots\!92\)\(q^{63} + \)\(31\!\cdots\!08\)\(q^{64} + \)\(13\!\cdots\!00\)\(q^{65} - \)\(12\!\cdots\!16\)\(q^{66} - \)\(10\!\cdots\!44\)\(q^{67} - \)\(95\!\cdots\!64\)\(q^{68} - \)\(57\!\cdots\!12\)\(q^{69} - \)\(64\!\cdots\!00\)\(q^{70} - \)\(11\!\cdots\!24\)\(q^{71} - \)\(24\!\cdots\!04\)\(q^{72} - \)\(15\!\cdots\!82\)\(q^{73} - \)\(65\!\cdots\!76\)\(q^{74} - \)\(95\!\cdots\!00\)\(q^{75} - \)\(19\!\cdots\!80\)\(q^{76} - \)\(18\!\cdots\!28\)\(q^{77} - \)\(13\!\cdots\!68\)\(q^{78} - \)\(40\!\cdots\!00\)\(q^{79} - \)\(72\!\cdots\!00\)\(q^{80} - \)\(15\!\cdots\!17\)\(q^{81} - \)\(10\!\cdots\!56\)\(q^{82} + \)\(48\!\cdots\!88\)\(q^{83} + \)\(10\!\cdots\!56\)\(q^{84} + \)\(64\!\cdots\!00\)\(q^{85} + \)\(32\!\cdots\!12\)\(q^{86} + \)\(14\!\cdots\!60\)\(q^{87} + \)\(26\!\cdots\!64\)\(q^{88} + \)\(45\!\cdots\!70\)\(q^{89} + \)\(23\!\cdots\!00\)\(q^{90} + \)\(42\!\cdots\!56\)\(q^{91} + \)\(37\!\cdots\!48\)\(q^{92} + \)\(57\!\cdots\!96\)\(q^{93} + \)\(86\!\cdots\!04\)\(q^{94} - \)\(51\!\cdots\!00\)\(q^{95} + \)\(46\!\cdots\!72\)\(q^{96} + \)\(76\!\cdots\!26\)\(q^{97} - \)\(55\!\cdots\!96\)\(q^{98} + \)\(32\!\cdots\!68\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 413501459186944860372404680 x - 2966140105783309949999568694815716833028\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 17280 \nu - 5760 \)
\(\beta_{2}\)\(=\)\((\)\( 40960 \nu^{2} - 440723106595950720 \nu - 11291346512198027412866931825280 \)\()/633843\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 5760\)\()/17280\)
\(\nu^{2}\)\(=\)\((\)\(5704587 \beta_{2} + 229543284685391 \beta_{1} + 101622118609783568885122174279680\)\()/368640\)

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
−1.44110e13
−8.84892e12
2.32599e13
−6.87195e10 −3.50368e17 4.72237e21 −5.44340e25 2.40771e28 −1.20967e31 −3.24519e32 5.51727e34 3.74068e36
1.2 −6.87195e10 −2.54256e17 4.72237e21 3.27596e25 1.74723e28 8.99592e30 −3.24519e32 −2.93926e33 −2.25122e36
1.3 −6.87195e10 3.00585e17 4.72237e21 −1.10375e25 −2.06560e28 7.83423e29 −3.24519e32 2.27660e34 7.58489e35
\(n\): 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 \( T_{3}^{3} + \)\(30\!\cdots\!72\)\( T_{3}^{2} - \)\(92\!\cdots\!72\)\( T_{3} - \)\(26\!\cdots\!76\)\( \) acting on \(S_{74}^{\mathrm{new}}(\Gamma_0(2))\).