Properties

Label 1575.4.a.m.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.56155 q^{2} +22.9309 q^{4} +7.00000 q^{7} -83.0388 q^{8} +O(q^{10})\) \(q-5.56155 q^{2} +22.9309 q^{4} +7.00000 q^{7} -83.0388 q^{8} +33.6155 q^{11} +38.3542 q^{13} -38.9309 q^{14} +278.378 q^{16} +65.7235 q^{17} +33.3996 q^{19} -186.955 q^{22} +207.447 q^{23} -213.309 q^{26} +160.516 q^{28} +189.170 q^{29} +202.108 q^{31} -883.902 q^{32} -365.525 q^{34} +16.5227 q^{37} -185.754 q^{38} -388.617 q^{41} -41.8144 q^{43} +770.833 q^{44} -1153.73 q^{46} +368.648 q^{47} +49.0000 q^{49} +879.494 q^{52} +458.172 q^{53} -581.272 q^{56} -1052.08 q^{58} -256.216 q^{59} -123.511 q^{61} -1124.03 q^{62} +2688.85 q^{64} +336.277 q^{67} +1507.10 q^{68} +453.312 q^{71} -22.0436 q^{73} -91.8920 q^{74} +765.882 q^{76} +235.309 q^{77} +385.417 q^{79} +2161.32 q^{82} +23.7501 q^{83} +232.553 q^{86} -2791.39 q^{88} +1482.81 q^{89} +268.479 q^{91} +4756.94 q^{92} -2050.25 q^{94} -51.9867 q^{97} -272.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 7 q^{2} + 17 q^{4} + 14 q^{7} - 63 q^{8} + 26 q^{11} - 14 q^{13} - 49 q^{14} + 297 q^{16} + 16 q^{17} + 174 q^{19} - 176 q^{22} + 184 q^{23} - 138 q^{26} + 119 q^{28} + 32 q^{29} + 330 q^{31} - 1071 q^{32} - 294 q^{34} + 132 q^{37} - 388 q^{38} - 200 q^{41} - 364 q^{43} + 816 q^{44} - 1120 q^{46} + 292 q^{47} + 98 q^{49} + 1190 q^{52} + 34 q^{53} - 441 q^{56} - 826 q^{58} - 364 q^{59} + 792 q^{61} - 1308 q^{62} + 2809 q^{64} + 788 q^{67} + 1802 q^{68} - 454 q^{71} - 778 q^{73} - 258 q^{74} - 68 q^{76} + 182 q^{77} + 408 q^{79} + 1890 q^{82} + 1136 q^{83} + 696 q^{86} - 2944 q^{88} - 36 q^{89} - 98 q^{91} + 4896 q^{92} - 1940 q^{94} + 498 q^{97} - 343 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.56155 −1.96631 −0.983153 0.182785i \(-0.941489\pi\)
−0.983153 + 0.182785i \(0.941489\pi\)
\(3\) 0 0
\(4\) 22.9309 2.86636
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −83.0388 −3.66983
\(9\) 0 0
\(10\) 0 0
\(11\) 33.6155 0.921406 0.460703 0.887554i \(-0.347597\pi\)
0.460703 + 0.887554i \(0.347597\pi\)
\(12\) 0 0
\(13\) 38.3542 0.818272 0.409136 0.912474i \(-0.365830\pi\)
0.409136 + 0.912474i \(0.365830\pi\)
\(14\) −38.9309 −0.743194
\(15\) 0 0
\(16\) 278.378 4.34965
\(17\) 65.7235 0.937664 0.468832 0.883287i \(-0.344675\pi\)
0.468832 + 0.883287i \(0.344675\pi\)
\(18\) 0 0
\(19\) 33.3996 0.403284 0.201642 0.979459i \(-0.435372\pi\)
0.201642 + 0.979459i \(0.435372\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −186.955 −1.81177
\(23\) 207.447 1.88068 0.940341 0.340234i \(-0.110506\pi\)
0.940341 + 0.340234i \(0.110506\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −213.309 −1.60897
\(27\) 0 0
\(28\) 160.516 1.08338
\(29\) 189.170 1.21131 0.605656 0.795726i \(-0.292911\pi\)
0.605656 + 0.795726i \(0.292911\pi\)
\(30\) 0 0
\(31\) 202.108 1.17096 0.585478 0.810688i \(-0.300907\pi\)
0.585478 + 0.810688i \(0.300907\pi\)
\(32\) −883.902 −4.88292
\(33\) 0 0
\(34\) −365.525 −1.84373
\(35\) 0 0
\(36\) 0 0
\(37\) 16.5227 0.0734141 0.0367070 0.999326i \(-0.488313\pi\)
0.0367070 + 0.999326i \(0.488313\pi\)
\(38\) −185.754 −0.792980
\(39\) 0 0
\(40\) 0 0
\(41\) −388.617 −1.48029 −0.740144 0.672448i \(-0.765243\pi\)
−0.740144 + 0.672448i \(0.765243\pi\)
\(42\) 0 0
\(43\) −41.8144 −0.148294 −0.0741469 0.997247i \(-0.523623\pi\)
−0.0741469 + 0.997247i \(0.523623\pi\)
\(44\) 770.833 2.64108
\(45\) 0 0
\(46\) −1153.73 −3.69800
\(47\) 368.648 1.14410 0.572051 0.820218i \(-0.306148\pi\)
0.572051 + 0.820218i \(0.306148\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 879.494 2.34546
\(53\) 458.172 1.18745 0.593725 0.804668i \(-0.297657\pi\)
0.593725 + 0.804668i \(0.297657\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −581.272 −1.38707
\(57\) 0 0
\(58\) −1052.08 −2.38181
\(59\) −256.216 −0.565364 −0.282682 0.959214i \(-0.591224\pi\)
−0.282682 + 0.959214i \(0.591224\pi\)
\(60\) 0 0
\(61\) −123.511 −0.259246 −0.129623 0.991563i \(-0.541377\pi\)
−0.129623 + 0.991563i \(0.541377\pi\)
\(62\) −1124.03 −2.30246
\(63\) 0 0
\(64\) 2688.85 5.25166
\(65\) 0 0
\(66\) 0 0
\(67\) 336.277 0.613175 0.306587 0.951843i \(-0.400813\pi\)
0.306587 + 0.951843i \(0.400813\pi\)
\(68\) 1507.10 2.68768
\(69\) 0 0
\(70\) 0 0
\(71\) 453.312 0.757722 0.378861 0.925454i \(-0.376316\pi\)
0.378861 + 0.925454i \(0.376316\pi\)
\(72\) 0 0
\(73\) −22.0436 −0.0353426 −0.0176713 0.999844i \(-0.505625\pi\)
−0.0176713 + 0.999844i \(0.505625\pi\)
\(74\) −91.8920 −0.144355
\(75\) 0 0
\(76\) 765.882 1.15596
\(77\) 235.309 0.348259
\(78\) 0 0
\(79\) 385.417 0.548896 0.274448 0.961602i \(-0.411505\pi\)
0.274448 + 0.961602i \(0.411505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2161.32 2.91070
\(83\) 23.7501 0.0314085 0.0157043 0.999877i \(-0.495001\pi\)
0.0157043 + 0.999877i \(0.495001\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 232.553 0.291591
\(87\) 0 0
\(88\) −2791.39 −3.38140
\(89\) 1482.81 1.76604 0.883020 0.469335i \(-0.155506\pi\)
0.883020 + 0.469335i \(0.155506\pi\)
\(90\) 0 0
\(91\) 268.479 0.309278
\(92\) 4756.94 5.39071
\(93\) 0 0
\(94\) −2050.25 −2.24965
\(95\) 0 0
\(96\) 0 0
\(97\) −51.9867 −0.0544170 −0.0272085 0.999630i \(-0.508662\pi\)
−0.0272085 + 0.999630i \(0.508662\pi\)
\(98\) −272.516 −0.280901
\(99\) 0 0
\(100\) 0 0
\(101\) −1429.30 −1.40812 −0.704062 0.710138i \(-0.748632\pi\)
−0.704062 + 0.710138i \(0.748632\pi\)
\(102\) 0 0
\(103\) −434.212 −0.415381 −0.207690 0.978195i \(-0.566595\pi\)
−0.207690 + 0.978195i \(0.566595\pi\)
\(104\) −3184.88 −3.00292
\(105\) 0 0
\(106\) −2548.15 −2.33489
\(107\) 666.307 0.602003 0.301001 0.953624i \(-0.402679\pi\)
0.301001 + 0.953624i \(0.402679\pi\)
\(108\) 0 0
\(109\) −1199.51 −1.05406 −0.527029 0.849847i \(-0.676694\pi\)
−0.527029 + 0.849847i \(0.676694\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1948.64 1.64401
\(113\) −81.5171 −0.0678627 −0.0339314 0.999424i \(-0.510803\pi\)
−0.0339314 + 0.999424i \(0.510803\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4337.84 3.47206
\(117\) 0 0
\(118\) 1424.96 1.11168
\(119\) 460.064 0.354404
\(120\) 0 0
\(121\) −200.996 −0.151011
\(122\) 686.915 0.509757
\(123\) 0 0
\(124\) 4634.51 3.35638
\(125\) 0 0
\(126\) 0 0
\(127\) 336.985 0.235453 0.117727 0.993046i \(-0.462439\pi\)
0.117727 + 0.993046i \(0.462439\pi\)
\(128\) −7882.95 −5.44344
\(129\) 0 0
\(130\) 0 0
\(131\) −2931.15 −1.95493 −0.977465 0.211097i \(-0.932297\pi\)
−0.977465 + 0.211097i \(0.932297\pi\)
\(132\) 0 0
\(133\) 233.797 0.152427
\(134\) −1870.22 −1.20569
\(135\) 0 0
\(136\) −5457.60 −3.44107
\(137\) −1585.07 −0.988477 −0.494238 0.869326i \(-0.664553\pi\)
−0.494238 + 0.869326i \(0.664553\pi\)
\(138\) 0 0
\(139\) −1298.85 −0.792569 −0.396284 0.918128i \(-0.629701\pi\)
−0.396284 + 0.918128i \(0.629701\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2521.12 −1.48991
\(143\) 1289.30 0.753960
\(144\) 0 0
\(145\) 0 0
\(146\) 122.597 0.0694943
\(147\) 0 0
\(148\) 378.881 0.210431
\(149\) 2003.29 1.10145 0.550724 0.834687i \(-0.314352\pi\)
0.550724 + 0.834687i \(0.314352\pi\)
\(150\) 0 0
\(151\) 2740.96 1.47719 0.738596 0.674148i \(-0.235489\pi\)
0.738596 + 0.674148i \(0.235489\pi\)
\(152\) −2773.47 −1.47999
\(153\) 0 0
\(154\) −1308.68 −0.684783
\(155\) 0 0
\(156\) 0 0
\(157\) −3644.22 −1.85249 −0.926243 0.376928i \(-0.876981\pi\)
−0.926243 + 0.376928i \(0.876981\pi\)
\(158\) −2143.52 −1.07930
\(159\) 0 0
\(160\) 0 0
\(161\) 1452.13 0.710831
\(162\) 0 0
\(163\) −2774.27 −1.33311 −0.666557 0.745454i \(-0.732233\pi\)
−0.666557 + 0.745454i \(0.732233\pi\)
\(164\) −8911.33 −4.24304
\(165\) 0 0
\(166\) −132.087 −0.0617588
\(167\) 1154.91 0.535149 0.267574 0.963537i \(-0.413778\pi\)
0.267574 + 0.963537i \(0.413778\pi\)
\(168\) 0 0
\(169\) −725.958 −0.330432
\(170\) 0 0
\(171\) 0 0
\(172\) −958.841 −0.425064
\(173\) −3387.46 −1.48869 −0.744346 0.667794i \(-0.767239\pi\)
−0.744346 + 0.667794i \(0.767239\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 9357.82 4.00780
\(177\) 0 0
\(178\) −8246.73 −3.47258
\(179\) −1603.32 −0.669486 −0.334743 0.942309i \(-0.608650\pi\)
−0.334743 + 0.942309i \(0.608650\pi\)
\(180\) 0 0
\(181\) 544.220 0.223489 0.111745 0.993737i \(-0.464356\pi\)
0.111745 + 0.993737i \(0.464356\pi\)
\(182\) −1493.16 −0.608134
\(183\) 0 0
\(184\) −17226.2 −6.90179
\(185\) 0 0
\(186\) 0 0
\(187\) 2209.33 0.863969
\(188\) 8453.41 3.27941
\(189\) 0 0
\(190\) 0 0
\(191\) −2993.44 −1.13402 −0.567010 0.823711i \(-0.691900\pi\)
−0.567010 + 0.823711i \(0.691900\pi\)
\(192\) 0 0
\(193\) −1309.32 −0.488325 −0.244163 0.969734i \(-0.578513\pi\)
−0.244163 + 0.969734i \(0.578513\pi\)
\(194\) 289.127 0.107001
\(195\) 0 0
\(196\) 1123.61 0.409480
\(197\) −1141.38 −0.412790 −0.206395 0.978469i \(-0.566173\pi\)
−0.206395 + 0.978469i \(0.566173\pi\)
\(198\) 0 0
\(199\) 2370.23 0.844327 0.422164 0.906520i \(-0.361271\pi\)
0.422164 + 0.906520i \(0.361271\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7949.12 2.76880
\(203\) 1324.19 0.457833
\(204\) 0 0
\(205\) 0 0
\(206\) 2414.89 0.816765
\(207\) 0 0
\(208\) 10676.9 3.55920
\(209\) 1122.75 0.371588
\(210\) 0 0
\(211\) −687.159 −0.224199 −0.112099 0.993697i \(-0.535758\pi\)
−0.112099 + 0.993697i \(0.535758\pi\)
\(212\) 10506.3 3.40366
\(213\) 0 0
\(214\) −3705.70 −1.18372
\(215\) 0 0
\(216\) 0 0
\(217\) 1414.76 0.442580
\(218\) 6671.15 2.07260
\(219\) 0 0
\(220\) 0 0
\(221\) 2520.77 0.767264
\(222\) 0 0
\(223\) −990.496 −0.297437 −0.148719 0.988880i \(-0.547515\pi\)
−0.148719 + 0.988880i \(0.547515\pi\)
\(224\) −6187.32 −1.84557
\(225\) 0 0
\(226\) 453.362 0.133439
\(227\) 1479.25 0.432517 0.216258 0.976336i \(-0.430615\pi\)
0.216258 + 0.976336i \(0.430615\pi\)
\(228\) 0 0
\(229\) 6704.47 1.93469 0.967345 0.253463i \(-0.0815696\pi\)
0.967345 + 0.253463i \(0.0815696\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −15708.5 −4.44531
\(233\) −1749.09 −0.491789 −0.245895 0.969297i \(-0.579082\pi\)
−0.245895 + 0.969297i \(0.579082\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −5875.25 −1.62054
\(237\) 0 0
\(238\) −2558.67 −0.696866
\(239\) 6320.89 1.71073 0.855365 0.518027i \(-0.173333\pi\)
0.855365 + 0.518027i \(0.173333\pi\)
\(240\) 0 0
\(241\) 3359.62 0.897975 0.448988 0.893538i \(-0.351785\pi\)
0.448988 + 0.893538i \(0.351785\pi\)
\(242\) 1117.85 0.296935
\(243\) 0 0
\(244\) −2832.22 −0.743092
\(245\) 0 0
\(246\) 0 0
\(247\) 1281.01 0.329996
\(248\) −16782.8 −4.29721
\(249\) 0 0
\(250\) 0 0
\(251\) 1330.50 0.334582 0.167291 0.985908i \(-0.446498\pi\)
0.167291 + 0.985908i \(0.446498\pi\)
\(252\) 0 0
\(253\) 6973.44 1.73287
\(254\) −1874.16 −0.462973
\(255\) 0 0
\(256\) 22330.6 5.45182
\(257\) −2476.95 −0.601197 −0.300599 0.953751i \(-0.597186\pi\)
−0.300599 + 0.953751i \(0.597186\pi\)
\(258\) 0 0
\(259\) 115.659 0.0277479
\(260\) 0 0
\(261\) 0 0
\(262\) 16301.8 3.84399
\(263\) −5152.56 −1.20806 −0.604032 0.796960i \(-0.706440\pi\)
−0.604032 + 0.796960i \(0.706440\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1300.28 −0.299718
\(267\) 0 0
\(268\) 7711.11 1.75758
\(269\) 1150.97 0.260876 0.130438 0.991456i \(-0.458362\pi\)
0.130438 + 0.991456i \(0.458362\pi\)
\(270\) 0 0
\(271\) 1838.32 0.412067 0.206034 0.978545i \(-0.433944\pi\)
0.206034 + 0.978545i \(0.433944\pi\)
\(272\) 18296.0 4.07851
\(273\) 0 0
\(274\) 8815.43 1.94365
\(275\) 0 0
\(276\) 0 0
\(277\) −568.447 −0.123302 −0.0616510 0.998098i \(-0.519637\pi\)
−0.0616510 + 0.998098i \(0.519637\pi\)
\(278\) 7223.62 1.55843
\(279\) 0 0
\(280\) 0 0
\(281\) −6015.00 −1.27696 −0.638479 0.769640i \(-0.720436\pi\)
−0.638479 + 0.769640i \(0.720436\pi\)
\(282\) 0 0
\(283\) −3985.75 −0.837202 −0.418601 0.908170i \(-0.637479\pi\)
−0.418601 + 0.908170i \(0.637479\pi\)
\(284\) 10394.8 2.17190
\(285\) 0 0
\(286\) −7170.48 −1.48252
\(287\) −2720.32 −0.559497
\(288\) 0 0
\(289\) −593.424 −0.120787
\(290\) 0 0
\(291\) 0 0
\(292\) −505.479 −0.101305
\(293\) −2490.01 −0.496478 −0.248239 0.968699i \(-0.579852\pi\)
−0.248239 + 0.968699i \(0.579852\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1372.03 −0.269417
\(297\) 0 0
\(298\) −11141.4 −2.16578
\(299\) 7956.45 1.53891
\(300\) 0 0
\(301\) −292.701 −0.0560498
\(302\) −15244.0 −2.90461
\(303\) 0 0
\(304\) 9297.72 1.75415
\(305\) 0 0
\(306\) 0 0
\(307\) 141.853 0.0263712 0.0131856 0.999913i \(-0.495803\pi\)
0.0131856 + 0.999913i \(0.495803\pi\)
\(308\) 5395.83 0.998234
\(309\) 0 0
\(310\) 0 0
\(311\) −2091.92 −0.381420 −0.190710 0.981646i \(-0.561079\pi\)
−0.190710 + 0.981646i \(0.561079\pi\)
\(312\) 0 0
\(313\) −5521.44 −0.997094 −0.498547 0.866863i \(-0.666133\pi\)
−0.498547 + 0.866863i \(0.666133\pi\)
\(314\) 20267.5 3.64255
\(315\) 0 0
\(316\) 8837.94 1.57333
\(317\) 5351.63 0.948195 0.474097 0.880472i \(-0.342774\pi\)
0.474097 + 0.880472i \(0.342774\pi\)
\(318\) 0 0
\(319\) 6359.06 1.11611
\(320\) 0 0
\(321\) 0 0
\(322\) −8076.09 −1.39771
\(323\) 2195.14 0.378145
\(324\) 0 0
\(325\) 0 0
\(326\) 15429.2 2.62131
\(327\) 0 0
\(328\) 32270.3 5.43241
\(329\) 2580.53 0.432430
\(330\) 0 0
\(331\) −4383.52 −0.727916 −0.363958 0.931415i \(-0.618575\pi\)
−0.363958 + 0.931415i \(0.618575\pi\)
\(332\) 544.609 0.0900281
\(333\) 0 0
\(334\) −6423.11 −1.05227
\(335\) 0 0
\(336\) 0 0
\(337\) 7124.57 1.15163 0.575817 0.817579i \(-0.304684\pi\)
0.575817 + 0.817579i \(0.304684\pi\)
\(338\) 4037.46 0.649730
\(339\) 0 0
\(340\) 0 0
\(341\) 6793.97 1.07893
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 3472.22 0.544214
\(345\) 0 0
\(346\) 18839.5 2.92722
\(347\) 507.743 0.0785506 0.0392753 0.999228i \(-0.487495\pi\)
0.0392753 + 0.999228i \(0.487495\pi\)
\(348\) 0 0
\(349\) 6155.14 0.944060 0.472030 0.881582i \(-0.343521\pi\)
0.472030 + 0.881582i \(0.343521\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −29712.8 −4.49915
\(353\) −6429.56 −0.969437 −0.484718 0.874670i \(-0.661078\pi\)
−0.484718 + 0.874670i \(0.661078\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 34002.1 5.06211
\(357\) 0 0
\(358\) 8916.97 1.31641
\(359\) −10075.4 −1.48123 −0.740614 0.671931i \(-0.765465\pi\)
−0.740614 + 0.671931i \(0.765465\pi\)
\(360\) 0 0
\(361\) −5743.46 −0.837362
\(362\) −3026.71 −0.439448
\(363\) 0 0
\(364\) 6156.46 0.886501
\(365\) 0 0
\(366\) 0 0
\(367\) −816.898 −0.116190 −0.0580950 0.998311i \(-0.518503\pi\)
−0.0580950 + 0.998311i \(0.518503\pi\)
\(368\) 57748.6 8.18031
\(369\) 0 0
\(370\) 0 0
\(371\) 3207.21 0.448814
\(372\) 0 0
\(373\) 3737.85 0.518870 0.259435 0.965761i \(-0.416464\pi\)
0.259435 + 0.965761i \(0.416464\pi\)
\(374\) −12287.3 −1.69883
\(375\) 0 0
\(376\) −30612.1 −4.19866
\(377\) 7255.47 0.991183
\(378\) 0 0
\(379\) 1950.47 0.264351 0.132176 0.991226i \(-0.457804\pi\)
0.132176 + 0.991226i \(0.457804\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16648.2 2.22983
\(383\) 6762.06 0.902155 0.451077 0.892485i \(-0.351040\pi\)
0.451077 + 0.892485i \(0.351040\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7281.84 0.960197
\(387\) 0 0
\(388\) −1192.10 −0.155979
\(389\) 2551.98 0.332624 0.166312 0.986073i \(-0.446814\pi\)
0.166312 + 0.986073i \(0.446814\pi\)
\(390\) 0 0
\(391\) 13634.1 1.76345
\(392\) −4068.90 −0.524262
\(393\) 0 0
\(394\) 6347.83 0.811672
\(395\) 0 0
\(396\) 0 0
\(397\) 4097.93 0.518058 0.259029 0.965869i \(-0.416597\pi\)
0.259029 + 0.965869i \(0.416597\pi\)
\(398\) −13182.2 −1.66021
\(399\) 0 0
\(400\) 0 0
\(401\) 1046.81 0.130362 0.0651811 0.997873i \(-0.479238\pi\)
0.0651811 + 0.997873i \(0.479238\pi\)
\(402\) 0 0
\(403\) 7751.68 0.958161
\(404\) −32775.1 −4.03619
\(405\) 0 0
\(406\) −7364.57 −0.900240
\(407\) 555.420 0.0676441
\(408\) 0 0
\(409\) 6516.92 0.787876 0.393938 0.919137i \(-0.371113\pi\)
0.393938 + 0.919137i \(0.371113\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −9956.86 −1.19063
\(413\) −1793.51 −0.213687
\(414\) 0 0
\(415\) 0 0
\(416\) −33901.3 −3.99555
\(417\) 0 0
\(418\) −6244.21 −0.730656
\(419\) −12279.1 −1.43168 −0.715838 0.698267i \(-0.753955\pi\)
−0.715838 + 0.698267i \(0.753955\pi\)
\(420\) 0 0
\(421\) 10146.9 1.17465 0.587325 0.809351i \(-0.300181\pi\)
0.587325 + 0.809351i \(0.300181\pi\)
\(422\) 3821.67 0.440844
\(423\) 0 0
\(424\) −38046.1 −4.35774
\(425\) 0 0
\(426\) 0 0
\(427\) −864.579 −0.0979858
\(428\) 15279.0 1.72556
\(429\) 0 0
\(430\) 0 0
\(431\) −7059.04 −0.788914 −0.394457 0.918914i \(-0.629067\pi\)
−0.394457 + 0.918914i \(0.629067\pi\)
\(432\) 0 0
\(433\) −6468.98 −0.717966 −0.358983 0.933344i \(-0.616876\pi\)
−0.358983 + 0.933344i \(0.616876\pi\)
\(434\) −7868.24 −0.870248
\(435\) 0 0
\(436\) −27505.8 −3.02131
\(437\) 6928.65 0.758449
\(438\) 0 0
\(439\) 4767.13 0.518275 0.259137 0.965840i \(-0.416562\pi\)
0.259137 + 0.965840i \(0.416562\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −14019.4 −1.50868
\(443\) 2366.55 0.253810 0.126905 0.991915i \(-0.459496\pi\)
0.126905 + 0.991915i \(0.459496\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5508.70 0.584853
\(447\) 0 0
\(448\) 18821.9 1.98494
\(449\) −1814.17 −0.190681 −0.0953406 0.995445i \(-0.530394\pi\)
−0.0953406 + 0.995445i \(0.530394\pi\)
\(450\) 0 0
\(451\) −13063.6 −1.36395
\(452\) −1869.26 −0.194519
\(453\) 0 0
\(454\) −8226.93 −0.850460
\(455\) 0 0
\(456\) 0 0
\(457\) −8284.13 −0.847955 −0.423977 0.905673i \(-0.639366\pi\)
−0.423977 + 0.905673i \(0.639366\pi\)
\(458\) −37287.3 −3.80419
\(459\) 0 0
\(460\) 0 0
\(461\) −1384.62 −0.139888 −0.0699439 0.997551i \(-0.522282\pi\)
−0.0699439 + 0.997551i \(0.522282\pi\)
\(462\) 0 0
\(463\) 13210.3 1.32599 0.662994 0.748624i \(-0.269285\pi\)
0.662994 + 0.748624i \(0.269285\pi\)
\(464\) 52660.9 5.26879
\(465\) 0 0
\(466\) 9727.67 0.967008
\(467\) −4574.24 −0.453256 −0.226628 0.973981i \(-0.572770\pi\)
−0.226628 + 0.973981i \(0.572770\pi\)
\(468\) 0 0
\(469\) 2353.94 0.231758
\(470\) 0 0
\(471\) 0 0
\(472\) 21275.9 2.07479
\(473\) −1405.61 −0.136639
\(474\) 0 0
\(475\) 0 0
\(476\) 10549.7 1.01585
\(477\) 0 0
\(478\) −35154.0 −3.36382
\(479\) 11031.8 1.05231 0.526154 0.850389i \(-0.323634\pi\)
0.526154 + 0.850389i \(0.323634\pi\)
\(480\) 0 0
\(481\) 633.716 0.0600726
\(482\) −18684.7 −1.76569
\(483\) 0 0
\(484\) −4609.02 −0.432853
\(485\) 0 0
\(486\) 0 0
\(487\) 5194.06 0.483296 0.241648 0.970364i \(-0.422312\pi\)
0.241648 + 0.970364i \(0.422312\pi\)
\(488\) 10256.2 0.951389
\(489\) 0 0
\(490\) 0 0
\(491\) −11954.7 −1.09880 −0.549398 0.835561i \(-0.685143\pi\)
−0.549398 + 0.835561i \(0.685143\pi\)
\(492\) 0 0
\(493\) 12432.9 1.13580
\(494\) −7124.43 −0.648873
\(495\) 0 0
\(496\) 56262.4 5.09326
\(497\) 3173.19 0.286392
\(498\) 0 0
\(499\) 2566.05 0.230205 0.115102 0.993354i \(-0.463280\pi\)
0.115102 + 0.993354i \(0.463280\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −7399.62 −0.657891
\(503\) 21103.5 1.87069 0.935347 0.353731i \(-0.115087\pi\)
0.935347 + 0.353731i \(0.115087\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −38783.1 −3.40735
\(507\) 0 0
\(508\) 7727.36 0.674894
\(509\) 781.732 0.0680740 0.0340370 0.999421i \(-0.489164\pi\)
0.0340370 + 0.999421i \(0.489164\pi\)
\(510\) 0 0
\(511\) −154.305 −0.0133582
\(512\) −61129.5 −5.27650
\(513\) 0 0
\(514\) 13775.7 1.18214
\(515\) 0 0
\(516\) 0 0
\(517\) 12392.3 1.05418
\(518\) −643.244 −0.0545609
\(519\) 0 0
\(520\) 0 0
\(521\) −14013.0 −1.17835 −0.589176 0.808005i \(-0.700547\pi\)
−0.589176 + 0.808005i \(0.700547\pi\)
\(522\) 0 0
\(523\) 10310.7 0.862052 0.431026 0.902339i \(-0.358152\pi\)
0.431026 + 0.902339i \(0.358152\pi\)
\(524\) −67213.8 −5.60353
\(525\) 0 0
\(526\) 28656.3 2.37542
\(527\) 13283.2 1.09796
\(528\) 0 0
\(529\) 30867.2 2.53696
\(530\) 0 0
\(531\) 0 0
\(532\) 5361.18 0.436911
\(533\) −14905.1 −1.21128
\(534\) 0 0
\(535\) 0 0
\(536\) −27924.0 −2.25025
\(537\) 0 0
\(538\) −6401.16 −0.512962
\(539\) 1647.16 0.131629
\(540\) 0 0
\(541\) −17562.9 −1.39572 −0.697862 0.716232i \(-0.745865\pi\)
−0.697862 + 0.716232i \(0.745865\pi\)
\(542\) −10223.9 −0.810250
\(543\) 0 0
\(544\) −58093.1 −4.57853
\(545\) 0 0
\(546\) 0 0
\(547\) 19889.6 1.55469 0.777347 0.629072i \(-0.216565\pi\)
0.777347 + 0.629072i \(0.216565\pi\)
\(548\) −36346.9 −2.83333
\(549\) 0 0
\(550\) 0 0
\(551\) 6318.22 0.488503
\(552\) 0 0
\(553\) 2697.92 0.207463
\(554\) 3161.45 0.242450
\(555\) 0 0
\(556\) −29783.8 −2.27179
\(557\) 5579.54 0.424439 0.212219 0.977222i \(-0.431931\pi\)
0.212219 + 0.977222i \(0.431931\pi\)
\(558\) 0 0
\(559\) −1603.76 −0.121345
\(560\) 0 0
\(561\) 0 0
\(562\) 33452.8 2.51089
\(563\) −24463.2 −1.83126 −0.915630 0.402022i \(-0.868307\pi\)
−0.915630 + 0.402022i \(0.868307\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22167.0 1.64620
\(567\) 0 0
\(568\) −37642.5 −2.78071
\(569\) 8582.14 0.632306 0.316153 0.948708i \(-0.397609\pi\)
0.316153 + 0.948708i \(0.397609\pi\)
\(570\) 0 0
\(571\) 17580.8 1.28850 0.644248 0.764816i \(-0.277170\pi\)
0.644248 + 0.764816i \(0.277170\pi\)
\(572\) 29564.7 2.16112
\(573\) 0 0
\(574\) 15129.2 1.10014
\(575\) 0 0
\(576\) 0 0
\(577\) 8692.57 0.627169 0.313585 0.949560i \(-0.398470\pi\)
0.313585 + 0.949560i \(0.398470\pi\)
\(578\) 3300.36 0.237503
\(579\) 0 0
\(580\) 0 0
\(581\) 166.250 0.0118713
\(582\) 0 0
\(583\) 15401.7 1.09412
\(584\) 1830.47 0.129701
\(585\) 0 0
\(586\) 13848.3 0.976227
\(587\) 3584.61 0.252049 0.126024 0.992027i \(-0.459778\pi\)
0.126024 + 0.992027i \(0.459778\pi\)
\(588\) 0 0
\(589\) 6750.33 0.472228
\(590\) 0 0
\(591\) 0 0
\(592\) 4599.56 0.319326
\(593\) −21853.6 −1.51335 −0.756676 0.653790i \(-0.773178\pi\)
−0.756676 + 0.653790i \(0.773178\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 45937.1 3.15714
\(597\) 0 0
\(598\) −44250.2 −3.02596
\(599\) −9090.48 −0.620078 −0.310039 0.950724i \(-0.600342\pi\)
−0.310039 + 0.950724i \(0.600342\pi\)
\(600\) 0 0
\(601\) −19546.1 −1.32663 −0.663314 0.748341i \(-0.730851\pi\)
−0.663314 + 0.748341i \(0.730851\pi\)
\(602\) 1627.87 0.110211
\(603\) 0 0
\(604\) 62852.6 4.23416
\(605\) 0 0
\(606\) 0 0
\(607\) 15726.0 1.05157 0.525783 0.850619i \(-0.323773\pi\)
0.525783 + 0.850619i \(0.323773\pi\)
\(608\) −29522.0 −1.96920
\(609\) 0 0
\(610\) 0 0
\(611\) 14139.2 0.936186
\(612\) 0 0
\(613\) 13572.5 0.894269 0.447135 0.894467i \(-0.352444\pi\)
0.447135 + 0.894467i \(0.352444\pi\)
\(614\) −788.921 −0.0518538
\(615\) 0 0
\(616\) −19539.8 −1.27805
\(617\) 17378.5 1.13393 0.566964 0.823743i \(-0.308118\pi\)
0.566964 + 0.823743i \(0.308118\pi\)
\(618\) 0 0
\(619\) −25113.3 −1.63068 −0.815338 0.578985i \(-0.803449\pi\)
−0.815338 + 0.578985i \(0.803449\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 11634.3 0.749989
\(623\) 10379.7 0.667501
\(624\) 0 0
\(625\) 0 0
\(626\) 30707.8 1.96059
\(627\) 0 0
\(628\) −83565.1 −5.30989
\(629\) 1085.93 0.0688377
\(630\) 0 0
\(631\) −10814.4 −0.682276 −0.341138 0.940013i \(-0.610812\pi\)
−0.341138 + 0.940013i \(0.610812\pi\)
\(632\) −32004.5 −2.01436
\(633\) 0 0
\(634\) −29763.4 −1.86444
\(635\) 0 0
\(636\) 0 0
\(637\) 1879.35 0.116896
\(638\) −35366.3 −2.19461
\(639\) 0 0
\(640\) 0 0
\(641\) 16359.0 1.00802 0.504010 0.863698i \(-0.331857\pi\)
0.504010 + 0.863698i \(0.331857\pi\)
\(642\) 0 0
\(643\) −8819.47 −0.540911 −0.270456 0.962732i \(-0.587174\pi\)
−0.270456 + 0.962732i \(0.587174\pi\)
\(644\) 33298.6 2.03750
\(645\) 0 0
\(646\) −12208.4 −0.743549
\(647\) −13828.8 −0.840290 −0.420145 0.907457i \(-0.638021\pi\)
−0.420145 + 0.907457i \(0.638021\pi\)
\(648\) 0 0
\(649\) −8612.83 −0.520930
\(650\) 0 0
\(651\) 0 0
\(652\) −63616.4 −3.82118
\(653\) 23988.7 1.43760 0.718798 0.695219i \(-0.244693\pi\)
0.718798 + 0.695219i \(0.244693\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −108182. −6.43874
\(657\) 0 0
\(658\) −14351.8 −0.850289
\(659\) −3109.28 −0.183794 −0.0918972 0.995769i \(-0.529293\pi\)
−0.0918972 + 0.995769i \(0.529293\pi\)
\(660\) 0 0
\(661\) 22695.0 1.33545 0.667726 0.744407i \(-0.267268\pi\)
0.667726 + 0.744407i \(0.267268\pi\)
\(662\) 24379.2 1.43131
\(663\) 0 0
\(664\) −1972.18 −0.115264
\(665\) 0 0
\(666\) 0 0
\(667\) 39242.8 2.27809
\(668\) 26483.2 1.53393
\(669\) 0 0
\(670\) 0 0
\(671\) −4151.90 −0.238871
\(672\) 0 0
\(673\) −22073.4 −1.26429 −0.632145 0.774850i \(-0.717825\pi\)
−0.632145 + 0.774850i \(0.717825\pi\)
\(674\) −39623.7 −2.26446
\(675\) 0 0
\(676\) −16646.9 −0.947136
\(677\) −2489.50 −0.141328 −0.0706642 0.997500i \(-0.522512\pi\)
−0.0706642 + 0.997500i \(0.522512\pi\)
\(678\) 0 0
\(679\) −363.907 −0.0205677
\(680\) 0 0
\(681\) 0 0
\(682\) −37785.0 −2.12150
\(683\) 7970.98 0.446561 0.223280 0.974754i \(-0.428323\pi\)
0.223280 + 0.974754i \(0.428323\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1907.61 −0.106171
\(687\) 0 0
\(688\) −11640.2 −0.645027
\(689\) 17572.8 0.971656
\(690\) 0 0
\(691\) −23892.7 −1.31537 −0.657687 0.753292i \(-0.728465\pi\)
−0.657687 + 0.753292i \(0.728465\pi\)
\(692\) −77677.4 −4.26712
\(693\) 0 0
\(694\) −2823.84 −0.154454
\(695\) 0 0
\(696\) 0 0
\(697\) −25541.3 −1.38801
\(698\) −34232.1 −1.85631
\(699\) 0 0
\(700\) 0 0
\(701\) −12197.0 −0.657170 −0.328585 0.944474i \(-0.606572\pi\)
−0.328585 + 0.944474i \(0.606572\pi\)
\(702\) 0 0
\(703\) 551.853 0.0296067
\(704\) 90387.0 4.83891
\(705\) 0 0
\(706\) 35758.4 1.90621
\(707\) −10005.1 −0.532221
\(708\) 0 0
\(709\) −8982.28 −0.475792 −0.237896 0.971291i \(-0.576458\pi\)
−0.237896 + 0.971291i \(0.576458\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −123131. −6.48107
\(713\) 41926.7 2.20220
\(714\) 0 0
\(715\) 0 0
\(716\) −36765.6 −1.91899
\(717\) 0 0
\(718\) 56035.0 2.91255
\(719\) 6501.61 0.337231 0.168616 0.985682i \(-0.446070\pi\)
0.168616 + 0.985682i \(0.446070\pi\)
\(720\) 0 0
\(721\) −3039.49 −0.156999
\(722\) 31942.6 1.64651
\(723\) 0 0
\(724\) 12479.4 0.640600
\(725\) 0 0
\(726\) 0 0
\(727\) 24228.7 1.23603 0.618013 0.786168i \(-0.287938\pi\)
0.618013 + 0.786168i \(0.287938\pi\)
\(728\) −22294.2 −1.13500
\(729\) 0 0
\(730\) 0 0
\(731\) −2748.19 −0.139050
\(732\) 0 0
\(733\) −36719.1 −1.85027 −0.925136 0.379636i \(-0.876049\pi\)
−0.925136 + 0.379636i \(0.876049\pi\)
\(734\) 4543.22 0.228465
\(735\) 0 0
\(736\) −183363. −9.18321
\(737\) 11304.1 0.564983
\(738\) 0 0
\(739\) 23304.5 1.16004 0.580021 0.814602i \(-0.303044\pi\)
0.580021 + 0.814602i \(0.303044\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −17837.0 −0.882505
\(743\) 6875.35 0.339478 0.169739 0.985489i \(-0.445708\pi\)
0.169739 + 0.985489i \(0.445708\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20788.3 −1.02026
\(747\) 0 0
\(748\) 50661.8 2.47644
\(749\) 4664.15 0.227536
\(750\) 0 0
\(751\) 1182.65 0.0574640 0.0287320 0.999587i \(-0.490853\pi\)
0.0287320 + 0.999587i \(0.490853\pi\)
\(752\) 102623. 4.97645
\(753\) 0 0
\(754\) −40351.7 −1.94897
\(755\) 0 0
\(756\) 0 0
\(757\) −25226.8 −1.21121 −0.605604 0.795766i \(-0.707068\pi\)
−0.605604 + 0.795766i \(0.707068\pi\)
\(758\) −10847.7 −0.519795
\(759\) 0 0
\(760\) 0 0
\(761\) 10909.2 0.519655 0.259827 0.965655i \(-0.416334\pi\)
0.259827 + 0.965655i \(0.416334\pi\)
\(762\) 0 0
\(763\) −8396.58 −0.398397
\(764\) −68642.2 −3.25051
\(765\) 0 0
\(766\) −37607.6 −1.77391
\(767\) −9826.95 −0.462621
\(768\) 0 0
\(769\) −12771.6 −0.598903 −0.299451 0.954112i \(-0.596804\pi\)
−0.299451 + 0.954112i \(0.596804\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −30023.8 −1.39972
\(773\) 2199.06 0.102322 0.0511610 0.998690i \(-0.483708\pi\)
0.0511610 + 0.998690i \(0.483708\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 4316.92 0.199701
\(777\) 0 0
\(778\) −14193.0 −0.654041
\(779\) −12979.7 −0.596977
\(780\) 0 0
\(781\) 15238.3 0.698170
\(782\) −75827.0 −3.46748
\(783\) 0 0
\(784\) 13640.5 0.621379
\(785\) 0 0
\(786\) 0 0
\(787\) 19587.7 0.887201 0.443601 0.896225i \(-0.353701\pi\)
0.443601 + 0.896225i \(0.353701\pi\)
\(788\) −26172.8 −1.18321
\(789\) 0 0
\(790\) 0 0
\(791\) −570.620 −0.0256497
\(792\) 0 0
\(793\) −4737.17 −0.212134
\(794\) −22790.9 −1.01866
\(795\) 0 0
\(796\) 54351.4 2.42014
\(797\) 21699.3 0.964401 0.482200 0.876061i \(-0.339838\pi\)
0.482200 + 0.876061i \(0.339838\pi\)
\(798\) 0 0
\(799\) 24228.8 1.07278
\(800\) 0 0
\(801\) 0 0
\(802\) −5821.89 −0.256332
\(803\) −741.007 −0.0325649
\(804\) 0 0
\(805\) 0 0
\(806\) −43111.4 −1.88404
\(807\) 0 0
\(808\) 118687. 5.16758
\(809\) 15649.4 0.680103 0.340051 0.940407i \(-0.389556\pi\)
0.340051 + 0.940407i \(0.389556\pi\)
\(810\) 0 0
\(811\) −33267.3 −1.44041 −0.720206 0.693760i \(-0.755953\pi\)
−0.720206 + 0.693760i \(0.755953\pi\)
\(812\) 30364.9 1.31231
\(813\) 0 0
\(814\) −3089.00 −0.133009
\(815\) 0 0
\(816\) 0 0
\(817\) −1396.59 −0.0598046
\(818\) −36244.2 −1.54920
\(819\) 0 0
\(820\) 0 0
\(821\) −5158.58 −0.219288 −0.109644 0.993971i \(-0.534971\pi\)
−0.109644 + 0.993971i \(0.534971\pi\)
\(822\) 0 0
\(823\) −26333.6 −1.11535 −0.557674 0.830060i \(-0.688306\pi\)
−0.557674 + 0.830060i \(0.688306\pi\)
\(824\) 36056.5 1.52438
\(825\) 0 0
\(826\) 9974.71 0.420175
\(827\) 19572.7 0.822988 0.411494 0.911413i \(-0.365007\pi\)
0.411494 + 0.911413i \(0.365007\pi\)
\(828\) 0 0
\(829\) 9642.26 0.403968 0.201984 0.979389i \(-0.435261\pi\)
0.201984 + 0.979389i \(0.435261\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 103128. 4.29728
\(833\) 3220.45 0.133952
\(834\) 0 0
\(835\) 0 0
\(836\) 25745.5 1.06511
\(837\) 0 0
\(838\) 68290.7 2.81511
\(839\) 31081.1 1.27895 0.639475 0.768812i \(-0.279152\pi\)
0.639475 + 0.768812i \(0.279152\pi\)
\(840\) 0 0
\(841\) 11396.5 0.467278
\(842\) −56432.3 −2.30972
\(843\) 0 0
\(844\) −15757.1 −0.642634
\(845\) 0 0
\(846\) 0 0
\(847\) −1406.97 −0.0570770
\(848\) 127545. 5.16499
\(849\) 0 0
\(850\) 0 0
\(851\) 3427.59 0.138068
\(852\) 0 0
\(853\) −25780.9 −1.03484 −0.517421 0.855731i \(-0.673108\pi\)
−0.517421 + 0.855731i \(0.673108\pi\)
\(854\) 4808.40 0.192670
\(855\) 0 0
\(856\) −55329.3 −2.20925
\(857\) 14452.6 0.576069 0.288035 0.957620i \(-0.406998\pi\)
0.288035 + 0.957620i \(0.406998\pi\)
\(858\) 0 0
\(859\) 889.366 0.0353257 0.0176628 0.999844i \(-0.494377\pi\)
0.0176628 + 0.999844i \(0.494377\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 39259.2 1.55125
\(863\) 41460.3 1.63537 0.817685 0.575665i \(-0.195257\pi\)
0.817685 + 0.575665i \(0.195257\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 35977.6 1.41174
\(867\) 0 0
\(868\) 32441.6 1.26859
\(869\) 12956.0 0.505756
\(870\) 0 0
\(871\) 12897.6 0.501744
\(872\) 99606.0 3.86822
\(873\) 0 0
\(874\) −38534.1 −1.49134
\(875\) 0 0
\(876\) 0 0
\(877\) −21173.0 −0.815236 −0.407618 0.913153i \(-0.633640\pi\)
−0.407618 + 0.913153i \(0.633640\pi\)
\(878\) −26512.6 −1.01909
\(879\) 0 0
\(880\) 0 0
\(881\) −9883.39 −0.377957 −0.188978 0.981981i \(-0.560518\pi\)
−0.188978 + 0.981981i \(0.560518\pi\)
\(882\) 0 0
\(883\) 45273.9 1.72547 0.862734 0.505658i \(-0.168750\pi\)
0.862734 + 0.505658i \(0.168750\pi\)
\(884\) 57803.4 2.19925
\(885\) 0 0
\(886\) −13161.7 −0.499069
\(887\) 644.388 0.0243928 0.0121964 0.999926i \(-0.496118\pi\)
0.0121964 + 0.999926i \(0.496118\pi\)
\(888\) 0 0
\(889\) 2358.89 0.0889930
\(890\) 0 0
\(891\) 0 0
\(892\) −22712.9 −0.852562
\(893\) 12312.7 0.461398
\(894\) 0 0
\(895\) 0 0
\(896\) −55180.6 −2.05743
\(897\) 0 0
\(898\) 10089.6 0.374937
\(899\) 38232.8 1.41839
\(900\) 0 0
\(901\) 30112.7 1.11343
\(902\) 72653.8 2.68194
\(903\) 0 0
\(904\) 6769.09 0.249045
\(905\) 0 0
\(906\) 0 0
\(907\) 15065.2 0.551522 0.275761 0.961226i \(-0.411070\pi\)
0.275761 + 0.961226i \(0.411070\pi\)
\(908\) 33920.5 1.23975
\(909\) 0 0
\(910\) 0 0
\(911\) −28789.9 −1.04704 −0.523520 0.852014i \(-0.675381\pi\)
−0.523520 + 0.852014i \(0.675381\pi\)
\(912\) 0 0
\(913\) 798.371 0.0289400
\(914\) 46072.6 1.66734
\(915\) 0 0
\(916\) 153739. 5.54552
\(917\) −20518.1 −0.738894
\(918\) 0 0
\(919\) −24163.8 −0.867345 −0.433673 0.901070i \(-0.642783\pi\)
−0.433673 + 0.901070i \(0.642783\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 7700.65 0.275062
\(923\) 17386.4 0.620023
\(924\) 0 0
\(925\) 0 0
\(926\) −73469.5 −2.60730
\(927\) 0 0
\(928\) −167208. −5.91474
\(929\) 35115.4 1.24015 0.620075 0.784542i \(-0.287102\pi\)
0.620075 + 0.784542i \(0.287102\pi\)
\(930\) 0 0
\(931\) 1636.58 0.0576120
\(932\) −40108.2 −1.40964
\(933\) 0 0
\(934\) 25439.9 0.891239
\(935\) 0 0
\(936\) 0 0
\(937\) 15512.6 0.540849 0.270424 0.962741i \(-0.412836\pi\)
0.270424 + 0.962741i \(0.412836\pi\)
\(938\) −13091.5 −0.455708
\(939\) 0 0
\(940\) 0 0
\(941\) 53283.8 1.84591 0.922956 0.384905i \(-0.125766\pi\)
0.922956 + 0.384905i \(0.125766\pi\)
\(942\) 0 0
\(943\) −80617.5 −2.78395
\(944\) −71324.8 −2.45914
\(945\) 0 0
\(946\) 7817.39 0.268674
\(947\) −55509.7 −1.90478 −0.952388 0.304890i \(-0.901380\pi\)
−0.952388 + 0.304890i \(0.901380\pi\)
\(948\) 0 0
\(949\) −845.464 −0.0289198
\(950\) 0 0
\(951\) 0 0
\(952\) −38203.2 −1.30060
\(953\) −28080.6 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 144943. 4.90356
\(957\) 0 0
\(958\) −61353.9 −2.06916
\(959\) −11095.5 −0.373609
\(960\) 0 0
\(961\) 11056.6 0.371140
\(962\) −3524.44 −0.118121
\(963\) 0 0
\(964\) 77039.0 2.57392
\(965\) 0 0
\(966\) 0 0
\(967\) −56609.3 −1.88256 −0.941278 0.337634i \(-0.890374\pi\)
−0.941278 + 0.337634i \(0.890374\pi\)
\(968\) 16690.5 0.554187
\(969\) 0 0
\(970\) 0 0
\(971\) 6782.17 0.224151 0.112075 0.993700i \(-0.464250\pi\)
0.112075 + 0.993700i \(0.464250\pi\)
\(972\) 0 0
\(973\) −9091.95 −0.299563
\(974\) −28887.0 −0.950309
\(975\) 0 0
\(976\) −34382.8 −1.12763
\(977\) −45655.0 −1.49502 −0.747509 0.664252i \(-0.768750\pi\)
−0.747509 + 0.664252i \(0.768750\pi\)
\(978\) 0 0
\(979\) 49845.5 1.62724
\(980\) 0 0
\(981\) 0 0
\(982\) 66486.8 2.16057
\(983\) −10102.3 −0.327785 −0.163893 0.986478i \(-0.552405\pi\)
−0.163893 + 0.986478i \(0.552405\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −69146.4 −2.23334
\(987\) 0 0
\(988\) 29374.8 0.945887
\(989\) −8674.27 −0.278894
\(990\) 0 0
\(991\) 25416.2 0.814705 0.407353 0.913271i \(-0.366452\pi\)
0.407353 + 0.913271i \(0.366452\pi\)
\(992\) −178644. −5.71768
\(993\) 0 0
\(994\) −17647.8 −0.563135
\(995\) 0 0
\(996\) 0 0
\(997\) −48152.5 −1.52959 −0.764797 0.644271i \(-0.777161\pi\)
−0.764797 + 0.644271i \(0.777161\pi\)
\(998\) −14271.2 −0.452653
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1575.4.a.m.1.1 2
3.2 odd 2 525.4.a.p.1.2 2
5.4 even 2 315.4.a.m.1.2 2
15.2 even 4 525.4.d.i.274.4 4
15.8 even 4 525.4.d.i.274.1 4
15.14 odd 2 105.4.a.c.1.1 2
35.34 odd 2 2205.4.a.bh.1.2 2
60.59 even 2 1680.4.a.bk.1.2 2
105.104 even 2 735.4.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.c.1.1 2 15.14 odd 2
315.4.a.m.1.2 2 5.4 even 2
525.4.a.p.1.2 2 3.2 odd 2
525.4.d.i.274.1 4 15.8 even 4
525.4.d.i.274.4 4 15.2 even 4
735.4.a.k.1.1 2 105.104 even 2
1575.4.a.m.1.1 2 1.1 even 1 trivial
1680.4.a.bk.1.2 2 60.59 even 2
2205.4.a.bh.1.2 2 35.34 odd 2