Properties

Label 294.4.a.m.1.1
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1345}) \)
Defining polynomial: \(x^{2} - x - 336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(18.8371\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -20.8371 q^{5} -6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -20.8371 q^{5} -6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} -41.6742 q^{10} +15.1629 q^{11} -12.0000 q^{12} -2.16288 q^{13} +62.5114 q^{15} +16.0000 q^{16} +119.348 q^{17} +18.0000 q^{18} +33.5114 q^{19} -83.3485 q^{20} +30.3258 q^{22} +0.651517 q^{23} -24.0000 q^{24} +309.186 q^{25} -4.32576 q^{26} -27.0000 q^{27} -163.208 q^{29} +125.023 q^{30} +223.326 q^{31} +32.0000 q^{32} -45.4886 q^{33} +238.697 q^{34} +36.0000 q^{36} +168.534 q^{37} +67.0227 q^{38} +6.48864 q^{39} -166.697 q^{40} +323.023 q^{41} +221.557 q^{43} +60.6515 q^{44} -187.534 q^{45} +1.30303 q^{46} -508.045 q^{47} -48.0000 q^{48} +618.371 q^{50} -358.045 q^{51} -8.65152 q^{52} -176.511 q^{53} -54.0000 q^{54} -315.951 q^{55} -100.534 q^{57} -326.417 q^{58} -454.928 q^{59} +250.045 q^{60} -38.6515 q^{61} +446.652 q^{62} +64.0000 q^{64} +45.0682 q^{65} -90.9773 q^{66} +141.792 q^{67} +477.394 q^{68} -1.95455 q^{69} +602.742 q^{71} +72.0000 q^{72} +1102.30 q^{73} +337.068 q^{74} -927.557 q^{75} +134.045 q^{76} +12.9773 q^{78} -116.303 q^{79} -333.394 q^{80} +81.0000 q^{81} +646.045 q^{82} +568.928 q^{83} -2486.88 q^{85} +443.114 q^{86} +489.625 q^{87} +121.303 q^{88} +383.159 q^{89} -375.068 q^{90} +2.60607 q^{92} -669.977 q^{93} -1016.09 q^{94} -698.280 q^{95} -96.0000 q^{96} -334.701 q^{97} +136.466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} - 6q^{3} + 8q^{4} - 5q^{5} - 12q^{6} + 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 4q^{2} - 6q^{3} + 8q^{4} - 5q^{5} - 12q^{6} + 16q^{8} + 18q^{9} - 10q^{10} + 67q^{11} - 24q^{12} - 41q^{13} + 15q^{15} + 32q^{16} + 92q^{17} + 36q^{18} - 43q^{19} - 20q^{20} + 134q^{22} + 148q^{23} - 48q^{24} + 435q^{25} - 82q^{26} - 54q^{27} + 77q^{29} + 30q^{30} + 520q^{31} + 64q^{32} - 201q^{33} + 184q^{34} + 72q^{36} + 7q^{37} - 86q^{38} + 123q^{39} - 40q^{40} + 426q^{41} - 107q^{43} + 268q^{44} - 45q^{45} + 296q^{46} - 576q^{47} - 96q^{48} + 870q^{50} - 276q^{51} - 164q^{52} - 243q^{53} - 108q^{54} + 505q^{55} + 129q^{57} + 154q^{58} + 7q^{59} + 60q^{60} - 224q^{61} + 1040q^{62} + 128q^{64} - 570q^{65} - 402q^{66} + 687q^{67} + 368q^{68} - 444q^{69} + 472q^{71} + 144q^{72} + 921q^{73} + 14q^{74} - 1305q^{75} - 172q^{76} + 246q^{78} - 526q^{79} - 80q^{80} + 162q^{81} + 852q^{82} + 221q^{83} - 2920q^{85} - 214q^{86} - 231q^{87} + 536q^{88} - 774q^{89} - 90q^{90} + 592q^{92} - 1560q^{93} - 1152q^{94} - 1910q^{95} - 192q^{96} - 1953q^{97} + 603q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −20.8371 −1.86373 −0.931864 0.362807i \(-0.881818\pi\)
−0.931864 + 0.362807i \(0.881818\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −41.6742 −1.31786
\(11\) 15.1629 0.415616 0.207808 0.978170i \(-0.433367\pi\)
0.207808 + 0.978170i \(0.433367\pi\)
\(12\) −12.0000 −0.288675
\(13\) −2.16288 −0.0461442 −0.0230721 0.999734i \(-0.507345\pi\)
−0.0230721 + 0.999734i \(0.507345\pi\)
\(14\) 0 0
\(15\) 62.5114 1.07602
\(16\) 16.0000 0.250000
\(17\) 119.348 1.70272 0.851361 0.524581i \(-0.175778\pi\)
0.851361 + 0.524581i \(0.175778\pi\)
\(18\) 18.0000 0.235702
\(19\) 33.5114 0.404633 0.202317 0.979320i \(-0.435153\pi\)
0.202317 + 0.979320i \(0.435153\pi\)
\(20\) −83.3485 −0.931864
\(21\) 0 0
\(22\) 30.3258 0.293885
\(23\) 0.651517 0.00590655 0.00295327 0.999996i \(-0.499060\pi\)
0.00295327 + 0.999996i \(0.499060\pi\)
\(24\) −24.0000 −0.204124
\(25\) 309.186 2.47348
\(26\) −4.32576 −0.0326289
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −163.208 −1.04507 −0.522535 0.852618i \(-0.675014\pi\)
−0.522535 + 0.852618i \(0.675014\pi\)
\(30\) 125.023 0.760864
\(31\) 223.326 1.29389 0.646943 0.762538i \(-0.276047\pi\)
0.646943 + 0.762538i \(0.276047\pi\)
\(32\) 32.0000 0.176777
\(33\) −45.4886 −0.239956
\(34\) 238.697 1.20401
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 168.534 0.748833 0.374417 0.927261i \(-0.377843\pi\)
0.374417 + 0.927261i \(0.377843\pi\)
\(38\) 67.0227 0.286119
\(39\) 6.48864 0.0266414
\(40\) −166.697 −0.658928
\(41\) 323.023 1.23043 0.615216 0.788359i \(-0.289069\pi\)
0.615216 + 0.788359i \(0.289069\pi\)
\(42\) 0 0
\(43\) 221.557 0.785746 0.392873 0.919593i \(-0.371481\pi\)
0.392873 + 0.919593i \(0.371481\pi\)
\(44\) 60.6515 0.207808
\(45\) −187.534 −0.621243
\(46\) 1.30303 0.00417656
\(47\) −508.045 −1.57672 −0.788362 0.615211i \(-0.789071\pi\)
−0.788362 + 0.615211i \(0.789071\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) 618.371 1.74902
\(51\) −358.045 −0.983066
\(52\) −8.65152 −0.0230721
\(53\) −176.511 −0.457466 −0.228733 0.973489i \(-0.573458\pi\)
−0.228733 + 0.973489i \(0.573458\pi\)
\(54\) −54.0000 −0.136083
\(55\) −315.951 −0.774596
\(56\) 0 0
\(57\) −100.534 −0.233615
\(58\) −326.417 −0.738976
\(59\) −454.928 −1.00384 −0.501920 0.864914i \(-0.667373\pi\)
−0.501920 + 0.864914i \(0.667373\pi\)
\(60\) 250.045 0.538012
\(61\) −38.6515 −0.0811282 −0.0405641 0.999177i \(-0.512915\pi\)
−0.0405641 + 0.999177i \(0.512915\pi\)
\(62\) 446.652 0.914916
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 45.0682 0.0860003
\(66\) −90.9773 −0.169675
\(67\) 141.792 0.258546 0.129273 0.991609i \(-0.458736\pi\)
0.129273 + 0.991609i \(0.458736\pi\)
\(68\) 477.394 0.851361
\(69\) −1.95455 −0.00341015
\(70\) 0 0
\(71\) 602.742 1.00750 0.503749 0.863850i \(-0.331954\pi\)
0.503749 + 0.863850i \(0.331954\pi\)
\(72\) 72.0000 0.117851
\(73\) 1102.30 1.76732 0.883660 0.468129i \(-0.155072\pi\)
0.883660 + 0.468129i \(0.155072\pi\)
\(74\) 337.068 0.529505
\(75\) −927.557 −1.42807
\(76\) 134.045 0.202317
\(77\) 0 0
\(78\) 12.9773 0.0188383
\(79\) −116.303 −0.165634 −0.0828172 0.996565i \(-0.526392\pi\)
−0.0828172 + 0.996565i \(0.526392\pi\)
\(80\) −333.394 −0.465932
\(81\) 81.0000 0.111111
\(82\) 646.045 0.870046
\(83\) 568.928 0.752385 0.376193 0.926542i \(-0.377233\pi\)
0.376193 + 0.926542i \(0.377233\pi\)
\(84\) 0 0
\(85\) −2486.88 −3.17341
\(86\) 443.114 0.555607
\(87\) 489.625 0.603371
\(88\) 121.303 0.146943
\(89\) 383.159 0.456346 0.228173 0.973621i \(-0.426725\pi\)
0.228173 + 0.973621i \(0.426725\pi\)
\(90\) −375.068 −0.439285
\(91\) 0 0
\(92\) 2.60607 0.00295327
\(93\) −669.977 −0.747026
\(94\) −1016.09 −1.11491
\(95\) −698.280 −0.754127
\(96\) −96.0000 −0.102062
\(97\) −334.701 −0.350348 −0.175174 0.984538i \(-0.556049\pi\)
−0.175174 + 0.984538i \(0.556049\pi\)
\(98\) 0 0
\(99\) 136.466 0.138539
\(100\) 1236.74 1.23674
\(101\) 14.7424 0.0145240 0.00726201 0.999974i \(-0.497688\pi\)
0.00726201 + 0.999974i \(0.497688\pi\)
\(102\) −716.091 −0.695133
\(103\) 841.420 0.804928 0.402464 0.915436i \(-0.368154\pi\)
0.402464 + 0.915436i \(0.368154\pi\)
\(104\) −17.3030 −0.0163144
\(105\) 0 0
\(106\) −353.023 −0.323477
\(107\) −715.670 −0.646603 −0.323301 0.946296i \(-0.604793\pi\)
−0.323301 + 0.946296i \(0.604793\pi\)
\(108\) −108.000 −0.0962250
\(109\) 600.019 0.527260 0.263630 0.964624i \(-0.415080\pi\)
0.263630 + 0.964624i \(0.415080\pi\)
\(110\) −631.901 −0.547722
\(111\) −505.602 −0.432339
\(112\) 0 0
\(113\) 622.644 0.518349 0.259174 0.965831i \(-0.416550\pi\)
0.259174 + 0.965831i \(0.416550\pi\)
\(114\) −201.068 −0.165191
\(115\) −13.5757 −0.0110082
\(116\) −652.833 −0.522535
\(117\) −19.4659 −0.0153814
\(118\) −909.856 −0.709822
\(119\) 0 0
\(120\) 500.091 0.380432
\(121\) −1101.09 −0.827263
\(122\) −77.3030 −0.0573663
\(123\) −969.068 −0.710390
\(124\) 893.303 0.646943
\(125\) −3837.90 −2.74618
\(126\) 0 0
\(127\) −180.076 −0.125820 −0.0629100 0.998019i \(-0.520038\pi\)
−0.0629100 + 0.998019i \(0.520038\pi\)
\(128\) 128.000 0.0883883
\(129\) −664.670 −0.453651
\(130\) 90.1363 0.0608114
\(131\) −217.860 −0.145302 −0.0726508 0.997357i \(-0.523146\pi\)
−0.0726508 + 0.997357i \(0.523146\pi\)
\(132\) −181.955 −0.119978
\(133\) 0 0
\(134\) 283.583 0.182820
\(135\) 562.602 0.358675
\(136\) 954.788 0.602003
\(137\) −2601.86 −1.62257 −0.811283 0.584654i \(-0.801230\pi\)
−0.811283 + 0.584654i \(0.801230\pi\)
\(138\) −3.90910 −0.00241134
\(139\) 2651.55 1.61800 0.808998 0.587811i \(-0.200010\pi\)
0.808998 + 0.587811i \(0.200010\pi\)
\(140\) 0 0
\(141\) 1524.14 0.910322
\(142\) 1205.48 0.712409
\(143\) −32.7955 −0.0191783
\(144\) 144.000 0.0833333
\(145\) 3400.79 1.94773
\(146\) 2204.60 1.24968
\(147\) 0 0
\(148\) 674.136 0.374417
\(149\) 581.023 0.319458 0.159729 0.987161i \(-0.448938\pi\)
0.159729 + 0.987161i \(0.448938\pi\)
\(150\) −1855.11 −1.00980
\(151\) −615.390 −0.331654 −0.165827 0.986155i \(-0.553029\pi\)
−0.165827 + 0.986155i \(0.553029\pi\)
\(152\) 268.091 0.143059
\(153\) 1074.14 0.567574
\(154\) 0 0
\(155\) −4653.47 −2.41145
\(156\) 25.9546 0.0133207
\(157\) 306.932 0.156024 0.0780122 0.996952i \(-0.475143\pi\)
0.0780122 + 0.996952i \(0.475143\pi\)
\(158\) −232.606 −0.117121
\(159\) 529.534 0.264118
\(160\) −666.788 −0.329464
\(161\) 0 0
\(162\) 162.000 0.0785674
\(163\) 3514.50 1.68882 0.844408 0.535701i \(-0.179953\pi\)
0.844408 + 0.535701i \(0.179953\pi\)
\(164\) 1292.09 0.615216
\(165\) 947.852 0.447213
\(166\) 1137.86 0.532017
\(167\) −1123.30 −0.520502 −0.260251 0.965541i \(-0.583805\pi\)
−0.260251 + 0.965541i \(0.583805\pi\)
\(168\) 0 0
\(169\) −2192.32 −0.997871
\(170\) −4973.76 −2.24394
\(171\) 301.602 0.134878
\(172\) 886.227 0.392873
\(173\) −1530.60 −0.672655 −0.336327 0.941745i \(-0.609185\pi\)
−0.336327 + 0.941745i \(0.609185\pi\)
\(174\) 979.250 0.426648
\(175\) 0 0
\(176\) 242.606 0.103904
\(177\) 1364.78 0.579568
\(178\) 766.318 0.322685
\(179\) 3413.43 1.42532 0.712659 0.701511i \(-0.247491\pi\)
0.712659 + 0.701511i \(0.247491\pi\)
\(180\) −750.136 −0.310621
\(181\) −1286.71 −0.528399 −0.264200 0.964468i \(-0.585108\pi\)
−0.264200 + 0.964468i \(0.585108\pi\)
\(182\) 0 0
\(183\) 115.955 0.0468394
\(184\) 5.21213 0.00208828
\(185\) −3511.77 −1.39562
\(186\) −1339.95 −0.528227
\(187\) 1809.67 0.707679
\(188\) −2032.18 −0.788362
\(189\) 0 0
\(190\) −1396.56 −0.533248
\(191\) 1055.30 0.399783 0.199891 0.979818i \(-0.435941\pi\)
0.199891 + 0.979818i \(0.435941\pi\)
\(192\) −192.000 −0.0721688
\(193\) −4770.84 −1.77934 −0.889670 0.456604i \(-0.849066\pi\)
−0.889670 + 0.456604i \(0.849066\pi\)
\(194\) −669.402 −0.247733
\(195\) −135.205 −0.0496523
\(196\) 0 0
\(197\) 1622.31 0.586725 0.293363 0.956001i \(-0.405226\pi\)
0.293363 + 0.956001i \(0.405226\pi\)
\(198\) 272.932 0.0979617
\(199\) 3550.14 1.26464 0.632318 0.774709i \(-0.282104\pi\)
0.632318 + 0.774709i \(0.282104\pi\)
\(200\) 2473.48 0.874509
\(201\) −425.375 −0.149272
\(202\) 29.4848 0.0102700
\(203\) 0 0
\(204\) −1432.18 −0.491533
\(205\) −6730.86 −2.29319
\(206\) 1682.84 0.569170
\(207\) 5.86365 0.00196885
\(208\) −34.6061 −0.0115361
\(209\) 508.129 0.168172
\(210\) 0 0
\(211\) 4653.39 1.51826 0.759129 0.650941i \(-0.225625\pi\)
0.759129 + 0.650941i \(0.225625\pi\)
\(212\) −706.045 −0.228733
\(213\) −1808.23 −0.581679
\(214\) −1431.34 −0.457217
\(215\) −4616.61 −1.46442
\(216\) −216.000 −0.0680414
\(217\) 0 0
\(218\) 1200.04 0.372829
\(219\) −3306.90 −1.02036
\(220\) −1263.80 −0.387298
\(221\) −258.136 −0.0785707
\(222\) −1011.20 −0.305710
\(223\) 4649.53 1.39621 0.698107 0.715993i \(-0.254026\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(224\) 0 0
\(225\) 2782.67 0.824495
\(226\) 1245.29 0.366528
\(227\) −4151.72 −1.21392 −0.606958 0.794734i \(-0.707611\pi\)
−0.606958 + 0.794734i \(0.707611\pi\)
\(228\) −402.136 −0.116808
\(229\) −4263.63 −1.23034 −0.615172 0.788393i \(-0.710913\pi\)
−0.615172 + 0.788393i \(0.710913\pi\)
\(230\) −27.1515 −0.00778398
\(231\) 0 0
\(232\) −1305.67 −0.369488
\(233\) 3049.90 0.857535 0.428768 0.903415i \(-0.358948\pi\)
0.428768 + 0.903415i \(0.358948\pi\)
\(234\) −38.9318 −0.0108763
\(235\) 10586.2 2.93859
\(236\) −1819.71 −0.501920
\(237\) 348.909 0.0956290
\(238\) 0 0
\(239\) 3987.20 1.07912 0.539562 0.841946i \(-0.318590\pi\)
0.539562 + 0.841946i \(0.318590\pi\)
\(240\) 1000.18 0.269006
\(241\) 624.648 0.166959 0.0834795 0.996509i \(-0.473397\pi\)
0.0834795 + 0.996509i \(0.473397\pi\)
\(242\) −2202.17 −0.584963
\(243\) −243.000 −0.0641500
\(244\) −154.606 −0.0405641
\(245\) 0 0
\(246\) −1938.14 −0.502321
\(247\) −72.4810 −0.0186715
\(248\) 1786.61 0.457458
\(249\) −1706.78 −0.434390
\(250\) −7675.80 −1.94184
\(251\) 1328.78 0.334152 0.167076 0.985944i \(-0.446568\pi\)
0.167076 + 0.985944i \(0.446568\pi\)
\(252\) 0 0
\(253\) 9.87887 0.00245486
\(254\) −360.152 −0.0889682
\(255\) 7460.64 1.83217
\(256\) 256.000 0.0625000
\(257\) 3226.18 0.783049 0.391525 0.920168i \(-0.371948\pi\)
0.391525 + 0.920168i \(0.371948\pi\)
\(258\) −1329.34 −0.320780
\(259\) 0 0
\(260\) 180.273 0.0430001
\(261\) −1468.87 −0.348357
\(262\) −435.720 −0.102744
\(263\) 3250.61 0.762135 0.381067 0.924547i \(-0.375557\pi\)
0.381067 + 0.924547i \(0.375557\pi\)
\(264\) −363.909 −0.0848373
\(265\) 3677.99 0.852593
\(266\) 0 0
\(267\) −1149.48 −0.263471
\(268\) 567.167 0.129273
\(269\) −2826.04 −0.640546 −0.320273 0.947325i \(-0.603775\pi\)
−0.320273 + 0.947325i \(0.603775\pi\)
\(270\) 1125.20 0.253621
\(271\) 2396.77 0.537245 0.268622 0.963246i \(-0.413432\pi\)
0.268622 + 0.963246i \(0.413432\pi\)
\(272\) 1909.58 0.425680
\(273\) 0 0
\(274\) −5203.71 −1.14733
\(275\) 4688.14 1.02802
\(276\) −7.81820 −0.00170507
\(277\) 1820.47 0.394878 0.197439 0.980315i \(-0.436738\pi\)
0.197439 + 0.980315i \(0.436738\pi\)
\(278\) 5303.10 1.14410
\(279\) 2009.93 0.431296
\(280\) 0 0
\(281\) 3083.81 0.654679 0.327339 0.944907i \(-0.393848\pi\)
0.327339 + 0.944907i \(0.393848\pi\)
\(282\) 3048.27 0.643695
\(283\) −2554.77 −0.536626 −0.268313 0.963332i \(-0.586466\pi\)
−0.268313 + 0.963332i \(0.586466\pi\)
\(284\) 2410.97 0.503749
\(285\) 2094.84 0.435395
\(286\) −65.5910 −0.0135611
\(287\) 0 0
\(288\) 288.000 0.0589256
\(289\) 9331.06 1.89926
\(290\) 6801.58 1.37725
\(291\) 1004.10 0.202273
\(292\) 4409.20 0.883660
\(293\) 1846.47 0.368163 0.184081 0.982911i \(-0.441069\pi\)
0.184081 + 0.982911i \(0.441069\pi\)
\(294\) 0 0
\(295\) 9479.39 1.87089
\(296\) 1348.27 0.264753
\(297\) −409.398 −0.0799854
\(298\) 1162.05 0.225891
\(299\) −1.40915 −0.000272553 0
\(300\) −3710.23 −0.714034
\(301\) 0 0
\(302\) −1230.78 −0.234515
\(303\) −44.2272 −0.00838544
\(304\) 536.182 0.101158
\(305\) 805.386 0.151201
\(306\) 2148.27 0.401335
\(307\) −7041.50 −1.30905 −0.654527 0.756039i \(-0.727132\pi\)
−0.654527 + 0.756039i \(0.727132\pi\)
\(308\) 0 0
\(309\) −2524.26 −0.464726
\(310\) −9306.93 −1.70516
\(311\) 2685.99 0.489738 0.244869 0.969556i \(-0.421255\pi\)
0.244869 + 0.969556i \(0.421255\pi\)
\(312\) 51.9091 0.00941915
\(313\) −2219.19 −0.400754 −0.200377 0.979719i \(-0.564217\pi\)
−0.200377 + 0.979719i \(0.564217\pi\)
\(314\) 613.864 0.110326
\(315\) 0 0
\(316\) −465.212 −0.0828172
\(317\) 2221.26 0.393560 0.196780 0.980448i \(-0.436952\pi\)
0.196780 + 0.980448i \(0.436952\pi\)
\(318\) 1059.07 0.186760
\(319\) −2474.71 −0.434348
\(320\) −1333.58 −0.232966
\(321\) 2147.01 0.373316
\(322\) 0 0
\(323\) 3999.53 0.688978
\(324\) 324.000 0.0555556
\(325\) −668.731 −0.114137
\(326\) 7029.00 1.19417
\(327\) −1800.06 −0.304414
\(328\) 2584.18 0.435023
\(329\) 0 0
\(330\) 1895.70 0.316228
\(331\) 4154.06 0.689812 0.344906 0.938637i \(-0.387911\pi\)
0.344906 + 0.938637i \(0.387911\pi\)
\(332\) 2275.71 0.376193
\(333\) 1516.81 0.249611
\(334\) −2246.61 −0.368050
\(335\) −2954.53 −0.481860
\(336\) 0 0
\(337\) −254.167 −0.0410841 −0.0205420 0.999789i \(-0.506539\pi\)
−0.0205420 + 0.999789i \(0.506539\pi\)
\(338\) −4384.64 −0.705601
\(339\) −1867.93 −0.299269
\(340\) −9947.52 −1.58671
\(341\) 3386.26 0.537761
\(342\) 603.205 0.0953730
\(343\) 0 0
\(344\) 1772.45 0.277803
\(345\) 40.7272 0.00635559
\(346\) −3061.20 −0.475639
\(347\) −6224.64 −0.962986 −0.481493 0.876450i \(-0.659905\pi\)
−0.481493 + 0.876450i \(0.659905\pi\)
\(348\) 1958.50 0.301686
\(349\) −9732.21 −1.49270 −0.746352 0.665552i \(-0.768196\pi\)
−0.746352 + 0.665552i \(0.768196\pi\)
\(350\) 0 0
\(351\) 58.3977 0.00888046
\(352\) 485.212 0.0734713
\(353\) −1425.61 −0.214951 −0.107476 0.994208i \(-0.534277\pi\)
−0.107476 + 0.994208i \(0.534277\pi\)
\(354\) 2729.57 0.409816
\(355\) −12559.4 −1.87770
\(356\) 1532.64 0.228173
\(357\) 0 0
\(358\) 6826.86 1.00785
\(359\) −5766.49 −0.847754 −0.423877 0.905720i \(-0.639331\pi\)
−0.423877 + 0.905720i \(0.639331\pi\)
\(360\) −1500.27 −0.219643
\(361\) −5735.99 −0.836272
\(362\) −2573.42 −0.373635
\(363\) 3303.26 0.477621
\(364\) 0 0
\(365\) −22968.7 −3.29381
\(366\) 231.909 0.0331204
\(367\) −11545.3 −1.64213 −0.821065 0.570834i \(-0.806620\pi\)
−0.821065 + 0.570834i \(0.806620\pi\)
\(368\) 10.4243 0.00147664
\(369\) 2907.20 0.410144
\(370\) −7023.53 −0.986854
\(371\) 0 0
\(372\) −2679.91 −0.373513
\(373\) −6479.57 −0.899463 −0.449731 0.893164i \(-0.648480\pi\)
−0.449731 + 0.893164i \(0.648480\pi\)
\(374\) 3619.33 0.500404
\(375\) 11513.7 1.58551
\(376\) −4064.36 −0.557456
\(377\) 353.000 0.0482239
\(378\) 0 0
\(379\) 611.996 0.0829449 0.0414725 0.999140i \(-0.486795\pi\)
0.0414725 + 0.999140i \(0.486795\pi\)
\(380\) −2793.12 −0.377063
\(381\) 540.227 0.0726422
\(382\) 2110.59 0.282689
\(383\) −4360.81 −0.581794 −0.290897 0.956754i \(-0.593954\pi\)
−0.290897 + 0.956754i \(0.593954\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −9541.68 −1.25818
\(387\) 1994.01 0.261915
\(388\) −1338.80 −0.175174
\(389\) −13146.9 −1.71356 −0.856781 0.515681i \(-0.827539\pi\)
−0.856781 + 0.515681i \(0.827539\pi\)
\(390\) −270.409 −0.0351095
\(391\) 77.7575 0.0100572
\(392\) 0 0
\(393\) 653.580 0.0838899
\(394\) 3244.62 0.414877
\(395\) 2423.42 0.308697
\(396\) 545.864 0.0692694
\(397\) 8478.04 1.07179 0.535895 0.844285i \(-0.319974\pi\)
0.535895 + 0.844285i \(0.319974\pi\)
\(398\) 7100.27 0.894232
\(399\) 0 0
\(400\) 4946.97 0.618371
\(401\) 2803.00 0.349065 0.174533 0.984651i \(-0.444159\pi\)
0.174533 + 0.984651i \(0.444159\pi\)
\(402\) −850.750 −0.105551
\(403\) −483.027 −0.0597054
\(404\) 58.9697 0.00726201
\(405\) −1687.81 −0.207081
\(406\) 0 0
\(407\) 2555.46 0.311227
\(408\) −2864.36 −0.347566
\(409\) 6385.39 0.771973 0.385987 0.922504i \(-0.373861\pi\)
0.385987 + 0.922504i \(0.373861\pi\)
\(410\) −13461.7 −1.62153
\(411\) 7805.57 0.936789
\(412\) 3365.68 0.402464
\(413\) 0 0
\(414\) 11.7273 0.00139219
\(415\) −11854.8 −1.40224
\(416\) −69.2121 −0.00815722
\(417\) −7954.65 −0.934151
\(418\) 1016.26 0.118916
\(419\) −4831.66 −0.563346 −0.281673 0.959510i \(-0.590889\pi\)
−0.281673 + 0.959510i \(0.590889\pi\)
\(420\) 0 0
\(421\) 7475.37 0.865385 0.432693 0.901542i \(-0.357564\pi\)
0.432693 + 0.901542i \(0.357564\pi\)
\(422\) 9306.77 1.07357
\(423\) −4572.41 −0.525575
\(424\) −1412.09 −0.161739
\(425\) 36900.8 4.21165
\(426\) −3616.45 −0.411309
\(427\) 0 0
\(428\) −2862.68 −0.323301
\(429\) 98.3864 0.0110726
\(430\) −9233.21 −1.03550
\(431\) 6991.93 0.781414 0.390707 0.920515i \(-0.372231\pi\)
0.390707 + 0.920515i \(0.372231\pi\)
\(432\) −432.000 −0.0481125
\(433\) 7699.26 0.854510 0.427255 0.904131i \(-0.359481\pi\)
0.427255 + 0.904131i \(0.359481\pi\)
\(434\) 0 0
\(435\) −10202.4 −1.12452
\(436\) 2400.08 0.263630
\(437\) 21.8332 0.00238999
\(438\) −6613.80 −0.721505
\(439\) −9412.32 −1.02329 −0.511646 0.859196i \(-0.670964\pi\)
−0.511646 + 0.859196i \(0.670964\pi\)
\(440\) −2527.61 −0.273861
\(441\) 0 0
\(442\) −516.273 −0.0555579
\(443\) −6258.18 −0.671185 −0.335593 0.942007i \(-0.608937\pi\)
−0.335593 + 0.942007i \(0.608937\pi\)
\(444\) −2022.41 −0.216170
\(445\) −7983.93 −0.850505
\(446\) 9299.07 0.987273
\(447\) −1743.07 −0.184439
\(448\) 0 0
\(449\) −11633.8 −1.22279 −0.611396 0.791325i \(-0.709392\pi\)
−0.611396 + 0.791325i \(0.709392\pi\)
\(450\) 5565.34 0.583006
\(451\) 4897.95 0.511387
\(452\) 2490.58 0.259174
\(453\) 1846.17 0.191480
\(454\) −8303.43 −0.858369
\(455\) 0 0
\(456\) −804.273 −0.0825954
\(457\) −13104.6 −1.34138 −0.670688 0.741740i \(-0.734001\pi\)
−0.670688 + 0.741740i \(0.734001\pi\)
\(458\) −8527.26 −0.869985
\(459\) −3222.41 −0.327689
\(460\) −54.3029 −0.00550410
\(461\) 2594.63 0.262134 0.131067 0.991373i \(-0.458160\pi\)
0.131067 + 0.991373i \(0.458160\pi\)
\(462\) 0 0
\(463\) −14136.2 −1.41893 −0.709465 0.704741i \(-0.751063\pi\)
−0.709465 + 0.704741i \(0.751063\pi\)
\(464\) −2611.33 −0.261267
\(465\) 13960.4 1.39225
\(466\) 6099.80 0.606369
\(467\) −15590.2 −1.54482 −0.772409 0.635125i \(-0.780948\pi\)
−0.772409 + 0.635125i \(0.780948\pi\)
\(468\) −77.8637 −0.00769070
\(469\) 0 0
\(470\) 21172.4 2.07789
\(471\) −920.795 −0.0900807
\(472\) −3639.42 −0.354911
\(473\) 3359.44 0.326569
\(474\) 697.818 0.0676199
\(475\) 10361.2 1.00085
\(476\) 0 0
\(477\) −1588.60 −0.152489
\(478\) 7974.41 0.763056
\(479\) 8453.51 0.806369 0.403184 0.915119i \(-0.367903\pi\)
0.403184 + 0.915119i \(0.367903\pi\)
\(480\) 2000.36 0.190216
\(481\) −364.519 −0.0345543
\(482\) 1249.30 0.118058
\(483\) 0 0
\(484\) −4404.35 −0.413632
\(485\) 6974.20 0.652953
\(486\) −486.000 −0.0453609
\(487\) −4011.07 −0.373221 −0.186611 0.982434i \(-0.559750\pi\)
−0.186611 + 0.982434i \(0.559750\pi\)
\(488\) −309.212 −0.0286831
\(489\) −10543.5 −0.975038
\(490\) 0 0
\(491\) 13927.9 1.28016 0.640079 0.768309i \(-0.278902\pi\)
0.640079 + 0.768309i \(0.278902\pi\)
\(492\) −3876.27 −0.355195
\(493\) −19478.7 −1.77946
\(494\) −144.962 −0.0132027
\(495\) −2843.56 −0.258199
\(496\) 3573.21 0.323472
\(497\) 0 0
\(498\) −3413.57 −0.307160
\(499\) −3947.55 −0.354141 −0.177071 0.984198i \(-0.556662\pi\)
−0.177071 + 0.984198i \(0.556662\pi\)
\(500\) −15351.6 −1.37309
\(501\) 3369.91 0.300512
\(502\) 2657.57 0.236281
\(503\) 13725.3 1.21666 0.608331 0.793684i \(-0.291839\pi\)
0.608331 + 0.793684i \(0.291839\pi\)
\(504\) 0 0
\(505\) −307.190 −0.0270688
\(506\) 19.7577 0.00173585
\(507\) 6576.97 0.576121
\(508\) −720.303 −0.0629100
\(509\) −7830.10 −0.681853 −0.340926 0.940090i \(-0.610741\pi\)
−0.340926 + 0.940090i \(0.610741\pi\)
\(510\) 14921.3 1.29554
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) −904.807 −0.0778717
\(514\) 6452.36 0.553700
\(515\) −17532.8 −1.50017
\(516\) −2658.68 −0.226825
\(517\) −7703.43 −0.655312
\(518\) 0 0
\(519\) 4591.80 0.388357
\(520\) 360.545 0.0304057
\(521\) 5907.39 0.496751 0.248376 0.968664i \(-0.420103\pi\)
0.248376 + 0.968664i \(0.420103\pi\)
\(522\) −2937.75 −0.246325
\(523\) −7908.06 −0.661176 −0.330588 0.943775i \(-0.607247\pi\)
−0.330588 + 0.943775i \(0.607247\pi\)
\(524\) −871.439 −0.0726508
\(525\) 0 0
\(526\) 6501.23 0.538911
\(527\) 26653.6 2.20313
\(528\) −727.818 −0.0599891
\(529\) −12166.6 −0.999965
\(530\) 7355.98 0.602874
\(531\) −4094.35 −0.334613
\(532\) 0 0
\(533\) −698.659 −0.0567773
\(534\) −2298.95 −0.186302
\(535\) 14912.5 1.20509
\(536\) 1134.33 0.0914100
\(537\) −10240.3 −0.822908
\(538\) −5652.08 −0.452934
\(539\) 0 0
\(540\) 2250.41 0.179337
\(541\) −3941.04 −0.313195 −0.156598 0.987662i \(-0.550053\pi\)
−0.156598 + 0.987662i \(0.550053\pi\)
\(542\) 4793.54 0.379889
\(543\) 3860.12 0.305071
\(544\) 3819.15 0.301001
\(545\) −12502.7 −0.982670
\(546\) 0 0
\(547\) −1828.71 −0.142943 −0.0714717 0.997443i \(-0.522770\pi\)
−0.0714717 + 0.997443i \(0.522770\pi\)
\(548\) −10407.4 −0.811283
\(549\) −347.864 −0.0270427
\(550\) 9376.29 0.726920
\(551\) −5469.33 −0.422870
\(552\) −15.6364 −0.00120567
\(553\) 0 0
\(554\) 3640.93 0.279221
\(555\) 10535.3 0.805763
\(556\) 10606.2 0.808998
\(557\) 22532.0 1.71402 0.857011 0.515298i \(-0.172319\pi\)
0.857011 + 0.515298i \(0.172319\pi\)
\(558\) 4019.86 0.304972
\(559\) −479.201 −0.0362577
\(560\) 0 0
\(561\) −5429.00 −0.408579
\(562\) 6167.62 0.462928
\(563\) −23355.7 −1.74836 −0.874179 0.485604i \(-0.838600\pi\)
−0.874179 + 0.485604i \(0.838600\pi\)
\(564\) 6096.55 0.455161
\(565\) −12974.1 −0.966062
\(566\) −5109.54 −0.379452
\(567\) 0 0
\(568\) 4821.94 0.356204
\(569\) −20887.6 −1.53894 −0.769468 0.638686i \(-0.779478\pi\)
−0.769468 + 0.638686i \(0.779478\pi\)
\(570\) 4189.68 0.307871
\(571\) 23745.3 1.74029 0.870147 0.492792i \(-0.164024\pi\)
0.870147 + 0.492792i \(0.164024\pi\)
\(572\) −131.182 −0.00958914
\(573\) −3165.89 −0.230815
\(574\) 0 0
\(575\) 201.440 0.0146098
\(576\) 576.000 0.0416667
\(577\) −2454.39 −0.177084 −0.0885422 0.996072i \(-0.528221\pi\)
−0.0885422 + 0.996072i \(0.528221\pi\)
\(578\) 18662.1 1.34298
\(579\) 14312.5 1.02730
\(580\) 13603.2 0.973863
\(581\) 0 0
\(582\) 2008.20 0.143029
\(583\) −2676.42 −0.190130
\(584\) 8818.39 0.624842
\(585\) 405.614 0.0286668
\(586\) 3692.93 0.260330
\(587\) −18567.5 −1.30556 −0.652780 0.757547i \(-0.726397\pi\)
−0.652780 + 0.757547i \(0.726397\pi\)
\(588\) 0 0
\(589\) 7483.95 0.523550
\(590\) 18958.8 1.32292
\(591\) −4866.93 −0.338746
\(592\) 2696.55 0.187208
\(593\) −17112.9 −1.18507 −0.592533 0.805546i \(-0.701872\pi\)
−0.592533 + 0.805546i \(0.701872\pi\)
\(594\) −818.795 −0.0565582
\(595\) 0 0
\(596\) 2324.09 0.159729
\(597\) −10650.4 −0.730138
\(598\) −2.81830 −0.000192724 0
\(599\) 23264.8 1.58694 0.793469 0.608611i \(-0.208273\pi\)
0.793469 + 0.608611i \(0.208273\pi\)
\(600\) −7420.45 −0.504898
\(601\) −25322.3 −1.71867 −0.859334 0.511416i \(-0.829121\pi\)
−0.859334 + 0.511416i \(0.829121\pi\)
\(602\) 0 0
\(603\) 1276.13 0.0861821
\(604\) −2461.56 −0.165827
\(605\) 22943.5 1.54179
\(606\) −88.4545 −0.00592940
\(607\) 21734.4 1.45333 0.726665 0.686992i \(-0.241069\pi\)
0.726665 + 0.686992i \(0.241069\pi\)
\(608\) 1072.36 0.0715297
\(609\) 0 0
\(610\) 1610.77 0.106915
\(611\) 1098.84 0.0727567
\(612\) 4296.55 0.283787
\(613\) −13572.4 −0.894262 −0.447131 0.894468i \(-0.647554\pi\)
−0.447131 + 0.894468i \(0.647554\pi\)
\(614\) −14083.0 −0.925641
\(615\) 20192.6 1.32397
\(616\) 0 0
\(617\) −8497.12 −0.554427 −0.277213 0.960808i \(-0.589411\pi\)
−0.277213 + 0.960808i \(0.589411\pi\)
\(618\) −5048.52 −0.328611
\(619\) 22982.9 1.49235 0.746173 0.665752i \(-0.231889\pi\)
0.746173 + 0.665752i \(0.231889\pi\)
\(620\) −18613.9 −1.20573
\(621\) −17.5910 −0.00113672
\(622\) 5371.98 0.346297
\(623\) 0 0
\(624\) 103.818 0.00666034
\(625\) 41322.5 2.64464
\(626\) −4438.38 −0.283376
\(627\) −1524.39 −0.0970943
\(628\) 1227.73 0.0780122
\(629\) 20114.3 1.27505
\(630\) 0 0
\(631\) −15717.9 −0.991635 −0.495817 0.868427i \(-0.665131\pi\)
−0.495817 + 0.868427i \(0.665131\pi\)
\(632\) −930.424 −0.0585606
\(633\) −13960.2 −0.876566
\(634\) 4442.52 0.278289
\(635\) 3752.26 0.234494
\(636\) 2118.14 0.132059
\(637\) 0 0
\(638\) −4949.42 −0.307131
\(639\) 5424.68 0.335833
\(640\) −2667.15 −0.164732
\(641\) 29107.4 1.79356 0.896780 0.442478i \(-0.145900\pi\)
0.896780 + 0.442478i \(0.145900\pi\)
\(642\) 4294.02 0.263974
\(643\) 3112.26 0.190880 0.0954398 0.995435i \(-0.469574\pi\)
0.0954398 + 0.995435i \(0.469574\pi\)
\(644\) 0 0
\(645\) 13849.8 0.845482
\(646\) 7999.06 0.487181
\(647\) 7857.59 0.477456 0.238728 0.971087i \(-0.423270\pi\)
0.238728 + 0.971087i \(0.423270\pi\)
\(648\) 648.000 0.0392837
\(649\) −6898.02 −0.417213
\(650\) −1337.46 −0.0807071
\(651\) 0 0
\(652\) 14058.0 0.844408
\(653\) −19522.0 −1.16992 −0.584958 0.811063i \(-0.698889\pi\)
−0.584958 + 0.811063i \(0.698889\pi\)
\(654\) −3600.11 −0.215253
\(655\) 4539.57 0.270803
\(656\) 5168.36 0.307608
\(657\) 9920.69 0.589107
\(658\) 0 0
\(659\) 664.061 0.0392536 0.0196268 0.999807i \(-0.493752\pi\)
0.0196268 + 0.999807i \(0.493752\pi\)
\(660\) 3791.41 0.223607
\(661\) −15921.6 −0.936883 −0.468442 0.883494i \(-0.655184\pi\)
−0.468442 + 0.883494i \(0.655184\pi\)
\(662\) 8308.11 0.487770
\(663\) 774.409 0.0453628
\(664\) 4551.42 0.266008
\(665\) 0 0
\(666\) 3033.61 0.176502
\(667\) −106.333 −0.00617276
\(668\) −4493.21 −0.260251
\(669\) −13948.6 −0.806105
\(670\) −5909.06 −0.340727
\(671\) −586.068 −0.0337182
\(672\) 0 0
\(673\) 24631.0 1.41078 0.705391 0.708819i \(-0.250771\pi\)
0.705391 + 0.708819i \(0.250771\pi\)
\(674\) −508.333 −0.0290508
\(675\) −8348.01 −0.476022
\(676\) −8769.29 −0.498935
\(677\) −17092.8 −0.970353 −0.485177 0.874416i \(-0.661245\pi\)
−0.485177 + 0.874416i \(0.661245\pi\)
\(678\) −3735.86 −0.211615
\(679\) 0 0
\(680\) −19895.0 −1.12197
\(681\) 12455.1 0.700855
\(682\) 6772.52 0.380254
\(683\) −19163.6 −1.07361 −0.536804 0.843707i \(-0.680368\pi\)
−0.536804 + 0.843707i \(0.680368\pi\)
\(684\) 1206.41 0.0674389
\(685\) 54215.2 3.02402
\(686\) 0 0
\(687\) 12790.9 0.710339
\(688\) 3544.91 0.196437
\(689\) 381.773 0.0211094
\(690\) 81.4544 0.00449408
\(691\) −8095.87 −0.445704 −0.222852 0.974852i \(-0.571537\pi\)
−0.222852 + 0.974852i \(0.571537\pi\)
\(692\) −6122.39 −0.336327
\(693\) 0 0
\(694\) −12449.3 −0.680934
\(695\) −55250.7 −3.01551
\(696\) 3917.00 0.213324
\(697\) 38552.3 2.09508
\(698\) −19464.4 −1.05550
\(699\) −9149.70 −0.495098
\(700\) 0 0
\(701\) 12354.7 0.665664 0.332832 0.942986i \(-0.391996\pi\)
0.332832 + 0.942986i \(0.391996\pi\)
\(702\) 116.795 0.00627943
\(703\) 5647.81 0.303003
\(704\) 970.424 0.0519520
\(705\) −31758.6 −1.69659
\(706\) −2851.23 −0.151993
\(707\) 0 0
\(708\) 5459.14 0.289784
\(709\) 3828.82 0.202813 0.101406 0.994845i \(-0.467666\pi\)
0.101406 + 0.994845i \(0.467666\pi\)
\(710\) −25118.8 −1.32774
\(711\) −1046.73 −0.0552114
\(712\) 3065.27 0.161343
\(713\) 145.500 0.00764241
\(714\) 0 0
\(715\) 683.363 0.0357431
\(716\) 13653.7 0.712659
\(717\) −11961.6 −0.623033
\(718\) −11533.0 −0.599453
\(719\) −1223.00 −0.0634356 −0.0317178 0.999497i \(-0.510098\pi\)
−0.0317178 + 0.999497i \(0.510098\pi\)
\(720\) −3000.55 −0.155311
\(721\) 0 0
\(722\) −11472.0 −0.591333
\(723\) −1873.94 −0.0963938
\(724\) −5146.83 −0.264200
\(725\) −50461.7 −2.58496
\(726\) 6606.52 0.337729
\(727\) −6368.21 −0.324875 −0.162437 0.986719i \(-0.551936\pi\)
−0.162437 + 0.986719i \(0.551936\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −45937.5 −2.32907
\(731\) 26442.5 1.33791
\(732\) 463.818 0.0234197
\(733\) 25154.0 1.26751 0.633753 0.773535i \(-0.281513\pi\)
0.633753 + 0.773535i \(0.281513\pi\)
\(734\) −23090.7 −1.16116
\(735\) 0 0
\(736\) 20.8485 0.00104414
\(737\) 2149.97 0.107456
\(738\) 5814.41 0.290015
\(739\) −10739.1 −0.534566 −0.267283 0.963618i \(-0.586126\pi\)
−0.267283 + 0.963618i \(0.586126\pi\)
\(740\) −14047.1 −0.697811
\(741\) 217.443 0.0107800
\(742\) 0 0
\(743\) 28166.3 1.39074 0.695370 0.718652i \(-0.255240\pi\)
0.695370 + 0.718652i \(0.255240\pi\)
\(744\) −5359.82 −0.264114
\(745\) −12106.8 −0.595383
\(746\) −12959.1 −0.636016
\(747\) 5120.35 0.250795
\(748\) 7238.67 0.353839
\(749\) 0 0
\(750\) 23027.4 1.12112
\(751\) 28657.0 1.39242 0.696211 0.717837i \(-0.254868\pi\)
0.696211 + 0.717837i \(0.254868\pi\)
\(752\) −8128.73 −0.394181
\(753\) −3986.35 −0.192923
\(754\) 706.000 0.0340995
\(755\) 12823.0 0.618113
\(756\) 0 0
\(757\) −23604.1 −1.13330 −0.566648 0.823960i \(-0.691760\pi\)
−0.566648 + 0.823960i \(0.691760\pi\)
\(758\) 1223.99 0.0586509
\(759\) −29.6366 −0.00141731
\(760\) −5586.24 −0.266624
\(761\) −4630.97 −0.220595 −0.110297 0.993899i \(-0.535180\pi\)
−0.110297 + 0.993899i \(0.535180\pi\)
\(762\) 1080.45 0.0513658
\(763\) 0 0
\(764\) 4221.18 0.199891
\(765\) −22381.9 −1.05780
\(766\) −8721.62 −0.411390
\(767\) 983.954 0.0463214
\(768\) −768.000 −0.0360844
\(769\) −33276.8 −1.56046 −0.780228 0.625495i \(-0.784897\pi\)
−0.780228 + 0.625495i \(0.784897\pi\)
\(770\) 0 0
\(771\) −9678.55 −0.452094
\(772\) −19083.4 −0.889670
\(773\) 22938.8 1.06734 0.533668 0.845694i \(-0.320813\pi\)
0.533668 + 0.845694i \(0.320813\pi\)
\(774\) 3988.02 0.185202
\(775\) 69049.1 3.20041
\(776\) −2677.61 −0.123867
\(777\) 0 0
\(778\) −26293.8 −1.21167
\(779\) 10824.9 0.497873
\(780\) −540.818 −0.0248261
\(781\) 9139.31 0.418733
\(782\) 155.515 0.00711152
\(783\) 4406.62 0.201124
\(784\) 0 0
\(785\) −6395.58 −0.290787
\(786\) 1307.16 0.0593191
\(787\) 13514.5 0.612120 0.306060 0.952012i \(-0.400989\pi\)
0.306060 + 0.952012i \(0.400989\pi\)
\(788\) 6489.24 0.293363
\(789\) −9751.84 −0.440019
\(790\) 4846.84 0.218282
\(791\) 0 0
\(792\) 1091.73 0.0489809
\(793\) 83.5986 0.00374360
\(794\) 16956.1 0.757870
\(795\) −11034.0 −0.492245
\(796\) 14200.5 0.632318
\(797\) −10473.4 −0.465480 −0.232740 0.972539i \(-0.574769\pi\)
−0.232740 + 0.972539i \(0.574769\pi\)
\(798\) 0 0
\(799\) −60634.5 −2.68472
\(800\) 9893.94 0.437254
\(801\) 3448.43 0.152115
\(802\) 5606.00 0.246826
\(803\) 16714.0 0.734527
\(804\) −1701.50 −0.0746359
\(805\) 0 0
\(806\) −966.053 −0.0422181
\(807\) 8478.12 0.369819
\(808\) 117.939 0.00513501
\(809\) 23568.0 1.02423 0.512117 0.858916i \(-0.328861\pi\)
0.512117 + 0.858916i \(0.328861\pi\)
\(810\) −3375.61 −0.146428
\(811\) 6704.22 0.290280 0.145140 0.989411i \(-0.453637\pi\)
0.145140 + 0.989411i \(0.453637\pi\)
\(812\) 0 0
\(813\) −7190.31 −0.310178
\(814\) 5110.92 0.220071
\(815\) −73232.1 −3.14749
\(816\) −5728.73 −0.245767
\(817\) 7424.67 0.317939
\(818\) 12770.8 0.545867
\(819\) 0 0
\(820\) −26923.5 −1.14659
\(821\) 24539.4 1.04316 0.521579 0.853203i \(-0.325343\pi\)
0.521579 + 0.853203i \(0.325343\pi\)
\(822\) 15611.1 0.662410
\(823\) −31117.0 −1.31795 −0.658973 0.752167i \(-0.729009\pi\)
−0.658973 + 0.752167i \(0.729009\pi\)
\(824\) 6731.36 0.284585
\(825\) −14064.4 −0.593528
\(826\) 0 0
\(827\) 31244.9 1.31377 0.656887 0.753989i \(-0.271873\pi\)
0.656887 + 0.753989i \(0.271873\pi\)
\(828\) 23.4546 0.000984425 0
\(829\) −4231.50 −0.177281 −0.0886405 0.996064i \(-0.528252\pi\)
−0.0886405 + 0.996064i \(0.528252\pi\)
\(830\) −23709.6 −0.991535
\(831\) −5461.40 −0.227983
\(832\) −138.424 −0.00576803
\(833\) 0 0
\(834\) −15909.3 −0.660544
\(835\) 23406.4 0.970074
\(836\) 2032.51 0.0840861
\(837\) −6029.80 −0.249009
\(838\) −9663.32 −0.398346
\(839\) 38670.4 1.59124 0.795621 0.605795i \(-0.207145\pi\)
0.795621 + 0.605795i \(0.207145\pi\)
\(840\) 0 0
\(841\) 2247.96 0.0921710
\(842\) 14950.7 0.611920
\(843\) −9251.43 −0.377979
\(844\) 18613.5 0.759129
\(845\) 45681.7 1.85976
\(846\) −9144.82 −0.371637
\(847\) 0 0
\(848\) −2824.18 −0.114367
\(849\) 7664.31 0.309821
\(850\) 73801.7 2.97809
\(851\) 109.803 0.00442302
\(852\) −7232.91 −0.290840
\(853\) 19944.4 0.800565 0.400282 0.916392i \(-0.368912\pi\)
0.400282 + 0.916392i \(0.368912\pi\)
\(854\) 0 0
\(855\) −6284.52 −0.251376
\(856\) −5725.36 −0.228609
\(857\) −13882.2 −0.553334 −0.276667 0.960966i \(-0.589230\pi\)
−0.276667 + 0.960966i \(0.589230\pi\)
\(858\) 196.773 0.00782950
\(859\) 4157.16 0.165123 0.0825614 0.996586i \(-0.473690\pi\)
0.0825614 + 0.996586i \(0.473690\pi\)
\(860\) −18466.4 −0.732209
\(861\) 0 0
\(862\) 13983.9 0.552543
\(863\) −16237.9 −0.640493 −0.320246 0.947334i \(-0.603766\pi\)
−0.320246 + 0.947334i \(0.603766\pi\)
\(864\) −864.000 −0.0340207
\(865\) 31893.3 1.25365
\(866\) 15398.5 0.604230
\(867\) −27993.2 −1.09654
\(868\) 0 0
\(869\) −1763.49 −0.0688403
\(870\) −20404.8 −0.795156
\(871\) −306.678 −0.0119304
\(872\) 4800.15 0.186415
\(873\) −3012.31 −0.116783
\(874\) 43.6664 0.00168998
\(875\) 0 0
\(876\) −13227.6 −0.510181
\(877\) −16489.4 −0.634900 −0.317450 0.948275i \(-0.602827\pi\)
−0.317450 + 0.948275i \(0.602827\pi\)
\(878\) −18824.6 −0.723577
\(879\) −5539.40 −0.212559
\(880\) −5055.21 −0.193649
\(881\) −45411.7 −1.73662 −0.868309 0.496023i \(-0.834793\pi\)
−0.868309 + 0.496023i \(0.834793\pi\)
\(882\) 0 0
\(883\) −2206.85 −0.0841070 −0.0420535 0.999115i \(-0.513390\pi\)
−0.0420535 + 0.999115i \(0.513390\pi\)
\(884\) −1032.55 −0.0392854
\(885\) −28438.2 −1.08016
\(886\) −12516.4 −0.474600
\(887\) −28146.2 −1.06545 −0.532727 0.846287i \(-0.678833\pi\)
−0.532727 + 0.846287i \(0.678833\pi\)
\(888\) −4044.82 −0.152855
\(889\) 0 0
\(890\) −15967.9 −0.601398
\(891\) 1228.19 0.0461796
\(892\) 18598.1 0.698107
\(893\) −17025.3 −0.637995
\(894\) −3486.14 −0.130418
\(895\) −71126.1 −2.65641
\(896\) 0 0
\(897\) 4.22746 0.000157359 0
\(898\) −23267.6 −0.864644
\(899\) −36448.6 −1.35220
\(900\) 11130.7 0.412247
\(901\) −21066.4 −0.778937
\(902\) 9795.91 0.361605
\(903\) 0 0
\(904\) 4981.15 0.183264
\(905\) 26811.3 0.984793
\(906\) 3692.34 0.135397
\(907\) −5042.25 −0.184592 −0.0922960 0.995732i \(-0.529421\pi\)
−0.0922960 + 0.995732i \(0.529421\pi\)
\(908\) −16606.9 −0.606958
\(909\) 132.682 0.00484134
\(910\) 0 0
\(911\) 29647.3 1.07822 0.539110 0.842235i \(-0.318761\pi\)
0.539110 + 0.842235i \(0.318761\pi\)
\(912\) −1608.55 −0.0584038
\(913\) 8626.59 0.312704
\(914\) −26209.3 −0.948496
\(915\) −2416.16 −0.0872959
\(916\) −17054.5 −0.615172
\(917\) 0 0
\(918\) −6444.82 −0.231711
\(919\) −11891.3 −0.426830 −0.213415 0.976962i \(-0.568459\pi\)
−0.213415 + 0.976962i \(0.568459\pi\)
\(920\) −108.606 −0.00389199
\(921\) 21124.5 0.755782
\(922\) 5189.26 0.185357
\(923\) −1303.66 −0.0464902
\(924\) 0 0
\(925\) 52108.3 1.85223
\(926\) −28272.4 −1.00333
\(927\) 7572.78 0.268309
\(928\) −5222.67 −0.184744
\(929\) 39188.5 1.38400 0.691999 0.721898i \(-0.256730\pi\)
0.691999 + 0.721898i \(0.256730\pi\)
\(930\) 27920.8 0.984472
\(931\) 0 0
\(932\) 12199.6 0.428768
\(933\) −8057.98 −0.282751
\(934\) −31180.5 −1.09235
\(935\) −37708.2 −1.31892
\(936\) −155.727 −0.00543815
\(937\) −9716.23 −0.338757 −0.169379 0.985551i \(-0.554176\pi\)
−0.169379 + 0.985551i \(0.554176\pi\)
\(938\) 0 0
\(939\) 6657.57 0.231375
\(940\) 42344.8 1.46929
\(941\) 6995.87 0.242358 0.121179 0.992631i \(-0.461333\pi\)
0.121179 + 0.992631i \(0.461333\pi\)
\(942\) −1841.59 −0.0636967
\(943\) 210.455 0.00726760
\(944\) −7278.85 −0.250960
\(945\) 0 0
\(946\) 6718.88 0.230919
\(947\) 14979.2 0.514002 0.257001 0.966411i \(-0.417266\pi\)
0.257001 + 0.966411i \(0.417266\pi\)
\(948\) 1395.64 0.0478145
\(949\) −2384.14 −0.0815516
\(950\) 20722.5 0.707711
\(951\) −6663.78 −0.227222
\(952\) 0 0
\(953\) 29393.3 0.999100 0.499550 0.866285i \(-0.333499\pi\)
0.499550 + 0.866285i \(0.333499\pi\)
\(954\) −3177.20 −0.107826
\(955\) −21989.3 −0.745087
\(956\) 15948.8 0.539562
\(957\) 7424.12 0.250771
\(958\) 16907.0 0.570189
\(959\) 0 0
\(960\) 4000.73 0.134503
\(961\) 20083.4 0.674143
\(962\) −729.038 −0.0244336
\(963\) −6441.03 −0.215534
\(964\) 2498.59 0.0834795
\(965\) 99410.6 3.31621
\(966\) 0 0
\(967\) −7133.95 −0.237241 −0.118621 0.992940i \(-0.537847\pi\)
−0.118621 + 0.992940i \(0.537847\pi\)
\(968\) −8808.70 −0.292482
\(969\) −11998.6 −0.397781
\(970\) 13948.4 0.461707
\(971\) 9688.13 0.320192 0.160096 0.987101i \(-0.448820\pi\)
0.160096 + 0.987101i \(0.448820\pi\)
\(972\) −972.000 −0.0320750
\(973\) 0 0
\(974\) −8022.14 −0.263907
\(975\) 2006.19 0.0658970
\(976\) −618.424 −0.0202820
\(977\) 21305.7 0.697676 0.348838 0.937183i \(-0.386576\pi\)
0.348838 + 0.937183i \(0.386576\pi\)
\(978\) −21087.0 −0.689456
\(979\) 5809.79 0.189665
\(980\) 0 0
\(981\) 5400.17 0.175753
\(982\) 27855.8 0.905208
\(983\) 37280.8 1.20964 0.604818 0.796364i \(-0.293246\pi\)
0.604818 + 0.796364i \(0.293246\pi\)
\(984\) −7752.55 −0.251161
\(985\) −33804.3 −1.09350
\(986\) −38957.3 −1.25827
\(987\) 0 0
\(988\) −289.924 −0.00933574
\(989\) 144.348 0.00464105
\(990\) −5687.11 −0.182574
\(991\) −51397.1 −1.64751 −0.823755 0.566946i \(-0.808125\pi\)
−0.823755 + 0.566946i \(0.808125\pi\)
\(992\) 7146.42 0.228729
\(993\) −12462.2 −0.398263
\(994\) 0 0
\(995\) −73974.6 −2.35694
\(996\) −6827.14 −0.217195
\(997\) 34373.8 1.09191 0.545953 0.837816i \(-0.316168\pi\)
0.545953 + 0.837816i \(0.316168\pi\)
\(998\) −7895.10 −0.250416
\(999\) −4550.42 −0.144113
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 294.4.a.m.1.1 2
3.2 odd 2 882.4.a.z.1.2 2
4.3 odd 2 2352.4.a.ca.1.1 2
7.2 even 3 294.4.e.l.67.2 4
7.3 odd 6 42.4.e.c.37.1 yes 4
7.4 even 3 294.4.e.l.79.2 4
7.5 odd 6 42.4.e.c.25.1 4
7.6 odd 2 294.4.a.n.1.2 2
21.2 odd 6 882.4.g.bf.361.1 4
21.5 even 6 126.4.g.g.109.2 4
21.11 odd 6 882.4.g.bf.667.1 4
21.17 even 6 126.4.g.g.37.2 4
21.20 even 2 882.4.a.v.1.1 2
28.3 even 6 336.4.q.j.289.1 4
28.19 even 6 336.4.q.j.193.1 4
28.27 even 2 2352.4.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.c.25.1 4 7.5 odd 6
42.4.e.c.37.1 yes 4 7.3 odd 6
126.4.g.g.37.2 4 21.17 even 6
126.4.g.g.109.2 4 21.5 even 6
294.4.a.m.1.1 2 1.1 even 1 trivial
294.4.a.n.1.2 2 7.6 odd 2
294.4.e.l.67.2 4 7.2 even 3
294.4.e.l.79.2 4 7.4 even 3
336.4.q.j.193.1 4 28.19 even 6
336.4.q.j.289.1 4 28.3 even 6
882.4.a.v.1.1 2 21.20 even 2
882.4.a.z.1.2 2 3.2 odd 2
882.4.g.bf.361.1 4 21.2 odd 6
882.4.g.bf.667.1 4 21.11 odd 6
2352.4.a.bq.1.2 2 28.27 even 2
2352.4.a.ca.1.1 2 4.3 odd 2