Properties

Label 29.18.a.b.1.8
Level $29$
Weight $18$
Character 29.1
Self dual yes
Analytic conductor $53.134$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.1344053299\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-237.844 q^{2} -14184.6 q^{3} -74502.2 q^{4} +1.22166e6 q^{5} +3.37373e6 q^{6} +1.05564e7 q^{7} +4.88946e7 q^{8} +7.20633e7 q^{9} +O(q^{10})\) \(q-237.844 q^{2} -14184.6 q^{3} -74502.2 q^{4} +1.22166e6 q^{5} +3.37373e6 q^{6} +1.05564e7 q^{7} +4.88946e7 q^{8} +7.20633e7 q^{9} -2.90566e8 q^{10} +9.82200e8 q^{11} +1.05678e9 q^{12} +5.20942e9 q^{13} -2.51077e9 q^{14} -1.73289e10 q^{15} -1.86415e9 q^{16} -2.48405e10 q^{17} -1.71398e10 q^{18} +7.35139e10 q^{19} -9.10167e10 q^{20} -1.49738e11 q^{21} -2.33611e11 q^{22} -4.63391e11 q^{23} -6.93552e11 q^{24} +7.29526e11 q^{25} -1.23903e12 q^{26} +8.09613e11 q^{27} -7.86473e11 q^{28} +5.00246e11 q^{29} +4.12157e12 q^{30} +4.00808e12 q^{31} -5.96534e12 q^{32} -1.39321e13 q^{33} +5.90816e12 q^{34} +1.28964e13 q^{35} -5.36887e12 q^{36} +9.74612e12 q^{37} -1.74849e13 q^{38} -7.38937e13 q^{39} +5.97328e13 q^{40} +5.79656e13 q^{41} +3.56144e13 q^{42} -8.74679e13 q^{43} -7.31760e13 q^{44} +8.80372e13 q^{45} +1.10215e14 q^{46} +8.94917e13 q^{47} +2.64423e13 q^{48} -1.21193e14 q^{49} -1.73513e14 q^{50} +3.52353e14 q^{51} -3.88113e14 q^{52} +6.03303e14 q^{53} -1.92562e14 q^{54} +1.19992e15 q^{55} +5.16150e14 q^{56} -1.04277e15 q^{57} -1.18981e14 q^{58} +9.08899e14 q^{59} +1.29104e15 q^{60} -2.23542e15 q^{61} -9.53298e14 q^{62} +7.60728e14 q^{63} +1.66316e15 q^{64} +6.36417e15 q^{65} +3.31368e15 q^{66} +1.04268e15 q^{67} +1.85067e15 q^{68} +6.57303e15 q^{69} -3.06732e15 q^{70} +2.03035e15 q^{71} +3.52351e15 q^{72} -3.65531e15 q^{73} -2.31806e15 q^{74} -1.03480e16 q^{75} -5.47695e15 q^{76} +1.03685e16 q^{77} +1.75752e16 q^{78} +6.95275e15 q^{79} -2.27737e15 q^{80} -2.07903e16 q^{81} -1.37868e16 q^{82} -1.05156e16 q^{83} +1.11558e16 q^{84} -3.03467e16 q^{85} +2.08037e16 q^{86} -7.09581e15 q^{87} +4.80243e16 q^{88} -3.63273e16 q^{89} -2.09391e16 q^{90} +5.49926e16 q^{91} +3.45236e16 q^{92} -5.68531e16 q^{93} -2.12851e16 q^{94} +8.98094e16 q^{95} +8.46160e16 q^{96} +1.37559e17 q^{97} +2.88251e16 q^{98} +7.07806e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 256q^{2} + 23966q^{3} + 1452522q^{4} + 998272q^{5} + 3411526q^{6} + 2193368q^{7} - 138137226q^{8} + 1264832799q^{9} + O(q^{10}) \) \( 21q + 256q^{2} + 23966q^{3} + 1452522q^{4} + 998272q^{5} + 3411526q^{6} + 2193368q^{7} - 138137226q^{8} + 1264832799q^{9} - 224469478q^{10} + 1203139534q^{11} - 5164251122q^{12} + 3854339312q^{13} + 25262272904q^{14} + 28324474306q^{15} + 196520815922q^{16} + 76444714794q^{17} + 75758949126q^{18} + 246497292428q^{19} - 46900976670q^{20} + 360937126704q^{21} - 275001533522q^{22} + 213498528140q^{23} - 451123453870q^{24} + 3898884886997q^{25} - 3609347694206q^{26} - 2718903745978q^{27} - 5946174617200q^{28} + 10505174672181q^{29} - 20237658929454q^{30} + 16670029895798q^{31} - 42141001912046q^{32} - 7157109761394q^{33} + 12785761151136q^{34} + 46677934312888q^{35} + 132137824374868q^{36} + 53445659988410q^{37} + 76581637956388q^{38} + 79233849032530q^{39} + 193617444734146q^{40} - 20814769309298q^{41} + 76690667258352q^{42} + 185498647364454q^{43} + 315429066899678q^{44} - 486270821438526q^{45} + 261474367677132q^{46} + 389503471719450q^{47} - 101509672247630q^{48} + 730079062141437q^{49} + 1482269666368354q^{50} + 718238208473988q^{51} + 1966802817157170q^{52} + 747441265526156q^{53} + 5692893333117030q^{54} + 1639109418219546q^{55} + 5657219329125240q^{56} + 4694352396864932q^{57} + 128063081718016q^{58} + 5280258638332960q^{59} + 15251367906033378q^{60} + 5813675353074254q^{61} + 6242066590947250q^{62} + 10947760075450368q^{63} + 24583792057508902q^{64} + 19190799243789974q^{65} + 41877805444482390q^{66} + 13420580230958268q^{67} + 24771837384165388q^{68} + 30973047049935252q^{69} + 8505088080182440q^{70} + 4824462822979508q^{71} + 1180071997284592q^{72} + 11228916281304662q^{73} - 89132715356772q^{74} + 59161419576630296q^{75} + 57466858643173460q^{76} + 58741564492720064q^{77} + 142050530910210210q^{78} + 71718598015696758q^{79} + 48350023652407550q^{80} + 75805931446703569q^{81} + 188661890754420812q^{82} + 50769377111735608q^{83} + 198832046985593048q^{84} + 53422044849490784q^{85} + 35014892323844118q^{86} + 11988905533023326q^{87} + 37459283979085258q^{88} - 70981414576978018q^{89} + 57211029866143724q^{90} + 112933943315157320q^{91} - 103019729095759724q^{92} - 350358290646906646q^{93} - 150286322409612578q^{94} - 102561321856584476q^{95} - 213770098354021866q^{96} - 130930167251505210q^{97} - 537387515497557296q^{98} - 95267700931431064q^{99} + 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −237.844 −0.656958 −0.328479 0.944511i \(-0.606536\pi\)
−0.328479 + 0.944511i \(0.606536\pi\)
\(3\) −14184.6 −1.24821 −0.624104 0.781341i \(-0.714536\pi\)
−0.624104 + 0.781341i \(0.714536\pi\)
\(4\) −74502.2 −0.568406
\(5\) 1.22166e6 1.39864 0.699322 0.714807i \(-0.253485\pi\)
0.699322 + 0.714807i \(0.253485\pi\)
\(6\) 3.37373e6 0.820020
\(7\) 1.05564e7 0.692120 0.346060 0.938212i \(-0.387519\pi\)
0.346060 + 0.938212i \(0.387519\pi\)
\(8\) 4.88946e7 1.03038
\(9\) 7.20633e7 0.558024
\(10\) −2.90566e8 −0.918850
\(11\) 9.82200e8 1.38154 0.690768 0.723076i \(-0.257273\pi\)
0.690768 + 0.723076i \(0.257273\pi\)
\(12\) 1.05678e9 0.709490
\(13\) 5.20942e9 1.77121 0.885607 0.464435i \(-0.153743\pi\)
0.885607 + 0.464435i \(0.153743\pi\)
\(14\) −2.51077e9 −0.454694
\(15\) −1.73289e10 −1.74580
\(16\) −1.86415e9 −0.108508
\(17\) −2.48405e10 −0.863663 −0.431831 0.901954i \(-0.642132\pi\)
−0.431831 + 0.901954i \(0.642132\pi\)
\(18\) −1.71398e10 −0.366598
\(19\) 7.35139e10 0.993034 0.496517 0.868027i \(-0.334612\pi\)
0.496517 + 0.868027i \(0.334612\pi\)
\(20\) −9.10167e10 −0.794998
\(21\) −1.49738e11 −0.863910
\(22\) −2.33611e11 −0.907611
\(23\) −4.63391e11 −1.23385 −0.616923 0.787023i \(-0.711621\pi\)
−0.616923 + 0.787023i \(0.711621\pi\)
\(24\) −6.93552e11 −1.28613
\(25\) 7.29526e11 0.956204
\(26\) −1.23903e12 −1.16361
\(27\) 8.09613e11 0.551678
\(28\) −7.86473e11 −0.393406
\(29\) 5.00246e11 0.185695
\(30\) 4.12157e12 1.14692
\(31\) 4.00808e12 0.844038 0.422019 0.906587i \(-0.361322\pi\)
0.422019 + 0.906587i \(0.361322\pi\)
\(32\) −5.96534e12 −0.959092
\(33\) −1.39321e13 −1.72444
\(34\) 5.90816e12 0.567390
\(35\) 1.28964e13 0.968030
\(36\) −5.36887e12 −0.317184
\(37\) 9.74612e12 0.456160 0.228080 0.973642i \(-0.426755\pi\)
0.228080 + 0.973642i \(0.426755\pi\)
\(38\) −1.74849e13 −0.652381
\(39\) −7.38937e13 −2.21084
\(40\) 5.97328e13 1.44113
\(41\) 5.79656e13 1.13372 0.566862 0.823813i \(-0.308157\pi\)
0.566862 + 0.823813i \(0.308157\pi\)
\(42\) 3.56144e13 0.567553
\(43\) −8.74679e13 −1.14121 −0.570607 0.821223i \(-0.693292\pi\)
−0.570607 + 0.821223i \(0.693292\pi\)
\(44\) −7.31760e13 −0.785274
\(45\) 8.80372e13 0.780477
\(46\) 1.10215e14 0.810585
\(47\) 8.94917e13 0.548215 0.274107 0.961699i \(-0.411618\pi\)
0.274107 + 0.961699i \(0.411618\pi\)
\(48\) 2.64423e13 0.135440
\(49\) −1.21193e14 −0.520970
\(50\) −1.73513e14 −0.628186
\(51\) 3.52353e14 1.07803
\(52\) −3.88113e14 −1.00677
\(53\) 6.03303e14 1.33104 0.665519 0.746381i \(-0.268210\pi\)
0.665519 + 0.746381i \(0.268210\pi\)
\(54\) −1.92562e14 −0.362429
\(55\) 1.19992e15 1.93228
\(56\) 5.16150e14 0.713145
\(57\) −1.04277e15 −1.23951
\(58\) −1.18981e14 −0.121994
\(59\) 9.08899e14 0.805886 0.402943 0.915225i \(-0.367987\pi\)
0.402943 + 0.915225i \(0.367987\pi\)
\(60\) 1.29104e15 0.992323
\(61\) −2.23542e15 −1.49298 −0.746492 0.665395i \(-0.768263\pi\)
−0.746492 + 0.665395i \(0.768263\pi\)
\(62\) −9.53298e14 −0.554497
\(63\) 7.60728e14 0.386220
\(64\) 1.66316e15 0.738591
\(65\) 6.36417e15 2.47730
\(66\) 3.31368e15 1.13289
\(67\) 1.04268e15 0.313702 0.156851 0.987622i \(-0.449866\pi\)
0.156851 + 0.987622i \(0.449866\pi\)
\(68\) 1.85067e15 0.490911
\(69\) 6.57303e15 1.54010
\(70\) −3.06732e15 −0.635955
\(71\) 2.03035e15 0.373143 0.186571 0.982441i \(-0.440262\pi\)
0.186571 + 0.982441i \(0.440262\pi\)
\(72\) 3.52351e15 0.574975
\(73\) −3.65531e15 −0.530493 −0.265246 0.964181i \(-0.585453\pi\)
−0.265246 + 0.964181i \(0.585453\pi\)
\(74\) −2.31806e15 −0.299678
\(75\) −1.03480e16 −1.19354
\(76\) −5.47695e15 −0.564447
\(77\) 1.03685e16 0.956189
\(78\) 1.75752e16 1.45243
\(79\) 6.95275e15 0.515616 0.257808 0.966196i \(-0.417000\pi\)
0.257808 + 0.966196i \(0.417000\pi\)
\(80\) −2.27737e15 −0.151764
\(81\) −2.07903e16 −1.24663
\(82\) −1.37868e16 −0.744809
\(83\) −1.05156e16 −0.512470 −0.256235 0.966615i \(-0.582482\pi\)
−0.256235 + 0.966615i \(0.582482\pi\)
\(84\) 1.11558e16 0.491052
\(85\) −3.03467e16 −1.20796
\(86\) 2.08037e16 0.749729
\(87\) −7.09581e15 −0.231786
\(88\) 4.80243e16 1.42350
\(89\) −3.63273e16 −0.978179 −0.489089 0.872234i \(-0.662671\pi\)
−0.489089 + 0.872234i \(0.662671\pi\)
\(90\) −2.09391e16 −0.512740
\(91\) 5.49926e16 1.22589
\(92\) 3.45236e16 0.701326
\(93\) −5.68531e16 −1.05354
\(94\) −2.12851e16 −0.360154
\(95\) 8.98094e16 1.38890
\(96\) 8.46160e16 1.19715
\(97\) 1.37559e17 1.78209 0.891043 0.453919i \(-0.149975\pi\)
0.891043 + 0.453919i \(0.149975\pi\)
\(98\) 2.88251e16 0.342255
\(99\) 7.07806e16 0.770930
\(100\) −5.43512e16 −0.543512
\(101\) −2.04386e17 −1.87810 −0.939050 0.343781i \(-0.888292\pi\)
−0.939050 + 0.343781i \(0.888292\pi\)
\(102\) −8.38050e16 −0.708221
\(103\) −1.18180e16 −0.0919234 −0.0459617 0.998943i \(-0.514635\pi\)
−0.0459617 + 0.998943i \(0.514635\pi\)
\(104\) 2.54713e17 1.82502
\(105\) −1.82930e17 −1.20830
\(106\) −1.43492e17 −0.874436
\(107\) −2.47034e17 −1.38994 −0.694969 0.719040i \(-0.744582\pi\)
−0.694969 + 0.719040i \(0.744582\pi\)
\(108\) −6.03179e16 −0.313577
\(109\) −5.51412e15 −0.0265064 −0.0132532 0.999912i \(-0.504219\pi\)
−0.0132532 + 0.999912i \(0.504219\pi\)
\(110\) −2.85394e17 −1.26942
\(111\) −1.38245e17 −0.569383
\(112\) −1.96787e16 −0.0751005
\(113\) 2.51475e16 0.0889872 0.0444936 0.999010i \(-0.485833\pi\)
0.0444936 + 0.999010i \(0.485833\pi\)
\(114\) 2.48016e17 0.814308
\(115\) −5.66109e17 −1.72571
\(116\) −3.72694e16 −0.105550
\(117\) 3.75408e17 0.988380
\(118\) −2.16176e17 −0.529433
\(119\) −2.62225e17 −0.597758
\(120\) −8.47288e17 −1.79883
\(121\) 4.59270e17 0.908642
\(122\) 5.31681e17 0.980827
\(123\) −8.22220e17 −1.41512
\(124\) −2.98611e17 −0.479757
\(125\) −4.08205e16 −0.0612552
\(126\) −1.80935e17 −0.253730
\(127\) −4.74031e17 −0.621549 −0.310774 0.950484i \(-0.600588\pi\)
−0.310774 + 0.950484i \(0.600588\pi\)
\(128\) 3.86316e17 0.473869
\(129\) 1.24070e18 1.42447
\(130\) −1.51368e18 −1.62748
\(131\) 1.16841e18 1.17703 0.588517 0.808485i \(-0.299712\pi\)
0.588517 + 0.808485i \(0.299712\pi\)
\(132\) 1.03797e18 0.980185
\(133\) 7.76041e17 0.687299
\(134\) −2.47996e17 −0.206089
\(135\) 9.89076e17 0.771601
\(136\) −1.21457e18 −0.889898
\(137\) 2.80722e18 1.93264 0.966322 0.257335i \(-0.0828443\pi\)
0.966322 + 0.257335i \(0.0828443\pi\)
\(138\) −1.56336e18 −1.01178
\(139\) −2.89479e17 −0.176194 −0.0880971 0.996112i \(-0.528079\pi\)
−0.0880971 + 0.996112i \(0.528079\pi\)
\(140\) −9.60806e17 −0.550234
\(141\) −1.26941e18 −0.684286
\(142\) −4.82907e17 −0.245139
\(143\) 5.11670e18 2.44700
\(144\) −1.34337e17 −0.0605500
\(145\) 6.11133e17 0.259722
\(146\) 8.69394e17 0.348512
\(147\) 1.71908e18 0.650278
\(148\) −7.26107e17 −0.259284
\(149\) −9.75114e17 −0.328830 −0.164415 0.986391i \(-0.552574\pi\)
−0.164415 + 0.986391i \(0.552574\pi\)
\(150\) 2.46122e18 0.784107
\(151\) −1.42811e18 −0.429988 −0.214994 0.976615i \(-0.568973\pi\)
−0.214994 + 0.976615i \(0.568973\pi\)
\(152\) 3.59444e18 1.02320
\(153\) −1.79009e18 −0.481945
\(154\) −2.46608e18 −0.628176
\(155\) 4.89653e18 1.18051
\(156\) 5.50524e18 1.25666
\(157\) 8.21807e18 1.77673 0.888367 0.459134i \(-0.151840\pi\)
0.888367 + 0.459134i \(0.151840\pi\)
\(158\) −1.65367e18 −0.338738
\(159\) −8.55762e18 −1.66141
\(160\) −7.28764e18 −1.34143
\(161\) −4.89173e18 −0.853970
\(162\) 4.94486e18 0.818986
\(163\) 7.38540e18 1.16086 0.580429 0.814311i \(-0.302885\pi\)
0.580429 + 0.814311i \(0.302885\pi\)
\(164\) −4.31856e18 −0.644416
\(165\) −1.70204e19 −2.41188
\(166\) 2.50106e18 0.336671
\(167\) −3.96033e17 −0.0506572 −0.0253286 0.999679i \(-0.508063\pi\)
−0.0253286 + 0.999679i \(0.508063\pi\)
\(168\) −7.32139e18 −0.890153
\(169\) 1.84877e19 2.13720
\(170\) 7.21779e18 0.793576
\(171\) 5.29766e18 0.554137
\(172\) 6.51655e18 0.648673
\(173\) −1.63389e19 −1.54821 −0.774106 0.633056i \(-0.781800\pi\)
−0.774106 + 0.633056i \(0.781800\pi\)
\(174\) 1.68770e18 0.152274
\(175\) 7.70115e18 0.661808
\(176\) −1.83097e18 −0.149908
\(177\) −1.28924e19 −1.00591
\(178\) 8.64024e18 0.642622
\(179\) 2.73924e19 1.94258 0.971289 0.237901i \(-0.0764595\pi\)
0.971289 + 0.237901i \(0.0764595\pi\)
\(180\) −6.55896e18 −0.443628
\(181\) −8.93353e18 −0.576441 −0.288221 0.957564i \(-0.593064\pi\)
−0.288221 + 0.957564i \(0.593064\pi\)
\(182\) −1.30797e19 −0.805360
\(183\) 3.17085e19 1.86355
\(184\) −2.26573e19 −1.27133
\(185\) 1.19065e19 0.638005
\(186\) 1.35222e19 0.692128
\(187\) −2.43983e19 −1.19318
\(188\) −6.66732e18 −0.311609
\(189\) 8.54659e18 0.381828
\(190\) −2.13606e19 −0.912449
\(191\) 7.52613e18 0.307460 0.153730 0.988113i \(-0.450871\pi\)
0.153730 + 0.988113i \(0.450871\pi\)
\(192\) −2.35913e19 −0.921915
\(193\) −4.31029e19 −1.61165 −0.805823 0.592156i \(-0.798277\pi\)
−0.805823 + 0.592156i \(0.798277\pi\)
\(194\) −3.27175e19 −1.17076
\(195\) −9.02733e19 −3.09218
\(196\) 9.02917e18 0.296122
\(197\) −1.81537e19 −0.570167 −0.285083 0.958503i \(-0.592021\pi\)
−0.285083 + 0.958503i \(0.592021\pi\)
\(198\) −1.68348e19 −0.506469
\(199\) 2.25070e19 0.648733 0.324367 0.945931i \(-0.394849\pi\)
0.324367 + 0.945931i \(0.394849\pi\)
\(200\) 3.56699e19 0.985250
\(201\) −1.47901e19 −0.391565
\(202\) 4.86119e19 1.23383
\(203\) 5.28079e18 0.128524
\(204\) −2.62510e19 −0.612760
\(205\) 7.08145e19 1.58568
\(206\) 2.81083e18 0.0603898
\(207\) −3.33935e19 −0.688516
\(208\) −9.71115e18 −0.192191
\(209\) 7.22054e19 1.37191
\(210\) 4.35088e19 0.793804
\(211\) 7.39746e19 1.29623 0.648115 0.761543i \(-0.275558\pi\)
0.648115 + 0.761543i \(0.275558\pi\)
\(212\) −4.49474e19 −0.756570
\(213\) −2.87997e19 −0.465760
\(214\) 5.87557e19 0.913131
\(215\) −1.06856e20 −1.59615
\(216\) 3.95857e19 0.568436
\(217\) 4.23108e19 0.584176
\(218\) 1.31150e18 0.0174136
\(219\) 5.18492e19 0.662166
\(220\) −8.93966e19 −1.09832
\(221\) −1.29405e20 −1.52973
\(222\) 3.28808e19 0.374061
\(223\) 5.06723e19 0.554854 0.277427 0.960747i \(-0.410518\pi\)
0.277427 + 0.960747i \(0.410518\pi\)
\(224\) −6.29724e19 −0.663807
\(225\) 5.25720e19 0.533585
\(226\) −5.98118e18 −0.0584608
\(227\) 6.30244e19 0.593320 0.296660 0.954983i \(-0.404127\pi\)
0.296660 + 0.954983i \(0.404127\pi\)
\(228\) 7.76884e19 0.704547
\(229\) −1.57688e20 −1.37784 −0.688918 0.724840i \(-0.741914\pi\)
−0.688918 + 0.724840i \(0.741914\pi\)
\(230\) 1.34646e20 1.13372
\(231\) −1.47073e20 −1.19352
\(232\) 2.44594e19 0.191336
\(233\) 3.00771e19 0.226836 0.113418 0.993547i \(-0.463820\pi\)
0.113418 + 0.993547i \(0.463820\pi\)
\(234\) −8.92887e19 −0.649324
\(235\) 1.09329e20 0.766757
\(236\) −6.77150e19 −0.458071
\(237\) −9.86221e19 −0.643596
\(238\) 6.23688e19 0.392702
\(239\) −1.38253e20 −0.840026 −0.420013 0.907518i \(-0.637974\pi\)
−0.420013 + 0.907518i \(0.637974\pi\)
\(240\) 3.23036e19 0.189433
\(241\) 1.57658e20 0.892422 0.446211 0.894928i \(-0.352773\pi\)
0.446211 + 0.894928i \(0.352773\pi\)
\(242\) −1.09235e20 −0.596939
\(243\) 1.90349e20 1.00438
\(244\) 1.66543e20 0.848621
\(245\) −1.48058e20 −0.728651
\(246\) 1.95560e20 0.929677
\(247\) 3.82965e20 1.75888
\(248\) 1.95974e20 0.869677
\(249\) 1.49159e20 0.639669
\(250\) 9.70891e18 0.0402421
\(251\) 2.64777e19 0.106085 0.0530424 0.998592i \(-0.483108\pi\)
0.0530424 + 0.998592i \(0.483108\pi\)
\(252\) −5.66758e19 −0.219530
\(253\) −4.55143e20 −1.70460
\(254\) 1.12746e20 0.408331
\(255\) 4.30457e20 1.50778
\(256\) −3.09877e20 −1.04990
\(257\) 2.73778e19 0.0897360 0.0448680 0.998993i \(-0.485713\pi\)
0.0448680 + 0.998993i \(0.485713\pi\)
\(258\) −2.95093e20 −0.935818
\(259\) 1.02884e20 0.315718
\(260\) −4.74144e20 −1.40811
\(261\) 3.60494e19 0.103622
\(262\) −2.77899e20 −0.773261
\(263\) 6.19094e20 1.66776 0.833878 0.551949i \(-0.186116\pi\)
0.833878 + 0.551949i \(0.186116\pi\)
\(264\) −6.81207e20 −1.77683
\(265\) 7.37034e20 1.86165
\(266\) −1.84577e20 −0.451526
\(267\) 5.15289e20 1.22097
\(268\) −7.76822e19 −0.178310
\(269\) 4.77375e20 1.06161 0.530805 0.847494i \(-0.321890\pi\)
0.530805 + 0.847494i \(0.321890\pi\)
\(270\) −2.35246e20 −0.506909
\(271\) 8.30074e20 1.73332 0.866658 0.498903i \(-0.166264\pi\)
0.866658 + 0.498903i \(0.166264\pi\)
\(272\) 4.63064e19 0.0937142
\(273\) −7.80050e20 −1.53017
\(274\) −6.67681e20 −1.26967
\(275\) 7.16540e20 1.32103
\(276\) −4.89705e20 −0.875401
\(277\) −2.63424e20 −0.456644 −0.228322 0.973586i \(-0.573324\pi\)
−0.228322 + 0.973586i \(0.573324\pi\)
\(278\) 6.88510e19 0.115752
\(279\) 2.88836e20 0.470994
\(280\) 6.30562e20 0.997435
\(281\) 6.96762e20 1.06925 0.534627 0.845088i \(-0.320452\pi\)
0.534627 + 0.845088i \(0.320452\pi\)
\(282\) 3.01921e20 0.449547
\(283\) −9.79407e20 −1.41507 −0.707536 0.706678i \(-0.750193\pi\)
−0.707536 + 0.706678i \(0.750193\pi\)
\(284\) −1.51265e20 −0.212097
\(285\) −1.27391e21 −1.73364
\(286\) −1.21698e21 −1.60757
\(287\) 6.11907e20 0.784674
\(288\) −4.29882e20 −0.535196
\(289\) −2.10191e20 −0.254087
\(290\) −1.45355e20 −0.170626
\(291\) −1.95122e21 −2.22441
\(292\) 2.72328e20 0.301536
\(293\) −7.92943e20 −0.852839 −0.426420 0.904525i \(-0.640225\pi\)
−0.426420 + 0.904525i \(0.640225\pi\)
\(294\) −4.08874e20 −0.427206
\(295\) 1.11037e21 1.12715
\(296\) 4.76533e20 0.470017
\(297\) 7.95203e20 0.762163
\(298\) 2.31925e20 0.216028
\(299\) −2.41400e21 −2.18541
\(300\) 7.70952e20 0.678417
\(301\) −9.23344e20 −0.789857
\(302\) 3.39667e20 0.282484
\(303\) 2.89913e21 2.34426
\(304\) −1.37041e20 −0.107752
\(305\) −2.73093e21 −2.08815
\(306\) 4.25762e20 0.316617
\(307\) 6.85552e20 0.495866 0.247933 0.968777i \(-0.420249\pi\)
0.247933 + 0.968777i \(0.420249\pi\)
\(308\) −7.72474e20 −0.543504
\(309\) 1.67633e20 0.114740
\(310\) −1.16461e21 −0.775544
\(311\) 6.30454e20 0.408498 0.204249 0.978919i \(-0.434525\pi\)
0.204249 + 0.978919i \(0.434525\pi\)
\(312\) −3.61300e21 −2.27800
\(313\) −1.36573e21 −0.837988 −0.418994 0.907989i \(-0.637617\pi\)
−0.418994 + 0.907989i \(0.637617\pi\)
\(314\) −1.95462e21 −1.16724
\(315\) 9.29354e20 0.540184
\(316\) −5.17994e20 −0.293079
\(317\) −1.46033e21 −0.804352 −0.402176 0.915562i \(-0.631746\pi\)
−0.402176 + 0.915562i \(0.631746\pi\)
\(318\) 2.03538e21 1.09148
\(319\) 4.91342e20 0.256545
\(320\) 2.03182e21 1.03303
\(321\) 3.50409e21 1.73493
\(322\) 1.16347e21 0.561023
\(323\) −1.82612e21 −0.857646
\(324\) 1.54892e21 0.708594
\(325\) 3.80041e21 1.69364
\(326\) −1.75657e21 −0.762635
\(327\) 7.82157e19 0.0330855
\(328\) 2.83421e21 1.16816
\(329\) 9.44708e20 0.379431
\(330\) 4.04820e21 1.58451
\(331\) 1.56688e21 0.597721 0.298860 0.954297i \(-0.403394\pi\)
0.298860 + 0.954297i \(0.403394\pi\)
\(332\) 7.83432e20 0.291291
\(333\) 7.02338e20 0.254548
\(334\) 9.41941e19 0.0332796
\(335\) 1.27381e21 0.438757
\(336\) 2.79135e20 0.0937411
\(337\) 2.53384e21 0.829708 0.414854 0.909888i \(-0.363833\pi\)
0.414854 + 0.909888i \(0.363833\pi\)
\(338\) −4.39718e21 −1.40405
\(339\) −3.56707e20 −0.111075
\(340\) 2.26090e21 0.686610
\(341\) 3.93674e21 1.16607
\(342\) −1.26002e21 −0.364045
\(343\) −3.73510e21 −1.05269
\(344\) −4.27671e21 −1.17588
\(345\) 8.03004e21 2.15405
\(346\) 3.88611e21 1.01711
\(347\) −3.01289e21 −0.769453 −0.384727 0.923031i \(-0.625704\pi\)
−0.384727 + 0.923031i \(0.625704\pi\)
\(348\) 5.28653e20 0.131749
\(349\) −1.08133e21 −0.262992 −0.131496 0.991317i \(-0.541978\pi\)
−0.131496 + 0.991317i \(0.541978\pi\)
\(350\) −1.83167e21 −0.434780
\(351\) 4.21762e21 0.977140
\(352\) −5.85916e21 −1.32502
\(353\) −6.80308e21 −1.50183 −0.750915 0.660399i \(-0.770387\pi\)
−0.750915 + 0.660399i \(0.770387\pi\)
\(354\) 3.06638e21 0.660843
\(355\) 2.48041e21 0.521894
\(356\) 2.70646e21 0.556003
\(357\) 3.71957e21 0.746127
\(358\) −6.51511e21 −1.27619
\(359\) −6.03010e21 −1.15351 −0.576755 0.816917i \(-0.695681\pi\)
−0.576755 + 0.816917i \(0.695681\pi\)
\(360\) 4.30455e21 0.804185
\(361\) −7.60878e19 −0.0138837
\(362\) 2.12479e21 0.378698
\(363\) −6.51458e21 −1.13417
\(364\) −4.09707e21 −0.696806
\(365\) −4.46556e21 −0.741971
\(366\) −7.54169e21 −1.22428
\(367\) −5.73509e21 −0.909658 −0.454829 0.890579i \(-0.650300\pi\)
−0.454829 + 0.890579i \(0.650300\pi\)
\(368\) 8.63831e20 0.133882
\(369\) 4.17719e21 0.632646
\(370\) −2.83189e21 −0.419143
\(371\) 6.36869e21 0.921238
\(372\) 4.23568e21 0.598836
\(373\) −6.10804e21 −0.844067 −0.422033 0.906580i \(-0.638684\pi\)
−0.422033 + 0.906580i \(0.638684\pi\)
\(374\) 5.80300e21 0.783870
\(375\) 5.79023e20 0.0764592
\(376\) 4.37566e21 0.564868
\(377\) 2.60600e21 0.328906
\(378\) −2.03276e21 −0.250845
\(379\) 5.55696e21 0.670508 0.335254 0.942128i \(-0.391178\pi\)
0.335254 + 0.942128i \(0.391178\pi\)
\(380\) −6.69099e21 −0.789460
\(381\) 6.72395e21 0.775822
\(382\) −1.79005e21 −0.201988
\(383\) 4.05111e21 0.447079 0.223540 0.974695i \(-0.428239\pi\)
0.223540 + 0.974695i \(0.428239\pi\)
\(384\) −5.47975e21 −0.591487
\(385\) 1.26668e22 1.33737
\(386\) 1.02518e22 1.05878
\(387\) −6.30323e21 −0.636825
\(388\) −1.02484e22 −1.01295
\(389\) −1.55408e22 −1.50280 −0.751399 0.659848i \(-0.770621\pi\)
−0.751399 + 0.659848i \(0.770621\pi\)
\(390\) 2.14710e22 2.03143
\(391\) 1.15109e22 1.06563
\(392\) −5.92570e21 −0.536795
\(393\) −1.65734e22 −1.46918
\(394\) 4.31775e21 0.374575
\(395\) 8.49392e21 0.721162
\(396\) −5.27331e21 −0.438202
\(397\) −1.02208e22 −0.831314 −0.415657 0.909521i \(-0.636448\pi\)
−0.415657 + 0.909521i \(0.636448\pi\)
\(398\) −5.35316e21 −0.426191
\(399\) −1.10078e22 −0.857892
\(400\) −1.35995e21 −0.103756
\(401\) −8.17444e21 −0.610563 −0.305282 0.952262i \(-0.598751\pi\)
−0.305282 + 0.952262i \(0.598751\pi\)
\(402\) 3.51773e21 0.257242
\(403\) 2.08798e22 1.49497
\(404\) 1.52272e22 1.06752
\(405\) −2.53988e22 −1.74360
\(406\) −1.25601e21 −0.0844345
\(407\) 9.57265e21 0.630202
\(408\) 1.72282e22 1.11078
\(409\) −1.63861e22 −1.03473 −0.517364 0.855765i \(-0.673087\pi\)
−0.517364 + 0.855765i \(0.673087\pi\)
\(410\) −1.68428e22 −1.04172
\(411\) −3.98194e22 −2.41234
\(412\) 8.80463e20 0.0522498
\(413\) 9.59468e21 0.557770
\(414\) 7.94245e21 0.452326
\(415\) −1.28465e22 −0.716763
\(416\) −3.10760e22 −1.69876
\(417\) 4.10615e21 0.219927
\(418\) −1.71736e22 −0.901288
\(419\) 2.91601e22 1.49958 0.749789 0.661677i \(-0.230155\pi\)
0.749789 + 0.661677i \(0.230155\pi\)
\(420\) 1.36287e22 0.686807
\(421\) 1.82872e22 0.903128 0.451564 0.892239i \(-0.350866\pi\)
0.451564 + 0.892239i \(0.350866\pi\)
\(422\) −1.75944e22 −0.851568
\(423\) 6.44907e21 0.305917
\(424\) 2.94983e22 1.37147
\(425\) −1.81218e22 −0.825838
\(426\) 6.84985e21 0.305985
\(427\) −2.35979e22 −1.03332
\(428\) 1.84046e22 0.790050
\(429\) −7.25784e22 −3.05436
\(430\) 2.54152e22 1.04860
\(431\) −1.56184e21 −0.0631801 −0.0315900 0.999501i \(-0.510057\pi\)
−0.0315900 + 0.999501i \(0.510057\pi\)
\(432\) −1.50924e21 −0.0598614
\(433\) 4.19456e22 1.63132 0.815660 0.578532i \(-0.196374\pi\)
0.815660 + 0.578532i \(0.196374\pi\)
\(434\) −1.00634e22 −0.383779
\(435\) −8.66870e21 −0.324187
\(436\) 4.10814e20 0.0150664
\(437\) −3.40657e22 −1.22525
\(438\) −1.23320e22 −0.435015
\(439\) 1.98023e22 0.685121 0.342561 0.939496i \(-0.388706\pi\)
0.342561 + 0.939496i \(0.388706\pi\)
\(440\) 5.86696e22 1.99097
\(441\) −8.73360e21 −0.290714
\(442\) 3.07781e22 1.00497
\(443\) 1.66103e22 0.532043 0.266021 0.963967i \(-0.414291\pi\)
0.266021 + 0.963967i \(0.414291\pi\)
\(444\) 1.02996e22 0.323641
\(445\) −4.43798e22 −1.36812
\(446\) −1.20521e22 −0.364516
\(447\) 1.38316e22 0.410449
\(448\) 1.75569e22 0.511194
\(449\) 5.25193e22 1.50046 0.750231 0.661176i \(-0.229942\pi\)
0.750231 + 0.661176i \(0.229942\pi\)
\(450\) −1.25040e22 −0.350543
\(451\) 5.69338e22 1.56628
\(452\) −1.87354e21 −0.0505809
\(453\) 2.02572e22 0.536715
\(454\) −1.49900e22 −0.389786
\(455\) 6.71826e22 1.71459
\(456\) −5.09857e22 −1.27717
\(457\) −3.05676e22 −0.751577 −0.375788 0.926706i \(-0.622628\pi\)
−0.375788 + 0.926706i \(0.622628\pi\)
\(458\) 3.75052e22 0.905180
\(459\) −2.01112e22 −0.476464
\(460\) 4.21763e22 0.980906
\(461\) 5.88809e21 0.134436 0.0672182 0.997738i \(-0.478588\pi\)
0.0672182 + 0.997738i \(0.478588\pi\)
\(462\) 3.49804e22 0.784095
\(463\) −3.11045e22 −0.684517 −0.342259 0.939606i \(-0.611192\pi\)
−0.342259 + 0.939606i \(0.611192\pi\)
\(464\) −9.32535e20 −0.0201494
\(465\) −6.94554e22 −1.47352
\(466\) −7.15367e21 −0.149021
\(467\) 3.18998e22 0.652522 0.326261 0.945280i \(-0.394211\pi\)
0.326261 + 0.945280i \(0.394211\pi\)
\(468\) −2.79687e22 −0.561802
\(469\) 1.10070e22 0.217119
\(470\) −2.60032e22 −0.503727
\(471\) −1.16570e23 −2.21773
\(472\) 4.44403e22 0.830366
\(473\) −8.59110e22 −1.57663
\(474\) 2.34567e22 0.422815
\(475\) 5.36303e22 0.949543
\(476\) 1.95364e22 0.339770
\(477\) 4.34760e22 0.742751
\(478\) 3.28827e22 0.551862
\(479\) −6.35415e22 −1.04762 −0.523812 0.851834i \(-0.675491\pi\)
−0.523812 + 0.851834i \(0.675491\pi\)
\(480\) 1.03372e23 1.67438
\(481\) 5.07717e22 0.807957
\(482\) −3.74979e22 −0.586283
\(483\) 6.93874e22 1.06593
\(484\) −3.42166e22 −0.516478
\(485\) 1.68051e23 2.49250
\(486\) −4.52735e22 −0.659835
\(487\) 1.72595e22 0.247191 0.123596 0.992333i \(-0.460557\pi\)
0.123596 + 0.992333i \(0.460557\pi\)
\(488\) −1.09300e23 −1.53834
\(489\) −1.04759e23 −1.44899
\(490\) 3.52147e22 0.478693
\(491\) 3.48189e22 0.465181 0.232591 0.972575i \(-0.425280\pi\)
0.232591 + 0.972575i \(0.425280\pi\)
\(492\) 6.12572e22 0.804366
\(493\) −1.24264e22 −0.160378
\(494\) −9.10860e22 −1.15551
\(495\) 8.64702e22 1.07826
\(496\) −7.47167e21 −0.0915848
\(497\) 2.14331e22 0.258260
\(498\) −3.54766e22 −0.420236
\(499\) 5.70356e22 0.664189 0.332094 0.943246i \(-0.392245\pi\)
0.332094 + 0.943246i \(0.392245\pi\)
\(500\) 3.04121e21 0.0348178
\(501\) 5.61757e21 0.0632307
\(502\) −6.29756e21 −0.0696933
\(503\) −1.92954e22 −0.209955 −0.104977 0.994475i \(-0.533477\pi\)
−0.104977 + 0.994475i \(0.533477\pi\)
\(504\) 3.71955e22 0.397952
\(505\) −2.49691e23 −2.62679
\(506\) 1.08253e23 1.11985
\(507\) −2.62241e23 −2.66767
\(508\) 3.53163e22 0.353292
\(509\) −1.21292e23 −1.19325 −0.596627 0.802519i \(-0.703493\pi\)
−0.596627 + 0.802519i \(0.703493\pi\)
\(510\) −1.02382e23 −0.990549
\(511\) −3.85868e22 −0.367165
\(512\) 2.30671e22 0.215873
\(513\) 5.95179e22 0.547835
\(514\) −6.51165e21 −0.0589528
\(515\) −1.44376e22 −0.128568
\(516\) −9.24348e22 −0.809679
\(517\) 8.78987e22 0.757379
\(518\) −2.44703e22 −0.207413
\(519\) 2.31761e23 1.93249
\(520\) 3.11174e23 2.55255
\(521\) 2.27742e23 1.83790 0.918951 0.394371i \(-0.129038\pi\)
0.918951 + 0.394371i \(0.129038\pi\)
\(522\) −8.57414e21 −0.0680756
\(523\) −1.00048e23 −0.781525 −0.390763 0.920492i \(-0.627789\pi\)
−0.390763 + 0.920492i \(0.627789\pi\)
\(524\) −8.70490e22 −0.669033
\(525\) −1.09238e23 −0.826074
\(526\) −1.47248e23 −1.09565
\(527\) −9.95626e22 −0.728964
\(528\) 2.59716e22 0.187116
\(529\) 7.36814e22 0.522378
\(530\) −1.75299e23 −1.22302
\(531\) 6.54983e22 0.449704
\(532\) −5.78167e22 −0.390665
\(533\) 3.01967e23 2.00807
\(534\) −1.22559e23 −0.802126
\(535\) −3.01793e23 −1.94403
\(536\) 5.09816e22 0.323231
\(537\) −3.88550e23 −2.42474
\(538\) −1.13541e23 −0.697433
\(539\) −1.19036e23 −0.719738
\(540\) −7.36883e22 −0.438583
\(541\) 1.99451e23 1.16858 0.584292 0.811543i \(-0.301372\pi\)
0.584292 + 0.811543i \(0.301372\pi\)
\(542\) −1.97428e23 −1.13872
\(543\) 1.26719e23 0.719519
\(544\) 1.48182e23 0.828332
\(545\) −6.73641e21 −0.0370730
\(546\) 1.85530e23 1.00526
\(547\) 3.31665e23 1.76932 0.884662 0.466232i \(-0.154389\pi\)
0.884662 + 0.466232i \(0.154389\pi\)
\(548\) −2.09144e23 −1.09853
\(549\) −1.61092e23 −0.833121
\(550\) −1.70425e23 −0.867861
\(551\) 3.67751e22 0.184402
\(552\) 3.21386e23 1.58688
\(553\) 7.33958e22 0.356868
\(554\) 6.26540e22 0.299996
\(555\) −1.68889e23 −0.796364
\(556\) 2.15668e22 0.100150
\(557\) 1.81484e23 0.829982 0.414991 0.909825i \(-0.363785\pi\)
0.414991 + 0.909825i \(0.363785\pi\)
\(558\) −6.86979e22 −0.309423
\(559\) −4.55657e23 −2.02133
\(560\) −2.40408e22 −0.105039
\(561\) 3.46081e23 1.48934
\(562\) −1.65721e23 −0.702455
\(563\) −4.35433e23 −1.81802 −0.909012 0.416770i \(-0.863162\pi\)
−0.909012 + 0.416770i \(0.863162\pi\)
\(564\) 9.45734e22 0.388953
\(565\) 3.07218e22 0.124461
\(566\) 2.32946e23 0.929642
\(567\) −2.19471e23 −0.862820
\(568\) 9.92732e22 0.384478
\(569\) −1.29059e23 −0.492417 −0.246208 0.969217i \(-0.579185\pi\)
−0.246208 + 0.969217i \(0.579185\pi\)
\(570\) 3.02993e23 1.13893
\(571\) 4.64417e23 1.71989 0.859946 0.510386i \(-0.170497\pi\)
0.859946 + 0.510386i \(0.170497\pi\)
\(572\) −3.81205e23 −1.39089
\(573\) −1.06755e23 −0.383774
\(574\) −1.45538e23 −0.515498
\(575\) −3.38056e23 −1.17981
\(576\) 1.19853e23 0.412151
\(577\) −6.87036e22 −0.232801 −0.116400 0.993202i \(-0.537136\pi\)
−0.116400 + 0.993202i \(0.537136\pi\)
\(578\) 4.99927e22 0.166924
\(579\) 6.11399e23 2.01167
\(580\) −4.55308e22 −0.147627
\(581\) −1.11006e23 −0.354691
\(582\) 4.64086e23 1.46135
\(583\) 5.92564e23 1.83888
\(584\) −1.78725e23 −0.546608
\(585\) 4.58623e23 1.38239
\(586\) 1.88597e23 0.560279
\(587\) 3.83769e23 1.12369 0.561844 0.827243i \(-0.310092\pi\)
0.561844 + 0.827243i \(0.310092\pi\)
\(588\) −1.28075e23 −0.369622
\(589\) 2.94650e23 0.838158
\(590\) −2.64095e23 −0.740488
\(591\) 2.57503e23 0.711687
\(592\) −1.81682e22 −0.0494970
\(593\) 4.18329e22 0.112345 0.0561724 0.998421i \(-0.482110\pi\)
0.0561724 + 0.998421i \(0.482110\pi\)
\(594\) −1.89134e23 −0.500709
\(595\) −3.20352e23 −0.836051
\(596\) 7.26481e22 0.186909
\(597\) −3.19253e23 −0.809754
\(598\) 5.74156e23 1.43572
\(599\) 6.00244e23 1.47979 0.739894 0.672723i \(-0.234876\pi\)
0.739894 + 0.672723i \(0.234876\pi\)
\(600\) −5.05964e23 −1.22980
\(601\) 7.80759e23 1.87104 0.935522 0.353267i \(-0.114929\pi\)
0.935522 + 0.353267i \(0.114929\pi\)
\(602\) 2.19612e23 0.518903
\(603\) 7.51392e22 0.175053
\(604\) 1.06397e23 0.244408
\(605\) 5.61074e23 1.27087
\(606\) −6.89542e23 −1.54008
\(607\) −1.52608e23 −0.336103 −0.168051 0.985778i \(-0.553747\pi\)
−0.168051 + 0.985778i \(0.553747\pi\)
\(608\) −4.38535e23 −0.952411
\(609\) −7.49060e22 −0.160424
\(610\) 6.49536e23 1.37183
\(611\) 4.66200e23 0.971006
\(612\) 1.33365e23 0.273940
\(613\) 1.75688e23 0.355899 0.177950 0.984040i \(-0.443054\pi\)
0.177950 + 0.984040i \(0.443054\pi\)
\(614\) −1.63054e23 −0.325763
\(615\) −1.00448e24 −1.97925
\(616\) 5.06963e23 0.985235
\(617\) 1.85521e23 0.355606 0.177803 0.984066i \(-0.443101\pi\)
0.177803 + 0.984066i \(0.443101\pi\)
\(618\) −3.98706e22 −0.0753791
\(619\) −6.46969e23 −1.20646 −0.603230 0.797567i \(-0.706120\pi\)
−0.603230 + 0.797567i \(0.706120\pi\)
\(620\) −3.64802e23 −0.671008
\(621\) −3.75168e23 −0.680686
\(622\) −1.49950e23 −0.268366
\(623\) −3.83485e23 −0.677017
\(624\) 1.37749e23 0.239894
\(625\) −6.06453e23 −1.04188
\(626\) 3.24831e23 0.550523
\(627\) −1.02421e24 −1.71243
\(628\) −6.12264e23 −1.00991
\(629\) −2.42098e23 −0.393968
\(630\) −2.21041e23 −0.354878
\(631\) 5.43759e23 0.861305 0.430653 0.902518i \(-0.358283\pi\)
0.430653 + 0.902518i \(0.358283\pi\)
\(632\) 3.39952e23 0.531278
\(633\) −1.04930e24 −1.61796
\(634\) 3.47330e23 0.528425
\(635\) −5.79107e23 −0.869325
\(636\) 6.37561e23 0.944357
\(637\) −6.31348e23 −0.922749
\(638\) −1.16863e23 −0.168539
\(639\) 1.46314e23 0.208223
\(640\) 4.71949e23 0.662773
\(641\) −2.60005e23 −0.360321 −0.180160 0.983637i \(-0.557662\pi\)
−0.180160 + 0.983637i \(0.557662\pi\)
\(642\) −8.33427e23 −1.13978
\(643\) −3.09562e23 −0.417787 −0.208893 0.977938i \(-0.566986\pi\)
−0.208893 + 0.977938i \(0.566986\pi\)
\(644\) 3.64445e23 0.485402
\(645\) 1.51572e24 1.99233
\(646\) 4.34332e23 0.563437
\(647\) −1.34080e24 −1.71664 −0.858319 0.513116i \(-0.828491\pi\)
−0.858319 + 0.513116i \(0.828491\pi\)
\(648\) −1.01654e24 −1.28450
\(649\) 8.92721e23 1.11336
\(650\) −9.03905e23 −1.11265
\(651\) −6.00163e23 −0.729173
\(652\) −5.50228e23 −0.659840
\(653\) 6.55237e23 0.775597 0.387798 0.921744i \(-0.373236\pi\)
0.387798 + 0.921744i \(0.373236\pi\)
\(654\) −1.86032e22 −0.0217358
\(655\) 1.42740e24 1.64625
\(656\) −1.08057e23 −0.123018
\(657\) −2.63414e23 −0.296028
\(658\) −2.24693e23 −0.249270
\(659\) −2.46802e23 −0.270286 −0.135143 0.990826i \(-0.543149\pi\)
−0.135143 + 0.990826i \(0.543149\pi\)
\(660\) 1.26806e24 1.37093
\(661\) 1.79192e24 1.91252 0.956261 0.292516i \(-0.0944925\pi\)
0.956261 + 0.292516i \(0.0944925\pi\)
\(662\) −3.72674e23 −0.392677
\(663\) 1.83555e24 1.90942
\(664\) −5.14154e23 −0.528037
\(665\) 9.48062e23 0.961286
\(666\) −1.67047e23 −0.167228
\(667\) −2.31810e23 −0.229120
\(668\) 2.95053e22 0.0287939
\(669\) −7.18767e23 −0.692574
\(670\) −3.02968e23 −0.288245
\(671\) −2.19563e24 −2.06261
\(672\) 8.93239e23 0.828569
\(673\) −1.61280e24 −1.47725 −0.738623 0.674119i \(-0.764524\pi\)
−0.738623 + 0.674119i \(0.764524\pi\)
\(674\) −6.02659e23 −0.545083
\(675\) 5.90634e23 0.527517
\(676\) −1.37737e24 −1.21480
\(677\) −5.39428e23 −0.469818 −0.234909 0.972017i \(-0.575479\pi\)
−0.234909 + 0.972017i \(0.575479\pi\)
\(678\) 8.48407e22 0.0729713
\(679\) 1.45212e24 1.23342
\(680\) −1.48379e24 −1.24465
\(681\) −8.93977e23 −0.740587
\(682\) −9.36330e23 −0.766058
\(683\) −1.34863e24 −1.08973 −0.544863 0.838525i \(-0.683418\pi\)
−0.544863 + 0.838525i \(0.683418\pi\)
\(684\) −3.94687e23 −0.314975
\(685\) 3.42949e24 2.70308
\(686\) 8.88371e23 0.691576
\(687\) 2.23675e24 1.71983
\(688\) 1.63053e23 0.123831
\(689\) 3.14286e24 2.35755
\(690\) −1.90990e24 −1.41512
\(691\) −1.23627e24 −0.904792 −0.452396 0.891817i \(-0.649431\pi\)
−0.452396 + 0.891817i \(0.649431\pi\)
\(692\) 1.21728e24 0.880013
\(693\) 7.47187e23 0.533577
\(694\) 7.16597e23 0.505498
\(695\) −3.53647e23 −0.246433
\(696\) −3.46947e23 −0.238827
\(697\) −1.43989e24 −0.979156
\(698\) 2.57188e23 0.172775
\(699\) −4.26633e23 −0.283138
\(700\) −5.73752e23 −0.376176
\(701\) −1.61438e24 −1.04569 −0.522846 0.852427i \(-0.675130\pi\)
−0.522846 + 0.852427i \(0.675130\pi\)
\(702\) −1.00314e24 −0.641940
\(703\) 7.16476e23 0.452982
\(704\) 1.63355e24 1.02039
\(705\) −1.55079e24 −0.957073
\(706\) 1.61807e24 0.986639
\(707\) −2.15757e24 −1.29987
\(708\) 9.60511e23 0.571768
\(709\) 6.23711e23 0.366852 0.183426 0.983034i \(-0.441281\pi\)
0.183426 + 0.983034i \(0.441281\pi\)
\(710\) −5.89950e23 −0.342862
\(711\) 5.01038e23 0.287726
\(712\) −1.77621e24 −1.00789
\(713\) −1.85731e24 −1.04141
\(714\) −8.84678e23 −0.490174
\(715\) 6.25089e24 3.42248
\(716\) −2.04079e24 −1.10417
\(717\) 1.96107e24 1.04853
\(718\) 1.43422e24 0.757808
\(719\) −1.75531e24 −0.916557 −0.458278 0.888809i \(-0.651534\pi\)
−0.458278 + 0.888809i \(0.651534\pi\)
\(720\) −1.64115e23 −0.0846879
\(721\) −1.24755e23 −0.0636221
\(722\) 1.80970e22 0.00912098
\(723\) −2.23631e24 −1.11393
\(724\) 6.65567e23 0.327653
\(725\) 3.64943e23 0.177563
\(726\) 1.54945e24 0.745105
\(727\) 1.18905e24 0.565142 0.282571 0.959246i \(-0.408813\pi\)
0.282571 + 0.959246i \(0.408813\pi\)
\(728\) 2.68884e24 1.26313
\(729\) −1.51667e22 −0.00704217
\(730\) 1.06211e24 0.487443
\(731\) 2.17274e24 0.985623
\(732\) −2.36235e24 −1.05926
\(733\) −1.24090e24 −0.549986 −0.274993 0.961446i \(-0.588675\pi\)
−0.274993 + 0.961446i \(0.588675\pi\)
\(734\) 1.36406e24 0.597607
\(735\) 2.10014e24 0.909508
\(736\) 2.76429e24 1.18337
\(737\) 1.02412e24 0.433390
\(738\) −9.93521e23 −0.415622
\(739\) 2.80759e24 1.16106 0.580532 0.814237i \(-0.302845\pi\)
0.580532 + 0.814237i \(0.302845\pi\)
\(740\) −8.87060e23 −0.362646
\(741\) −5.43222e24 −2.19544
\(742\) −1.51476e24 −0.605215
\(743\) −3.17587e24 −1.25446 −0.627231 0.778833i \(-0.715812\pi\)
−0.627231 + 0.778833i \(0.715812\pi\)
\(744\) −2.77981e24 −1.08554
\(745\) −1.19126e24 −0.459917
\(746\) 1.45276e24 0.554516
\(747\) −7.57786e23 −0.285971
\(748\) 1.81773e24 0.678212
\(749\) −2.60779e24 −0.962004
\(750\) −1.37717e23 −0.0502305
\(751\) 2.22510e24 0.802435 0.401217 0.915983i \(-0.368587\pi\)
0.401217 + 0.915983i \(0.368587\pi\)
\(752\) −1.66826e23 −0.0594856
\(753\) −3.75576e23 −0.132416
\(754\) −6.19821e23 −0.216078
\(755\) −1.74467e24 −0.601400
\(756\) −6.36739e23 −0.217033
\(757\) 2.90700e24 0.979783 0.489892 0.871783i \(-0.337036\pi\)
0.489892 + 0.871783i \(0.337036\pi\)
\(758\) −1.32169e24 −0.440496
\(759\) 6.45603e24 2.12770
\(760\) 4.39120e24 1.43109
\(761\) −5.65138e24 −1.82131 −0.910657 0.413164i \(-0.864424\pi\)
−0.910657 + 0.413164i \(0.864424\pi\)
\(762\) −1.59925e24 −0.509683
\(763\) −5.82092e22 −0.0183456
\(764\) −5.60713e23 −0.174762
\(765\) −2.18689e24 −0.674069
\(766\) −9.63533e23 −0.293712
\(767\) 4.73484e24 1.42740
\(768\) 4.39548e24 1.31050
\(769\) −1.93750e24 −0.571306 −0.285653 0.958333i \(-0.592210\pi\)
−0.285653 + 0.958333i \(0.592210\pi\)
\(770\) −3.01273e24 −0.878594
\(771\) −3.88344e23 −0.112009
\(772\) 3.21126e24 0.916070
\(773\) −1.33043e24 −0.375377 −0.187688 0.982229i \(-0.560099\pi\)
−0.187688 + 0.982229i \(0.560099\pi\)
\(774\) 1.49919e24 0.418367
\(775\) 2.92400e24 0.807072
\(776\) 6.72588e24 1.83622
\(777\) −1.45937e24 −0.394081
\(778\) 3.69628e24 0.987275
\(779\) 4.26128e24 1.12583
\(780\) 6.72556e24 1.75762
\(781\) 1.99421e24 0.515510
\(782\) −2.73779e24 −0.700072
\(783\) 4.05006e23 0.102444
\(784\) 2.25923e23 0.0565293
\(785\) 1.00397e25 2.48502
\(786\) 3.94189e24 0.965191
\(787\) −3.58217e24 −0.867683 −0.433842 0.900989i \(-0.642842\pi\)
−0.433842 + 0.900989i \(0.642842\pi\)
\(788\) 1.35249e24 0.324086
\(789\) −8.78161e24 −2.08171
\(790\) −2.02023e24 −0.473773
\(791\) 2.65466e23 0.0615898
\(792\) 3.46079e24 0.794349
\(793\) −1.16452e25 −2.64439
\(794\) 2.43095e24 0.546138
\(795\) −1.04545e25 −2.32372
\(796\) −1.67682e24 −0.368744
\(797\) −4.17640e24 −0.908670 −0.454335 0.890831i \(-0.650123\pi\)
−0.454335 + 0.890831i \(0.650123\pi\)
\(798\) 2.61815e24 0.563599
\(799\) −2.22302e24 −0.473473
\(800\) −4.35187e24 −0.917087
\(801\) −2.61787e24 −0.545847
\(802\) 1.94424e24 0.401114
\(803\) −3.59024e24 −0.732895
\(804\) 1.10189e24 0.222568
\(805\) −5.97606e24 −1.19440
\(806\) −4.96613e24 −0.982134
\(807\) −6.77138e24 −1.32511
\(808\) −9.99336e24 −1.93515
\(809\) −1.19044e24 −0.228111 −0.114055 0.993474i \(-0.536384\pi\)
−0.114055 + 0.993474i \(0.536384\pi\)
\(810\) 6.04096e24 1.14547
\(811\) 1.26895e24 0.238105 0.119052 0.992888i \(-0.462014\pi\)
0.119052 + 0.992888i \(0.462014\pi\)
\(812\) −3.93430e23 −0.0730536
\(813\) −1.17743e25 −2.16354
\(814\) −2.27680e24 −0.414016
\(815\) 9.02248e24 1.62363
\(816\) −6.56839e23 −0.116975
\(817\) −6.43011e24 −1.13326
\(818\) 3.89733e24 0.679773
\(819\) 3.96295e24 0.684078
\(820\) −5.27584e24 −0.901309
\(821\) 2.08061e24 0.351782 0.175891 0.984410i \(-0.443719\pi\)
0.175891 + 0.984410i \(0.443719\pi\)
\(822\) 9.47081e24 1.58481
\(823\) 2.64902e24 0.438720 0.219360 0.975644i \(-0.429603\pi\)
0.219360 + 0.975644i \(0.429603\pi\)
\(824\) −5.77835e23 −0.0947157
\(825\) −1.01639e25 −1.64892
\(826\) −2.28204e24 −0.366431
\(827\) −6.65948e24 −1.05838 −0.529192 0.848502i \(-0.677505\pi\)
−0.529192 + 0.848502i \(0.677505\pi\)
\(828\) 2.48789e24 0.391357
\(829\) 9.74066e23 0.151661 0.0758306 0.997121i \(-0.475839\pi\)
0.0758306 + 0.997121i \(0.475839\pi\)
\(830\) 3.05546e24 0.470883
\(831\) 3.73658e24 0.569987
\(832\) 8.66410e24 1.30820
\(833\) 3.01050e24 0.449942
\(834\) −9.76625e23 −0.144483
\(835\) −4.83819e23 −0.0708514
\(836\) −5.37946e24 −0.779804
\(837\) 3.24500e24 0.465637
\(838\) −6.93555e24 −0.985160
\(839\) 3.43808e24 0.483436 0.241718 0.970347i \(-0.422289\pi\)
0.241718 + 0.970347i \(0.422289\pi\)
\(840\) −8.94429e24 −1.24501
\(841\) 2.50246e23 0.0344828
\(842\) −4.34950e24 −0.593317
\(843\) −9.88331e24 −1.33465
\(844\) −5.51127e24 −0.736785
\(845\) 2.25857e25 2.98918
\(846\) −1.53387e24 −0.200975
\(847\) 4.84823e24 0.628889
\(848\) −1.12465e24 −0.144428
\(849\) 1.38925e25 1.76630
\(850\) 4.31016e24 0.542540
\(851\) −4.51627e24 −0.562832
\(852\) 2.14564e24 0.264741
\(853\) 8.21174e24 1.00316 0.501578 0.865112i \(-0.332753\pi\)
0.501578 + 0.865112i \(0.332753\pi\)
\(854\) 5.61262e24 0.678850
\(855\) 6.47196e24 0.775040
\(856\) −1.20787e25 −1.43216
\(857\) 1.49858e24 0.175931 0.0879656 0.996124i \(-0.471963\pi\)
0.0879656 + 0.996124i \(0.471963\pi\)
\(858\) 1.72623e25 2.00659
\(859\) −5.73359e24 −0.659911 −0.329955 0.943997i \(-0.607034\pi\)
−0.329955 + 0.943997i \(0.607034\pi\)
\(860\) 7.96104e24 0.907262
\(861\) −8.67967e24 −0.979436
\(862\) 3.71475e23 0.0415067
\(863\) 9.31180e23 0.103025 0.0515124 0.998672i \(-0.483596\pi\)
0.0515124 + 0.998672i \(0.483596\pi\)
\(864\) −4.82962e24 −0.529110
\(865\) −1.99606e25 −2.16540
\(866\) −9.97651e24 −1.07171
\(867\) 2.98148e24 0.317153
\(868\) −3.15225e24 −0.332049
\(869\) 6.82899e24 0.712342
\(870\) 2.06180e24 0.212977
\(871\) 5.43178e24 0.555633
\(872\) −2.69611e23 −0.0273116
\(873\) 9.91294e24 0.994447
\(874\) 8.10233e24 0.804939
\(875\) −4.30916e23 −0.0423959
\(876\) −3.86287e24 −0.376379
\(877\) 1.12860e25 1.08904 0.544518 0.838749i \(-0.316712\pi\)
0.544518 + 0.838749i \(0.316712\pi\)
\(878\) −4.70986e24 −0.450096
\(879\) 1.12476e25 1.06452
\(880\) −2.23683e24 −0.209667
\(881\) −1.33735e25 −1.24151 −0.620756 0.784004i \(-0.713174\pi\)
−0.620756 + 0.784004i \(0.713174\pi\)
\(882\) 2.07724e24 0.190987
\(883\) 1.98397e25 1.80663 0.903316 0.428976i \(-0.141126\pi\)
0.903316 + 0.428976i \(0.141126\pi\)
\(884\) 9.64092e24 0.869509
\(885\) −1.57502e25 −1.40691
\(886\) −3.95067e24 −0.349530
\(887\) −2.20564e24 −0.193279 −0.0966395 0.995319i \(-0.530809\pi\)
−0.0966395 + 0.995319i \(0.530809\pi\)
\(888\) −6.75944e24 −0.586679
\(889\) −5.00405e24 −0.430186
\(890\) 1.05555e25 0.898799
\(891\) −2.04203e25 −1.72227
\(892\) −3.77519e24 −0.315383
\(893\) 6.57889e24 0.544396
\(894\) −3.28977e24 −0.269648
\(895\) 3.34643e25 2.71698
\(896\) 4.07810e24 0.327974
\(897\) 3.42417e25 2.72784
\(898\) −1.24914e25 −0.985740
\(899\) 2.00503e24 0.156734
\(900\) −3.91673e24 −0.303293
\(901\) −1.49863e25 −1.14957
\(902\) −1.35414e25 −1.02898
\(903\) 1.30973e25 0.985906
\(904\) 1.22958e24 0.0916903
\(905\) −1.09138e25 −0.806236
\(906\) −4.81805e24 −0.352599
\(907\) 3.48076e24 0.252355 0.126177 0.992008i \(-0.459729\pi\)
0.126177 + 0.992008i \(0.459729\pi\)
\(908\) −4.69545e24 −0.337247
\(909\) −1.47287e25 −1.04802
\(910\) −1.59790e25 −1.12641
\(911\) 2.83850e24 0.198236 0.0991181 0.995076i \(-0.468398\pi\)
0.0991181 + 0.995076i \(0.468398\pi\)
\(912\) 1.94388e24 0.134497
\(913\) −1.03284e25 −0.707996
\(914\) 7.27032e24 0.493754
\(915\) 3.87372e25 2.60645
\(916\) 1.17481e25 0.783170
\(917\) 1.23342e25 0.814649
\(918\) 4.78333e24 0.313017
\(919\) −1.30046e25 −0.843173 −0.421587 0.906788i \(-0.638527\pi\)
−0.421587 + 0.906788i \(0.638527\pi\)
\(920\) −2.76797e25 −1.77813
\(921\) −9.72429e24 −0.618943
\(922\) −1.40045e24 −0.0883190
\(923\) 1.05770e25 0.660916
\(924\) 1.09572e25 0.678406
\(925\) 7.11005e24 0.436182
\(926\) 7.39802e24 0.449699
\(927\) −8.51641e23 −0.0512955
\(928\) −2.98414e24 −0.178099
\(929\) 1.92619e25 1.13911 0.569555 0.821954i \(-0.307116\pi\)
0.569555 + 0.821954i \(0.307116\pi\)
\(930\) 1.65196e25 0.968041
\(931\) −8.90940e24 −0.517340
\(932\) −2.24081e24 −0.128935
\(933\) −8.94274e24 −0.509890
\(934\) −7.58719e24 −0.428679
\(935\) −2.98066e25 −1.66884
\(936\) 1.83554e25 1.01840
\(937\) 3.06535e25 1.68536 0.842681 0.538413i \(-0.180976\pi\)
0.842681 + 0.538413i \(0.180976\pi\)
\(938\) −2.61794e24 −0.142638
\(939\) 1.93724e25 1.04598
\(940\) −8.14523e24 −0.435830
\(941\) −5.57666e23 −0.0295708 −0.0147854 0.999891i \(-0.504707\pi\)
−0.0147854 + 0.999891i \(0.504707\pi\)
\(942\) 2.77255e25 1.45696
\(943\) −2.68608e25 −1.39884
\(944\) −1.69433e24 −0.0874450
\(945\) 1.04411e25 0.534041
\(946\) 2.04334e25 1.03578
\(947\) −2.97873e25 −1.49643 −0.748215 0.663456i \(-0.769089\pi\)
−0.748215 + 0.663456i \(0.769089\pi\)
\(948\) 7.34756e24 0.365824
\(949\) −1.90420e25 −0.939617
\(950\) −1.27557e25 −0.623810
\(951\) 2.07142e25 1.00400
\(952\) −1.28214e25 −0.615916
\(953\) −2.28962e25 −1.09012 −0.545059 0.838398i \(-0.683493\pi\)
−0.545059 + 0.838398i \(0.683493\pi\)
\(954\) −1.03405e25 −0.487956
\(955\) 9.19441e24 0.430027
\(956\) 1.03002e25 0.477476
\(957\) −6.96950e24 −0.320221
\(958\) 1.51130e25 0.688245
\(959\) 2.96341e25 1.33762
\(960\) −2.88206e25 −1.28943
\(961\) −6.48541e24 −0.287600
\(962\) −1.20757e25 −0.530794
\(963\) −1.78021e25 −0.775619
\(964\) −1.17458e25 −0.507258
\(965\) −5.26573e25 −2.25412
\(966\) −1.65034e25 −0.700273
\(967\) 3.33756e25 1.40380 0.701898 0.712278i \(-0.252336\pi\)
0.701898 + 0.712278i \(0.252336\pi\)
\(968\) 2.24558e25 0.936244
\(969\) 2.59028e25 1.07052
\(970\) −3.99699e25 −1.63747
\(971\) 3.17068e25 1.28762 0.643812 0.765184i \(-0.277352\pi\)
0.643812 + 0.765184i \(0.277352\pi\)
\(972\) −1.41814e25 −0.570896
\(973\) −3.05585e24 −0.121948
\(974\) −4.10508e24 −0.162394
\(975\) −5.39073e25 −2.11402
\(976\) 4.16716e24 0.162000
\(977\) −2.62945e25 −1.01335 −0.506676 0.862136i \(-0.669126\pi\)
−0.506676 + 0.862136i \(0.669126\pi\)
\(978\) 2.49163e25 0.951928
\(979\) −3.56807e25 −1.35139
\(980\) 1.10306e25 0.414170
\(981\) −3.97366e23 −0.0147912
\(982\) −8.28147e24 −0.305604
\(983\) 1.91222e25 0.699572 0.349786 0.936830i \(-0.386254\pi\)
0.349786 + 0.936830i \(0.386254\pi\)
\(984\) −4.02021e25 −1.45811
\(985\) −2.21777e25 −0.797460
\(986\) 2.95554e24 0.105362
\(987\) −1.34003e25 −0.473609
\(988\) −2.85317e25 −0.999756
\(989\) 4.05319e25 1.40808
\(990\) −2.05664e25 −0.708369
\(991\) −1.64728e24 −0.0562524 −0.0281262 0.999604i \(-0.508954\pi\)
−0.0281262 + 0.999604i \(0.508954\pi\)
\(992\) −2.39095e25 −0.809510
\(993\) −2.22256e25 −0.746080
\(994\) −5.09775e24 −0.169666
\(995\) 2.74960e25 0.907347
\(996\) −1.11127e25 −0.363592
\(997\) −9.66652e24 −0.313589 −0.156795 0.987631i \(-0.550116\pi\)
−0.156795 + 0.987631i \(0.550116\pi\)
\(998\) −1.35656e25 −0.436344
\(999\) 7.89059e24 0.251654
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 29.18.a.b.1.8 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.18.a.b.1.8 21 1.1 even 1 trivial