Properties

Label 15.10.a.a.1.1
Level 15
Weight 10
Character 15.1
Self dual yes
Analytic conductor 7.726
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.72553754246\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 15.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +81.0000 q^{3} -496.000 q^{4} +625.000 q^{5} -324.000 q^{6} -7680.00 q^{7} +4032.00 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +81.0000 q^{3} -496.000 q^{4} +625.000 q^{5} -324.000 q^{6} -7680.00 q^{7} +4032.00 q^{8} +6561.00 q^{9} -2500.00 q^{10} -86404.0 q^{11} -40176.0 q^{12} -149978. q^{13} +30720.0 q^{14} +50625.0 q^{15} +237824. q^{16} -207622. q^{17} -26244.0 q^{18} +716284. q^{19} -310000. q^{20} -622080. q^{21} +345616. q^{22} +1.36992e6 q^{23} +326592. q^{24} +390625. q^{25} +599912. q^{26} +531441. q^{27} +3.80928e6 q^{28} -3.19440e6 q^{29} -202500. q^{30} -2.34900e6 q^{31} -3.01568e6 q^{32} -6.99872e6 q^{33} +830488. q^{34} -4.80000e6 q^{35} -3.25426e6 q^{36} +1.87357e7 q^{37} -2.86514e6 q^{38} -1.21482e7 q^{39} +2.52000e6 q^{40} -2.92826e7 q^{41} +2.48832e6 q^{42} -1.51672e6 q^{43} +4.28564e7 q^{44} +4.10063e6 q^{45} -5.47968e6 q^{46} +615752. q^{47} +1.92637e7 q^{48} +1.86288e7 q^{49} -1.56250e6 q^{50} -1.68174e7 q^{51} +7.43891e7 q^{52} +4.74743e6 q^{53} -2.12576e6 q^{54} -5.40025e7 q^{55} -3.09658e7 q^{56} +5.80190e7 q^{57} +1.27776e7 q^{58} +6.06161e7 q^{59} -2.51100e7 q^{60} -1.26746e8 q^{61} +9.39600e6 q^{62} -5.03885e7 q^{63} -1.09703e8 q^{64} -9.37363e7 q^{65} +2.79949e7 q^{66} -1.11183e8 q^{67} +1.02981e8 q^{68} +1.10964e8 q^{69} +1.92000e7 q^{70} -1.75552e8 q^{71} +2.64540e7 q^{72} -6.12334e7 q^{73} -7.49428e7 q^{74} +3.16406e7 q^{75} -3.55277e8 q^{76} +6.63583e8 q^{77} +4.85929e7 q^{78} +2.34431e8 q^{79} +1.48640e8 q^{80} +4.30467e7 q^{81} +1.17131e8 q^{82} +1.18910e8 q^{83} +3.08552e8 q^{84} -1.29764e8 q^{85} +6.06690e6 q^{86} -2.58747e8 q^{87} -3.48381e8 q^{88} -3.16534e8 q^{89} -1.64025e7 q^{90} +1.15183e9 q^{91} -6.79480e8 q^{92} -1.90269e8 q^{93} -2.46301e6 q^{94} +4.47678e8 q^{95} -2.44270e8 q^{96} +2.42912e8 q^{97} -7.45152e7 q^{98} -5.66897e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.176777 −0.0883883 0.996086i \(-0.528172\pi\)
−0.0883883 + 0.996086i \(0.528172\pi\)
\(3\) 81.0000 0.577350
\(4\) −496.000 −0.968750
\(5\) 625.000 0.447214
\(6\) −324.000 −0.102062
\(7\) −7680.00 −1.20898 −0.604491 0.796612i \(-0.706624\pi\)
−0.604491 + 0.796612i \(0.706624\pi\)
\(8\) 4032.00 0.348029
\(9\) 6561.00 0.333333
\(10\) −2500.00 −0.0790569
\(11\) −86404.0 −1.77937 −0.889686 0.456573i \(-0.849077\pi\)
−0.889686 + 0.456573i \(0.849077\pi\)
\(12\) −40176.0 −0.559308
\(13\) −149978. −1.45641 −0.728203 0.685361i \(-0.759644\pi\)
−0.728203 + 0.685361i \(0.759644\pi\)
\(14\) 30720.0 0.213720
\(15\) 50625.0 0.258199
\(16\) 237824. 0.907227
\(17\) −207622. −0.602911 −0.301456 0.953480i \(-0.597472\pi\)
−0.301456 + 0.953480i \(0.597472\pi\)
\(18\) −26244.0 −0.0589256
\(19\) 716284. 1.26094 0.630469 0.776214i \(-0.282862\pi\)
0.630469 + 0.776214i \(0.282862\pi\)
\(20\) −310000. −0.433238
\(21\) −622080. −0.698006
\(22\) 345616. 0.314552
\(23\) 1.36992e6 1.02075 0.510376 0.859952i \(-0.329506\pi\)
0.510376 + 0.859952i \(0.329506\pi\)
\(24\) 326592. 0.200935
\(25\) 390625. 0.200000
\(26\) 599912. 0.257459
\(27\) 531441. 0.192450
\(28\) 3.80928e6 1.17120
\(29\) −3.19440e6 −0.838684 −0.419342 0.907828i \(-0.637739\pi\)
−0.419342 + 0.907828i \(0.637739\pi\)
\(30\) −202500. −0.0456435
\(31\) −2.34900e6 −0.456831 −0.228415 0.973564i \(-0.573354\pi\)
−0.228415 + 0.973564i \(0.573354\pi\)
\(32\) −3.01568e6 −0.508406
\(33\) −6.99872e6 −1.02732
\(34\) 830488. 0.106581
\(35\) −4.80000e6 −0.540673
\(36\) −3.25426e6 −0.322917
\(37\) 1.87357e7 1.64347 0.821736 0.569868i \(-0.193006\pi\)
0.821736 + 0.569868i \(0.193006\pi\)
\(38\) −2.86514e6 −0.222905
\(39\) −1.21482e7 −0.840856
\(40\) 2.52000e6 0.155643
\(41\) −2.92826e7 −1.61839 −0.809194 0.587541i \(-0.800096\pi\)
−0.809194 + 0.587541i \(0.800096\pi\)
\(42\) 2.48832e6 0.123391
\(43\) −1.51672e6 −0.0676548 −0.0338274 0.999428i \(-0.510770\pi\)
−0.0338274 + 0.999428i \(0.510770\pi\)
\(44\) 4.28564e7 1.72377
\(45\) 4.10063e6 0.149071
\(46\) −5.47968e6 −0.180445
\(47\) 615752. 0.0184063 0.00920313 0.999958i \(-0.497071\pi\)
0.00920313 + 0.999958i \(0.497071\pi\)
\(48\) 1.92637e7 0.523788
\(49\) 1.86288e7 0.461639
\(50\) −1.56250e6 −0.0353553
\(51\) −1.68174e7 −0.348091
\(52\) 7.43891e7 1.41089
\(53\) 4.74743e6 0.0826451 0.0413226 0.999146i \(-0.486843\pi\)
0.0413226 + 0.999146i \(0.486843\pi\)
\(54\) −2.12576e6 −0.0340207
\(55\) −5.40025e7 −0.795759
\(56\) −3.09658e7 −0.420761
\(57\) 5.80190e7 0.728003
\(58\) 1.27776e7 0.148260
\(59\) 6.06161e7 0.651259 0.325630 0.945497i \(-0.394424\pi\)
0.325630 + 0.945497i \(0.394424\pi\)
\(60\) −2.51100e7 −0.250130
\(61\) −1.26746e8 −1.17206 −0.586029 0.810290i \(-0.699309\pi\)
−0.586029 + 0.810290i \(0.699309\pi\)
\(62\) 9.39600e6 0.0807570
\(63\) −5.03885e7 −0.402994
\(64\) −1.09703e8 −0.817352
\(65\) −9.37363e7 −0.651325
\(66\) 2.79949e7 0.181606
\(67\) −1.11183e8 −0.674063 −0.337031 0.941493i \(-0.609423\pi\)
−0.337031 + 0.941493i \(0.609423\pi\)
\(68\) 1.02981e8 0.584070
\(69\) 1.10964e8 0.589331
\(70\) 1.92000e7 0.0955785
\(71\) −1.75552e8 −0.819865 −0.409932 0.912116i \(-0.634448\pi\)
−0.409932 + 0.912116i \(0.634448\pi\)
\(72\) 2.64540e7 0.116010
\(73\) −6.12334e7 −0.252369 −0.126184 0.992007i \(-0.540273\pi\)
−0.126184 + 0.992007i \(0.540273\pi\)
\(74\) −7.49428e7 −0.290528
\(75\) 3.16406e7 0.115470
\(76\) −3.55277e8 −1.22153
\(77\) 6.63583e8 2.15123
\(78\) 4.85929e7 0.148644
\(79\) 2.34431e8 0.677163 0.338582 0.940937i \(-0.390053\pi\)
0.338582 + 0.940937i \(0.390053\pi\)
\(80\) 1.48640e8 0.405724
\(81\) 4.30467e7 0.111111
\(82\) 1.17131e8 0.286093
\(83\) 1.18910e8 0.275023 0.137511 0.990500i \(-0.456090\pi\)
0.137511 + 0.990500i \(0.456090\pi\)
\(84\) 3.08552e8 0.676194
\(85\) −1.29764e8 −0.269630
\(86\) 6.06690e6 0.0119598
\(87\) −2.58747e8 −0.484215
\(88\) −3.48381e8 −0.619273
\(89\) −3.16534e8 −0.534768 −0.267384 0.963590i \(-0.586159\pi\)
−0.267384 + 0.963590i \(0.586159\pi\)
\(90\) −1.64025e7 −0.0263523
\(91\) 1.15183e9 1.76077
\(92\) −6.79480e8 −0.988853
\(93\) −1.90269e8 −0.263751
\(94\) −2.46301e6 −0.00325380
\(95\) 4.47678e8 0.563909
\(96\) −2.44270e8 −0.293528
\(97\) 2.42912e8 0.278597 0.139299 0.990250i \(-0.455515\pi\)
0.139299 + 0.990250i \(0.455515\pi\)
\(98\) −7.45152e7 −0.0816070
\(99\) −5.66897e8 −0.593124
\(100\) −1.93750e8 −0.193750
\(101\) −6.53803e8 −0.625173 −0.312587 0.949889i \(-0.601195\pi\)
−0.312587 + 0.949889i \(0.601195\pi\)
\(102\) 6.72695e7 0.0615343
\(103\) 1.40420e9 1.22931 0.614656 0.788795i \(-0.289295\pi\)
0.614656 + 0.788795i \(0.289295\pi\)
\(104\) −6.04711e8 −0.506872
\(105\) −3.88800e8 −0.312158
\(106\) −1.89897e7 −0.0146097
\(107\) −1.83854e9 −1.35595 −0.677977 0.735083i \(-0.737143\pi\)
−0.677977 + 0.735083i \(0.737143\pi\)
\(108\) −2.63595e8 −0.186436
\(109\) −9.33452e8 −0.633392 −0.316696 0.948527i \(-0.602574\pi\)
−0.316696 + 0.948527i \(0.602574\pi\)
\(110\) 2.16010e8 0.140672
\(111\) 1.51759e9 0.948859
\(112\) −1.82649e9 −1.09682
\(113\) −9.28534e7 −0.0535728 −0.0267864 0.999641i \(-0.508527\pi\)
−0.0267864 + 0.999641i \(0.508527\pi\)
\(114\) −2.32076e8 −0.128694
\(115\) 8.56200e8 0.456494
\(116\) 1.58442e9 0.812476
\(117\) −9.84006e8 −0.485469
\(118\) −2.42464e8 −0.115127
\(119\) 1.59454e9 0.728909
\(120\) 2.04120e8 0.0898607
\(121\) 5.10770e9 2.16616
\(122\) 5.06983e8 0.207192
\(123\) −2.37189e9 −0.934377
\(124\) 1.16510e9 0.442555
\(125\) 2.44141e8 0.0894427
\(126\) 2.01554e8 0.0712400
\(127\) 1.73819e9 0.592900 0.296450 0.955048i \(-0.404197\pi\)
0.296450 + 0.955048i \(0.404197\pi\)
\(128\) 1.98284e9 0.652894
\(129\) −1.22855e8 −0.0390605
\(130\) 3.74945e8 0.115139
\(131\) −2.49730e9 −0.740882 −0.370441 0.928856i \(-0.620793\pi\)
−0.370441 + 0.928856i \(0.620793\pi\)
\(132\) 3.47137e9 0.995217
\(133\) −5.50106e9 −1.52445
\(134\) 4.44731e8 0.119159
\(135\) 3.32151e8 0.0860663
\(136\) −8.37132e8 −0.209831
\(137\) −7.96226e9 −1.93105 −0.965526 0.260306i \(-0.916177\pi\)
−0.965526 + 0.260306i \(0.916177\pi\)
\(138\) −4.43854e8 −0.104180
\(139\) −2.85565e9 −0.648842 −0.324421 0.945913i \(-0.605169\pi\)
−0.324421 + 0.945913i \(0.605169\pi\)
\(140\) 2.38080e9 0.523777
\(141\) 4.98759e7 0.0106269
\(142\) 7.02206e8 0.144933
\(143\) 1.29587e10 2.59149
\(144\) 1.56036e9 0.302409
\(145\) −1.99650e9 −0.375071
\(146\) 2.44933e8 0.0446129
\(147\) 1.50893e9 0.266527
\(148\) −9.29291e9 −1.59211
\(149\) −9.63383e9 −1.60126 −0.800628 0.599161i \(-0.795501\pi\)
−0.800628 + 0.599161i \(0.795501\pi\)
\(150\) −1.26563e8 −0.0204124
\(151\) −5.38292e9 −0.842601 −0.421300 0.906921i \(-0.638426\pi\)
−0.421300 + 0.906921i \(0.638426\pi\)
\(152\) 2.88806e9 0.438843
\(153\) −1.36221e9 −0.200970
\(154\) −2.65433e9 −0.380287
\(155\) −1.46812e9 −0.204301
\(156\) 6.02552e9 0.814580
\(157\) 5.19434e8 0.0682310 0.0341155 0.999418i \(-0.489139\pi\)
0.0341155 + 0.999418i \(0.489139\pi\)
\(158\) −9.37725e8 −0.119707
\(159\) 3.84542e8 0.0477152
\(160\) −1.88480e9 −0.227366
\(161\) −1.05210e10 −1.23407
\(162\) −1.72187e8 −0.0196419
\(163\) 9.41239e9 1.04437 0.522187 0.852831i \(-0.325116\pi\)
0.522187 + 0.852831i \(0.325116\pi\)
\(164\) 1.45242e10 1.56781
\(165\) −4.37420e9 −0.459432
\(166\) −4.75642e8 −0.0486176
\(167\) 9.37241e9 0.932453 0.466227 0.884665i \(-0.345613\pi\)
0.466227 + 0.884665i \(0.345613\pi\)
\(168\) −2.50823e9 −0.242927
\(169\) 1.18889e10 1.12112
\(170\) 5.19055e8 0.0476643
\(171\) 4.69954e9 0.420313
\(172\) 7.52295e8 0.0655406
\(173\) 1.23573e10 1.04886 0.524428 0.851455i \(-0.324279\pi\)
0.524428 + 0.851455i \(0.324279\pi\)
\(174\) 1.03499e9 0.0855979
\(175\) −3.00000e9 −0.241797
\(176\) −2.05489e10 −1.61429
\(177\) 4.90990e9 0.376005
\(178\) 1.26614e9 0.0945346
\(179\) −6.66040e8 −0.0484910 −0.0242455 0.999706i \(-0.507718\pi\)
−0.0242455 + 0.999706i \(0.507718\pi\)
\(180\) −2.03391e9 −0.144413
\(181\) 5.27207e9 0.365113 0.182557 0.983195i \(-0.441563\pi\)
0.182557 + 0.983195i \(0.441563\pi\)
\(182\) −4.60732e9 −0.311263
\(183\) −1.02664e10 −0.676688
\(184\) 5.52352e9 0.355251
\(185\) 1.17098e10 0.734983
\(186\) 7.61076e8 0.0466251
\(187\) 1.79394e10 1.07280
\(188\) −3.05413e8 −0.0178311
\(189\) −4.08147e9 −0.232669
\(190\) −1.79071e9 −0.0996860
\(191\) −2.93896e10 −1.59788 −0.798939 0.601412i \(-0.794605\pi\)
−0.798939 + 0.601412i \(0.794605\pi\)
\(192\) −8.88596e9 −0.471899
\(193\) −1.48746e10 −0.771681 −0.385841 0.922565i \(-0.626089\pi\)
−0.385841 + 0.922565i \(0.626089\pi\)
\(194\) −9.71649e8 −0.0492495
\(195\) −7.59264e9 −0.376042
\(196\) −9.23988e9 −0.447213
\(197\) 4.98675e9 0.235895 0.117948 0.993020i \(-0.462368\pi\)
0.117948 + 0.993020i \(0.462368\pi\)
\(198\) 2.26759e9 0.104851
\(199\) 1.45527e10 0.657816 0.328908 0.944362i \(-0.393319\pi\)
0.328908 + 0.944362i \(0.393319\pi\)
\(200\) 1.57500e9 0.0696058
\(201\) −9.00579e9 −0.389170
\(202\) 2.61521e9 0.110516
\(203\) 2.45330e10 1.01395
\(204\) 8.34142e9 0.337213
\(205\) −1.83016e10 −0.723765
\(206\) −5.61681e9 −0.217314
\(207\) 8.98805e9 0.340250
\(208\) −3.56684e10 −1.32129
\(209\) −6.18898e10 −2.24368
\(210\) 1.55520e9 0.0551823
\(211\) 5.15407e10 1.79011 0.895054 0.445959i \(-0.147137\pi\)
0.895054 + 0.445959i \(0.147137\pi\)
\(212\) −2.35473e9 −0.0800624
\(213\) −1.42197e10 −0.473349
\(214\) 7.35414e9 0.239701
\(215\) −9.47952e8 −0.0302561
\(216\) 2.14277e9 0.0669782
\(217\) 1.80403e10 0.552300
\(218\) 3.73381e9 0.111969
\(219\) −4.95990e9 −0.145705
\(220\) 2.67852e10 0.770892
\(221\) 3.11387e10 0.878083
\(222\) −6.07037e9 −0.167736
\(223\) 4.61272e10 1.24907 0.624533 0.780998i \(-0.285289\pi\)
0.624533 + 0.780998i \(0.285289\pi\)
\(224\) 2.31604e10 0.614654
\(225\) 2.56289e9 0.0666667
\(226\) 3.71414e8 0.00947043
\(227\) −3.75833e10 −0.939460 −0.469730 0.882810i \(-0.655649\pi\)
−0.469730 + 0.882810i \(0.655649\pi\)
\(228\) −2.87774e10 −0.705253
\(229\) −6.41082e10 −1.54047 −0.770236 0.637759i \(-0.779862\pi\)
−0.770236 + 0.637759i \(0.779862\pi\)
\(230\) −3.42480e9 −0.0806975
\(231\) 5.37502e10 1.24201
\(232\) −1.28798e10 −0.291887
\(233\) 6.96578e10 1.54835 0.774174 0.632973i \(-0.218166\pi\)
0.774174 + 0.632973i \(0.218166\pi\)
\(234\) 3.93602e9 0.0858195
\(235\) 3.84845e8 0.00823153
\(236\) −3.00656e10 −0.630907
\(237\) 1.89889e10 0.390960
\(238\) −6.37815e9 −0.128854
\(239\) 6.65825e10 1.31999 0.659993 0.751272i \(-0.270559\pi\)
0.659993 + 0.751272i \(0.270559\pi\)
\(240\) 1.20398e10 0.234245
\(241\) −4.41659e10 −0.843354 −0.421677 0.906746i \(-0.638558\pi\)
−0.421677 + 0.906746i \(0.638558\pi\)
\(242\) −2.04308e10 −0.382927
\(243\) 3.48678e9 0.0641500
\(244\) 6.28659e10 1.13543
\(245\) 1.16430e10 0.206451
\(246\) 9.48757e9 0.165176
\(247\) −1.07427e11 −1.83644
\(248\) −9.47117e9 −0.158990
\(249\) 9.63174e9 0.158784
\(250\) −9.76563e8 −0.0158114
\(251\) 8.36236e10 1.32983 0.664916 0.746918i \(-0.268467\pi\)
0.664916 + 0.746918i \(0.268467\pi\)
\(252\) 2.49927e10 0.390401
\(253\) −1.18367e11 −1.81630
\(254\) −6.95277e9 −0.104811
\(255\) −1.05109e10 −0.155671
\(256\) 4.82367e10 0.701936
\(257\) −8.65274e10 −1.23724 −0.618621 0.785690i \(-0.712308\pi\)
−0.618621 + 0.785690i \(0.712308\pi\)
\(258\) 4.91419e8 0.00690499
\(259\) −1.43890e11 −1.98693
\(260\) 4.64932e10 0.630971
\(261\) −2.09585e10 −0.279561
\(262\) 9.98919e9 0.130971
\(263\) −9.61535e10 −1.23927 −0.619633 0.784892i \(-0.712718\pi\)
−0.619633 + 0.784892i \(0.712718\pi\)
\(264\) −2.82189e10 −0.357538
\(265\) 2.96714e9 0.0369600
\(266\) 2.20042e10 0.269488
\(267\) −2.56393e10 −0.308749
\(268\) 5.51466e10 0.652998
\(269\) −1.09505e10 −0.127511 −0.0637557 0.997966i \(-0.520308\pi\)
−0.0637557 + 0.997966i \(0.520308\pi\)
\(270\) −1.32860e9 −0.0152145
\(271\) 7.80287e10 0.878805 0.439403 0.898290i \(-0.355190\pi\)
0.439403 + 0.898290i \(0.355190\pi\)
\(272\) −4.93775e10 −0.546977
\(273\) 9.32983e10 1.01658
\(274\) 3.18491e10 0.341365
\(275\) −3.37516e10 −0.355874
\(276\) −5.50379e10 −0.570914
\(277\) 6.56840e10 0.670349 0.335174 0.942156i \(-0.391205\pi\)
0.335174 + 0.942156i \(0.391205\pi\)
\(278\) 1.14226e10 0.114700
\(279\) −1.54118e10 −0.152277
\(280\) −1.93536e10 −0.188170
\(281\) −6.44906e10 −0.617046 −0.308523 0.951217i \(-0.599835\pi\)
−0.308523 + 0.951217i \(0.599835\pi\)
\(282\) −1.99504e8 −0.00187858
\(283\) 9.63133e10 0.892580 0.446290 0.894888i \(-0.352745\pi\)
0.446290 + 0.894888i \(0.352745\pi\)
\(284\) 8.70736e10 0.794244
\(285\) 3.62619e10 0.325573
\(286\) −5.18348e10 −0.458115
\(287\) 2.24891e11 1.95660
\(288\) −1.97859e10 −0.169469
\(289\) −7.54810e10 −0.636498
\(290\) 7.98601e9 0.0663038
\(291\) 1.96759e10 0.160848
\(292\) 3.03717e10 0.244482
\(293\) 8.16308e10 0.647068 0.323534 0.946217i \(-0.395129\pi\)
0.323534 + 0.946217i \(0.395129\pi\)
\(294\) −6.03573e9 −0.0471158
\(295\) 3.78850e10 0.291252
\(296\) 7.55424e10 0.571976
\(297\) −4.59186e10 −0.342440
\(298\) 3.85353e10 0.283065
\(299\) −2.05458e11 −1.48663
\(300\) −1.56938e10 −0.111862
\(301\) 1.16484e10 0.0817935
\(302\) 2.15317e10 0.148952
\(303\) −5.29580e10 −0.360944
\(304\) 1.70350e11 1.14396
\(305\) −7.92161e10 −0.524160
\(306\) 5.44883e9 0.0355269
\(307\) −2.95582e10 −0.189914 −0.0949568 0.995481i \(-0.530271\pi\)
−0.0949568 + 0.995481i \(0.530271\pi\)
\(308\) −3.29137e11 −2.08400
\(309\) 1.13740e11 0.709744
\(310\) 5.87250e9 0.0361156
\(311\) −3.99071e10 −0.241896 −0.120948 0.992659i \(-0.538593\pi\)
−0.120948 + 0.992659i \(0.538593\pi\)
\(312\) −4.89816e10 −0.292643
\(313\) 1.85371e11 1.09167 0.545836 0.837892i \(-0.316212\pi\)
0.545836 + 0.837892i \(0.316212\pi\)
\(314\) −2.07774e9 −0.0120617
\(315\) −3.14928e10 −0.180224
\(316\) −1.16278e11 −0.656002
\(317\) 2.68895e11 1.49560 0.747800 0.663924i \(-0.231110\pi\)
0.747800 + 0.663924i \(0.231110\pi\)
\(318\) −1.53817e9 −0.00843493
\(319\) 2.76009e11 1.49233
\(320\) −6.85645e10 −0.365531
\(321\) −1.48921e11 −0.782860
\(322\) 4.20839e10 0.218155
\(323\) −1.48716e11 −0.760234
\(324\) −2.13512e10 −0.107639
\(325\) −5.85852e10 −0.291281
\(326\) −3.76496e10 −0.184621
\(327\) −7.56096e10 −0.365689
\(328\) −1.18068e11 −0.563246
\(329\) −4.72898e9 −0.0222528
\(330\) 1.74968e10 0.0812169
\(331\) −4.29099e11 −1.96486 −0.982430 0.186629i \(-0.940244\pi\)
−0.982430 + 0.186629i \(0.940244\pi\)
\(332\) −5.89796e10 −0.266428
\(333\) 1.22925e11 0.547824
\(334\) −3.74896e10 −0.164836
\(335\) −6.94892e10 −0.301450
\(336\) −1.47946e11 −0.633250
\(337\) −2.02598e10 −0.0855657 −0.0427828 0.999084i \(-0.513622\pi\)
−0.0427828 + 0.999084i \(0.513622\pi\)
\(338\) −4.75556e10 −0.198188
\(339\) −7.52112e9 −0.0309303
\(340\) 6.43628e10 0.261204
\(341\) 2.02963e11 0.812872
\(342\) −1.87982e10 −0.0743015
\(343\) 1.66847e11 0.650869
\(344\) −6.11543e9 −0.0235458
\(345\) 6.93522e10 0.263557
\(346\) −4.94292e10 −0.185413
\(347\) 2.92783e10 0.108409 0.0542043 0.998530i \(-0.482738\pi\)
0.0542043 + 0.998530i \(0.482738\pi\)
\(348\) 1.28338e11 0.469083
\(349\) 7.05132e10 0.254423 0.127211 0.991876i \(-0.459397\pi\)
0.127211 + 0.991876i \(0.459397\pi\)
\(350\) 1.20000e10 0.0427440
\(351\) −7.97045e10 −0.280285
\(352\) 2.60567e11 0.904643
\(353\) −6.57350e10 −0.225325 −0.112663 0.993633i \(-0.535938\pi\)
−0.112663 + 0.993633i \(0.535938\pi\)
\(354\) −1.96396e10 −0.0664688
\(355\) −1.09720e11 −0.366655
\(356\) 1.57001e11 0.518057
\(357\) 1.29157e11 0.420836
\(358\) 2.66416e9 0.00857209
\(359\) 5.81702e11 1.84831 0.924157 0.382013i \(-0.124769\pi\)
0.924157 + 0.382013i \(0.124769\pi\)
\(360\) 1.65337e10 0.0518811
\(361\) 1.90375e11 0.589967
\(362\) −2.10883e10 −0.0645435
\(363\) 4.13724e11 1.25064
\(364\) −5.71308e11 −1.70575
\(365\) −3.82708e10 −0.112863
\(366\) 4.10656e10 0.119623
\(367\) −4.17070e11 −1.20008 −0.600042 0.799969i \(-0.704849\pi\)
−0.600042 + 0.799969i \(0.704849\pi\)
\(368\) 3.25800e11 0.926053
\(369\) −1.92123e11 −0.539463
\(370\) −4.68393e10 −0.129928
\(371\) −3.64603e10 −0.0999165
\(372\) 9.43734e10 0.255509
\(373\) −7.60417e10 −0.203405 −0.101703 0.994815i \(-0.532429\pi\)
−0.101703 + 0.994815i \(0.532429\pi\)
\(374\) −7.17575e10 −0.189647
\(375\) 1.97754e10 0.0516398
\(376\) 2.48271e9 0.00640591
\(377\) 4.79090e11 1.22147
\(378\) 1.63259e10 0.0411304
\(379\) −1.79180e11 −0.446080 −0.223040 0.974809i \(-0.571598\pi\)
−0.223040 + 0.974809i \(0.571598\pi\)
\(380\) −2.22048e11 −0.546287
\(381\) 1.40794e11 0.342311
\(382\) 1.17558e11 0.282467
\(383\) −7.95018e11 −1.88792 −0.943958 0.330066i \(-0.892929\pi\)
−0.943958 + 0.330066i \(0.892929\pi\)
\(384\) 1.60610e11 0.376949
\(385\) 4.14739e11 0.962059
\(386\) 5.94985e10 0.136415
\(387\) −9.95123e9 −0.0225516
\(388\) −1.20484e11 −0.269891
\(389\) −1.79533e11 −0.397532 −0.198766 0.980047i \(-0.563693\pi\)
−0.198766 + 0.980047i \(0.563693\pi\)
\(390\) 3.03705e10 0.0664755
\(391\) −2.84426e11 −0.615422
\(392\) 7.51113e10 0.160664
\(393\) −2.02281e11 −0.427749
\(394\) −1.99470e10 −0.0417008
\(395\) 1.46519e11 0.302837
\(396\) 2.81181e11 0.574589
\(397\) −3.43730e11 −0.694480 −0.347240 0.937776i \(-0.612881\pi\)
−0.347240 + 0.937776i \(0.612881\pi\)
\(398\) −5.82108e10 −0.116287
\(399\) −4.45586e11 −0.880143
\(400\) 9.29000e10 0.181445
\(401\) 7.72080e11 1.49112 0.745560 0.666438i \(-0.232182\pi\)
0.745560 + 0.666438i \(0.232182\pi\)
\(402\) 3.60232e10 0.0687963
\(403\) 3.52298e11 0.665331
\(404\) 3.24286e11 0.605637
\(405\) 2.69042e10 0.0496904
\(406\) −9.81320e10 −0.179244
\(407\) −1.61884e12 −2.92435
\(408\) −6.78077e10 −0.121146
\(409\) 2.60632e11 0.460546 0.230273 0.973126i \(-0.426038\pi\)
0.230273 + 0.973126i \(0.426038\pi\)
\(410\) 7.32066e10 0.127945
\(411\) −6.44943e11 −1.11489
\(412\) −6.96484e11 −1.19090
\(413\) −4.65531e11 −0.787361
\(414\) −3.59522e10 −0.0601483
\(415\) 7.43190e10 0.122994
\(416\) 4.52286e11 0.740445
\(417\) −2.31308e11 −0.374609
\(418\) 2.47559e11 0.396630
\(419\) −5.60166e11 −0.887879 −0.443939 0.896057i \(-0.646419\pi\)
−0.443939 + 0.896057i \(0.646419\pi\)
\(420\) 1.92845e11 0.302403
\(421\) 1.68321e11 0.261137 0.130569 0.991439i \(-0.458320\pi\)
0.130569 + 0.991439i \(0.458320\pi\)
\(422\) −2.06163e11 −0.316449
\(423\) 4.03995e9 0.00613542
\(424\) 1.91416e10 0.0287629
\(425\) −8.11023e10 −0.120582
\(426\) 5.68787e10 0.0836771
\(427\) 9.73407e11 1.41700
\(428\) 9.11913e11 1.31358
\(429\) 1.04965e12 1.49620
\(430\) 3.79181e9 0.00534858
\(431\) 4.48383e11 0.625895 0.312948 0.949770i \(-0.398684\pi\)
0.312948 + 0.949770i \(0.398684\pi\)
\(432\) 1.26389e11 0.174596
\(433\) −1.08485e12 −1.48311 −0.741556 0.670891i \(-0.765912\pi\)
−0.741556 + 0.670891i \(0.765912\pi\)
\(434\) −7.21613e10 −0.0976339
\(435\) −1.61717e11 −0.216547
\(436\) 4.62992e11 0.613599
\(437\) 9.81252e11 1.28711
\(438\) 1.98396e10 0.0257573
\(439\) 4.60548e11 0.591814 0.295907 0.955217i \(-0.404378\pi\)
0.295907 + 0.955217i \(0.404378\pi\)
\(440\) −2.17738e11 −0.276947
\(441\) 1.22224e11 0.153880
\(442\) −1.24555e11 −0.155225
\(443\) −1.32095e10 −0.0162956 −0.00814779 0.999967i \(-0.502594\pi\)
−0.00814779 + 0.999967i \(0.502594\pi\)
\(444\) −7.52726e11 −0.919207
\(445\) −1.97834e11 −0.239156
\(446\) −1.84509e11 −0.220806
\(447\) −7.80341e11 −0.924486
\(448\) 8.42520e11 0.988165
\(449\) −6.91889e11 −0.803393 −0.401696 0.915773i \(-0.631579\pi\)
−0.401696 + 0.915773i \(0.631579\pi\)
\(450\) −1.02516e10 −0.0117851
\(451\) 2.53014e12 2.87971
\(452\) 4.60553e10 0.0518987
\(453\) −4.36017e11 −0.486476
\(454\) 1.50333e11 0.166075
\(455\) 7.19894e11 0.787440
\(456\) 2.33933e11 0.253366
\(457\) −3.73135e11 −0.400168 −0.200084 0.979779i \(-0.564122\pi\)
−0.200084 + 0.979779i \(0.564122\pi\)
\(458\) 2.56433e11 0.272320
\(459\) −1.10339e11 −0.116030
\(460\) −4.24675e11 −0.442228
\(461\) −1.45940e12 −1.50494 −0.752470 0.658627i \(-0.771138\pi\)
−0.752470 + 0.658627i \(0.771138\pi\)
\(462\) −2.15001e11 −0.219559
\(463\) 1.34213e11 0.135732 0.0678658 0.997694i \(-0.478381\pi\)
0.0678658 + 0.997694i \(0.478381\pi\)
\(464\) −7.59705e11 −0.760877
\(465\) −1.18918e11 −0.117953
\(466\) −2.78631e11 −0.273712
\(467\) −3.64531e10 −0.0354657 −0.0177329 0.999843i \(-0.505645\pi\)
−0.0177329 + 0.999843i \(0.505645\pi\)
\(468\) 4.88067e11 0.470298
\(469\) 8.53883e11 0.814930
\(470\) −1.53938e9 −0.00145514
\(471\) 4.20741e10 0.0393932
\(472\) 2.44404e11 0.226657
\(473\) 1.31051e11 0.120383
\(474\) −7.59557e10 −0.0691127
\(475\) 2.79798e11 0.252188
\(476\) −7.90890e11 −0.706131
\(477\) 3.11479e10 0.0275484
\(478\) −2.66330e11 −0.233343
\(479\) 8.82280e11 0.765767 0.382883 0.923797i \(-0.374931\pi\)
0.382883 + 0.923797i \(0.374931\pi\)
\(480\) −1.52669e11 −0.131270
\(481\) −2.80994e12 −2.39356
\(482\) 1.76663e11 0.149085
\(483\) −8.52200e11 −0.712491
\(484\) −2.53342e12 −2.09847
\(485\) 1.51820e11 0.124592
\(486\) −1.39471e10 −0.0113402
\(487\) 5.09840e11 0.410727 0.205364 0.978686i \(-0.434162\pi\)
0.205364 + 0.978686i \(0.434162\pi\)
\(488\) −5.11039e11 −0.407910
\(489\) 7.62404e11 0.602969
\(490\) −4.65720e10 −0.0364958
\(491\) −1.52131e12 −1.18128 −0.590638 0.806937i \(-0.701124\pi\)
−0.590638 + 0.806937i \(0.701124\pi\)
\(492\) 1.17646e12 0.905178
\(493\) 6.63228e11 0.505652
\(494\) 4.29707e11 0.324640
\(495\) −3.54310e11 −0.265253
\(496\) −5.58649e11 −0.414449
\(497\) 1.34824e12 0.991202
\(498\) −3.85270e10 −0.0280694
\(499\) 7.03413e11 0.507876 0.253938 0.967220i \(-0.418274\pi\)
0.253938 + 0.967220i \(0.418274\pi\)
\(500\) −1.21094e11 −0.0866476
\(501\) 7.59165e11 0.538352
\(502\) −3.34494e11 −0.235083
\(503\) 3.78018e8 0.000263304 0 0.000131652 1.00000i \(-0.499958\pi\)
0.000131652 1.00000i \(0.499958\pi\)
\(504\) −2.03166e11 −0.140254
\(505\) −4.08627e11 −0.279586
\(506\) 4.73466e11 0.321079
\(507\) 9.63001e11 0.647278
\(508\) −8.62144e11 −0.574372
\(509\) 1.32057e12 0.872027 0.436013 0.899940i \(-0.356390\pi\)
0.436013 + 0.899940i \(0.356390\pi\)
\(510\) 4.20435e10 0.0275190
\(511\) 4.70272e11 0.305109
\(512\) −1.20816e12 −0.776980
\(513\) 3.80663e11 0.242668
\(514\) 3.46110e11 0.218715
\(515\) 8.77627e11 0.549765
\(516\) 6.09359e10 0.0378399
\(517\) −5.32034e10 −0.0327516
\(518\) 5.75561e11 0.351243
\(519\) 1.00094e12 0.605558
\(520\) −3.77945e11 −0.226680
\(521\) −1.31853e12 −0.784009 −0.392005 0.919963i \(-0.628218\pi\)
−0.392005 + 0.919963i \(0.628218\pi\)
\(522\) 8.38339e10 0.0494200
\(523\) −1.69211e12 −0.988945 −0.494472 0.869193i \(-0.664639\pi\)
−0.494472 + 0.869193i \(0.664639\pi\)
\(524\) 1.23866e12 0.717730
\(525\) −2.43000e11 −0.139601
\(526\) 3.84614e11 0.219073
\(527\) 4.87704e11 0.275428
\(528\) −1.66446e12 −0.932013
\(529\) 7.55281e10 0.0419332
\(530\) −1.18686e10 −0.00653367
\(531\) 3.97702e11 0.217086
\(532\) 2.72853e12 1.47681
\(533\) 4.39175e12 2.35703
\(534\) 1.02557e11 0.0545796
\(535\) −1.14908e12 −0.606401
\(536\) −4.48288e11 −0.234594
\(537\) −5.39492e10 −0.0279963
\(538\) 4.38021e10 0.0225411
\(539\) −1.60960e12 −0.821427
\(540\) −1.64747e11 −0.0833767
\(541\) −1.86369e12 −0.935373 −0.467687 0.883894i \(-0.654912\pi\)
−0.467687 + 0.883894i \(0.654912\pi\)
\(542\) −3.12115e11 −0.155352
\(543\) 4.27037e11 0.210798
\(544\) 6.26122e11 0.306523
\(545\) −5.83408e11 −0.283262
\(546\) −3.73193e11 −0.179708
\(547\) 4.37242e11 0.208823 0.104412 0.994534i \(-0.466704\pi\)
0.104412 + 0.994534i \(0.466704\pi\)
\(548\) 3.94928e12 1.87071
\(549\) −8.31578e11 −0.390686
\(550\) 1.35006e11 0.0629103
\(551\) −2.28810e12 −1.05753
\(552\) 4.47405e11 0.205104
\(553\) −1.80043e12 −0.818679
\(554\) −2.62736e11 −0.118502
\(555\) 9.48495e11 0.424343
\(556\) 1.41640e12 0.628565
\(557\) −7.09146e11 −0.312167 −0.156084 0.987744i \(-0.549887\pi\)
−0.156084 + 0.987744i \(0.549887\pi\)
\(558\) 6.16472e10 0.0269190
\(559\) 2.27475e11 0.0985328
\(560\) −1.14156e12 −0.490513
\(561\) 1.45309e12 0.619383
\(562\) 2.57962e11 0.109079
\(563\) 3.77472e10 0.0158342 0.00791711 0.999969i \(-0.497480\pi\)
0.00791711 + 0.999969i \(0.497480\pi\)
\(564\) −2.47385e10 −0.0102948
\(565\) −5.80334e10 −0.0239585
\(566\) −3.85253e11 −0.157787
\(567\) −3.30599e11 −0.134331
\(568\) −7.07824e11 −0.285337
\(569\) −3.56270e11 −0.142487 −0.0712433 0.997459i \(-0.522697\pi\)
−0.0712433 + 0.997459i \(0.522697\pi\)
\(570\) −1.45048e11 −0.0575537
\(571\) 3.87932e12 1.52719 0.763596 0.645694i \(-0.223432\pi\)
0.763596 + 0.645694i \(0.223432\pi\)
\(572\) −6.42751e12 −2.51050
\(573\) −2.38056e12 −0.922535
\(574\) −8.99562e11 −0.345882
\(575\) 5.35125e11 0.204150
\(576\) −7.19762e11 −0.272451
\(577\) −4.28876e12 −1.61080 −0.805399 0.592734i \(-0.798049\pi\)
−0.805399 + 0.592734i \(0.798049\pi\)
\(578\) 3.01924e11 0.112518
\(579\) −1.20484e12 −0.445530
\(580\) 9.90265e11 0.363350
\(581\) −9.13232e11 −0.332498
\(582\) −7.87036e10 −0.0284342
\(583\) −4.10197e11 −0.147056
\(584\) −2.46893e11 −0.0878316
\(585\) −6.15004e11 −0.217108
\(586\) −3.26523e11 −0.114387
\(587\) 4.43245e12 1.54089 0.770447 0.637504i \(-0.220033\pi\)
0.770447 + 0.637504i \(0.220033\pi\)
\(588\) −7.48430e11 −0.258198
\(589\) −1.68255e12 −0.576036
\(590\) −1.51540e11 −0.0514865
\(591\) 4.03927e11 0.136194
\(592\) 4.45580e12 1.49100
\(593\) −5.10104e12 −1.69400 −0.846998 0.531596i \(-0.821593\pi\)
−0.846998 + 0.531596i \(0.821593\pi\)
\(594\) 1.83675e11 0.0605355
\(595\) 9.96586e11 0.325978
\(596\) 4.77838e12 1.55122
\(597\) 1.17877e12 0.379790
\(598\) 8.21831e11 0.262801
\(599\) −7.04599e11 −0.223626 −0.111813 0.993729i \(-0.535666\pi\)
−0.111813 + 0.993729i \(0.535666\pi\)
\(600\) 1.27575e11 0.0401869
\(601\) −1.73879e12 −0.543641 −0.271821 0.962348i \(-0.587626\pi\)
−0.271821 + 0.962348i \(0.587626\pi\)
\(602\) −4.65938e10 −0.0144592
\(603\) −7.29469e11 −0.224688
\(604\) 2.66993e12 0.816269
\(605\) 3.19231e12 0.968738
\(606\) 2.11832e11 0.0638065
\(607\) −5.78292e11 −0.172901 −0.0864507 0.996256i \(-0.527553\pi\)
−0.0864507 + 0.996256i \(0.527553\pi\)
\(608\) −2.16008e12 −0.641068
\(609\) 1.98717e12 0.585407
\(610\) 3.16864e11 0.0926593
\(611\) −9.23493e10 −0.0268070
\(612\) 6.75655e11 0.194690
\(613\) 3.74595e12 1.07150 0.535748 0.844378i \(-0.320030\pi\)
0.535748 + 0.844378i \(0.320030\pi\)
\(614\) 1.18233e11 0.0335723
\(615\) −1.48243e12 −0.417866
\(616\) 2.67557e12 0.748691
\(617\) −3.94875e12 −1.09692 −0.548461 0.836176i \(-0.684786\pi\)
−0.548461 + 0.836176i \(0.684786\pi\)
\(618\) −4.54962e11 −0.125466
\(619\) 3.42253e12 0.937000 0.468500 0.883463i \(-0.344795\pi\)
0.468500 + 0.883463i \(0.344795\pi\)
\(620\) 7.28190e11 0.197917
\(621\) 7.28032e11 0.196444
\(622\) 1.59628e11 0.0427615
\(623\) 2.43098e12 0.646526
\(624\) −2.88914e12 −0.762847
\(625\) 1.52588e11 0.0400000
\(626\) −7.41484e11 −0.192982
\(627\) −5.01307e12 −1.29539
\(628\) −2.57639e11 −0.0660988
\(629\) −3.88995e12 −0.990868
\(630\) 1.25971e11 0.0318595
\(631\) 5.84755e12 1.46839 0.734196 0.678938i \(-0.237560\pi\)
0.734196 + 0.678938i \(0.237560\pi\)
\(632\) 9.45226e11 0.235673
\(633\) 4.17479e12 1.03352
\(634\) −1.07558e12 −0.264387
\(635\) 1.08637e12 0.265153
\(636\) −1.90733e11 −0.0462241
\(637\) −2.79391e12 −0.672334
\(638\) −1.10404e12 −0.263809
\(639\) −1.15179e12 −0.273288
\(640\) 1.23928e12 0.291983
\(641\) 2.66671e12 0.623899 0.311950 0.950099i \(-0.399018\pi\)
0.311950 + 0.950099i \(0.399018\pi\)
\(642\) 5.95685e11 0.138391
\(643\) 9.68716e10 0.0223484 0.0111742 0.999938i \(-0.496443\pi\)
0.0111742 + 0.999938i \(0.496443\pi\)
\(644\) 5.21841e12 1.19551
\(645\) −7.67842e10 −0.0174684
\(646\) 5.94865e11 0.134392
\(647\) 4.47368e10 0.0100368 0.00501840 0.999987i \(-0.498403\pi\)
0.00501840 + 0.999987i \(0.498403\pi\)
\(648\) 1.73564e11 0.0386699
\(649\) −5.23747e12 −1.15883
\(650\) 2.34341e11 0.0514917
\(651\) 1.46127e12 0.318871
\(652\) −4.66855e12 −1.01174
\(653\) 4.95385e12 1.06619 0.533094 0.846056i \(-0.321029\pi\)
0.533094 + 0.846056i \(0.321029\pi\)
\(654\) 3.02438e11 0.0646453
\(655\) −1.56081e12 −0.331333
\(656\) −6.96411e12 −1.46824
\(657\) −4.01752e11 −0.0841228
\(658\) 1.89159e10 0.00393378
\(659\) −5.85077e12 −1.20845 −0.604225 0.796814i \(-0.706517\pi\)
−0.604225 + 0.796814i \(0.706517\pi\)
\(660\) 2.16960e12 0.445075
\(661\) −8.81007e11 −0.179503 −0.0897517 0.995964i \(-0.528607\pi\)
−0.0897517 + 0.995964i \(0.528607\pi\)
\(662\) 1.71640e12 0.347342
\(663\) 2.52224e12 0.506962
\(664\) 4.79447e11 0.0957159
\(665\) −3.43816e12 −0.681756
\(666\) −4.91700e11 −0.0968425
\(667\) −4.37608e12 −0.856088
\(668\) −4.64872e12 −0.903314
\(669\) 3.73630e12 0.721149
\(670\) 2.77957e11 0.0532894
\(671\) 1.09513e13 2.08553
\(672\) 1.87599e12 0.354870
\(673\) −8.66521e12 −1.62821 −0.814107 0.580715i \(-0.802773\pi\)
−0.814107 + 0.580715i \(0.802773\pi\)
\(674\) 8.10390e10 0.0151260
\(675\) 2.07594e11 0.0384900
\(676\) −5.89689e12 −1.08608
\(677\) 8.98549e12 1.64397 0.821983 0.569512i \(-0.192868\pi\)
0.821983 + 0.569512i \(0.192868\pi\)
\(678\) 3.00845e10 0.00546776
\(679\) −1.86557e12 −0.336819
\(680\) −5.23207e11 −0.0938391
\(681\) −3.04425e12 −0.542397
\(682\) −8.11852e11 −0.143697
\(683\) −3.86477e12 −0.679564 −0.339782 0.940504i \(-0.610353\pi\)
−0.339782 + 0.940504i \(0.610353\pi\)
\(684\) −2.33097e12 −0.407178
\(685\) −4.97642e12 −0.863593
\(686\) −6.67386e11 −0.115059
\(687\) −5.19276e12 −0.889392
\(688\) −3.60713e11 −0.0613782
\(689\) −7.12010e11 −0.120365
\(690\) −2.77409e11 −0.0465907
\(691\) −7.08564e12 −1.18230 −0.591150 0.806561i \(-0.701326\pi\)
−0.591150 + 0.806561i \(0.701326\pi\)
\(692\) −6.12922e12 −1.01608
\(693\) 4.35377e12 0.717077
\(694\) −1.17113e11 −0.0191641
\(695\) −1.78478e12 −0.290171
\(696\) −1.04327e12 −0.168521
\(697\) 6.07972e12 0.975744
\(698\) −2.82053e11 −0.0449760
\(699\) 5.64229e12 0.893939
\(700\) 1.48800e12 0.234240
\(701\) 4.69380e12 0.734165 0.367083 0.930188i \(-0.380357\pi\)
0.367083 + 0.930188i \(0.380357\pi\)
\(702\) 3.18818e11 0.0495479
\(703\) 1.34201e13 2.07232
\(704\) 9.47879e12 1.45437
\(705\) 3.11724e10 0.00475247
\(706\) 2.62940e11 0.0398323
\(707\) 5.02120e12 0.755824
\(708\) −2.43531e12 −0.364254
\(709\) 1.06645e13 1.58501 0.792503 0.609868i \(-0.208777\pi\)
0.792503 + 0.609868i \(0.208777\pi\)
\(710\) 4.38879e11 0.0648160
\(711\) 1.53810e12 0.225721
\(712\) −1.27627e12 −0.186115
\(713\) −3.21794e12 −0.466311
\(714\) −5.16630e11 −0.0743940
\(715\) 8.09919e12 1.15895
\(716\) 3.30356e11 0.0469757
\(717\) 5.39318e12 0.762095
\(718\) −2.32681e12 −0.326739
\(719\) −8.15663e12 −1.13823 −0.569116 0.822257i \(-0.692715\pi\)
−0.569116 + 0.822257i \(0.692715\pi\)
\(720\) 9.75227e11 0.135241
\(721\) −1.07843e13 −1.48622
\(722\) −7.61500e11 −0.104292
\(723\) −3.57743e12 −0.486911
\(724\) −2.61495e12 −0.353703
\(725\) −1.24781e12 −0.167737
\(726\) −1.65490e12 −0.221083
\(727\) 6.64771e12 0.882606 0.441303 0.897358i \(-0.354516\pi\)
0.441303 + 0.897358i \(0.354516\pi\)
\(728\) 4.64418e12 0.612799
\(729\) 2.82430e11 0.0370370
\(730\) 1.53083e11 0.0199515
\(731\) 3.14905e11 0.0407898
\(732\) 5.09213e12 0.655541
\(733\) 7.07821e12 0.905640 0.452820 0.891602i \(-0.350418\pi\)
0.452820 + 0.891602i \(0.350418\pi\)
\(734\) 1.66828e12 0.212147
\(735\) 9.43083e11 0.119195
\(736\) −4.13124e12 −0.518956
\(737\) 9.60663e12 1.19941
\(738\) 7.68493e11 0.0953644
\(739\) −2.61052e12 −0.321979 −0.160989 0.986956i \(-0.551468\pi\)
−0.160989 + 0.986956i \(0.551468\pi\)
\(740\) −5.80807e12 −0.712015
\(741\) −8.70157e12 −1.06027
\(742\) 1.45841e11 0.0176629
\(743\) −1.41841e13 −1.70747 −0.853734 0.520709i \(-0.825667\pi\)
−0.853734 + 0.520709i \(0.825667\pi\)
\(744\) −7.67165e11 −0.0917932
\(745\) −6.02115e12 −0.716104
\(746\) 3.04167e11 0.0359573
\(747\) 7.80171e11 0.0916742
\(748\) −8.89793e12 −1.03928
\(749\) 1.41199e13 1.63932
\(750\) −7.91016e10 −0.00912871
\(751\) 8.44355e11 0.0968603 0.0484301 0.998827i \(-0.484578\pi\)
0.0484301 + 0.998827i \(0.484578\pi\)
\(752\) 1.46441e11 0.0166986
\(753\) 6.77351e12 0.767779
\(754\) −1.91636e12 −0.215927
\(755\) −3.36433e12 −0.376822
\(756\) 2.02441e12 0.225398
\(757\) 9.05305e12 1.00199 0.500995 0.865450i \(-0.332968\pi\)
0.500995 + 0.865450i \(0.332968\pi\)
\(758\) 7.16719e11 0.0788565
\(759\) −9.58769e12 −1.04864
\(760\) 1.80504e12 0.196257
\(761\) −6.97701e12 −0.754116 −0.377058 0.926190i \(-0.623064\pi\)
−0.377058 + 0.926190i \(0.623064\pi\)
\(762\) −5.63174e11 −0.0605126
\(763\) 7.16891e12 0.765760
\(764\) 1.45772e13 1.54794
\(765\) −8.51380e11 −0.0898767
\(766\) 3.18007e12 0.333739
\(767\) −9.09108e12 −0.948498
\(768\) 3.90717e12 0.405263
\(769\) −1.07233e13 −1.10576 −0.552879 0.833261i \(-0.686471\pi\)
−0.552879 + 0.833261i \(0.686471\pi\)
\(770\) −1.65896e12 −0.170070
\(771\) −7.00872e12 −0.714322
\(772\) 7.37781e12 0.747566
\(773\) −1.37568e13 −1.38583 −0.692916 0.721019i \(-0.743674\pi\)
−0.692916 + 0.721019i \(0.743674\pi\)
\(774\) 3.98049e10 0.00398660
\(775\) −9.17578e11 −0.0913662
\(776\) 9.79422e11 0.0969599
\(777\) −1.16551e13 −1.14715
\(778\) 7.18134e11 0.0702744
\(779\) −2.09747e13 −2.04069
\(780\) 3.76595e12 0.364291
\(781\) 1.51684e13 1.45884
\(782\) 1.13770e12 0.108792
\(783\) −1.69764e12 −0.161405
\(784\) 4.43037e12 0.418811
\(785\) 3.24646e11 0.0305138
\(786\) 8.09124e11 0.0756160
\(787\) −1.27539e13 −1.18510 −0.592551 0.805533i \(-0.701879\pi\)
−0.592551 + 0.805533i \(0.701879\pi\)
\(788\) −2.47343e12 −0.228524
\(789\) −7.78843e12 −0.715490
\(790\) −5.86078e11 −0.0535345
\(791\) 7.13114e11 0.0647686
\(792\) −2.28573e12 −0.206424
\(793\) 1.90091e13 1.70699
\(794\) 1.37492e12 0.122768
\(795\) 2.40339e11 0.0213389
\(796\) −7.21814e12 −0.637260
\(797\) 1.27845e13 1.12233 0.561164 0.827704i \(-0.310354\pi\)
0.561164 + 0.827704i \(0.310354\pi\)
\(798\) 1.78234e12 0.155589
\(799\) −1.27844e11 −0.0110973
\(800\) −1.17800e12 −0.101681
\(801\) −2.07678e12 −0.178256
\(802\) −3.08832e12 −0.263595
\(803\) 5.29081e12 0.449057
\(804\) 4.46687e12 0.377009
\(805\) −6.57562e12 −0.551893
\(806\) −1.40919e12 −0.117615
\(807\) −8.86992e11 −0.0736188
\(808\) −2.63613e12 −0.217579
\(809\) 6.03746e12 0.495548 0.247774 0.968818i \(-0.420301\pi\)
0.247774 + 0.968818i \(0.420301\pi\)
\(810\) −1.07617e11 −0.00878410
\(811\) −2.50043e12 −0.202965 −0.101482 0.994837i \(-0.532359\pi\)
−0.101482 + 0.994837i \(0.532359\pi\)
\(812\) −1.21684e13 −0.982269
\(813\) 6.32033e12 0.507379
\(814\) 6.47536e12 0.516957
\(815\) 5.88275e12 0.467058
\(816\) −3.99958e12 −0.315797
\(817\) −1.08641e12 −0.0853085
\(818\) −1.04253e12 −0.0814138
\(819\) 7.55716e12 0.586923
\(820\) 9.07762e12 0.701148
\(821\) −4.21082e12 −0.323461 −0.161731 0.986835i \(-0.551708\pi\)
−0.161731 + 0.986835i \(0.551708\pi\)
\(822\) 2.57977e12 0.197087
\(823\) 2.08206e11 0.0158196 0.00790978 0.999969i \(-0.497482\pi\)
0.00790978 + 0.999969i \(0.497482\pi\)
\(824\) 5.66174e12 0.427836
\(825\) −2.73388e12 −0.205464
\(826\) 1.86213e12 0.139187
\(827\) −9.26106e12 −0.688472 −0.344236 0.938883i \(-0.611862\pi\)
−0.344236 + 0.938883i \(0.611862\pi\)
\(828\) −4.45807e12 −0.329618
\(829\) 2.42762e13 1.78519 0.892597 0.450856i \(-0.148881\pi\)
0.892597 + 0.450856i \(0.148881\pi\)
\(830\) −2.97276e11 −0.0217424
\(831\) 5.32041e12 0.387026
\(832\) 1.64531e13 1.19040
\(833\) −3.86775e12 −0.278327
\(834\) 9.25231e11 0.0662221
\(835\) 5.85776e12 0.417006
\(836\) 3.06973e13 2.17356
\(837\) −1.24835e12 −0.0879171
\(838\) 2.24066e12 0.156956
\(839\) −1.25546e13 −0.874728 −0.437364 0.899285i \(-0.644088\pi\)
−0.437364 + 0.899285i \(0.644088\pi\)
\(840\) −1.56764e12 −0.108640
\(841\) −4.30294e12 −0.296608
\(842\) −6.73284e11 −0.0461630
\(843\) −5.22374e12 −0.356252
\(844\) −2.55642e13 −1.73417
\(845\) 7.43056e12 0.501379
\(846\) −1.61598e10 −0.00108460
\(847\) −3.92272e13 −2.61886
\(848\) 1.12905e12 0.0749778
\(849\) 7.80137e12 0.515331
\(850\) 3.24409e11 0.0213161
\(851\) 2.56664e13 1.67758
\(852\) 7.05296e12 0.458557
\(853\) −1.36320e13 −0.881632 −0.440816 0.897597i \(-0.645311\pi\)
−0.440816 + 0.897597i \(0.645311\pi\)
\(854\) −3.89363e12 −0.250492
\(855\) 2.93721e12 0.187970
\(856\) −7.41297e12 −0.471911
\(857\) 1.73987e13 1.10180 0.550901 0.834570i \(-0.314284\pi\)
0.550901 + 0.834570i \(0.314284\pi\)
\(858\) −4.19862e12 −0.264493
\(859\) −6.43355e10 −0.00403163 −0.00201582 0.999998i \(-0.500642\pi\)
−0.00201582 + 0.999998i \(0.500642\pi\)
\(860\) 4.70184e11 0.0293106
\(861\) 1.82161e13 1.12965
\(862\) −1.79353e12 −0.110644
\(863\) 3.32120e12 0.203820 0.101910 0.994794i \(-0.467505\pi\)
0.101910 + 0.994794i \(0.467505\pi\)
\(864\) −1.60266e12 −0.0978427
\(865\) 7.72331e12 0.469063
\(866\) 4.33940e12 0.262180
\(867\) −6.11396e12 −0.367482
\(868\) −8.94800e12 −0.535041
\(869\) −2.02558e13 −1.20493
\(870\) 6.46866e11 0.0382805
\(871\) 1.66750e13 0.981709
\(872\) −3.76368e12 −0.220439
\(873\) 1.59375e12 0.0928657
\(874\) −3.92501e12 −0.227530
\(875\) −1.87500e12 −0.108135
\(876\) 2.46011e12 0.141152
\(877\) 1.95832e12 0.111786 0.0558928 0.998437i \(-0.482200\pi\)
0.0558928 + 0.998437i \(0.482200\pi\)
\(878\) −1.84219e12 −0.104619
\(879\) 6.61210e12 0.373585
\(880\) −1.28431e13 −0.721934
\(881\) −3.02253e13 −1.69036 −0.845179 0.534483i \(-0.820506\pi\)
−0.845179 + 0.534483i \(0.820506\pi\)
\(882\) −4.88894e11 −0.0272023
\(883\) −9.30097e12 −0.514879 −0.257439 0.966294i \(-0.582879\pi\)
−0.257439 + 0.966294i \(0.582879\pi\)
\(884\) −1.54448e13 −0.850643
\(885\) 3.06869e12 0.168154
\(886\) 5.28380e10 0.00288068
\(887\) 9.27171e12 0.502925 0.251463 0.967867i \(-0.419088\pi\)
0.251463 + 0.967867i \(0.419088\pi\)
\(888\) 6.11893e12 0.330231
\(889\) −1.33493e13 −0.716805
\(890\) 7.91336e11 0.0422772
\(891\) −3.71941e12 −0.197708
\(892\) −2.28791e13 −1.21003
\(893\) 4.41053e11 0.0232092
\(894\) 3.12136e12 0.163428
\(895\) −4.16275e11 −0.0216859
\(896\) −1.52282e13 −0.789338
\(897\) −1.66421e13 −0.858305
\(898\) 2.76756e12 0.142021
\(899\) 7.50365e12 0.383137
\(900\) −1.27119e12 −0.0645833
\(901\) −9.85671e11 −0.0498276
\(902\) −1.01205e13 −0.509066
\(903\) 9.43524e11 0.0472235
\(904\) −3.74385e11 −0.0186449
\(905\) 3.29504e12 0.163284
\(906\) 1.74407e12 0.0859976
\(907\) 1.32868e12 0.0651908 0.0325954 0.999469i \(-0.489623\pi\)
0.0325954 + 0.999469i \(0.489623\pi\)
\(908\) 1.86413e13 0.910102
\(909\) −4.28960e12 −0.208391
\(910\) −2.87958e12 −0.139201
\(911\) 2.71297e12 0.130501 0.0652503 0.997869i \(-0.479215\pi\)
0.0652503 + 0.997869i \(0.479215\pi\)
\(912\) 1.37983e13 0.660464
\(913\) −1.02743e13 −0.489368
\(914\) 1.49254e12 0.0707404
\(915\) −6.41650e12 −0.302624
\(916\) 3.17977e13 1.49233
\(917\) 1.91792e13 0.895714
\(918\) 4.41355e11 0.0205114
\(919\) 1.32139e12 0.0611100 0.0305550 0.999533i \(-0.490273\pi\)
0.0305550 + 0.999533i \(0.490273\pi\)
\(920\) 3.45220e12 0.158873
\(921\) −2.39422e12 −0.109647
\(922\) 5.83758e12 0.266038
\(923\) 2.63289e13 1.19406
\(924\) −2.66601e13 −1.20320
\(925\) 7.31864e12 0.328694
\(926\) −5.36853e11 −0.0239942
\(927\) 9.21297e12 0.409771
\(928\) 9.63329e12 0.426392
\(929\) 1.16124e12 0.0511507 0.0255754 0.999673i \(-0.491858\pi\)
0.0255754 + 0.999673i \(0.491858\pi\)
\(930\) 4.75673e11 0.0208514
\(931\) 1.33435e13 0.582098
\(932\) −3.45503e13 −1.49996
\(933\) −3.23247e12 −0.139659
\(934\) 1.45813e11 0.00626952
\(935\) 1.12121e13 0.479772
\(936\) −3.96751e12 −0.168957
\(937\) 3.40914e13 1.44483 0.722415 0.691460i \(-0.243032\pi\)
0.722415 + 0.691460i \(0.243032\pi\)
\(938\) −3.41553e12 −0.144061
\(939\) 1.50151e13 0.630277
\(940\) −1.90883e11 −0.00797429
\(941\) −1.60215e12 −0.0666114 −0.0333057 0.999445i \(-0.510603\pi\)
−0.0333057 + 0.999445i \(0.510603\pi\)
\(942\) −1.68297e11 −0.00696380
\(943\) −4.01149e13 −1.65197
\(944\) 1.44160e13 0.590840
\(945\) −2.55092e12 −0.104053
\(946\) −5.24204e11 −0.0212809
\(947\) −3.38850e13 −1.36909 −0.684546 0.728969i \(-0.740000\pi\)
−0.684546 + 0.728969i \(0.740000\pi\)
\(948\) −9.41851e12 −0.378743
\(949\) 9.18366e12 0.367551
\(950\) −1.11919e12 −0.0445809
\(951\) 2.17805e13 0.863485
\(952\) 6.42917e12 0.253682
\(953\) 2.15757e13 0.847320 0.423660 0.905821i \(-0.360745\pi\)
0.423660 + 0.905821i \(0.360745\pi\)
\(954\) −1.24592e11 −0.00486991
\(955\) −1.83685e13 −0.714592
\(956\) −3.30249e13 −1.27874
\(957\) 2.23567e13 0.861598
\(958\) −3.52912e12 −0.135370
\(959\) 6.11502e13 2.33461
\(960\) −5.55372e12 −0.211039
\(961\) −2.09218e13 −0.791306
\(962\) 1.12398e13 0.423126
\(963\) −1.20626e13 −0.451985
\(964\) 2.19063e13 0.816999
\(965\) −9.29664e12 −0.345106
\(966\) 3.40880e12 0.125952
\(967\) 3.06249e13 1.12630 0.563152 0.826353i \(-0.309589\pi\)
0.563152 + 0.826353i \(0.309589\pi\)
\(968\) 2.05943e13 0.753888
\(969\) −1.20460e13 −0.438921
\(970\) −6.07281e11 −0.0220250
\(971\) 1.92365e12 0.0694447 0.0347224 0.999397i \(-0.488945\pi\)
0.0347224 + 0.999397i \(0.488945\pi\)
\(972\) −1.72945e12 −0.0621453
\(973\) 2.19314e13 0.784438
\(974\) −2.03936e12 −0.0726070
\(975\) −4.74540e12 −0.168171
\(976\) −3.01432e13 −1.06332
\(977\) −1.83893e13 −0.645714 −0.322857 0.946448i \(-0.604643\pi\)
−0.322857 + 0.946448i \(0.604643\pi\)
\(978\) −3.04962e12 −0.106591
\(979\) 2.73498e13 0.951552
\(980\) −5.77493e12 −0.200000
\(981\) −6.12438e12 −0.211131
\(982\) 6.08524e12 0.208822
\(983\) −4.63273e13 −1.58251 −0.791254 0.611488i \(-0.790571\pi\)
−0.791254 + 0.611488i \(0.790571\pi\)
\(984\) −9.56347e12 −0.325190
\(985\) 3.11672e12 0.105496
\(986\) −2.65291e12 −0.0893875
\(987\) −3.83047e11 −0.0128477
\(988\) 5.32837e13 1.77905
\(989\) −2.07779e12 −0.0690587
\(990\) 1.41724e12 0.0468906
\(991\) 2.76252e12 0.0909857 0.0454929 0.998965i \(-0.485514\pi\)
0.0454929 + 0.998965i \(0.485514\pi\)
\(992\) 7.08383e12 0.232255
\(993\) −3.47570e13 −1.13441
\(994\) −5.39295e12 −0.175221
\(995\) 9.09544e12 0.294184
\(996\) −4.77734e12 −0.153822
\(997\) −1.74502e13 −0.559337 −0.279668 0.960097i \(-0.590224\pi\)
−0.279668 + 0.960097i \(0.590224\pi\)
\(998\) −2.81365e12 −0.0897807
\(999\) 9.95692e12 0.316286
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 15.10.a.a.1.1 1
3.2 odd 2 45.10.a.b.1.1 1
4.3 odd 2 240.10.a.c.1.1 1
5.2 odd 4 75.10.b.d.49.1 2
5.3 odd 4 75.10.b.d.49.2 2
5.4 even 2 75.10.a.c.1.1 1
15.2 even 4 225.10.b.e.199.2 2
15.8 even 4 225.10.b.e.199.1 2
15.14 odd 2 225.10.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.a.1.1 1 1.1 even 1 trivial
45.10.a.b.1.1 1 3.2 odd 2
75.10.a.c.1.1 1 5.4 even 2
75.10.b.d.49.1 2 5.2 odd 4
75.10.b.d.49.2 2 5.3 odd 4
225.10.a.c.1.1 1 15.14 odd 2
225.10.b.e.199.1 2 15.8 even 4
225.10.b.e.199.2 2 15.2 even 4
240.10.a.c.1.1 1 4.3 odd 2