Properties

Label 9.36.a.b.1.3
Level $9$
Weight $36$
Character 9.1
Self dual yes
Analytic conductor $69.836$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.8356175703\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 12422194 x - 2645665785\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-213.765\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+165109. q^{2} -7.09870e9 q^{4} +2.05014e12 q^{5} -1.25160e14 q^{7} -6.84517e15 q^{8} +O(q^{10})\) \(q+165109. q^{2} -7.09870e9 q^{4} +2.05014e12 q^{5} -1.25160e14 q^{7} -6.84517e15 q^{8} +3.38496e17 q^{10} +1.70842e18 q^{11} -4.94986e19 q^{13} -2.06651e19 q^{14} -8.86291e20 q^{16} -1.32045e21 q^{17} +3.94388e21 q^{19} -1.45533e22 q^{20} +2.82076e23 q^{22} +3.48881e23 q^{23} +1.29267e24 q^{25} -8.17268e24 q^{26} +8.88473e23 q^{28} -3.21628e25 q^{29} +3.41044e25 q^{31} +8.88635e25 q^{32} -2.18018e26 q^{34} -2.56595e26 q^{35} -4.03624e27 q^{37} +6.51171e26 q^{38} -1.40335e28 q^{40} -8.65857e27 q^{41} +9.89389e26 q^{43} -1.21276e28 q^{44} +5.76034e28 q^{46} -1.95852e29 q^{47} -3.63154e29 q^{49} +2.13432e29 q^{50} +3.51376e29 q^{52} +9.96909e29 q^{53} +3.50249e30 q^{55} +8.56741e29 q^{56} -5.31037e30 q^{58} -3.91953e30 q^{59} +7.64909e29 q^{61} +5.63096e30 q^{62} +4.51249e31 q^{64} -1.01479e32 q^{65} -1.64301e32 q^{67} +9.37346e30 q^{68} -4.23662e31 q^{70} -7.65930e31 q^{71} -7.08063e32 q^{73} -6.66421e32 q^{74} -2.79964e31 q^{76} -2.13826e32 q^{77} +2.34599e33 q^{79} -1.81702e33 q^{80} -1.42961e33 q^{82} -5.18971e33 q^{83} -2.70710e33 q^{85} +1.63357e32 q^{86} -1.16944e34 q^{88} -1.44578e34 q^{89} +6.19525e33 q^{91} -2.47660e33 q^{92} -3.23369e34 q^{94} +8.08549e33 q^{95} -2.87399e34 q^{97} -5.99600e34 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 139656q^{2} + 34841262144q^{4} - 892652054010q^{5} + 878422149346056q^{7} - 22336009925337600q^{8} + O(q^{10}) \) \( 3q - 139656q^{2} + 34841262144q^{4} - 892652054010q^{5} + 878422149346056q^{7} - 22336009925337600q^{8} + 1019870812729298160q^{10} + 1157945428549987044q^{11} - 62139610550998650558q^{13} - \)\(13\!\cdots\!28\)\(q^{14} + \)\(21\!\cdots\!48\)\(q^{16} + \)\(39\!\cdots\!94\)\(q^{17} - \)\(32\!\cdots\!40\)\(q^{19} - \)\(14\!\cdots\!80\)\(q^{20} + \)\(22\!\cdots\!12\)\(q^{22} + \)\(51\!\cdots\!08\)\(q^{23} + \)\(64\!\cdots\!25\)\(q^{25} - \)\(42\!\cdots\!36\)\(q^{26} + \)\(10\!\cdots\!08\)\(q^{28} + \)\(38\!\cdots\!10\)\(q^{29} + \)\(10\!\cdots\!56\)\(q^{31} + \)\(92\!\cdots\!44\)\(q^{32} - \)\(55\!\cdots\!12\)\(q^{34} - \)\(15\!\cdots\!60\)\(q^{35} + \)\(24\!\cdots\!06\)\(q^{37} + \)\(12\!\cdots\!00\)\(q^{38} + \)\(16\!\cdots\!00\)\(q^{40} - \)\(23\!\cdots\!06\)\(q^{41} - \)\(47\!\cdots\!08\)\(q^{43} + \)\(80\!\cdots\!12\)\(q^{44} + \)\(31\!\cdots\!56\)\(q^{46} - \)\(16\!\cdots\!56\)\(q^{47} - \)\(59\!\cdots\!21\)\(q^{49} - \)\(37\!\cdots\!00\)\(q^{50} - \)\(51\!\cdots\!44\)\(q^{52} + \)\(16\!\cdots\!58\)\(q^{53} + \)\(30\!\cdots\!20\)\(q^{55} - \)\(56\!\cdots\!40\)\(q^{56} - \)\(23\!\cdots\!00\)\(q^{58} - \)\(43\!\cdots\!80\)\(q^{59} + \)\(23\!\cdots\!06\)\(q^{61} - \)\(29\!\cdots\!12\)\(q^{62} + \)\(93\!\cdots\!84\)\(q^{64} - \)\(75\!\cdots\!20\)\(q^{65} - \)\(18\!\cdots\!44\)\(q^{67} - \)\(21\!\cdots\!08\)\(q^{68} + \)\(22\!\cdots\!60\)\(q^{70} - \)\(34\!\cdots\!56\)\(q^{71} - \)\(28\!\cdots\!58\)\(q^{73} - \)\(21\!\cdots\!08\)\(q^{74} - \)\(27\!\cdots\!20\)\(q^{76} - \)\(69\!\cdots\!12\)\(q^{77} - \)\(42\!\cdots\!60\)\(q^{79} - \)\(68\!\cdots\!60\)\(q^{80} - \)\(40\!\cdots\!88\)\(q^{82} - \)\(14\!\cdots\!92\)\(q^{83} - \)\(87\!\cdots\!40\)\(q^{85} - \)\(14\!\cdots\!96\)\(q^{86} - \)\(18\!\cdots\!00\)\(q^{88} - \)\(30\!\cdots\!70\)\(q^{89} + \)\(10\!\cdots\!96\)\(q^{91} - \)\(83\!\cdots\!56\)\(q^{92} + \)\(19\!\cdots\!68\)\(q^{94} + \)\(84\!\cdots\!00\)\(q^{95} - \)\(10\!\cdots\!94\)\(q^{97} + \)\(13\!\cdots\!92\)\(q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 165109. 0.890730 0.445365 0.895349i \(-0.353074\pi\)
0.445365 + 0.895349i \(0.353074\pi\)
\(3\) 0 0
\(4\) −7.09870e9 −0.206599
\(5\) 2.05014e12 1.20173 0.600866 0.799350i \(-0.294823\pi\)
0.600866 + 0.799350i \(0.294823\pi\)
\(6\) 0 0
\(7\) −1.25160e14 −0.203353 −0.101676 0.994818i \(-0.532421\pi\)
−0.101676 + 0.994818i \(0.532421\pi\)
\(8\) −6.84517e15 −1.07475
\(9\) 0 0
\(10\) 3.38496e17 1.07042
\(11\) 1.70842e18 1.01911 0.509557 0.860437i \(-0.329809\pi\)
0.509557 + 0.860437i \(0.329809\pi\)
\(12\) 0 0
\(13\) −4.94986e19 −1.58703 −0.793514 0.608552i \(-0.791751\pi\)
−0.793514 + 0.608552i \(0.791751\pi\)
\(14\) −2.06651e19 −0.181132
\(15\) 0 0
\(16\) −8.86291e20 −0.750717
\(17\) −1.32045e21 −0.387137 −0.193569 0.981087i \(-0.562006\pi\)
−0.193569 + 0.981087i \(0.562006\pi\)
\(18\) 0 0
\(19\) 3.94388e21 0.165096 0.0825479 0.996587i \(-0.473694\pi\)
0.0825479 + 0.996587i \(0.473694\pi\)
\(20\) −1.45533e22 −0.248277
\(21\) 0 0
\(22\) 2.82076e23 0.907756
\(23\) 3.48881e23 0.515750 0.257875 0.966178i \(-0.416978\pi\)
0.257875 + 0.966178i \(0.416978\pi\)
\(24\) 0 0
\(25\) 1.29267e24 0.444159
\(26\) −8.17268e24 −1.41361
\(27\) 0 0
\(28\) 8.88473e23 0.0420125
\(29\) −3.21628e25 −0.822979 −0.411489 0.911415i \(-0.634991\pi\)
−0.411489 + 0.911415i \(0.634991\pi\)
\(30\) 0 0
\(31\) 3.41044e25 0.271632 0.135816 0.990734i \(-0.456634\pi\)
0.135816 + 0.990734i \(0.456634\pi\)
\(32\) 8.88635e25 0.406068
\(33\) 0 0
\(34\) −2.18018e26 −0.344835
\(35\) −2.56595e26 −0.244375
\(36\) 0 0
\(37\) −4.03624e27 −1.45361 −0.726804 0.686845i \(-0.758995\pi\)
−0.726804 + 0.686845i \(0.758995\pi\)
\(38\) 6.51171e26 0.147056
\(39\) 0 0
\(40\) −1.40335e28 −1.29157
\(41\) −8.65857e27 −0.517283 −0.258642 0.965973i \(-0.583275\pi\)
−0.258642 + 0.965973i \(0.583275\pi\)
\(42\) 0 0
\(43\) 9.89389e26 0.0256844 0.0128422 0.999918i \(-0.495912\pi\)
0.0128422 + 0.999918i \(0.495912\pi\)
\(44\) −1.21276e28 −0.210548
\(45\) 0 0
\(46\) 5.76034e28 0.459395
\(47\) −1.95852e29 −1.07205 −0.536025 0.844202i \(-0.680075\pi\)
−0.536025 + 0.844202i \(0.680075\pi\)
\(48\) 0 0
\(49\) −3.63154e29 −0.958648
\(50\) 2.13432e29 0.395626
\(51\) 0 0
\(52\) 3.51376e29 0.327879
\(53\) 9.96909e29 0.666543 0.333271 0.942831i \(-0.391847\pi\)
0.333271 + 0.942831i \(0.391847\pi\)
\(54\) 0 0
\(55\) 3.50249e30 1.22470
\(56\) 8.56741e29 0.218554
\(57\) 0 0
\(58\) −5.31037e30 −0.733052
\(59\) −3.91953e30 −0.401166 −0.200583 0.979677i \(-0.564284\pi\)
−0.200583 + 0.979677i \(0.564284\pi\)
\(60\) 0 0
\(61\) 7.64909e29 0.0436855 0.0218428 0.999761i \(-0.493047\pi\)
0.0218428 + 0.999761i \(0.493047\pi\)
\(62\) 5.63096e30 0.241951
\(63\) 0 0
\(64\) 4.51249e31 1.11241
\(65\) −1.01479e32 −1.90718
\(66\) 0 0
\(67\) −1.64301e32 −1.81690 −0.908451 0.417992i \(-0.862734\pi\)
−0.908451 + 0.417992i \(0.862734\pi\)
\(68\) 9.37346e30 0.0799823
\(69\) 0 0
\(70\) −4.23662e31 −0.217672
\(71\) −7.65930e31 −0.307021 −0.153510 0.988147i \(-0.549058\pi\)
−0.153510 + 0.988147i \(0.549058\pi\)
\(72\) 0 0
\(73\) −7.08063e32 −1.74551 −0.872754 0.488160i \(-0.837668\pi\)
−0.872754 + 0.488160i \(0.837668\pi\)
\(74\) −6.66421e32 −1.29477
\(75\) 0 0
\(76\) −2.79964e31 −0.0341087
\(77\) −2.13826e32 −0.207239
\(78\) 0 0
\(79\) 2.34599e33 1.45162 0.725809 0.687896i \(-0.241466\pi\)
0.725809 + 0.687896i \(0.241466\pi\)
\(80\) −1.81702e33 −0.902161
\(81\) 0 0
\(82\) −1.42961e33 −0.460760
\(83\) −5.18971e33 −1.35293 −0.676467 0.736473i \(-0.736490\pi\)
−0.676467 + 0.736473i \(0.736490\pi\)
\(84\) 0 0
\(85\) −2.70710e33 −0.465235
\(86\) 1.63357e32 0.0228779
\(87\) 0 0
\(88\) −1.16944e34 −1.09530
\(89\) −1.44578e34 −1.11116 −0.555582 0.831461i \(-0.687505\pi\)
−0.555582 + 0.831461i \(0.687505\pi\)
\(90\) 0 0
\(91\) 6.19525e33 0.322726
\(92\) −2.47660e33 −0.106554
\(93\) 0 0
\(94\) −3.23369e34 −0.954907
\(95\) 8.08549e33 0.198401
\(96\) 0 0
\(97\) −2.87399e34 −0.489756 −0.244878 0.969554i \(-0.578748\pi\)
−0.244878 + 0.969554i \(0.578748\pi\)
\(98\) −5.99600e34 −0.853897
\(99\) 0 0
\(100\) −9.17629e33 −0.0917629
\(101\) −1.13198e35 −0.951079 −0.475539 0.879694i \(-0.657747\pi\)
−0.475539 + 0.879694i \(0.657747\pi\)
\(102\) 0 0
\(103\) 2.53635e35 1.51202 0.756011 0.654559i \(-0.227145\pi\)
0.756011 + 0.654559i \(0.227145\pi\)
\(104\) 3.38826e35 1.70567
\(105\) 0 0
\(106\) 1.64599e35 0.593710
\(107\) 3.63939e35 1.11381 0.556907 0.830575i \(-0.311988\pi\)
0.556907 + 0.830575i \(0.311988\pi\)
\(108\) 0 0
\(109\) −4.65371e34 −0.103000 −0.0514998 0.998673i \(-0.516400\pi\)
−0.0514998 + 0.998673i \(0.516400\pi\)
\(110\) 5.78294e35 1.09088
\(111\) 0 0
\(112\) 1.10928e35 0.152660
\(113\) 2.18346e35 0.257201 0.128601 0.991696i \(-0.458951\pi\)
0.128601 + 0.991696i \(0.458951\pi\)
\(114\) 0 0
\(115\) 7.15252e35 0.619794
\(116\) 2.28314e35 0.170027
\(117\) 0 0
\(118\) −6.47150e35 −0.357330
\(119\) 1.65267e35 0.0787254
\(120\) 0 0
\(121\) 1.08456e35 0.0385932
\(122\) 1.26293e35 0.0389120
\(123\) 0 0
\(124\) −2.42097e35 −0.0561191
\(125\) −3.31653e36 −0.667972
\(126\) 0 0
\(127\) 4.19558e36 0.640070 0.320035 0.947406i \(-0.396305\pi\)
0.320035 + 0.947406i \(0.396305\pi\)
\(128\) 4.39721e36 0.584793
\(129\) 0 0
\(130\) −1.67551e37 −1.69878
\(131\) −2.40631e36 −0.213355 −0.106678 0.994294i \(-0.534021\pi\)
−0.106678 + 0.994294i \(0.534021\pi\)
\(132\) 0 0
\(133\) −4.93616e35 −0.0335727
\(134\) −2.71276e37 −1.61837
\(135\) 0 0
\(136\) 9.03869e36 0.416078
\(137\) 2.14718e37 0.869477 0.434739 0.900557i \(-0.356841\pi\)
0.434739 + 0.900557i \(0.356841\pi\)
\(138\) 0 0
\(139\) 2.63209e37 0.827068 0.413534 0.910489i \(-0.364294\pi\)
0.413534 + 0.910489i \(0.364294\pi\)
\(140\) 1.82149e36 0.0504878
\(141\) 0 0
\(142\) −1.26462e37 −0.273473
\(143\) −8.45645e37 −1.61736
\(144\) 0 0
\(145\) −6.59381e37 −0.989000
\(146\) −1.16908e38 −1.55478
\(147\) 0 0
\(148\) 2.86521e37 0.300314
\(149\) 1.74389e38 1.62465 0.812324 0.583206i \(-0.198202\pi\)
0.812324 + 0.583206i \(0.198202\pi\)
\(150\) 0 0
\(151\) −1.40947e38 −1.03982 −0.519912 0.854220i \(-0.674035\pi\)
−0.519912 + 0.854220i \(0.674035\pi\)
\(152\) −2.69965e37 −0.177437
\(153\) 0 0
\(154\) −3.53046e37 −0.184595
\(155\) 6.99187e37 0.326429
\(156\) 0 0
\(157\) 1.49924e38 0.559277 0.279639 0.960105i \(-0.409785\pi\)
0.279639 + 0.960105i \(0.409785\pi\)
\(158\) 3.87345e38 1.29300
\(159\) 0 0
\(160\) 1.82182e38 0.487985
\(161\) −4.36659e37 −0.104879
\(162\) 0 0
\(163\) 8.60787e38 1.66576 0.832880 0.553453i \(-0.186690\pi\)
0.832880 + 0.553453i \(0.186690\pi\)
\(164\) 6.14646e37 0.106870
\(165\) 0 0
\(166\) −8.56869e38 −1.20510
\(167\) 9.23344e38 1.16903 0.584515 0.811383i \(-0.301285\pi\)
0.584515 + 0.811383i \(0.301285\pi\)
\(168\) 0 0
\(169\) 1.47733e39 1.51866
\(170\) −4.46966e38 −0.414399
\(171\) 0 0
\(172\) −7.02337e36 −0.00530638
\(173\) 1.33161e39 0.909012 0.454506 0.890744i \(-0.349816\pi\)
0.454506 + 0.890744i \(0.349816\pi\)
\(174\) 0 0
\(175\) −1.61791e38 −0.0903208
\(176\) −1.51416e39 −0.765066
\(177\) 0 0
\(178\) −2.38712e39 −0.989748
\(179\) −6.27202e38 −0.235765 −0.117883 0.993028i \(-0.537611\pi\)
−0.117883 + 0.993028i \(0.537611\pi\)
\(180\) 0 0
\(181\) −2.39965e39 −0.742629 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(182\) 1.02289e39 0.287462
\(183\) 0 0
\(184\) −2.38815e39 −0.554305
\(185\) −8.27484e39 −1.74685
\(186\) 0 0
\(187\) −2.25588e39 −0.394537
\(188\) 1.39029e39 0.221485
\(189\) 0 0
\(190\) 1.33499e39 0.176722
\(191\) −8.14518e39 −0.983596 −0.491798 0.870709i \(-0.663660\pi\)
−0.491798 + 0.870709i \(0.663660\pi\)
\(192\) 0 0
\(193\) 1.38549e40 1.39428 0.697139 0.716936i \(-0.254456\pi\)
0.697139 + 0.716936i \(0.254456\pi\)
\(194\) −4.74522e39 −0.436240
\(195\) 0 0
\(196\) 2.57792e39 0.198056
\(197\) 1.15539e40 0.812026 0.406013 0.913867i \(-0.366919\pi\)
0.406013 + 0.913867i \(0.366919\pi\)
\(198\) 0 0
\(199\) −4.76691e39 −0.280742 −0.140371 0.990099i \(-0.544830\pi\)
−0.140371 + 0.990099i \(0.544830\pi\)
\(200\) −8.84856e39 −0.477362
\(201\) 0 0
\(202\) −1.86901e40 −0.847155
\(203\) 4.02549e39 0.167355
\(204\) 0 0
\(205\) −1.77512e40 −0.621636
\(206\) 4.18775e40 1.34680
\(207\) 0 0
\(208\) 4.38702e40 1.19141
\(209\) 6.73781e39 0.168251
\(210\) 0 0
\(211\) −1.87354e40 −0.396022 −0.198011 0.980200i \(-0.563448\pi\)
−0.198011 + 0.980200i \(0.563448\pi\)
\(212\) −7.07676e39 −0.137707
\(213\) 0 0
\(214\) 6.00896e40 0.992107
\(215\) 2.02838e39 0.0308657
\(216\) 0 0
\(217\) −4.26851e39 −0.0552371
\(218\) −7.68370e39 −0.0917448
\(219\) 0 0
\(220\) −2.48631e40 −0.253023
\(221\) 6.53603e40 0.614398
\(222\) 0 0
\(223\) −7.12100e40 −0.571750 −0.285875 0.958267i \(-0.592284\pi\)
−0.285875 + 0.958267i \(0.592284\pi\)
\(224\) −1.11222e40 −0.0825750
\(225\) 0 0
\(226\) 3.60510e40 0.229097
\(227\) 1.04005e41 0.611787 0.305893 0.952066i \(-0.401045\pi\)
0.305893 + 0.952066i \(0.401045\pi\)
\(228\) 0 0
\(229\) 1.82859e41 0.922564 0.461282 0.887254i \(-0.347390\pi\)
0.461282 + 0.887254i \(0.347390\pi\)
\(230\) 1.18095e41 0.552069
\(231\) 0 0
\(232\) 2.20160e41 0.884501
\(233\) −1.57847e41 −0.588177 −0.294088 0.955778i \(-0.595016\pi\)
−0.294088 + 0.955778i \(0.595016\pi\)
\(234\) 0 0
\(235\) −4.01523e41 −1.28832
\(236\) 2.78236e40 0.0828806
\(237\) 0 0
\(238\) 2.72871e40 0.0701231
\(239\) −5.00984e41 −1.19636 −0.598178 0.801363i \(-0.704108\pi\)
−0.598178 + 0.801363i \(0.704108\pi\)
\(240\) 0 0
\(241\) −3.33164e41 −0.687638 −0.343819 0.939036i \(-0.611721\pi\)
−0.343819 + 0.939036i \(0.611721\pi\)
\(242\) 1.79072e40 0.0343762
\(243\) 0 0
\(244\) −5.42986e39 −0.00902540
\(245\) −7.44514e41 −1.15204
\(246\) 0 0
\(247\) −1.95217e41 −0.262011
\(248\) −2.33451e41 −0.291938
\(249\) 0 0
\(250\) −5.47589e41 −0.594983
\(251\) −1.52760e42 −1.54782 −0.773909 0.633297i \(-0.781701\pi\)
−0.773909 + 0.633297i \(0.781701\pi\)
\(252\) 0 0
\(253\) 5.96035e41 0.525608
\(254\) 6.92730e41 0.570130
\(255\) 0 0
\(256\) −8.24460e41 −0.591521
\(257\) 2.72482e42 1.82603 0.913016 0.407925i \(-0.133747\pi\)
0.913016 + 0.407925i \(0.133747\pi\)
\(258\) 0 0
\(259\) 5.05176e41 0.295595
\(260\) 7.20368e41 0.394022
\(261\) 0 0
\(262\) −3.97303e41 −0.190042
\(263\) −4.22055e40 −0.0188861 −0.00944307 0.999955i \(-0.503006\pi\)
−0.00944307 + 0.999955i \(0.503006\pi\)
\(264\) 0 0
\(265\) 2.04380e42 0.801005
\(266\) −8.15005e40 −0.0299042
\(267\) 0 0
\(268\) 1.16632e42 0.375371
\(269\) 1.30618e42 0.393858 0.196929 0.980418i \(-0.436903\pi\)
0.196929 + 0.980418i \(0.436903\pi\)
\(270\) 0 0
\(271\) −4.11918e41 −0.109106 −0.0545529 0.998511i \(-0.517373\pi\)
−0.0545529 + 0.998511i \(0.517373\pi\)
\(272\) 1.17030e42 0.290631
\(273\) 0 0
\(274\) 3.54519e42 0.774470
\(275\) 2.20843e42 0.452648
\(276\) 0 0
\(277\) 7.38232e42 1.33290 0.666449 0.745551i \(-0.267813\pi\)
0.666449 + 0.745551i \(0.267813\pi\)
\(278\) 4.34582e42 0.736694
\(279\) 0 0
\(280\) 1.75644e42 0.262643
\(281\) −4.08223e42 −0.573505 −0.286752 0.958005i \(-0.592576\pi\)
−0.286752 + 0.958005i \(0.592576\pi\)
\(282\) 0 0
\(283\) −1.06503e43 −1.32159 −0.660796 0.750565i \(-0.729781\pi\)
−0.660796 + 0.750565i \(0.729781\pi\)
\(284\) 5.43711e41 0.0634303
\(285\) 0 0
\(286\) −1.39624e43 −1.44063
\(287\) 1.08371e42 0.105191
\(288\) 0 0
\(289\) −9.88997e42 −0.850125
\(290\) −1.08870e43 −0.880932
\(291\) 0 0
\(292\) 5.02633e42 0.360621
\(293\) −1.90474e43 −1.28722 −0.643610 0.765353i \(-0.722564\pi\)
−0.643610 + 0.765353i \(0.722564\pi\)
\(294\) 0 0
\(295\) −8.03556e42 −0.482093
\(296\) 2.76288e43 1.56227
\(297\) 0 0
\(298\) 2.87932e43 1.44712
\(299\) −1.72691e43 −0.818510
\(300\) 0 0
\(301\) −1.23832e41 −0.00522298
\(302\) −2.32717e43 −0.926203
\(303\) 0 0
\(304\) −3.49543e42 −0.123940
\(305\) 1.56817e42 0.0524983
\(306\) 0 0
\(307\) −1.39752e43 −0.417288 −0.208644 0.977992i \(-0.566905\pi\)
−0.208644 + 0.977992i \(0.566905\pi\)
\(308\) 1.51789e42 0.0428156
\(309\) 0 0
\(310\) 1.15442e43 0.290760
\(311\) −2.22377e43 −0.529399 −0.264700 0.964331i \(-0.585273\pi\)
−0.264700 + 0.964331i \(0.585273\pi\)
\(312\) 0 0
\(313\) 3.63171e43 0.772833 0.386417 0.922324i \(-0.373713\pi\)
0.386417 + 0.922324i \(0.373713\pi\)
\(314\) 2.47538e43 0.498165
\(315\) 0 0
\(316\) −1.66535e43 −0.299903
\(317\) −5.29385e43 −0.902058 −0.451029 0.892509i \(-0.648943\pi\)
−0.451029 + 0.892509i \(0.648943\pi\)
\(318\) 0 0
\(319\) −5.49476e43 −0.838709
\(320\) 9.25121e43 1.33682
\(321\) 0 0
\(322\) −7.20964e42 −0.0934191
\(323\) −5.20769e42 −0.0639147
\(324\) 0 0
\(325\) −6.39855e43 −0.704892
\(326\) 1.42124e44 1.48374
\(327\) 0 0
\(328\) 5.92693e43 0.555953
\(329\) 2.45128e43 0.218004
\(330\) 0 0
\(331\) 6.25551e43 0.500349 0.250174 0.968201i \(-0.419512\pi\)
0.250174 + 0.968201i \(0.419512\pi\)
\(332\) 3.68402e43 0.279515
\(333\) 0 0
\(334\) 1.52453e44 1.04129
\(335\) −3.36840e44 −2.18343
\(336\) 0 0
\(337\) −1.04185e44 −0.608529 −0.304264 0.952588i \(-0.598411\pi\)
−0.304264 + 0.952588i \(0.598411\pi\)
\(338\) 2.43920e44 1.35271
\(339\) 0 0
\(340\) 1.92169e43 0.0961173
\(341\) 5.82647e43 0.276824
\(342\) 0 0
\(343\) 9.28652e43 0.398296
\(344\) −6.77253e42 −0.0276044
\(345\) 0 0
\(346\) 2.19861e44 0.809685
\(347\) 3.07657e44 1.07721 0.538606 0.842557i \(-0.318951\pi\)
0.538606 + 0.842557i \(0.318951\pi\)
\(348\) 0 0
\(349\) −2.62767e44 −0.832005 −0.416003 0.909363i \(-0.636569\pi\)
−0.416003 + 0.909363i \(0.636569\pi\)
\(350\) −2.67131e43 −0.0804515
\(351\) 0 0
\(352\) 1.51816e44 0.413830
\(353\) −7.06671e44 −1.83299 −0.916494 0.400048i \(-0.868994\pi\)
−0.916494 + 0.400048i \(0.868994\pi\)
\(354\) 0 0
\(355\) −1.57026e44 −0.368957
\(356\) 1.02632e44 0.229566
\(357\) 0 0
\(358\) −1.03557e44 −0.210003
\(359\) 5.35788e44 1.03476 0.517380 0.855756i \(-0.326907\pi\)
0.517380 + 0.855756i \(0.326907\pi\)
\(360\) 0 0
\(361\) −5.55104e44 −0.972743
\(362\) −3.96204e44 −0.661483
\(363\) 0 0
\(364\) −4.39782e43 −0.0666750
\(365\) −1.45163e45 −2.09763
\(366\) 0 0
\(367\) −3.00056e44 −0.394045 −0.197022 0.980399i \(-0.563127\pi\)
−0.197022 + 0.980399i \(0.563127\pi\)
\(368\) −3.09210e44 −0.387183
\(369\) 0 0
\(370\) −1.36625e45 −1.55597
\(371\) −1.24773e44 −0.135543
\(372\) 0 0
\(373\) 1.08201e45 1.06986 0.534932 0.844895i \(-0.320337\pi\)
0.534932 + 0.844895i \(0.320337\pi\)
\(374\) −3.72466e44 −0.351426
\(375\) 0 0
\(376\) 1.34064e45 1.15219
\(377\) 1.59201e45 1.30609
\(378\) 0 0
\(379\) 1.21847e45 0.911229 0.455614 0.890177i \(-0.349420\pi\)
0.455614 + 0.890177i \(0.349420\pi\)
\(380\) −5.73965e43 −0.0409895
\(381\) 0 0
\(382\) −1.34484e45 −0.876119
\(383\) 9.53201e43 0.0593207 0.0296603 0.999560i \(-0.490557\pi\)
0.0296603 + 0.999560i \(0.490557\pi\)
\(384\) 0 0
\(385\) −4.38372e44 −0.249046
\(386\) 2.28757e45 1.24193
\(387\) 0 0
\(388\) 2.04016e44 0.101183
\(389\) −1.90493e45 −0.903152 −0.451576 0.892233i \(-0.649138\pi\)
−0.451576 + 0.892233i \(0.649138\pi\)
\(390\) 0 0
\(391\) −4.60678e44 −0.199666
\(392\) 2.48585e45 1.03031
\(393\) 0 0
\(394\) 1.90766e45 0.723296
\(395\) 4.80960e45 1.74446
\(396\) 0 0
\(397\) 3.96399e45 1.31613 0.658065 0.752961i \(-0.271375\pi\)
0.658065 + 0.752961i \(0.271375\pi\)
\(398\) −7.87060e44 −0.250066
\(399\) 0 0
\(400\) −1.14568e45 −0.333438
\(401\) 5.53039e44 0.154074 0.0770369 0.997028i \(-0.475454\pi\)
0.0770369 + 0.997028i \(0.475454\pi\)
\(402\) 0 0
\(403\) −1.68812e45 −0.431088
\(404\) 8.03562e44 0.196492
\(405\) 0 0
\(406\) 6.64646e44 0.149068
\(407\) −6.89560e45 −1.48139
\(408\) 0 0
\(409\) 3.35495e45 0.661498 0.330749 0.943719i \(-0.392699\pi\)
0.330749 + 0.943719i \(0.392699\pi\)
\(410\) −2.93089e45 −0.553710
\(411\) 0 0
\(412\) −1.80048e45 −0.312383
\(413\) 4.90568e44 0.0815781
\(414\) 0 0
\(415\) −1.06396e46 −1.62586
\(416\) −4.39862e45 −0.644441
\(417\) 0 0
\(418\) 1.11247e45 0.149867
\(419\) 7.69138e45 0.993708 0.496854 0.867834i \(-0.334488\pi\)
0.496854 + 0.867834i \(0.334488\pi\)
\(420\) 0 0
\(421\) −2.77229e45 −0.329535 −0.164767 0.986332i \(-0.552687\pi\)
−0.164767 + 0.986332i \(0.552687\pi\)
\(422\) −3.09338e45 −0.352749
\(423\) 0 0
\(424\) −6.82401e45 −0.716370
\(425\) −1.70691e45 −0.171950
\(426\) 0 0
\(427\) −9.57360e43 −0.00888357
\(428\) −2.58349e45 −0.230113
\(429\) 0 0
\(430\) 3.34904e44 0.0274930
\(431\) −6.18484e45 −0.487502 −0.243751 0.969838i \(-0.578378\pi\)
−0.243751 + 0.969838i \(0.578378\pi\)
\(432\) 0 0
\(433\) 2.21783e46 1.61209 0.806047 0.591852i \(-0.201603\pi\)
0.806047 + 0.591852i \(0.201603\pi\)
\(434\) −7.04770e44 −0.0492014
\(435\) 0 0
\(436\) 3.30353e44 0.0212796
\(437\) 1.37594e45 0.0851482
\(438\) 0 0
\(439\) 1.91766e45 0.109557 0.0547787 0.998499i \(-0.482555\pi\)
0.0547787 + 0.998499i \(0.482555\pi\)
\(440\) −2.39752e46 −1.31625
\(441\) 0 0
\(442\) 1.07916e46 0.547263
\(443\) 3.00864e46 1.46658 0.733290 0.679916i \(-0.237984\pi\)
0.733290 + 0.679916i \(0.237984\pi\)
\(444\) 0 0
\(445\) −2.96405e46 −1.33532
\(446\) −1.17574e46 −0.509275
\(447\) 0 0
\(448\) −5.64783e45 −0.226212
\(449\) −5.68788e45 −0.219099 −0.109549 0.993981i \(-0.534941\pi\)
−0.109549 + 0.993981i \(0.534941\pi\)
\(450\) 0 0
\(451\) −1.47925e46 −0.527171
\(452\) −1.54997e45 −0.0531376
\(453\) 0 0
\(454\) 1.71721e46 0.544937
\(455\) 1.27011e46 0.387830
\(456\) 0 0
\(457\) 4.67626e46 1.32241 0.661204 0.750206i \(-0.270046\pi\)
0.661204 + 0.750206i \(0.270046\pi\)
\(458\) 3.01918e46 0.821755
\(459\) 0 0
\(460\) −5.07736e45 −0.128049
\(461\) 3.72978e46 0.905560 0.452780 0.891622i \(-0.350432\pi\)
0.452780 + 0.891622i \(0.350432\pi\)
\(462\) 0 0
\(463\) −3.67649e46 −0.827497 −0.413748 0.910391i \(-0.635781\pi\)
−0.413748 + 0.910391i \(0.635781\pi\)
\(464\) 2.85056e46 0.617825
\(465\) 0 0
\(466\) −2.60620e46 −0.523907
\(467\) 2.37743e46 0.460323 0.230161 0.973152i \(-0.426075\pi\)
0.230161 + 0.973152i \(0.426075\pi\)
\(468\) 0 0
\(469\) 2.05639e46 0.369472
\(470\) −6.62951e46 −1.14754
\(471\) 0 0
\(472\) 2.68298e46 0.431155
\(473\) 1.69029e45 0.0261753
\(474\) 0 0
\(475\) 5.09814e45 0.0733287
\(476\) −1.17318e45 −0.0162646
\(477\) 0 0
\(478\) −8.27170e46 −1.06563
\(479\) 8.80315e46 1.09337 0.546684 0.837339i \(-0.315890\pi\)
0.546684 + 0.837339i \(0.315890\pi\)
\(480\) 0 0
\(481\) 1.99788e47 2.30691
\(482\) −5.50084e46 −0.612500
\(483\) 0 0
\(484\) −7.69900e44 −0.00797334
\(485\) −5.89206e46 −0.588555
\(486\) 0 0
\(487\) 7.13443e46 0.663136 0.331568 0.943431i \(-0.392422\pi\)
0.331568 + 0.943431i \(0.392422\pi\)
\(488\) −5.23593e45 −0.0469512
\(489\) 0 0
\(490\) −1.22926e47 −1.02615
\(491\) −8.49115e46 −0.683975 −0.341988 0.939704i \(-0.611100\pi\)
−0.341988 + 0.939704i \(0.611100\pi\)
\(492\) 0 0
\(493\) 4.24693e46 0.318606
\(494\) −3.22321e46 −0.233382
\(495\) 0 0
\(496\) −3.02265e46 −0.203919
\(497\) 9.58638e45 0.0624335
\(498\) 0 0
\(499\) −1.75876e47 −1.06770 −0.533848 0.845581i \(-0.679255\pi\)
−0.533848 + 0.845581i \(0.679255\pi\)
\(500\) 2.35430e46 0.138003
\(501\) 0 0
\(502\) −2.52221e47 −1.37869
\(503\) −1.24051e47 −0.654880 −0.327440 0.944872i \(-0.606186\pi\)
−0.327440 + 0.944872i \(0.606186\pi\)
\(504\) 0 0
\(505\) −2.32072e47 −1.14294
\(506\) 9.84108e46 0.468175
\(507\) 0 0
\(508\) −2.97832e46 −0.132238
\(509\) −3.71512e47 −1.59372 −0.796859 0.604166i \(-0.793507\pi\)
−0.796859 + 0.604166i \(0.793507\pi\)
\(510\) 0 0
\(511\) 8.86212e46 0.354954
\(512\) −2.87213e47 −1.11168
\(513\) 0 0
\(514\) 4.49893e47 1.62650
\(515\) 5.19987e47 1.81705
\(516\) 0 0
\(517\) −3.34597e47 −1.09254
\(518\) 8.34092e46 0.263295
\(519\) 0 0
\(520\) 6.94640e47 2.04975
\(521\) −7.89659e45 −0.0225309 −0.0112655 0.999937i \(-0.503586\pi\)
−0.0112655 + 0.999937i \(0.503586\pi\)
\(522\) 0 0
\(523\) 1.18122e47 0.315175 0.157588 0.987505i \(-0.449628\pi\)
0.157588 + 0.987505i \(0.449628\pi\)
\(524\) 1.70816e46 0.0440790
\(525\) 0 0
\(526\) −6.96851e45 −0.0168225
\(527\) −4.50331e46 −0.105159
\(528\) 0 0
\(529\) −3.35870e47 −0.734001
\(530\) 3.37450e47 0.713480
\(531\) 0 0
\(532\) 3.50403e45 0.00693609
\(533\) 4.28587e47 0.820943
\(534\) 0 0
\(535\) 7.46124e47 1.33850
\(536\) 1.12467e48 1.95272
\(537\) 0 0
\(538\) 2.15663e47 0.350821
\(539\) −6.20419e47 −0.976971
\(540\) 0 0
\(541\) −6.62658e47 −0.977997 −0.488998 0.872285i \(-0.662638\pi\)
−0.488998 + 0.872285i \(0.662638\pi\)
\(542\) −6.80114e46 −0.0971839
\(543\) 0 0
\(544\) −1.17340e47 −0.157204
\(545\) −9.54073e46 −0.123778
\(546\) 0 0
\(547\) −1.12424e47 −0.136798 −0.0683990 0.997658i \(-0.521789\pi\)
−0.0683990 + 0.997658i \(0.521789\pi\)
\(548\) −1.52422e47 −0.179633
\(549\) 0 0
\(550\) 3.64631e47 0.403188
\(551\) −1.26846e47 −0.135870
\(552\) 0 0
\(553\) −2.93624e47 −0.295190
\(554\) 1.21889e48 1.18725
\(555\) 0 0
\(556\) −1.86844e47 −0.170872
\(557\) −6.68090e47 −0.592064 −0.296032 0.955178i \(-0.595664\pi\)
−0.296032 + 0.955178i \(0.595664\pi\)
\(558\) 0 0
\(559\) −4.89734e46 −0.0407618
\(560\) 2.27418e47 0.183457
\(561\) 0 0
\(562\) −6.74014e47 −0.510838
\(563\) −2.47733e48 −1.82007 −0.910033 0.414536i \(-0.863944\pi\)
−0.910033 + 0.414536i \(0.863944\pi\)
\(564\) 0 0
\(565\) 4.47639e47 0.309087
\(566\) −1.75846e48 −1.17718
\(567\) 0 0
\(568\) 5.24292e47 0.329972
\(569\) −8.59661e47 −0.524641 −0.262321 0.964981i \(-0.584488\pi\)
−0.262321 + 0.964981i \(0.584488\pi\)
\(570\) 0 0
\(571\) −3.39185e48 −1.94673 −0.973364 0.229266i \(-0.926367\pi\)
−0.973364 + 0.229266i \(0.926367\pi\)
\(572\) 6.00298e47 0.334146
\(573\) 0 0
\(574\) 1.78930e47 0.0936968
\(575\) 4.50988e47 0.229075
\(576\) 0 0
\(577\) 2.12534e48 1.01590 0.507951 0.861386i \(-0.330403\pi\)
0.507951 + 0.861386i \(0.330403\pi\)
\(578\) −1.63292e48 −0.757232
\(579\) 0 0
\(580\) 4.68075e47 0.204327
\(581\) 6.49544e47 0.275123
\(582\) 0 0
\(583\) 1.70314e48 0.679283
\(584\) 4.84681e48 1.87599
\(585\) 0 0
\(586\) −3.14491e48 −1.14657
\(587\) −1.06269e48 −0.376044 −0.188022 0.982165i \(-0.560208\pi\)
−0.188022 + 0.982165i \(0.560208\pi\)
\(588\) 0 0
\(589\) 1.34504e47 0.0448453
\(590\) −1.32675e48 −0.429415
\(591\) 0 0
\(592\) 3.57728e48 1.09125
\(593\) 3.98260e48 1.17953 0.589766 0.807574i \(-0.299220\pi\)
0.589766 + 0.807574i \(0.299220\pi\)
\(594\) 0 0
\(595\) 3.38820e47 0.0946068
\(596\) −1.23794e48 −0.335651
\(597\) 0 0
\(598\) −2.85129e48 −0.729072
\(599\) −5.14535e48 −1.27775 −0.638874 0.769311i \(-0.720599\pi\)
−0.638874 + 0.769311i \(0.720599\pi\)
\(600\) 0 0
\(601\) −1.83878e48 −0.430751 −0.215375 0.976531i \(-0.569098\pi\)
−0.215375 + 0.976531i \(0.569098\pi\)
\(602\) −2.04458e46 −0.00465227
\(603\) 0 0
\(604\) 1.00054e48 0.214827
\(605\) 2.22350e47 0.0463787
\(606\) 0 0
\(607\) 2.06846e48 0.407236 0.203618 0.979050i \(-0.434730\pi\)
0.203618 + 0.979050i \(0.434730\pi\)
\(608\) 3.50467e47 0.0670401
\(609\) 0 0
\(610\) 2.58919e47 0.0467618
\(611\) 9.69439e48 1.70137
\(612\) 0 0
\(613\) 1.02050e48 0.169143 0.0845717 0.996417i \(-0.473048\pi\)
0.0845717 + 0.996417i \(0.473048\pi\)
\(614\) −2.30743e48 −0.371691
\(615\) 0 0
\(616\) 1.46367e48 0.222732
\(617\) 3.46313e48 0.512246 0.256123 0.966644i \(-0.417555\pi\)
0.256123 + 0.966644i \(0.417555\pi\)
\(618\) 0 0
\(619\) 6.56428e48 0.917488 0.458744 0.888568i \(-0.348299\pi\)
0.458744 + 0.888568i \(0.348299\pi\)
\(620\) −4.96332e47 −0.0674400
\(621\) 0 0
\(622\) −3.67165e48 −0.471552
\(623\) 1.80954e48 0.225958
\(624\) 0 0
\(625\) −1.05615e49 −1.24688
\(626\) 5.99629e48 0.688386
\(627\) 0 0
\(628\) −1.06427e48 −0.115546
\(629\) 5.32965e48 0.562746
\(630\) 0 0
\(631\) 5.18692e48 0.518079 0.259039 0.965867i \(-0.416594\pi\)
0.259039 + 0.965867i \(0.416594\pi\)
\(632\) −1.60587e49 −1.56013
\(633\) 0 0
\(634\) −8.74063e48 −0.803490
\(635\) 8.60152e48 0.769192
\(636\) 0 0
\(637\) 1.79756e49 1.52140
\(638\) −9.07235e48 −0.747064
\(639\) 0 0
\(640\) 9.01487e48 0.702765
\(641\) 6.15569e48 0.466940 0.233470 0.972364i \(-0.424992\pi\)
0.233470 + 0.972364i \(0.424992\pi\)
\(642\) 0 0
\(643\) −6.50853e48 −0.467510 −0.233755 0.972296i \(-0.575101\pi\)
−0.233755 + 0.972296i \(0.575101\pi\)
\(644\) 3.09971e47 0.0216680
\(645\) 0 0
\(646\) −8.59837e47 −0.0569308
\(647\) −2.43533e49 −1.56940 −0.784698 0.619878i \(-0.787182\pi\)
−0.784698 + 0.619878i \(0.787182\pi\)
\(648\) 0 0
\(649\) −6.69620e48 −0.408833
\(650\) −1.05646e49 −0.627869
\(651\) 0 0
\(652\) −6.11047e48 −0.344145
\(653\) −4.33343e48 −0.237602 −0.118801 0.992918i \(-0.537905\pi\)
−0.118801 + 0.992918i \(0.537905\pi\)
\(654\) 0 0
\(655\) −4.93325e48 −0.256396
\(656\) 7.67401e48 0.388334
\(657\) 0 0
\(658\) 4.04729e48 0.194183
\(659\) −2.26798e49 −1.05960 −0.529802 0.848121i \(-0.677734\pi\)
−0.529802 + 0.848121i \(0.677734\pi\)
\(660\) 0 0
\(661\) −2.84364e49 −1.25994 −0.629969 0.776620i \(-0.716932\pi\)
−0.629969 + 0.776620i \(0.716932\pi\)
\(662\) 1.03284e49 0.445676
\(663\) 0 0
\(664\) 3.55245e49 1.45407
\(665\) −1.01198e48 −0.0403453
\(666\) 0 0
\(667\) −1.12210e49 −0.424452
\(668\) −6.55455e48 −0.241521
\(669\) 0 0
\(670\) −5.56153e49 −1.94485
\(671\) 1.30679e48 0.0445205
\(672\) 0 0
\(673\) 1.14434e49 0.370077 0.185039 0.982731i \(-0.440759\pi\)
0.185039 + 0.982731i \(0.440759\pi\)
\(674\) −1.72019e49 −0.542035
\(675\) 0 0
\(676\) −1.04871e49 −0.313753
\(677\) 3.00594e49 0.876351 0.438175 0.898890i \(-0.355625\pi\)
0.438175 + 0.898890i \(0.355625\pi\)
\(678\) 0 0
\(679\) 3.59708e48 0.0995931
\(680\) 1.85305e49 0.500014
\(681\) 0 0
\(682\) 9.62004e48 0.246576
\(683\) 7.78736e49 1.94549 0.972743 0.231885i \(-0.0744894\pi\)
0.972743 + 0.231885i \(0.0744894\pi\)
\(684\) 0 0
\(685\) 4.40201e49 1.04488
\(686\) 1.53329e49 0.354774
\(687\) 0 0
\(688\) −8.76886e47 −0.0192817
\(689\) −4.93456e49 −1.05782
\(690\) 0 0
\(691\) 9.61614e49 1.95945 0.979726 0.200340i \(-0.0642046\pi\)
0.979726 + 0.200340i \(0.0642046\pi\)
\(692\) −9.45269e48 −0.187801
\(693\) 0 0
\(694\) 5.07971e49 0.959506
\(695\) 5.39613e49 0.993913
\(696\) 0 0
\(697\) 1.14332e49 0.200260
\(698\) −4.33853e49 −0.741093
\(699\) 0 0
\(700\) 1.14850e48 0.0186602
\(701\) −1.19741e49 −0.189748 −0.0948742 0.995489i \(-0.530245\pi\)
−0.0948742 + 0.995489i \(0.530245\pi\)
\(702\) 0 0
\(703\) −1.59185e49 −0.239984
\(704\) 7.70923e49 1.13368
\(705\) 0 0
\(706\) −1.16678e50 −1.63270
\(707\) 1.41679e49 0.193404
\(708\) 0 0
\(709\) −4.80032e49 −0.623680 −0.311840 0.950135i \(-0.600945\pi\)
−0.311840 + 0.950135i \(0.600945\pi\)
\(710\) −2.59264e49 −0.328641
\(711\) 0 0
\(712\) 9.89664e49 1.19423
\(713\) 1.18984e49 0.140094
\(714\) 0 0
\(715\) −1.73369e50 −1.94363
\(716\) 4.45232e48 0.0487089
\(717\) 0 0
\(718\) 8.84635e49 0.921693
\(719\) 1.45387e49 0.147832 0.0739162 0.997264i \(-0.476450\pi\)
0.0739162 + 0.997264i \(0.476450\pi\)
\(720\) 0 0
\(721\) −3.17450e49 −0.307474
\(722\) −9.16528e49 −0.866452
\(723\) 0 0
\(724\) 1.70344e49 0.153427
\(725\) −4.15759e49 −0.365533
\(726\) 0 0
\(727\) 1.25503e50 1.05148 0.525742 0.850644i \(-0.323788\pi\)
0.525742 + 0.850644i \(0.323788\pi\)
\(728\) −4.24075e49 −0.346851
\(729\) 0 0
\(730\) −2.39677e50 −1.86843
\(731\) −1.30644e48 −0.00994338
\(732\) 0 0
\(733\) −1.40440e50 −1.01899 −0.509495 0.860473i \(-0.670168\pi\)
−0.509495 + 0.860473i \(0.670168\pi\)
\(734\) −4.95419e49 −0.350988
\(735\) 0 0
\(736\) 3.10027e49 0.209430
\(737\) −2.80695e50 −1.85163
\(738\) 0 0
\(739\) 2.93572e49 0.184687 0.0923435 0.995727i \(-0.470564\pi\)
0.0923435 + 0.995727i \(0.470564\pi\)
\(740\) 5.87406e49 0.360897
\(741\) 0 0
\(742\) −2.06012e49 −0.120732
\(743\) −1.10795e50 −0.634185 −0.317093 0.948395i \(-0.602707\pi\)
−0.317093 + 0.948395i \(0.602707\pi\)
\(744\) 0 0
\(745\) 3.57521e50 1.95239
\(746\) 1.78650e50 0.952960
\(747\) 0 0
\(748\) 1.60138e49 0.0815111
\(749\) −4.55506e49 −0.226497
\(750\) 0 0
\(751\) −4.13601e50 −1.96283 −0.981416 0.191892i \(-0.938538\pi\)
−0.981416 + 0.191892i \(0.938538\pi\)
\(752\) 1.73582e50 0.804806
\(753\) 0 0
\(754\) 2.62856e50 1.16337
\(755\) −2.88961e50 −1.24959
\(756\) 0 0
\(757\) 3.15399e50 1.30221 0.651106 0.758987i \(-0.274305\pi\)
0.651106 + 0.758987i \(0.274305\pi\)
\(758\) 2.01180e50 0.811659
\(759\) 0 0
\(760\) −5.53465e49 −0.213232
\(761\) 3.65841e50 1.37740 0.688702 0.725045i \(-0.258181\pi\)
0.688702 + 0.725045i \(0.258181\pi\)
\(762\) 0 0
\(763\) 5.82458e48 0.0209452
\(764\) 5.78202e49 0.203210
\(765\) 0 0
\(766\) 1.57382e49 0.0528387
\(767\) 1.94011e50 0.636661
\(768\) 0 0
\(769\) −2.07264e50 −0.649849 −0.324925 0.945740i \(-0.605339\pi\)
−0.324925 + 0.945740i \(0.605339\pi\)
\(770\) −7.23792e49 −0.221833
\(771\) 0 0
\(772\) −9.83516e49 −0.288057
\(773\) −2.04692e50 −0.586084 −0.293042 0.956100i \(-0.594668\pi\)
−0.293042 + 0.956100i \(0.594668\pi\)
\(774\) 0 0
\(775\) 4.40859e49 0.120648
\(776\) 1.96729e50 0.526367
\(777\) 0 0
\(778\) −3.14521e50 −0.804465
\(779\) −3.41484e49 −0.0854013
\(780\) 0 0
\(781\) −1.30853e50 −0.312889
\(782\) −7.60622e49 −0.177849
\(783\) 0 0
\(784\) 3.21860e50 0.719673
\(785\) 3.07365e50 0.672101
\(786\) 0 0
\(787\) −2.74398e50 −0.573882 −0.286941 0.957948i \(-0.592638\pi\)
−0.286941 + 0.957948i \(0.592638\pi\)
\(788\) −8.20177e49 −0.167764
\(789\) 0 0
\(790\) 7.94109e50 1.55384
\(791\) −2.73282e49 −0.0523025
\(792\) 0 0
\(793\) −3.78619e49 −0.0693301
\(794\) 6.54491e50 1.17232
\(795\) 0 0
\(796\) 3.38388e49 0.0580012
\(797\) −3.73868e50 −0.626899 −0.313450 0.949605i \(-0.601485\pi\)
−0.313450 + 0.949605i \(0.601485\pi\)
\(798\) 0 0
\(799\) 2.58612e50 0.415030
\(800\) 1.14871e50 0.180359
\(801\) 0 0
\(802\) 9.13117e49 0.137238
\(803\) −1.20967e51 −1.77887
\(804\) 0 0
\(805\) −8.95210e49 −0.126037
\(806\) −2.78725e50 −0.383983
\(807\) 0 0
\(808\) 7.74862e50 1.02218
\(809\) 3.87407e50 0.500114 0.250057 0.968231i \(-0.419551\pi\)
0.250057 + 0.968231i \(0.419551\pi\)
\(810\) 0 0
\(811\) −9.21816e50 −1.13967 −0.569835 0.821759i \(-0.692993\pi\)
−0.569835 + 0.821759i \(0.692993\pi\)
\(812\) −2.85758e49 −0.0345754
\(813\) 0 0
\(814\) −1.13853e51 −1.31952
\(815\) 1.76473e51 2.00180
\(816\) 0 0
\(817\) 3.90203e48 0.00424038
\(818\) 5.53933e50 0.589216
\(819\) 0 0
\(820\) 1.26011e50 0.128430
\(821\) −1.64737e51 −1.64357 −0.821783 0.569801i \(-0.807020\pi\)
−0.821783 + 0.569801i \(0.807020\pi\)
\(822\) 0 0
\(823\) −6.02163e50 −0.575727 −0.287864 0.957671i \(-0.592945\pi\)
−0.287864 + 0.957671i \(0.592945\pi\)
\(824\) −1.73618e51 −1.62505
\(825\) 0 0
\(826\) 8.09973e49 0.0726641
\(827\) 4.30476e50 0.378097 0.189048 0.981968i \(-0.439460\pi\)
0.189048 + 0.981968i \(0.439460\pi\)
\(828\) 0 0
\(829\) 8.88751e50 0.748300 0.374150 0.927368i \(-0.377935\pi\)
0.374150 + 0.927368i \(0.377935\pi\)
\(830\) −1.75670e51 −1.44821
\(831\) 0 0
\(832\) −2.23362e51 −1.76543
\(833\) 4.79525e50 0.371128
\(834\) 0 0
\(835\) 1.89298e51 1.40486
\(836\) −4.78297e49 −0.0347606
\(837\) 0 0
\(838\) 1.26992e51 0.885126
\(839\) −3.76912e50 −0.257280 −0.128640 0.991691i \(-0.541061\pi\)
−0.128640 + 0.991691i \(0.541061\pi\)
\(840\) 0 0
\(841\) −4.92875e50 −0.322706
\(842\) −4.57730e50 −0.293527
\(843\) 0 0
\(844\) 1.32997e50 0.0818179
\(845\) 3.02872e51 1.82502
\(846\) 0 0
\(847\) −1.35744e49 −0.00784804
\(848\) −8.83551e50 −0.500385
\(849\) 0 0
\(850\) −2.81826e50 −0.153161
\(851\) −1.40817e51 −0.749699
\(852\) 0 0
\(853\) −3.00248e51 −1.53417 −0.767083 0.641547i \(-0.778293\pi\)
−0.767083 + 0.641547i \(0.778293\pi\)
\(854\) −1.58069e49 −0.00791286
\(855\) 0 0
\(856\) −2.49122e51 −1.19708
\(857\) 2.46088e50 0.115858 0.0579291 0.998321i \(-0.481550\pi\)
0.0579291 + 0.998321i \(0.481550\pi\)
\(858\) 0 0
\(859\) 2.64506e51 1.19551 0.597757 0.801677i \(-0.296059\pi\)
0.597757 + 0.801677i \(0.296059\pi\)
\(860\) −1.43989e49 −0.00637684
\(861\) 0 0
\(862\) −1.02117e51 −0.434233
\(863\) 9.93861e50 0.414130 0.207065 0.978327i \(-0.433609\pi\)
0.207065 + 0.978327i \(0.433609\pi\)
\(864\) 0 0
\(865\) 2.72998e51 1.09239
\(866\) 3.66184e51 1.43594
\(867\) 0 0
\(868\) 3.03009e49 0.0114120
\(869\) 4.00794e51 1.47936
\(870\) 0 0
\(871\) 8.13268e51 2.88347
\(872\) 3.18554e50 0.110699
\(873\) 0 0
\(874\) 2.27181e50 0.0758441
\(875\) 4.15096e50 0.135834
\(876\) 0 0
\(877\) −2.79832e50 −0.0879842 −0.0439921 0.999032i \(-0.514008\pi\)
−0.0439921 + 0.999032i \(0.514008\pi\)
\(878\) 3.16623e50 0.0975861
\(879\) 0 0
\(880\) −3.10423e51 −0.919404
\(881\) 5.95050e50 0.172773 0.0863863 0.996262i \(-0.472468\pi\)
0.0863863 + 0.996262i \(0.472468\pi\)
\(882\) 0 0
\(883\) −3.85350e51 −1.07533 −0.537667 0.843157i \(-0.680694\pi\)
−0.537667 + 0.843157i \(0.680694\pi\)
\(884\) −4.63973e50 −0.126934
\(885\) 0 0
\(886\) 4.96755e51 1.30633
\(887\) 4.56338e50 0.117658 0.0588292 0.998268i \(-0.481263\pi\)
0.0588292 + 0.998268i \(0.481263\pi\)
\(888\) 0 0
\(889\) −5.25119e50 −0.130160
\(890\) −4.89393e51 −1.18941
\(891\) 0 0
\(892\) 5.05498e50 0.118123
\(893\) −7.72416e50 −0.176991
\(894\) 0 0
\(895\) −1.28585e51 −0.283326
\(896\) −5.50354e50 −0.118919
\(897\) 0 0
\(898\) −9.39122e50 −0.195158
\(899\) −1.09689e51 −0.223548
\(900\) 0 0
\(901\) −1.31637e51 −0.258044
\(902\) −2.44237e51 −0.469567
\(903\) 0 0
\(904\) −1.49462e51 −0.276428
\(905\) −4.91960e51 −0.892441
\(906\) 0 0
\(907\) −4.12286e51 −0.719566 −0.359783 0.933036i \(-0.617149\pi\)
−0.359783 + 0.933036i \(0.617149\pi\)
\(908\) −7.38298e50 −0.126395
\(909\) 0 0
\(910\) 2.09707e51 0.345452
\(911\) −6.95561e51 −1.12399 −0.561996 0.827140i \(-0.689966\pi\)
−0.561996 + 0.827140i \(0.689966\pi\)
\(912\) 0 0
\(913\) −8.86621e51 −1.37879
\(914\) 7.72094e51 1.17791
\(915\) 0 0
\(916\) −1.29806e51 −0.190601
\(917\) 3.01173e50 0.0433863
\(918\) 0 0
\(919\) 1.23335e51 0.171027 0.0855136 0.996337i \(-0.472747\pi\)
0.0855136 + 0.996337i \(0.472747\pi\)
\(920\) −4.89602e51 −0.666126
\(921\) 0 0
\(922\) 6.15821e51 0.806610
\(923\) 3.79125e51 0.487251
\(924\) 0 0
\(925\) −5.21754e51 −0.645632
\(926\) −6.07023e51 −0.737076
\(927\) 0 0
\(928\) −2.85810e51 −0.334185
\(929\) 9.97208e51 1.14422 0.572112 0.820176i \(-0.306124\pi\)
0.572112 + 0.820176i \(0.306124\pi\)
\(930\) 0 0
\(931\) −1.43224e51 −0.158269
\(932\) 1.12051e51 0.121517
\(933\) 0 0
\(934\) 3.92536e51 0.410024
\(935\) −4.62486e51 −0.474128
\(936\) 0 0
\(937\) 6.01334e51 0.593844 0.296922 0.954902i \(-0.404040\pi\)
0.296922 + 0.954902i \(0.404040\pi\)
\(938\) 3.39529e51 0.329100
\(939\) 0 0
\(940\) 2.85029e51 0.266165
\(941\) −1.38035e52 −1.26523 −0.632617 0.774465i \(-0.718019\pi\)
−0.632617 + 0.774465i \(0.718019\pi\)
\(942\) 0 0
\(943\) −3.02081e51 −0.266789
\(944\) 3.47384e51 0.301162
\(945\) 0 0
\(946\) 2.79083e50 0.0233151
\(947\) 9.01648e51 0.739456 0.369728 0.929140i \(-0.379451\pi\)
0.369728 + 0.929140i \(0.379451\pi\)
\(948\) 0 0
\(949\) 3.50482e52 2.77017
\(950\) 8.41750e50 0.0653161
\(951\) 0 0
\(952\) −1.13128e51 −0.0846105
\(953\) −4.32450e51 −0.317549 −0.158774 0.987315i \(-0.550754\pi\)
−0.158774 + 0.987315i \(0.550754\pi\)
\(954\) 0 0
\(955\) −1.66987e52 −1.18202
\(956\) 3.55633e51 0.247166
\(957\) 0 0
\(958\) 1.45348e52 0.973897
\(959\) −2.68741e51 −0.176810
\(960\) 0 0
\(961\) −1.46006e52 −0.926216
\(962\) 3.29869e52 2.05484
\(963\) 0 0
\(964\) 2.36503e51 0.142066
\(965\) 2.84044e52 1.67555
\(966\) 0 0
\(967\) −1.91085e51 −0.108709 −0.0543543 0.998522i \(-0.517310\pi\)
−0.0543543 + 0.998522i \(0.517310\pi\)
\(968\) −7.42403e50 −0.0414783
\(969\) 0 0
\(970\) −9.72834e51 −0.524244
\(971\) −2.54698e52 −1.34800 −0.674000 0.738732i \(-0.735425\pi\)
−0.674000 + 0.738732i \(0.735425\pi\)
\(972\) 0 0
\(973\) −3.29432e51 −0.168186
\(974\) 1.17796e52 0.590675
\(975\) 0 0
\(976\) −6.77932e50 −0.0327955
\(977\) −2.34465e52 −1.11410 −0.557048 0.830480i \(-0.688066\pi\)
−0.557048 + 0.830480i \(0.688066\pi\)
\(978\) 0 0
\(979\) −2.47001e52 −1.13240
\(980\) 5.28508e51 0.238010
\(981\) 0 0
\(982\) −1.40197e52 −0.609238
\(983\) −1.06892e52 −0.456308 −0.228154 0.973625i \(-0.573269\pi\)
−0.228154 + 0.973625i \(0.573269\pi\)
\(984\) 0 0
\(985\) 2.36871e52 0.975837
\(986\) 7.01207e51 0.283792
\(987\) 0 0
\(988\) 1.38579e51 0.0541314
\(989\) 3.45178e50 0.0132467
\(990\) 0 0
\(991\) 1.97227e52 0.730597 0.365298 0.930891i \(-0.380967\pi\)
0.365298 + 0.930891i \(0.380967\pi\)
\(992\) 3.03064e51 0.110301
\(993\) 0 0
\(994\) 1.58280e51 0.0556114
\(995\) −9.77280e51 −0.337377
\(996\) 0 0
\(997\) −1.44359e52 −0.481149 −0.240575 0.970631i \(-0.577336\pi\)
−0.240575 + 0.970631i \(0.577336\pi\)
\(998\) −2.90388e52 −0.951029
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.36.a.b.1.3 3
3.2 odd 2 1.36.a.a.1.1 3
12.11 even 2 16.36.a.d.1.3 3
15.2 even 4 25.36.b.a.24.2 6
15.8 even 4 25.36.b.a.24.5 6
15.14 odd 2 25.36.a.a.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.36.a.a.1.1 3 3.2 odd 2
9.36.a.b.1.3 3 1.1 even 1 trivial
16.36.a.d.1.3 3 12.11 even 2
25.36.a.a.1.3 3 15.14 odd 2
25.36.b.a.24.2 6 15.2 even 4
25.36.b.a.24.5 6 15.8 even 4