Properties

Label 1815.4.a.i.1.1
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(107.088466660\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} +5.00000 q^{5} -11.1047 q^{6} +9.10469 q^{7} +8.50781 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.70156 q^{2} +3.00000 q^{3} +5.70156 q^{4} +5.00000 q^{5} -11.1047 q^{6} +9.10469 q^{7} +8.50781 q^{8} +9.00000 q^{9} -18.5078 q^{10} +17.1047 q^{12} -20.2984 q^{13} -33.7016 q^{14} +15.0000 q^{15} -77.1047 q^{16} +61.8062 q^{17} -33.3141 q^{18} +8.68594 q^{19} +28.5078 q^{20} +27.3141 q^{21} -127.748 q^{23} +25.5234 q^{24} +25.0000 q^{25} +75.1359 q^{26} +27.0000 q^{27} +51.9109 q^{28} +85.9109 q^{29} -55.5234 q^{30} -158.733 q^{31} +217.345 q^{32} -228.780 q^{34} +45.5234 q^{35} +51.3141 q^{36} +138.423 q^{37} -32.1515 q^{38} -60.8953 q^{39} +42.5391 q^{40} -28.2094 q^{41} -101.105 q^{42} -265.884 q^{43} +45.0000 q^{45} +472.869 q^{46} -515.602 q^{47} -231.314 q^{48} -260.105 q^{49} -92.5391 q^{50} +185.419 q^{51} -115.733 q^{52} -466.702 q^{53} -99.9422 q^{54} +77.4609 q^{56} +26.0578 q^{57} -318.005 q^{58} +373.475 q^{59} +85.5234 q^{60} -84.8172 q^{61} +587.559 q^{62} +81.9422 q^{63} -187.680 q^{64} -101.492 q^{65} -88.7657 q^{67} +352.392 q^{68} -383.245 q^{69} -168.508 q^{70} -536.695 q^{71} +76.5703 q^{72} -322.450 q^{73} -512.383 q^{74} +75.0000 q^{75} +49.5234 q^{76} +225.408 q^{78} -265.592 q^{79} -385.523 q^{80} +81.0000 q^{81} +104.419 q^{82} +520.758 q^{83} +155.733 q^{84} +309.031 q^{85} +984.187 q^{86} +257.733 q^{87} -464.713 q^{89} -166.570 q^{90} -184.811 q^{91} -728.366 q^{92} -476.198 q^{93} +1908.53 q^{94} +43.4297 q^{95} +652.036 q^{96} -204.481 q^{97} +962.794 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 6q^{3} + 5q^{4} + 10q^{5} - 3q^{6} - q^{7} - 15q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - q^{2} + 6q^{3} + 5q^{4} + 10q^{5} - 3q^{6} - q^{7} - 15q^{8} + 18q^{9} - 5q^{10} + 15q^{12} - 47q^{13} - 61q^{14} + 30q^{15} - 135q^{16} + 98q^{17} - 9q^{18} + 75q^{19} + 25q^{20} - 3q^{21} - 57q^{23} - 45q^{24} + 50q^{25} + 3q^{26} + 54q^{27} + 59q^{28} + 127q^{29} - 15q^{30} - 183q^{31} + 249q^{32} - 131q^{34} - 5q^{35} + 45q^{36} - 229q^{37} + 147q^{38} - 141q^{39} - 75q^{40} - 18q^{41} - 183q^{42} - 186q^{43} + 90q^{45} + 664q^{46} - 615q^{47} - 405q^{48} - 501q^{49} - 25q^{50} + 294q^{51} - 97q^{52} - 927q^{53} - 27q^{54} + 315q^{56} + 225q^{57} - 207q^{58} - 380q^{59} + 75q^{60} - 509q^{61} + 522q^{62} - 9q^{63} + 361q^{64} - 235q^{65} - 1138q^{67} + 327q^{68} - 171q^{69} - 305q^{70} - 273q^{71} - 135q^{72} - 440q^{73} - 1505q^{74} + 150q^{75} + 3q^{76} + 9q^{78} - 973q^{79} - 675q^{80} + 162q^{81} + 132q^{82} - 15q^{83} + 177q^{84} + 490q^{85} + 1200q^{86} + 381q^{87} - 1288q^{89} - 45q^{90} + 85q^{91} - 778q^{92} - 549q^{93} + 1640q^{94} + 375q^{95} + 747q^{96} - 76q^{97} + 312q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.70156 −1.30870 −0.654350 0.756192i \(-0.727058\pi\)
−0.654350 + 0.756192i \(0.727058\pi\)
\(3\) 3.00000 0.577350
\(4\) 5.70156 0.712695
\(5\) 5.00000 0.447214
\(6\) −11.1047 −0.755578
\(7\) 9.10469 0.491607 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(8\) 8.50781 0.375996
\(9\) 9.00000 0.333333
\(10\) −18.5078 −0.585268
\(11\) 0 0
\(12\) 17.1047 0.411475
\(13\) −20.2984 −0.433060 −0.216530 0.976276i \(-0.569474\pi\)
−0.216530 + 0.976276i \(0.569474\pi\)
\(14\) −33.7016 −0.643366
\(15\) 15.0000 0.258199
\(16\) −77.1047 −1.20476
\(17\) 61.8062 0.881777 0.440889 0.897562i \(-0.354663\pi\)
0.440889 + 0.897562i \(0.354663\pi\)
\(18\) −33.3141 −0.436233
\(19\) 8.68594 0.104879 0.0524393 0.998624i \(-0.483300\pi\)
0.0524393 + 0.998624i \(0.483300\pi\)
\(20\) 28.5078 0.318727
\(21\) 27.3141 0.283829
\(22\) 0 0
\(23\) −127.748 −1.15815 −0.579074 0.815275i \(-0.696586\pi\)
−0.579074 + 0.815275i \(0.696586\pi\)
\(24\) 25.5234 0.217081
\(25\) 25.0000 0.200000
\(26\) 75.1359 0.566745
\(27\) 27.0000 0.192450
\(28\) 51.9109 0.350366
\(29\) 85.9109 0.550112 0.275056 0.961428i \(-0.411304\pi\)
0.275056 + 0.961428i \(0.411304\pi\)
\(30\) −55.5234 −0.337905
\(31\) −158.733 −0.919653 −0.459827 0.888009i \(-0.652088\pi\)
−0.459827 + 0.888009i \(0.652088\pi\)
\(32\) 217.345 1.20067
\(33\) 0 0
\(34\) −228.780 −1.15398
\(35\) 45.5234 0.219853
\(36\) 51.3141 0.237565
\(37\) 138.423 0.615045 0.307523 0.951541i \(-0.400500\pi\)
0.307523 + 0.951541i \(0.400500\pi\)
\(38\) −32.1515 −0.137254
\(39\) −60.8953 −0.250027
\(40\) 42.5391 0.168150
\(41\) −28.2094 −0.107453 −0.0537264 0.998556i \(-0.517110\pi\)
−0.0537264 + 0.998556i \(0.517110\pi\)
\(42\) −101.105 −0.371447
\(43\) −265.884 −0.942953 −0.471477 0.881879i \(-0.656279\pi\)
−0.471477 + 0.881879i \(0.656279\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 472.869 1.51567
\(47\) −515.602 −1.60017 −0.800087 0.599883i \(-0.795214\pi\)
−0.800087 + 0.599883i \(0.795214\pi\)
\(48\) −231.314 −0.695569
\(49\) −260.105 −0.758323
\(50\) −92.5391 −0.261740
\(51\) 185.419 0.509094
\(52\) −115.733 −0.308639
\(53\) −466.702 −1.20955 −0.604777 0.796395i \(-0.706738\pi\)
−0.604777 + 0.796395i \(0.706738\pi\)
\(54\) −99.9422 −0.251859
\(55\) 0 0
\(56\) 77.4609 0.184842
\(57\) 26.0578 0.0605516
\(58\) −318.005 −0.719932
\(59\) 373.475 0.824107 0.412053 0.911160i \(-0.364812\pi\)
0.412053 + 0.911160i \(0.364812\pi\)
\(60\) 85.5234 0.184017
\(61\) −84.8172 −0.178028 −0.0890142 0.996030i \(-0.528372\pi\)
−0.0890142 + 0.996030i \(0.528372\pi\)
\(62\) 587.559 1.20355
\(63\) 81.9422 0.163869
\(64\) −187.680 −0.366562
\(65\) −101.492 −0.193670
\(66\) 0 0
\(67\) −88.7657 −0.161858 −0.0809288 0.996720i \(-0.525789\pi\)
−0.0809288 + 0.996720i \(0.525789\pi\)
\(68\) 352.392 0.628439
\(69\) −383.245 −0.668657
\(70\) −168.508 −0.287722
\(71\) −536.695 −0.897099 −0.448549 0.893758i \(-0.648059\pi\)
−0.448549 + 0.893758i \(0.648059\pi\)
\(72\) 76.5703 0.125332
\(73\) −322.450 −0.516985 −0.258493 0.966013i \(-0.583226\pi\)
−0.258493 + 0.966013i \(0.583226\pi\)
\(74\) −512.383 −0.804909
\(75\) 75.0000 0.115470
\(76\) 49.5234 0.0747464
\(77\) 0 0
\(78\) 225.408 0.327210
\(79\) −265.592 −0.378246 −0.189123 0.981953i \(-0.560565\pi\)
−0.189123 + 0.981953i \(0.560565\pi\)
\(80\) −385.523 −0.538785
\(81\) 81.0000 0.111111
\(82\) 104.419 0.140623
\(83\) 520.758 0.688682 0.344341 0.938845i \(-0.388102\pi\)
0.344341 + 0.938845i \(0.388102\pi\)
\(84\) 155.733 0.202284
\(85\) 309.031 0.394343
\(86\) 984.187 1.23404
\(87\) 257.733 0.317608
\(88\) 0 0
\(89\) −464.713 −0.553477 −0.276738 0.960945i \(-0.589254\pi\)
−0.276738 + 0.960945i \(0.589254\pi\)
\(90\) −166.570 −0.195089
\(91\) −184.811 −0.212895
\(92\) −728.366 −0.825406
\(93\) −476.198 −0.530962
\(94\) 1908.53 2.09415
\(95\) 43.4297 0.0469031
\(96\) 652.036 0.693210
\(97\) −204.481 −0.214040 −0.107020 0.994257i \(-0.534131\pi\)
−0.107020 + 0.994257i \(0.534131\pi\)
\(98\) 962.794 0.992417
\(99\) 0 0
\(100\) 142.539 0.142539
\(101\) −27.2516 −0.0268479 −0.0134239 0.999910i \(-0.504273\pi\)
−0.0134239 + 0.999910i \(0.504273\pi\)
\(102\) −686.339 −0.666252
\(103\) 1062.80 1.01671 0.508353 0.861149i \(-0.330255\pi\)
0.508353 + 0.861149i \(0.330255\pi\)
\(104\) −172.695 −0.162828
\(105\) 136.570 0.126932
\(106\) 1727.52 1.58294
\(107\) −1702.96 −1.53861 −0.769306 0.638881i \(-0.779398\pi\)
−0.769306 + 0.638881i \(0.779398\pi\)
\(108\) 153.942 0.137158
\(109\) −556.334 −0.488873 −0.244437 0.969665i \(-0.578603\pi\)
−0.244437 + 0.969665i \(0.578603\pi\)
\(110\) 0 0
\(111\) 415.270 0.355096
\(112\) −702.014 −0.592269
\(113\) −2088.31 −1.73851 −0.869256 0.494363i \(-0.835401\pi\)
−0.869256 + 0.494363i \(0.835401\pi\)
\(114\) −96.4546 −0.0792439
\(115\) −638.742 −0.517939
\(116\) 489.827 0.392063
\(117\) −182.686 −0.144353
\(118\) −1382.44 −1.07851
\(119\) 562.727 0.433488
\(120\) 127.617 0.0970817
\(121\) 0 0
\(122\) 313.956 0.232986
\(123\) −84.6281 −0.0620379
\(124\) −905.025 −0.655433
\(125\) 125.000 0.0894427
\(126\) −303.314 −0.214455
\(127\) 2179.16 1.52259 0.761296 0.648405i \(-0.224564\pi\)
0.761296 + 0.648405i \(0.224564\pi\)
\(128\) −1044.05 −0.720955
\(129\) −797.653 −0.544414
\(130\) 375.680 0.253456
\(131\) 1951.11 1.30129 0.650645 0.759382i \(-0.274499\pi\)
0.650645 + 0.759382i \(0.274499\pi\)
\(132\) 0 0
\(133\) 79.0828 0.0515590
\(134\) 328.572 0.211823
\(135\) 135.000 0.0860663
\(136\) 525.836 0.331545
\(137\) −613.425 −0.382543 −0.191272 0.981537i \(-0.561261\pi\)
−0.191272 + 0.981537i \(0.561261\pi\)
\(138\) 1418.61 0.875071
\(139\) 2130.63 1.30013 0.650064 0.759880i \(-0.274742\pi\)
0.650064 + 0.759880i \(0.274742\pi\)
\(140\) 259.555 0.156688
\(141\) −1546.80 −0.923861
\(142\) 1986.61 1.17403
\(143\) 0 0
\(144\) −693.942 −0.401587
\(145\) 429.555 0.246018
\(146\) 1193.57 0.676578
\(147\) −780.314 −0.437818
\(148\) 789.230 0.438340
\(149\) 570.900 0.313892 0.156946 0.987607i \(-0.449835\pi\)
0.156946 + 0.987607i \(0.449835\pi\)
\(150\) −277.617 −0.151116
\(151\) −2154.78 −1.16128 −0.580641 0.814159i \(-0.697198\pi\)
−0.580641 + 0.814159i \(0.697198\pi\)
\(152\) 73.8983 0.0394339
\(153\) 556.256 0.293926
\(154\) 0 0
\(155\) −793.664 −0.411281
\(156\) −347.198 −0.178193
\(157\) −157.105 −0.0798619 −0.0399310 0.999202i \(-0.512714\pi\)
−0.0399310 + 0.999202i \(0.512714\pi\)
\(158\) 983.106 0.495011
\(159\) −1400.10 −0.698337
\(160\) 1086.73 0.536958
\(161\) −1163.11 −0.569353
\(162\) −299.827 −0.145411
\(163\) 1736.66 0.834512 0.417256 0.908789i \(-0.362992\pi\)
0.417256 + 0.908789i \(0.362992\pi\)
\(164\) −160.837 −0.0765811
\(165\) 0 0
\(166\) −1927.62 −0.901278
\(167\) 1466.55 0.679549 0.339775 0.940507i \(-0.389649\pi\)
0.339775 + 0.940507i \(0.389649\pi\)
\(168\) 232.383 0.106719
\(169\) −1784.97 −0.812459
\(170\) −1143.90 −0.516076
\(171\) 78.1735 0.0349595
\(172\) −1515.96 −0.672038
\(173\) 143.781 0.0631878 0.0315939 0.999501i \(-0.489942\pi\)
0.0315939 + 0.999501i \(0.489942\pi\)
\(174\) −954.014 −0.415653
\(175\) 227.617 0.0983214
\(176\) 0 0
\(177\) 1120.42 0.475798
\(178\) 1720.16 0.724335
\(179\) −1412.93 −0.589984 −0.294992 0.955500i \(-0.595317\pi\)
−0.294992 + 0.955500i \(0.595317\pi\)
\(180\) 256.570 0.106242
\(181\) −2239.25 −0.919570 −0.459785 0.888030i \(-0.652073\pi\)
−0.459785 + 0.888030i \(0.652073\pi\)
\(182\) 684.089 0.278616
\(183\) −254.452 −0.102785
\(184\) −1086.86 −0.435458
\(185\) 692.117 0.275057
\(186\) 1762.68 0.694870
\(187\) 0 0
\(188\) −2939.73 −1.14044
\(189\) 245.827 0.0946098
\(190\) −160.758 −0.0613821
\(191\) 1745.94 0.661425 0.330712 0.943732i \(-0.392711\pi\)
0.330712 + 0.943732i \(0.392711\pi\)
\(192\) −563.039 −0.211635
\(193\) 3985.04 1.48627 0.743134 0.669143i \(-0.233338\pi\)
0.743134 + 0.669143i \(0.233338\pi\)
\(194\) 756.900 0.280115
\(195\) −304.477 −0.111815
\(196\) −1483.00 −0.540453
\(197\) −2262.00 −0.818076 −0.409038 0.912517i \(-0.634136\pi\)
−0.409038 + 0.912517i \(0.634136\pi\)
\(198\) 0 0
\(199\) 2340.15 0.833613 0.416806 0.908995i \(-0.363149\pi\)
0.416806 + 0.908995i \(0.363149\pi\)
\(200\) 212.695 0.0751991
\(201\) −266.297 −0.0934485
\(202\) 100.873 0.0351358
\(203\) 782.192 0.270439
\(204\) 1057.18 0.362829
\(205\) −141.047 −0.0480543
\(206\) −3934.01 −1.33056
\(207\) −1149.74 −0.386049
\(208\) 1565.10 0.521733
\(209\) 0 0
\(210\) −505.523 −0.166116
\(211\) −1524.44 −0.497377 −0.248689 0.968583i \(-0.580000\pi\)
−0.248689 + 0.968583i \(0.580000\pi\)
\(212\) −2660.93 −0.862044
\(213\) −1610.09 −0.517940
\(214\) 6303.62 2.01358
\(215\) −1329.42 −0.421701
\(216\) 229.711 0.0723604
\(217\) −1445.21 −0.452108
\(218\) 2059.31 0.639788
\(219\) −967.350 −0.298482
\(220\) 0 0
\(221\) −1254.57 −0.381862
\(222\) −1537.15 −0.464715
\(223\) 5083.42 1.52651 0.763253 0.646100i \(-0.223601\pi\)
0.763253 + 0.646100i \(0.223601\pi\)
\(224\) 1978.86 0.590260
\(225\) 225.000 0.0666667
\(226\) 7730.01 2.27519
\(227\) −3910.66 −1.14343 −0.571717 0.820451i \(-0.693723\pi\)
−0.571717 + 0.820451i \(0.693723\pi\)
\(228\) 148.570 0.0431549
\(229\) 2733.31 0.788743 0.394371 0.918951i \(-0.370962\pi\)
0.394371 + 0.918951i \(0.370962\pi\)
\(230\) 2364.34 0.677827
\(231\) 0 0
\(232\) 730.914 0.206840
\(233\) −4355.46 −1.22462 −0.612308 0.790619i \(-0.709759\pi\)
−0.612308 + 0.790619i \(0.709759\pi\)
\(234\) 676.223 0.188915
\(235\) −2578.01 −0.715620
\(236\) 2129.39 0.587337
\(237\) −796.777 −0.218381
\(238\) −2082.97 −0.567305
\(239\) −4731.48 −1.28056 −0.640281 0.768141i \(-0.721182\pi\)
−0.640281 + 0.768141i \(0.721182\pi\)
\(240\) −1156.57 −0.311068
\(241\) 1833.12 0.489966 0.244983 0.969527i \(-0.421218\pi\)
0.244983 + 0.969527i \(0.421218\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) −483.591 −0.126880
\(245\) −1300.52 −0.339132
\(246\) 313.256 0.0811890
\(247\) −176.311 −0.0454186
\(248\) −1350.47 −0.345786
\(249\) 1562.27 0.397611
\(250\) −462.695 −0.117054
\(251\) −3168.31 −0.796741 −0.398370 0.917225i \(-0.630424\pi\)
−0.398370 + 0.917225i \(0.630424\pi\)
\(252\) 467.198 0.116789
\(253\) 0 0
\(254\) −8066.29 −1.99262
\(255\) 927.094 0.227674
\(256\) 5366.07 1.31008
\(257\) 5378.55 1.30547 0.652733 0.757588i \(-0.273622\pi\)
0.652733 + 0.757588i \(0.273622\pi\)
\(258\) 2952.56 0.712475
\(259\) 1260.30 0.302360
\(260\) −578.664 −0.138028
\(261\) 773.198 0.183371
\(262\) −7222.14 −1.70300
\(263\) 4584.03 1.07477 0.537383 0.843338i \(-0.319413\pi\)
0.537383 + 0.843338i \(0.319413\pi\)
\(264\) 0 0
\(265\) −2333.51 −0.540929
\(266\) −292.730 −0.0674752
\(267\) −1394.14 −0.319550
\(268\) −506.103 −0.115355
\(269\) −1741.03 −0.394618 −0.197309 0.980341i \(-0.563220\pi\)
−0.197309 + 0.980341i \(0.563220\pi\)
\(270\) −499.711 −0.112635
\(271\) 1340.58 0.300495 0.150248 0.988648i \(-0.451993\pi\)
0.150248 + 0.988648i \(0.451993\pi\)
\(272\) −4765.55 −1.06233
\(273\) −554.433 −0.122915
\(274\) 2270.63 0.500634
\(275\) 0 0
\(276\) −2185.10 −0.476548
\(277\) −5220.77 −1.13244 −0.566220 0.824254i \(-0.691595\pi\)
−0.566220 + 0.824254i \(0.691595\pi\)
\(278\) −7886.66 −1.70148
\(279\) −1428.60 −0.306551
\(280\) 387.305 0.0826639
\(281\) 2494.68 0.529609 0.264804 0.964302i \(-0.414693\pi\)
0.264804 + 0.964302i \(0.414693\pi\)
\(282\) 5725.59 1.20906
\(283\) 4329.43 0.909391 0.454696 0.890647i \(-0.349748\pi\)
0.454696 + 0.890647i \(0.349748\pi\)
\(284\) −3060.00 −0.639358
\(285\) 130.289 0.0270795
\(286\) 0 0
\(287\) −256.837 −0.0528245
\(288\) 1956.11 0.400225
\(289\) −1092.99 −0.222468
\(290\) −1590.02 −0.321963
\(291\) −613.444 −0.123576
\(292\) −1838.47 −0.368453
\(293\) −4330.93 −0.863534 −0.431767 0.901985i \(-0.642110\pi\)
−0.431767 + 0.901985i \(0.642110\pi\)
\(294\) 2888.38 0.572972
\(295\) 1867.37 0.368552
\(296\) 1177.68 0.231254
\(297\) 0 0
\(298\) −2113.22 −0.410791
\(299\) 2593.09 0.501547
\(300\) 427.617 0.0822950
\(301\) −2420.79 −0.463562
\(302\) 7976.06 1.51977
\(303\) −81.7547 −0.0155006
\(304\) −669.727 −0.126353
\(305\) −424.086 −0.0796167
\(306\) −2059.02 −0.384661
\(307\) −2052.56 −0.381582 −0.190791 0.981631i \(-0.561105\pi\)
−0.190791 + 0.981631i \(0.561105\pi\)
\(308\) 0 0
\(309\) 3188.39 0.586995
\(310\) 2937.80 0.538244
\(311\) 2627.76 0.479121 0.239561 0.970881i \(-0.422997\pi\)
0.239561 + 0.970881i \(0.422997\pi\)
\(312\) −518.086 −0.0940091
\(313\) 6878.40 1.24214 0.621070 0.783755i \(-0.286698\pi\)
0.621070 + 0.783755i \(0.286698\pi\)
\(314\) 581.533 0.104515
\(315\) 409.711 0.0732844
\(316\) −1514.29 −0.269574
\(317\) −3026.05 −0.536151 −0.268076 0.963398i \(-0.586388\pi\)
−0.268076 + 0.963398i \(0.586388\pi\)
\(318\) 5182.57 0.913913
\(319\) 0 0
\(320\) −938.398 −0.163931
\(321\) −5108.88 −0.888318
\(322\) 4305.32 0.745112
\(323\) 536.845 0.0924795
\(324\) 461.827 0.0791884
\(325\) −507.461 −0.0866119
\(326\) −6428.34 −1.09213
\(327\) −1669.00 −0.282251
\(328\) −240.000 −0.0404018
\(329\) −4694.39 −0.786657
\(330\) 0 0
\(331\) −4326.50 −0.718447 −0.359223 0.933252i \(-0.616958\pi\)
−0.359223 + 0.933252i \(0.616958\pi\)
\(332\) 2969.13 0.490820
\(333\) 1245.81 0.205015
\(334\) −5428.51 −0.889326
\(335\) −443.828 −0.0723849
\(336\) −2106.04 −0.341946
\(337\) −1759.36 −0.284387 −0.142194 0.989839i \(-0.545416\pi\)
−0.142194 + 0.989839i \(0.545416\pi\)
\(338\) 6607.19 1.06327
\(339\) −6264.93 −1.00373
\(340\) 1761.96 0.281046
\(341\) 0 0
\(342\) −289.364 −0.0457515
\(343\) −5491.08 −0.864403
\(344\) −2262.09 −0.354546
\(345\) −1916.23 −0.299032
\(346\) −532.215 −0.0826939
\(347\) −11730.7 −1.81481 −0.907404 0.420260i \(-0.861939\pi\)
−0.907404 + 0.420260i \(0.861939\pi\)
\(348\) 1469.48 0.226357
\(349\) −3152.38 −0.483504 −0.241752 0.970338i \(-0.577722\pi\)
−0.241752 + 0.970338i \(0.577722\pi\)
\(350\) −842.539 −0.128673
\(351\) −548.058 −0.0833423
\(352\) 0 0
\(353\) −6881.03 −1.03751 −0.518754 0.854923i \(-0.673604\pi\)
−0.518754 + 0.854923i \(0.673604\pi\)
\(354\) −4147.32 −0.622677
\(355\) −2683.48 −0.401195
\(356\) −2649.59 −0.394460
\(357\) 1688.18 0.250274
\(358\) 5230.04 0.772112
\(359\) −2858.96 −0.420307 −0.210154 0.977668i \(-0.567396\pi\)
−0.210154 + 0.977668i \(0.567396\pi\)
\(360\) 382.851 0.0560501
\(361\) −6783.55 −0.989001
\(362\) 8288.72 1.20344
\(363\) 0 0
\(364\) −1053.71 −0.151729
\(365\) −1612.25 −0.231203
\(366\) 941.868 0.134514
\(367\) −10973.4 −1.56079 −0.780393 0.625289i \(-0.784981\pi\)
−0.780393 + 0.625289i \(0.784981\pi\)
\(368\) 9850.00 1.39529
\(369\) −253.884 −0.0358176
\(370\) −2561.91 −0.359966
\(371\) −4249.17 −0.594625
\(372\) −2715.07 −0.378414
\(373\) 10443.4 1.44970 0.724852 0.688905i \(-0.241908\pi\)
0.724852 + 0.688905i \(0.241908\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) −4386.64 −0.601659
\(377\) −1743.86 −0.238231
\(378\) −909.942 −0.123816
\(379\) −7405.86 −1.00373 −0.501865 0.864946i \(-0.667352\pi\)
−0.501865 + 0.864946i \(0.667352\pi\)
\(380\) 247.617 0.0334276
\(381\) 6537.48 0.879069
\(382\) −6462.72 −0.865607
\(383\) 14155.0 1.88848 0.944238 0.329264i \(-0.106801\pi\)
0.944238 + 0.329264i \(0.106801\pi\)
\(384\) −3132.16 −0.416244
\(385\) 0 0
\(386\) −14750.9 −1.94508
\(387\) −2392.96 −0.314318
\(388\) −1165.86 −0.152546
\(389\) −13031.6 −1.69853 −0.849263 0.527970i \(-0.822953\pi\)
−0.849263 + 0.527970i \(0.822953\pi\)
\(390\) 1127.04 0.146333
\(391\) −7895.65 −1.02123
\(392\) −2212.92 −0.285126
\(393\) 5853.32 0.751300
\(394\) 8372.94 1.07062
\(395\) −1327.96 −0.169157
\(396\) 0 0
\(397\) 2281.02 0.288366 0.144183 0.989551i \(-0.453945\pi\)
0.144183 + 0.989551i \(0.453945\pi\)
\(398\) −8662.22 −1.09095
\(399\) 237.248 0.0297676
\(400\) −1927.62 −0.240952
\(401\) −13017.7 −1.62114 −0.810568 0.585645i \(-0.800841\pi\)
−0.810568 + 0.585645i \(0.800841\pi\)
\(402\) 985.715 0.122296
\(403\) 3222.03 0.398265
\(404\) −155.377 −0.0191343
\(405\) 405.000 0.0496904
\(406\) −2895.33 −0.353924
\(407\) 0 0
\(408\) 1577.51 0.191417
\(409\) −3548.46 −0.428998 −0.214499 0.976724i \(-0.568812\pi\)
−0.214499 + 0.976724i \(0.568812\pi\)
\(410\) 522.094 0.0628887
\(411\) −1840.27 −0.220861
\(412\) 6059.61 0.724601
\(413\) 3400.37 0.405136
\(414\) 4255.82 0.505222
\(415\) 2603.79 0.307988
\(416\) −4411.77 −0.519964
\(417\) 6391.89 0.750629
\(418\) 0 0
\(419\) −14801.3 −1.72575 −0.862876 0.505416i \(-0.831339\pi\)
−0.862876 + 0.505416i \(0.831339\pi\)
\(420\) 778.664 0.0904641
\(421\) 11542.8 1.33625 0.668123 0.744051i \(-0.267098\pi\)
0.668123 + 0.744051i \(0.267098\pi\)
\(422\) 5642.80 0.650918
\(423\) −4640.41 −0.533392
\(424\) −3970.61 −0.454787
\(425\) 1545.16 0.176355
\(426\) 5959.83 0.677828
\(427\) −772.234 −0.0875200
\(428\) −9709.54 −1.09656
\(429\) 0 0
\(430\) 4920.94 0.551881
\(431\) −12712.6 −1.42076 −0.710378 0.703820i \(-0.751476\pi\)
−0.710378 + 0.703820i \(0.751476\pi\)
\(432\) −2081.83 −0.231856
\(433\) −16385.4 −1.81855 −0.909273 0.416200i \(-0.863361\pi\)
−0.909273 + 0.416200i \(0.863361\pi\)
\(434\) 5349.54 0.591674
\(435\) 1288.66 0.142038
\(436\) −3171.97 −0.348418
\(437\) −1109.62 −0.121465
\(438\) 3580.71 0.390623
\(439\) −6603.62 −0.717935 −0.358968 0.933350i \(-0.616871\pi\)
−0.358968 + 0.933350i \(0.616871\pi\)
\(440\) 0 0
\(441\) −2340.94 −0.252774
\(442\) 4643.87 0.499743
\(443\) −11641.0 −1.24849 −0.624244 0.781230i \(-0.714593\pi\)
−0.624244 + 0.781230i \(0.714593\pi\)
\(444\) 2367.69 0.253076
\(445\) −2323.56 −0.247522
\(446\) −18816.6 −1.99774
\(447\) 1712.70 0.181226
\(448\) −1708.76 −0.180204
\(449\) 13886.8 1.45960 0.729799 0.683662i \(-0.239614\pi\)
0.729799 + 0.683662i \(0.239614\pi\)
\(450\) −832.851 −0.0872467
\(451\) 0 0
\(452\) −11906.6 −1.23903
\(453\) −6464.35 −0.670467
\(454\) 14475.6 1.49641
\(455\) −924.055 −0.0952096
\(456\) 221.695 0.0227672
\(457\) −3625.85 −0.371138 −0.185569 0.982631i \(-0.559413\pi\)
−0.185569 + 0.982631i \(0.559413\pi\)
\(458\) −10117.5 −1.03223
\(459\) 1668.77 0.169698
\(460\) −3641.83 −0.369133
\(461\) −18329.1 −1.85178 −0.925890 0.377794i \(-0.876683\pi\)
−0.925890 + 0.377794i \(0.876683\pi\)
\(462\) 0 0
\(463\) 10760.7 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(464\) −6624.14 −0.662754
\(465\) −2380.99 −0.237453
\(466\) 16122.0 1.60265
\(467\) −7964.29 −0.789172 −0.394586 0.918859i \(-0.629112\pi\)
−0.394586 + 0.918859i \(0.629112\pi\)
\(468\) −1041.60 −0.102880
\(469\) −808.184 −0.0795703
\(470\) 9542.66 0.936532
\(471\) −471.314 −0.0461083
\(472\) 3177.45 0.309861
\(473\) 0 0
\(474\) 2949.32 0.285795
\(475\) 217.149 0.0209757
\(476\) 3208.42 0.308945
\(477\) −4200.31 −0.403185
\(478\) 17513.9 1.67587
\(479\) −822.463 −0.0784536 −0.0392268 0.999230i \(-0.512489\pi\)
−0.0392268 + 0.999230i \(0.512489\pi\)
\(480\) 3260.18 0.310013
\(481\) −2809.78 −0.266351
\(482\) −6785.41 −0.641218
\(483\) −3489.33 −0.328716
\(484\) 0 0
\(485\) −1022.41 −0.0957218
\(486\) −899.480 −0.0839531
\(487\) 6709.58 0.624312 0.312156 0.950031i \(-0.398949\pi\)
0.312156 + 0.950031i \(0.398949\pi\)
\(488\) −721.609 −0.0669379
\(489\) 5209.97 0.481806
\(490\) 4813.97 0.443822
\(491\) −12689.3 −1.16631 −0.583156 0.812360i \(-0.698182\pi\)
−0.583156 + 0.812360i \(0.698182\pi\)
\(492\) −482.512 −0.0442141
\(493\) 5309.83 0.485077
\(494\) 652.626 0.0594394
\(495\) 0 0
\(496\) 12239.0 1.10796
\(497\) −4886.44 −0.441020
\(498\) −5782.85 −0.520353
\(499\) −9810.79 −0.880143 −0.440072 0.897963i \(-0.645047\pi\)
−0.440072 + 0.897963i \(0.645047\pi\)
\(500\) 712.695 0.0637454
\(501\) 4399.64 0.392338
\(502\) 11727.7 1.04269
\(503\) 12865.7 1.14047 0.570233 0.821483i \(-0.306853\pi\)
0.570233 + 0.821483i \(0.306853\pi\)
\(504\) 697.149 0.0616140
\(505\) −136.258 −0.0120067
\(506\) 0 0
\(507\) −5354.92 −0.469074
\(508\) 12424.6 1.08514
\(509\) 2551.78 0.222211 0.111106 0.993809i \(-0.464561\pi\)
0.111106 + 0.993809i \(0.464561\pi\)
\(510\) −3431.70 −0.297957
\(511\) −2935.81 −0.254153
\(512\) −11510.4 −0.993541
\(513\) 234.520 0.0201839
\(514\) −19909.0 −1.70846
\(515\) 5313.99 0.454684
\(516\) −4547.87 −0.388001
\(517\) 0 0
\(518\) −4665.09 −0.395699
\(519\) 431.344 0.0364815
\(520\) −863.476 −0.0728191
\(521\) 9761.13 0.820812 0.410406 0.911903i \(-0.365387\pi\)
0.410406 + 0.911903i \(0.365387\pi\)
\(522\) −2862.04 −0.239977
\(523\) −4548.79 −0.380315 −0.190157 0.981754i \(-0.560900\pi\)
−0.190157 + 0.981754i \(0.560900\pi\)
\(524\) 11124.4 0.927423
\(525\) 682.851 0.0567659
\(526\) −16968.1 −1.40655
\(527\) −9810.68 −0.810930
\(528\) 0 0
\(529\) 4152.66 0.341305
\(530\) 8637.62 0.707914
\(531\) 3361.27 0.274702
\(532\) 450.895 0.0367458
\(533\) 572.606 0.0465334
\(534\) 5160.49 0.418195
\(535\) −8514.80 −0.688088
\(536\) −755.202 −0.0608577
\(537\) −4238.78 −0.340628
\(538\) 6444.52 0.516437
\(539\) 0 0
\(540\) 769.711 0.0613390
\(541\) 9184.79 0.729917 0.364958 0.931024i \(-0.381083\pi\)
0.364958 + 0.931024i \(0.381083\pi\)
\(542\) −4962.23 −0.393258
\(543\) −6717.75 −0.530914
\(544\) 13433.3 1.05873
\(545\) −2781.67 −0.218631
\(546\) 2052.27 0.160859
\(547\) −20966.6 −1.63888 −0.819439 0.573167i \(-0.805715\pi\)
−0.819439 + 0.573167i \(0.805715\pi\)
\(548\) −3497.48 −0.272637
\(549\) −763.355 −0.0593428
\(550\) 0 0
\(551\) 746.217 0.0576950
\(552\) −3260.58 −0.251412
\(553\) −2418.13 −0.185948
\(554\) 19325.0 1.48202
\(555\) 2076.35 0.158804
\(556\) 12147.9 0.926595
\(557\) 4422.49 0.336422 0.168211 0.985751i \(-0.446201\pi\)
0.168211 + 0.985751i \(0.446201\pi\)
\(558\) 5288.03 0.401183
\(559\) 5397.04 0.408355
\(560\) −3510.07 −0.264871
\(561\) 0 0
\(562\) −9234.21 −0.693099
\(563\) 6066.46 0.454122 0.227061 0.973881i \(-0.427088\pi\)
0.227061 + 0.973881i \(0.427088\pi\)
\(564\) −8819.20 −0.658432
\(565\) −10441.6 −0.777486
\(566\) −16025.6 −1.19012
\(567\) 737.480 0.0546230
\(568\) −4566.10 −0.337305
\(569\) −17341.9 −1.27770 −0.638850 0.769331i \(-0.720590\pi\)
−0.638850 + 0.769331i \(0.720590\pi\)
\(570\) −482.273 −0.0354390
\(571\) −621.014 −0.0455142 −0.0227571 0.999741i \(-0.507244\pi\)
−0.0227571 + 0.999741i \(0.507244\pi\)
\(572\) 0 0
\(573\) 5237.83 0.381874
\(574\) 950.700 0.0691314
\(575\) −3193.71 −0.231629
\(576\) −1689.12 −0.122187
\(577\) 4458.77 0.321700 0.160850 0.986979i \(-0.448576\pi\)
0.160850 + 0.986979i \(0.448576\pi\)
\(578\) 4045.76 0.291144
\(579\) 11955.1 0.858097
\(580\) 2449.13 0.175336
\(581\) 4741.34 0.338561
\(582\) 2270.70 0.161724
\(583\) 0 0
\(584\) −2743.34 −0.194384
\(585\) −913.430 −0.0645567
\(586\) 16031.2 1.13011
\(587\) 1326.96 0.0933038 0.0466519 0.998911i \(-0.485145\pi\)
0.0466519 + 0.998911i \(0.485145\pi\)
\(588\) −4449.01 −0.312031
\(589\) −1378.74 −0.0964519
\(590\) −6912.20 −0.482324
\(591\) −6786.01 −0.472317
\(592\) −10673.1 −0.740982
\(593\) 13296.2 0.920756 0.460378 0.887723i \(-0.347714\pi\)
0.460378 + 0.887723i \(0.347714\pi\)
\(594\) 0 0
\(595\) 2813.63 0.193862
\(596\) 3255.02 0.223710
\(597\) 7020.45 0.481287
\(598\) −9598.50 −0.656374
\(599\) −7420.72 −0.506181 −0.253091 0.967443i \(-0.581447\pi\)
−0.253091 + 0.967443i \(0.581447\pi\)
\(600\) 638.086 0.0434162
\(601\) −5189.31 −0.352207 −0.176104 0.984372i \(-0.556349\pi\)
−0.176104 + 0.984372i \(0.556349\pi\)
\(602\) 8960.72 0.606664
\(603\) −798.891 −0.0539525
\(604\) −12285.6 −0.827641
\(605\) 0 0
\(606\) 302.620 0.0202857
\(607\) 26545.6 1.77505 0.887523 0.460763i \(-0.152424\pi\)
0.887523 + 0.460763i \(0.152424\pi\)
\(608\) 1887.85 0.125925
\(609\) 2346.58 0.156138
\(610\) 1569.78 0.104194
\(611\) 10465.9 0.692971
\(612\) 3171.53 0.209480
\(613\) 26097.9 1.71955 0.859776 0.510671i \(-0.170603\pi\)
0.859776 + 0.510671i \(0.170603\pi\)
\(614\) 7597.67 0.499377
\(615\) −423.141 −0.0277442
\(616\) 0 0
\(617\) 12275.1 0.800933 0.400466 0.916311i \(-0.368848\pi\)
0.400466 + 0.916311i \(0.368848\pi\)
\(618\) −11802.0 −0.768200
\(619\) 6067.21 0.393961 0.196980 0.980407i \(-0.436886\pi\)
0.196980 + 0.980407i \(0.436886\pi\)
\(620\) −4525.12 −0.293118
\(621\) −3449.21 −0.222886
\(622\) −9726.82 −0.627026
\(623\) −4231.06 −0.272093
\(624\) 4695.31 0.301223
\(625\) 625.000 0.0400000
\(626\) −25460.8 −1.62559
\(627\) 0 0
\(628\) −895.742 −0.0569172
\(629\) 8555.43 0.542333
\(630\) −1516.57 −0.0959073
\(631\) 3074.16 0.193947 0.0969733 0.995287i \(-0.469084\pi\)
0.0969733 + 0.995287i \(0.469084\pi\)
\(632\) −2259.61 −0.142219
\(633\) −4573.31 −0.287161
\(634\) 11201.1 0.701661
\(635\) 10895.8 0.680924
\(636\) −7982.78 −0.497701
\(637\) 5279.72 0.328399
\(638\) 0 0
\(639\) −4830.26 −0.299033
\(640\) −5220.27 −0.322421
\(641\) −10825.9 −0.667080 −0.333540 0.942736i \(-0.608243\pi\)
−0.333540 + 0.942736i \(0.608243\pi\)
\(642\) 18910.8 1.16254
\(643\) −117.142 −0.00718450 −0.00359225 0.999994i \(-0.501143\pi\)
−0.00359225 + 0.999994i \(0.501143\pi\)
\(644\) −6631.54 −0.405775
\(645\) −3988.27 −0.243469
\(646\) −1987.17 −0.121028
\(647\) 23811.6 1.44688 0.723439 0.690388i \(-0.242560\pi\)
0.723439 + 0.690388i \(0.242560\pi\)
\(648\) 689.133 0.0417773
\(649\) 0 0
\(650\) 1878.40 0.113349
\(651\) −4335.64 −0.261025
\(652\) 9901.65 0.594753
\(653\) −17773.3 −1.06512 −0.532560 0.846392i \(-0.678770\pi\)
−0.532560 + 0.846392i \(0.678770\pi\)
\(654\) 6177.92 0.369382
\(655\) 9755.53 0.581954
\(656\) 2175.07 0.129455
\(657\) −2902.05 −0.172328
\(658\) 17376.6 1.02950
\(659\) 13015.6 0.769370 0.384685 0.923048i \(-0.374310\pi\)
0.384685 + 0.923048i \(0.374310\pi\)
\(660\) 0 0
\(661\) 15769.6 0.927935 0.463968 0.885852i \(-0.346425\pi\)
0.463968 + 0.885852i \(0.346425\pi\)
\(662\) 16014.8 0.940231
\(663\) −3763.71 −0.220468
\(664\) 4430.51 0.258941
\(665\) 395.414 0.0230579
\(666\) −4611.45 −0.268303
\(667\) −10975.0 −0.637111
\(668\) 8361.60 0.484311
\(669\) 15250.3 0.881329
\(670\) 1642.86 0.0947301
\(671\) 0 0
\(672\) 5936.58 0.340787
\(673\) 31519.1 1.80531 0.902653 0.430370i \(-0.141617\pi\)
0.902653 + 0.430370i \(0.141617\pi\)
\(674\) 6512.38 0.372177
\(675\) 675.000 0.0384900
\(676\) −10177.1 −0.579036
\(677\) −12418.6 −0.705000 −0.352500 0.935812i \(-0.614668\pi\)
−0.352500 + 0.935812i \(0.614668\pi\)
\(678\) 23190.0 1.31358
\(679\) −1861.74 −0.105224
\(680\) 2629.18 0.148271
\(681\) −11732.0 −0.660162
\(682\) 0 0
\(683\) 23543.0 1.31896 0.659478 0.751724i \(-0.270777\pi\)
0.659478 + 0.751724i \(0.270777\pi\)
\(684\) 445.711 0.0249155
\(685\) −3067.12 −0.171079
\(686\) 20325.6 1.13124
\(687\) 8199.92 0.455381
\(688\) 20500.9 1.13603
\(689\) 9473.31 0.523809
\(690\) 7093.03 0.391344
\(691\) −9642.84 −0.530870 −0.265435 0.964129i \(-0.585516\pi\)
−0.265435 + 0.964129i \(0.585516\pi\)
\(692\) 819.778 0.0450336
\(693\) 0 0
\(694\) 43422.0 2.37504
\(695\) 10653.2 0.581435
\(696\) 2192.74 0.119419
\(697\) −1743.52 −0.0947494
\(698\) 11668.7 0.632762
\(699\) −13066.4 −0.707032
\(700\) 1297.77 0.0700732
\(701\) −30593.3 −1.64835 −0.824176 0.566334i \(-0.808361\pi\)
−0.824176 + 0.566334i \(0.808361\pi\)
\(702\) 2028.67 0.109070
\(703\) 1202.34 0.0645050
\(704\) 0 0
\(705\) −7734.02 −0.413163
\(706\) 25470.6 1.35779
\(707\) −248.117 −0.0131986
\(708\) 6388.17 0.339099
\(709\) −30248.4 −1.60226 −0.801130 0.598491i \(-0.795767\pi\)
−0.801130 + 0.598491i \(0.795767\pi\)
\(710\) 9933.05 0.525044
\(711\) −2390.33 −0.126082
\(712\) −3953.69 −0.208105
\(713\) 20277.9 1.06509
\(714\) −6248.90 −0.327534
\(715\) 0 0
\(716\) −8055.90 −0.420479
\(717\) −14194.4 −0.739332
\(718\) 10582.6 0.550056
\(719\) 30471.8 1.58054 0.790269 0.612761i \(-0.209941\pi\)
0.790269 + 0.612761i \(0.209941\pi\)
\(720\) −3469.71 −0.179595
\(721\) 9676.45 0.499819
\(722\) 25109.7 1.29430
\(723\) 5499.37 0.282882
\(724\) −12767.2 −0.655373
\(725\) 2147.77 0.110022
\(726\) 0 0
\(727\) 25149.9 1.28302 0.641511 0.767114i \(-0.278308\pi\)
0.641511 + 0.767114i \(0.278308\pi\)
\(728\) −1572.34 −0.0800476
\(729\) 729.000 0.0370370
\(730\) 5967.84 0.302575
\(731\) −16433.3 −0.831475
\(732\) −1450.77 −0.0732542
\(733\) 9974.73 0.502626 0.251313 0.967906i \(-0.419138\pi\)
0.251313 + 0.967906i \(0.419138\pi\)
\(734\) 40618.8 2.04260
\(735\) −3901.57 −0.195798
\(736\) −27765.5 −1.39056
\(737\) 0 0
\(738\) 939.769 0.0468745
\(739\) 11579.9 0.576418 0.288209 0.957568i \(-0.406940\pi\)
0.288209 + 0.957568i \(0.406940\pi\)
\(740\) 3946.15 0.196031
\(741\) −528.933 −0.0262225
\(742\) 15728.6 0.778186
\(743\) 20323.7 1.00351 0.501753 0.865011i \(-0.332689\pi\)
0.501753 + 0.865011i \(0.332689\pi\)
\(744\) −4051.41 −0.199639
\(745\) 2854.50 0.140377
\(746\) −38656.9 −1.89723
\(747\) 4686.82 0.229561
\(748\) 0 0
\(749\) −15504.9 −0.756392
\(750\) −1388.09 −0.0675810
\(751\) 8023.94 0.389877 0.194939 0.980815i \(-0.437549\pi\)
0.194939 + 0.980815i \(0.437549\pi\)
\(752\) 39755.3 1.92783
\(753\) −9504.93 −0.459998
\(754\) 6455.00 0.311773
\(755\) −10773.9 −0.519342
\(756\) 1401.60 0.0674279
\(757\) 542.713 0.0260571 0.0130285 0.999915i \(-0.495853\pi\)
0.0130285 + 0.999915i \(0.495853\pi\)
\(758\) 27413.2 1.31358
\(759\) 0 0
\(760\) 369.492 0.0176354
\(761\) −16292.2 −0.776073 −0.388037 0.921644i \(-0.626847\pi\)
−0.388037 + 0.921644i \(0.626847\pi\)
\(762\) −24198.9 −1.15044
\(763\) −5065.25 −0.240333
\(764\) 9954.61 0.471394
\(765\) 2781.28 0.131448
\(766\) −52395.6 −2.47145
\(767\) −7580.96 −0.356887
\(768\) 16098.2 0.756373
\(769\) 7107.01 0.333271 0.166636 0.986019i \(-0.446710\pi\)
0.166636 + 0.986019i \(0.446710\pi\)
\(770\) 0 0
\(771\) 16135.7 0.753711
\(772\) 22721.0 1.05926
\(773\) −19776.7 −0.920204 −0.460102 0.887866i \(-0.652187\pi\)
−0.460102 + 0.887866i \(0.652187\pi\)
\(774\) 8857.69 0.411348
\(775\) −3968.32 −0.183931
\(776\) −1739.69 −0.0804783
\(777\) 3780.91 0.174568
\(778\) 48237.1 2.22286
\(779\) −245.025 −0.0112695
\(780\) −1735.99 −0.0796904
\(781\) 0 0
\(782\) 29226.2 1.33648
\(783\) 2319.60 0.105869
\(784\) 20055.3 0.913597
\(785\) −785.523 −0.0357153
\(786\) −21666.4 −0.983226
\(787\) −19943.8 −0.903327 −0.451664 0.892188i \(-0.649169\pi\)
−0.451664 + 0.892188i \(0.649169\pi\)
\(788\) −12896.9 −0.583039
\(789\) 13752.1 0.620517
\(790\) 4915.53 0.221376
\(791\) −19013.4 −0.854664
\(792\) 0 0
\(793\) 1721.66 0.0770969
\(794\) −8443.34 −0.377384
\(795\) −7000.52 −0.312306
\(796\) 13342.5 0.594112
\(797\) −24333.8 −1.08149 −0.540744 0.841187i \(-0.681857\pi\)
−0.540744 + 0.841187i \(0.681857\pi\)
\(798\) −878.189 −0.0389568
\(799\) −31867.4 −1.41100
\(800\) 5433.63 0.240135
\(801\) −4182.41 −0.184492
\(802\) 48186.0 2.12158
\(803\) 0 0
\(804\) −1518.31 −0.0666003
\(805\) −5815.55 −0.254622
\(806\) −11926.5 −0.521209
\(807\) −5223.08 −0.227833
\(808\) −231.851 −0.0100947
\(809\) 15308.7 0.665298 0.332649 0.943051i \(-0.392058\pi\)
0.332649 + 0.943051i \(0.392058\pi\)
\(810\) −1499.13 −0.0650298
\(811\) −22639.1 −0.980229 −0.490115 0.871658i \(-0.663045\pi\)
−0.490115 + 0.871658i \(0.663045\pi\)
\(812\) 4459.72 0.192741
\(813\) 4021.73 0.173491
\(814\) 0 0
\(815\) 8683.28 0.373205
\(816\) −14296.7 −0.613337
\(817\) −2309.46 −0.0988955
\(818\) 13134.8 0.561429
\(819\) −1663.30 −0.0709650
\(820\) −804.187 −0.0342481
\(821\) 35341.0 1.50232 0.751162 0.660118i \(-0.229494\pi\)
0.751162 + 0.660118i \(0.229494\pi\)
\(822\) 6811.89 0.289041
\(823\) 28121.5 1.19107 0.595536 0.803329i \(-0.296940\pi\)
0.595536 + 0.803329i \(0.296940\pi\)
\(824\) 9042.09 0.382277
\(825\) 0 0
\(826\) −12586.7 −0.530202
\(827\) 5952.92 0.250306 0.125153 0.992137i \(-0.460058\pi\)
0.125153 + 0.992137i \(0.460058\pi\)
\(828\) −6555.29 −0.275135
\(829\) 18696.9 0.783316 0.391658 0.920111i \(-0.371902\pi\)
0.391658 + 0.920111i \(0.371902\pi\)
\(830\) −9638.09 −0.403064
\(831\) −15662.3 −0.653814
\(832\) 3809.60 0.158743
\(833\) −16076.1 −0.668672
\(834\) −23660.0 −0.982348
\(835\) 7332.73 0.303904
\(836\) 0 0
\(837\) −4285.79 −0.176987
\(838\) 54787.9 2.25849
\(839\) −25774.1 −1.06057 −0.530287 0.847818i \(-0.677916\pi\)
−0.530287 + 0.847818i \(0.677916\pi\)
\(840\) 1161.91 0.0477260
\(841\) −17008.3 −0.697376
\(842\) −42726.2 −1.74874
\(843\) 7484.04 0.305770
\(844\) −8691.68 −0.354478
\(845\) −8924.87 −0.363343
\(846\) 17176.8 0.698049
\(847\) 0 0
\(848\) 35984.9 1.45722
\(849\) 12988.3 0.525037
\(850\) −5719.49 −0.230796
\(851\) −17683.4 −0.712313
\(852\) −9180.00 −0.369134
\(853\) 18664.1 0.749174 0.374587 0.927192i \(-0.377785\pi\)
0.374587 + 0.927192i \(0.377785\pi\)
\(854\) 2858.47 0.114537
\(855\) 390.867 0.0156344
\(856\) −14488.5 −0.578511
\(857\) −37192.5 −1.48246 −0.741232 0.671249i \(-0.765758\pi\)
−0.741232 + 0.671249i \(0.765758\pi\)
\(858\) 0 0
\(859\) 46306.0 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(860\) −7579.78 −0.300545
\(861\) −770.512 −0.0304983
\(862\) 47056.6 1.85934
\(863\) −19587.1 −0.772599 −0.386299 0.922373i \(-0.626247\pi\)
−0.386299 + 0.922373i \(0.626247\pi\)
\(864\) 5868.32 0.231070
\(865\) 718.907 0.0282584
\(866\) 60651.4 2.37993
\(867\) −3278.96 −0.128442
\(868\) −8239.97 −0.322215
\(869\) 0 0
\(870\) −4770.07 −0.185886
\(871\) 1801.80 0.0700939
\(872\) −4733.19 −0.183814
\(873\) −1840.33 −0.0713468
\(874\) 4107.31 0.158961
\(875\) 1138.09 0.0439707
\(876\) −5515.41 −0.212726
\(877\) −26580.8 −1.02345 −0.511727 0.859148i \(-0.670994\pi\)
−0.511727 + 0.859148i \(0.670994\pi\)
\(878\) 24443.7 0.939562
\(879\) −12992.8 −0.498561
\(880\) 0 0
\(881\) −731.459 −0.0279722 −0.0139861 0.999902i \(-0.504452\pi\)
−0.0139861 + 0.999902i \(0.504452\pi\)
\(882\) 8665.14 0.330806
\(883\) −45291.1 −1.72612 −0.863062 0.505099i \(-0.831456\pi\)
−0.863062 + 0.505099i \(0.831456\pi\)
\(884\) −7153.01 −0.272151
\(885\) 5602.12 0.212783
\(886\) 43089.8 1.63390
\(887\) 37730.7 1.42827 0.714134 0.700009i \(-0.246821\pi\)
0.714134 + 0.700009i \(0.246821\pi\)
\(888\) 3533.04 0.133515
\(889\) 19840.6 0.748516
\(890\) 8600.81 0.323932
\(891\) 0 0
\(892\) 28983.4 1.08793
\(893\) −4478.48 −0.167824
\(894\) −6339.67 −0.237170
\(895\) −7064.64 −0.263849
\(896\) −9505.79 −0.354426
\(897\) 7779.28 0.289568
\(898\) −51402.9 −1.91017
\(899\) −13636.9 −0.505913
\(900\) 1282.85 0.0475130
\(901\) −28845.1 −1.06656
\(902\) 0 0
\(903\) −7262.38 −0.267638
\(904\) −17767.0 −0.653673
\(905\) −11196.2 −0.411244
\(906\) 23928.2 0.877440
\(907\) 25846.2 0.946208 0.473104 0.881007i \(-0.343134\pi\)
0.473104 + 0.881007i \(0.343134\pi\)
\(908\) −22296.9 −0.814920
\(909\) −245.264 −0.00894928
\(910\) 3420.45 0.124601
\(911\) −30349.8 −1.10377 −0.551885 0.833920i \(-0.686091\pi\)
−0.551885 + 0.833920i \(0.686091\pi\)
\(912\) −2009.18 −0.0729502
\(913\) 0 0
\(914\) 13421.3 0.485708
\(915\) −1272.26 −0.0459667
\(916\) 15584.1 0.562133
\(917\) 17764.2 0.639723
\(918\) −6177.05 −0.222084
\(919\) −41031.3 −1.47279 −0.736397 0.676550i \(-0.763474\pi\)
−0.736397 + 0.676550i \(0.763474\pi\)
\(920\) −5434.30 −0.194743
\(921\) −6157.68 −0.220307
\(922\) 67846.2 2.42342
\(923\) 10894.1 0.388497
\(924\) 0 0
\(925\) 3460.59 0.123009
\(926\) −39831.4 −1.41354
\(927\) 9565.18 0.338902
\(928\) 18672.3 0.660506
\(929\) −3859.82 −0.136315 −0.0681575 0.997675i \(-0.521712\pi\)
−0.0681575 + 0.997675i \(0.521712\pi\)
\(930\) 8813.39 0.310755
\(931\) −2259.25 −0.0795317
\(932\) −24832.9 −0.872778
\(933\) 7883.29 0.276621
\(934\) 29480.3 1.03279
\(935\) 0 0
\(936\) −1554.26 −0.0542762
\(937\) −22026.5 −0.767955 −0.383977 0.923343i \(-0.625446\pi\)
−0.383977 + 0.923343i \(0.625446\pi\)
\(938\) 2991.54 0.104134
\(939\) 20635.2 0.717150
\(940\) −14698.7 −0.510019
\(941\) 24778.0 0.858386 0.429193 0.903213i \(-0.358798\pi\)
0.429193 + 0.903213i \(0.358798\pi\)
\(942\) 1744.60 0.0603419
\(943\) 3603.70 0.124446
\(944\) −28796.7 −0.992851
\(945\) 1229.13 0.0423108
\(946\) 0 0
\(947\) 34055.7 1.16860 0.584299 0.811539i \(-0.301370\pi\)
0.584299 + 0.811539i \(0.301370\pi\)
\(948\) −4542.87 −0.155639
\(949\) 6545.23 0.223885
\(950\) −803.789 −0.0274509
\(951\) −9078.15 −0.309547
\(952\) 4787.57 0.162990
\(953\) 30211.9 1.02693 0.513463 0.858112i \(-0.328363\pi\)
0.513463 + 0.858112i \(0.328363\pi\)
\(954\) 15547.7 0.527648
\(955\) 8729.72 0.295798
\(956\) −26976.8 −0.912650
\(957\) 0 0
\(958\) 3044.40 0.102672
\(959\) −5585.04 −0.188061
\(960\) −2815.19 −0.0946459
\(961\) −4594.90 −0.154238
\(962\) 10400.6 0.348574
\(963\) −15326.6 −0.512871
\(964\) 10451.7 0.349196
\(965\) 19925.2 0.664679
\(966\) 12916.0 0.430191
\(967\) −34883.2 −1.16005 −0.580024 0.814599i \(-0.696957\pi\)
−0.580024 + 0.814599i \(0.696957\pi\)
\(968\) 0 0
\(969\) 1610.54 0.0533931
\(970\) 3784.50 0.125271
\(971\) 19753.0 0.652835 0.326418 0.945226i \(-0.394158\pi\)
0.326418 + 0.945226i \(0.394158\pi\)
\(972\) 1385.48 0.0457194
\(973\) 19398.7 0.639152
\(974\) −24835.9 −0.817038
\(975\) −1522.38 −0.0500054
\(976\) 6539.80 0.214482
\(977\) −2208.90 −0.0723326 −0.0361663 0.999346i \(-0.511515\pi\)
−0.0361663 + 0.999346i \(0.511515\pi\)
\(978\) −19285.0 −0.630539
\(979\) 0 0
\(980\) −7415.02 −0.241698
\(981\) −5007.01 −0.162958
\(982\) 46970.2 1.52635
\(983\) −40306.0 −1.30779 −0.653897 0.756583i \(-0.726867\pi\)
−0.653897 + 0.756583i \(0.726867\pi\)
\(984\) −720.000 −0.0233260
\(985\) −11310.0 −0.365855
\(986\) −19654.7 −0.634820
\(987\) −14083.2 −0.454177
\(988\) −1005.25 −0.0323696
\(989\) 33966.3 1.09208
\(990\) 0 0
\(991\) −15199.4 −0.487211 −0.243605 0.969874i \(-0.578330\pi\)
−0.243605 + 0.969874i \(0.578330\pi\)
\(992\) −34499.8 −1.10420
\(993\) −12979.5 −0.414796
\(994\) 18087.5 0.577163
\(995\) 11700.8 0.372803
\(996\) 8907.40 0.283375
\(997\) 48073.6 1.52709 0.763543 0.645757i \(-0.223458\pi\)
0.763543 + 0.645757i \(0.223458\pi\)
\(998\) 36315.3 1.15184
\(999\) 3737.43 0.118365
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 1815.4.a.i.1.1 2
11.10 odd 2 1815.4.a.o.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.4.a.i.1.1 2 1.1 even 1 trivial
1815.4.a.o.1.2 yes 2 11.10 odd 2