Properties

Label 525.4.d.l.274.2
Level 525
Weight 4
Character 525.274
Analytic conductor 30.976
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.l.274.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.82843i q^{2} -3.00000i q^{3} +4.65685 q^{4} -5.48528 q^{6} +7.00000i q^{7} -23.1421i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-1.82843i q^{2} -3.00000i q^{3} +4.65685 q^{4} -5.48528 q^{6} +7.00000i q^{7} -23.1421i q^{8} -9.00000 q^{9} -64.5685 q^{11} -13.9706i q^{12} -32.3431i q^{13} +12.7990 q^{14} -5.05887 q^{16} +56.3431i q^{17} +16.4558i q^{18} +2.74517 q^{19} +21.0000 q^{21} +118.059i q^{22} +88.1665i q^{23} -69.4264 q^{24} -59.1371 q^{26} +27.0000i q^{27} +32.5980i q^{28} -246.735 q^{29} -110.912 q^{31} -175.887i q^{32} +193.706i q^{33} +103.019 q^{34} -41.9117 q^{36} -120.676i q^{37} -5.01934i q^{38} -97.0294 q^{39} -176.274 q^{41} -38.3970i q^{42} -443.362i q^{43} -300.686 q^{44} +161.206 q^{46} +345.206i q^{47} +15.1766i q^{48} -49.0000 q^{49} +169.029 q^{51} -150.617i q^{52} +260.981i q^{53} +49.3675 q^{54} +161.995 q^{56} -8.23550i q^{57} +451.137i q^{58} -628.999 q^{59} -115.206 q^{61} +202.794i q^{62} -63.0000i q^{63} -362.068 q^{64} +354.177 q^{66} +951.480i q^{67} +262.382i q^{68} +264.500 q^{69} +356.264 q^{71} +208.279i q^{72} -656.754i q^{73} -220.648 q^{74} +12.7838 q^{76} -451.980i q^{77} +177.411i q^{78} -440.195 q^{79} +81.0000 q^{81} +322.304i q^{82} -54.4121i q^{83} +97.7939 q^{84} -810.656 q^{86} +740.205i q^{87} +1494.25i q^{88} +1018.78 q^{89} +226.402 q^{91} +410.579i q^{92} +332.735i q^{93} +631.184 q^{94} -527.662 q^{96} +724.108i q^{97} +89.5929i q^{98} +581.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 12q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 12q^{6} - 36q^{9} - 32q^{11} - 28q^{14} - 156q^{16} + 192q^{19} + 84q^{21} - 108q^{24} + 216q^{26} - 376q^{29} - 240q^{31} - 312q^{34} + 36q^{36} - 456q^{39} + 200q^{41} - 1248q^{44} + 1120q^{46} - 196q^{49} + 744q^{51} - 108q^{54} + 252q^{56} - 208q^{59} - 936q^{61} - 68q^{64} + 1824q^{66} - 96q^{69} - 272q^{71} - 2376q^{74} - 1216q^{76} + 864q^{79} + 324q^{81} - 84q^{84} - 912q^{86} + 2808q^{89} + 1064q^{91} - 3200q^{94} - 2484q^{96} + 288q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.82843i − 0.646447i −0.946323 0.323223i \(-0.895234\pi\)
0.946323 0.323223i \(-0.104766\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 4.65685 0.582107
\(5\) 0 0
\(6\) −5.48528 −0.373226
\(7\) 7.00000i 0.377964i
\(8\) − 23.1421i − 1.02275i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −64.5685 −1.76983 −0.884916 0.465751i \(-0.845784\pi\)
−0.884916 + 0.465751i \(0.845784\pi\)
\(12\) − 13.9706i − 0.336080i
\(13\) − 32.3431i − 0.690029i −0.938597 0.345014i \(-0.887874\pi\)
0.938597 0.345014i \(-0.112126\pi\)
\(14\) 12.7990 0.244334
\(15\) 0 0
\(16\) −5.05887 −0.0790449
\(17\) 56.3431i 0.803836i 0.915676 + 0.401918i \(0.131656\pi\)
−0.915676 + 0.401918i \(0.868344\pi\)
\(18\) 16.4558i 0.215482i
\(19\) 2.74517 0.0331465 0.0165733 0.999863i \(-0.494724\pi\)
0.0165733 + 0.999863i \(0.494724\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 118.059i 1.14410i
\(23\) 88.1665i 0.799304i 0.916667 + 0.399652i \(0.130869\pi\)
−0.916667 + 0.399652i \(0.869131\pi\)
\(24\) −69.4264 −0.590484
\(25\) 0 0
\(26\) −59.1371 −0.446067
\(27\) 27.0000i 0.192450i
\(28\) 32.5980i 0.220016i
\(29\) −246.735 −1.57992 −0.789958 0.613161i \(-0.789898\pi\)
−0.789958 + 0.613161i \(0.789898\pi\)
\(30\) 0 0
\(31\) −110.912 −0.642591 −0.321296 0.946979i \(-0.604118\pi\)
−0.321296 + 0.946979i \(0.604118\pi\)
\(32\) − 175.887i − 0.971649i
\(33\) 193.706i 1.02181i
\(34\) 103.019 0.519637
\(35\) 0 0
\(36\) −41.9117 −0.194036
\(37\) − 120.676i − 0.536190i −0.963392 0.268095i \(-0.913606\pi\)
0.963392 0.268095i \(-0.0863942\pi\)
\(38\) − 5.01934i − 0.0214275i
\(39\) −97.0294 −0.398388
\(40\) 0 0
\(41\) −176.274 −0.671449 −0.335724 0.941960i \(-0.608981\pi\)
−0.335724 + 0.941960i \(0.608981\pi\)
\(42\) − 38.3970i − 0.141066i
\(43\) − 443.362i − 1.57238i −0.617988 0.786188i \(-0.712052\pi\)
0.617988 0.786188i \(-0.287948\pi\)
\(44\) −300.686 −1.03023
\(45\) 0 0
\(46\) 161.206 0.516707
\(47\) 345.206i 1.07135i 0.844424 + 0.535675i \(0.179943\pi\)
−0.844424 + 0.535675i \(0.820057\pi\)
\(48\) 15.1766i 0.0456366i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 169.029 0.464095
\(52\) − 150.617i − 0.401670i
\(53\) 260.981i 0.676386i 0.941077 + 0.338193i \(0.109816\pi\)
−0.941077 + 0.338193i \(0.890184\pi\)
\(54\) 49.3675 0.124409
\(55\) 0 0
\(56\) 161.995 0.386562
\(57\) − 8.23550i − 0.0191372i
\(58\) 451.137i 1.02133i
\(59\) −628.999 −1.38794 −0.693972 0.720002i \(-0.744141\pi\)
−0.693972 + 0.720002i \(0.744141\pi\)
\(60\) 0 0
\(61\) −115.206 −0.241814 −0.120907 0.992664i \(-0.538580\pi\)
−0.120907 + 0.992664i \(0.538580\pi\)
\(62\) 202.794i 0.415401i
\(63\) − 63.0000i − 0.125988i
\(64\) −362.068 −0.707164
\(65\) 0 0
\(66\) 354.177 0.660547
\(67\) 951.480i 1.73495i 0.497479 + 0.867476i \(0.334259\pi\)
−0.497479 + 0.867476i \(0.665741\pi\)
\(68\) 262.382i 0.467919i
\(69\) 264.500 0.461478
\(70\) 0 0
\(71\) 356.264 0.595504 0.297752 0.954643i \(-0.403763\pi\)
0.297752 + 0.954643i \(0.403763\pi\)
\(72\) 208.279i 0.340916i
\(73\) − 656.754i − 1.05298i −0.850183 0.526488i \(-0.823509\pi\)
0.850183 0.526488i \(-0.176491\pi\)
\(74\) −220.648 −0.346618
\(75\) 0 0
\(76\) 12.7838 0.0192948
\(77\) − 451.980i − 0.668933i
\(78\) 177.411i 0.257537i
\(79\) −440.195 −0.626909 −0.313455 0.949603i \(-0.601486\pi\)
−0.313455 + 0.949603i \(0.601486\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 322.304i 0.434056i
\(83\) − 54.4121i − 0.0719579i −0.999353 0.0359790i \(-0.988545\pi\)
0.999353 0.0359790i \(-0.0114549\pi\)
\(84\) 97.7939 0.127026
\(85\) 0 0
\(86\) −810.656 −1.01646
\(87\) 740.205i 0.912165i
\(88\) 1494.25i 1.81009i
\(89\) 1018.78 1.21338 0.606690 0.794938i \(-0.292497\pi\)
0.606690 + 0.794938i \(0.292497\pi\)
\(90\) 0 0
\(91\) 226.402 0.260806
\(92\) 410.579i 0.465280i
\(93\) 332.735i 0.371000i
\(94\) 631.184 0.692571
\(95\) 0 0
\(96\) −527.662 −0.560982
\(97\) 724.108i 0.757959i 0.925405 + 0.378979i \(0.123725\pi\)
−0.925405 + 0.378979i \(0.876275\pi\)
\(98\) 89.5929i 0.0923495i
\(99\) 581.117 0.589944
\(100\) 0 0
\(101\) 268.725 0.264744 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(102\) − 309.058i − 0.300013i
\(103\) − 1840.63i − 1.76080i −0.474233 0.880399i \(-0.657275\pi\)
0.474233 0.880399i \(-0.342725\pi\)
\(104\) −748.489 −0.705725
\(105\) 0 0
\(106\) 477.184 0.437247
\(107\) 243.087i 0.219627i 0.993952 + 0.109813i \(0.0350253\pi\)
−0.993952 + 0.109813i \(0.964975\pi\)
\(108\) 125.735i 0.112027i
\(109\) 405.176 0.356044 0.178022 0.984027i \(-0.443030\pi\)
0.178022 + 0.984027i \(0.443030\pi\)
\(110\) 0 0
\(111\) −362.029 −0.309570
\(112\) − 35.4121i − 0.0298762i
\(113\) − 28.1766i − 0.0234569i −0.999931 0.0117285i \(-0.996267\pi\)
0.999931 0.0117285i \(-0.00373337\pi\)
\(114\) −15.0580 −0.0123712
\(115\) 0 0
\(116\) −1149.01 −0.919680
\(117\) 291.088i 0.230010i
\(118\) 1150.08i 0.897232i
\(119\) −394.402 −0.303822
\(120\) 0 0
\(121\) 2838.10 2.13230
\(122\) 210.646i 0.156320i
\(123\) 528.823i 0.387661i
\(124\) −516.500 −0.374057
\(125\) 0 0
\(126\) −115.191 −0.0814446
\(127\) 2740.90i 1.91508i 0.288298 + 0.957541i \(0.406911\pi\)
−0.288298 + 0.957541i \(0.593089\pi\)
\(128\) − 745.083i − 0.514505i
\(129\) −1330.09 −0.907811
\(130\) 0 0
\(131\) −1832.04 −1.22188 −0.610938 0.791678i \(-0.709208\pi\)
−0.610938 + 0.791678i \(0.709208\pi\)
\(132\) 902.059i 0.594804i
\(133\) 19.2162i 0.0125282i
\(134\) 1739.71 1.12155
\(135\) 0 0
\(136\) 1303.90 0.822122
\(137\) − 382.747i − 0.238688i −0.992853 0.119344i \(-0.961921\pi\)
0.992853 0.119344i \(-0.0380792\pi\)
\(138\) − 483.618i − 0.298321i
\(139\) −3053.60 −1.86333 −0.931667 0.363314i \(-0.881645\pi\)
−0.931667 + 0.363314i \(0.881645\pi\)
\(140\) 0 0
\(141\) 1035.62 0.618545
\(142\) − 651.403i − 0.384961i
\(143\) 2088.35i 1.22123i
\(144\) 45.5299 0.0263483
\(145\) 0 0
\(146\) −1200.83 −0.680692
\(147\) 147.000i 0.0824786i
\(148\) − 561.971i − 0.312120i
\(149\) −3560.60 −1.95769 −0.978843 0.204611i \(-0.934407\pi\)
−0.978843 + 0.204611i \(0.934407\pi\)
\(150\) 0 0
\(151\) 3261.80 1.75789 0.878945 0.476923i \(-0.158248\pi\)
0.878945 + 0.476923i \(0.158248\pi\)
\(152\) − 63.5290i − 0.0339005i
\(153\) − 507.088i − 0.267945i
\(154\) −826.412 −0.432430
\(155\) 0 0
\(156\) −451.852 −0.231905
\(157\) − 2878.46i − 1.46322i −0.681723 0.731611i \(-0.738769\pi\)
0.681723 0.731611i \(-0.261231\pi\)
\(158\) 804.865i 0.405263i
\(159\) 782.942 0.390512
\(160\) 0 0
\(161\) −617.166 −0.302108
\(162\) − 148.103i − 0.0718274i
\(163\) − 927.537i − 0.445708i −0.974852 0.222854i \(-0.928463\pi\)
0.974852 0.222854i \(-0.0715373\pi\)
\(164\) −820.883 −0.390855
\(165\) 0 0
\(166\) −99.4886 −0.0465169
\(167\) − 1094.52i − 0.507164i −0.967314 0.253582i \(-0.918391\pi\)
0.967314 0.253582i \(-0.0816087\pi\)
\(168\) − 485.985i − 0.223182i
\(169\) 1150.92 0.523860
\(170\) 0 0
\(171\) −24.7065 −0.0110488
\(172\) − 2064.67i − 0.915290i
\(173\) − 1713.25i − 0.752926i −0.926432 0.376463i \(-0.877140\pi\)
0.926432 0.376463i \(-0.122860\pi\)
\(174\) 1353.41 0.589666
\(175\) 0 0
\(176\) 326.644 0.139896
\(177\) 1887.00i 0.801330i
\(178\) − 1862.77i − 0.784386i
\(179\) 4065.58 1.69763 0.848816 0.528689i \(-0.177316\pi\)
0.848816 + 0.528689i \(0.177316\pi\)
\(180\) 0 0
\(181\) −2791.40 −1.14631 −0.573157 0.819445i \(-0.694282\pi\)
−0.573157 + 0.819445i \(0.694282\pi\)
\(182\) − 413.960i − 0.168597i
\(183\) 345.618i 0.139611i
\(184\) 2040.36 0.817486
\(185\) 0 0
\(186\) 608.382 0.239832
\(187\) − 3637.99i − 1.42266i
\(188\) 1607.57i 0.623640i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −634.185 −0.240251 −0.120126 0.992759i \(-0.538330\pi\)
−0.120126 + 0.992759i \(0.538330\pi\)
\(192\) 1086.20i 0.408281i
\(193\) − 254.999i − 0.0951049i −0.998869 0.0475524i \(-0.984858\pi\)
0.998869 0.0475524i \(-0.0151421\pi\)
\(194\) 1323.98 0.489980
\(195\) 0 0
\(196\) −228.186 −0.0831581
\(197\) − 4172.37i − 1.50898i −0.656311 0.754490i \(-0.727884\pi\)
0.656311 0.754490i \(-0.272116\pi\)
\(198\) − 1062.53i − 0.381367i
\(199\) 4626.48 1.64805 0.824026 0.566552i \(-0.191723\pi\)
0.824026 + 0.566552i \(0.191723\pi\)
\(200\) 0 0
\(201\) 2854.44 1.00168
\(202\) − 491.344i − 0.171143i
\(203\) − 1727.15i − 0.597152i
\(204\) 787.145 0.270153
\(205\) 0 0
\(206\) −3365.45 −1.13826
\(207\) − 793.499i − 0.266435i
\(208\) 163.620i 0.0545433i
\(209\) −177.251 −0.0586638
\(210\) 0 0
\(211\) −1562.64 −0.509843 −0.254921 0.966962i \(-0.582050\pi\)
−0.254921 + 0.966962i \(0.582050\pi\)
\(212\) 1215.35i 0.393729i
\(213\) − 1068.79i − 0.343814i
\(214\) 444.466 0.141977
\(215\) 0 0
\(216\) 624.838 0.196828
\(217\) − 776.382i − 0.242877i
\(218\) − 740.834i − 0.230163i
\(219\) −1970.26 −0.607935
\(220\) 0 0
\(221\) 1822.31 0.554670
\(222\) 661.943i 0.200120i
\(223\) − 1236.39i − 0.371278i −0.982618 0.185639i \(-0.940564\pi\)
0.982618 0.185639i \(-0.0594355\pi\)
\(224\) 1231.21 0.367249
\(225\) 0 0
\(226\) −51.5189 −0.0151637
\(227\) − 4181.82i − 1.22272i −0.791353 0.611359i \(-0.790623\pi\)
0.791353 0.611359i \(-0.209377\pi\)
\(228\) − 38.3515i − 0.0111399i
\(229\) 484.774 0.139890 0.0699449 0.997551i \(-0.477718\pi\)
0.0699449 + 0.997551i \(0.477718\pi\)
\(230\) 0 0
\(231\) −1355.94 −0.386209
\(232\) 5709.98i 1.61585i
\(233\) 2080.54i 0.584982i 0.956268 + 0.292491i \(0.0944842\pi\)
−0.956268 + 0.292491i \(0.905516\pi\)
\(234\) 532.234 0.148689
\(235\) 0 0
\(236\) −2929.16 −0.807932
\(237\) 1320.59i 0.361946i
\(238\) 721.135i 0.196404i
\(239\) −6814.10 −1.84422 −0.922108 0.386933i \(-0.873534\pi\)
−0.922108 + 0.386933i \(0.873534\pi\)
\(240\) 0 0
\(241\) −3921.84 −1.04825 −0.524125 0.851642i \(-0.675607\pi\)
−0.524125 + 0.851642i \(0.675607\pi\)
\(242\) − 5189.25i − 1.37842i
\(243\) − 243.000i − 0.0641500i
\(244\) −536.498 −0.140761
\(245\) 0 0
\(246\) 966.913 0.250602
\(247\) − 88.7873i − 0.0228721i
\(248\) 2566.73i 0.657209i
\(249\) −163.236 −0.0415449
\(250\) 0 0
\(251\) −5219.10 −1.31246 −0.656228 0.754562i \(-0.727849\pi\)
−0.656228 + 0.754562i \(0.727849\pi\)
\(252\) − 293.382i − 0.0733386i
\(253\) − 5692.78i − 1.41463i
\(254\) 5011.53 1.23800
\(255\) 0 0
\(256\) −4258.88 −1.03976
\(257\) − 6975.71i − 1.69312i −0.532289 0.846562i \(-0.678668\pi\)
0.532289 0.846562i \(-0.321332\pi\)
\(258\) 2431.97i 0.586852i
\(259\) 844.733 0.202661
\(260\) 0 0
\(261\) 2220.62 0.526639
\(262\) 3349.75i 0.789878i
\(263\) − 3607.36i − 0.845776i −0.906182 0.422888i \(-0.861016\pi\)
0.906182 0.422888i \(-0.138984\pi\)
\(264\) 4482.76 1.04506
\(265\) 0 0
\(266\) 35.1354 0.00809882
\(267\) − 3056.35i − 0.700546i
\(268\) 4430.90i 1.00993i
\(269\) −5.88572 −0.00133405 −0.000667023 1.00000i \(-0.500212\pi\)
−0.000667023 1.00000i \(0.500212\pi\)
\(270\) 0 0
\(271\) 6916.32 1.55032 0.775160 0.631765i \(-0.217669\pi\)
0.775160 + 0.631765i \(0.217669\pi\)
\(272\) − 285.033i − 0.0635392i
\(273\) − 679.206i − 0.150577i
\(274\) −699.825 −0.154299
\(275\) 0 0
\(276\) 1231.74 0.268630
\(277\) 2119.46i 0.459733i 0.973222 + 0.229867i \(0.0738290\pi\)
−0.973222 + 0.229867i \(0.926171\pi\)
\(278\) 5583.29i 1.20455i
\(279\) 998.205 0.214197
\(280\) 0 0
\(281\) −239.917 −0.0509334 −0.0254667 0.999676i \(-0.508107\pi\)
−0.0254667 + 0.999676i \(0.508107\pi\)
\(282\) − 1893.55i − 0.399856i
\(283\) − 4542.12i − 0.954067i −0.878885 0.477034i \(-0.841712\pi\)
0.878885 0.477034i \(-0.158288\pi\)
\(284\) 1659.07 0.346647
\(285\) 0 0
\(286\) 3818.40 0.789463
\(287\) − 1233.92i − 0.253784i
\(288\) 1582.99i 0.323883i
\(289\) 1738.45 0.353847
\(290\) 0 0
\(291\) 2172.32 0.437608
\(292\) − 3058.41i − 0.612944i
\(293\) − 2171.70i − 0.433010i −0.976281 0.216505i \(-0.930534\pi\)
0.976281 0.216505i \(-0.0694658\pi\)
\(294\) 268.779 0.0533180
\(295\) 0 0
\(296\) −2792.70 −0.548387
\(297\) − 1743.35i − 0.340604i
\(298\) 6510.29i 1.26554i
\(299\) 2851.58 0.551543
\(300\) 0 0
\(301\) 3103.54 0.594302
\(302\) − 5963.96i − 1.13638i
\(303\) − 806.175i − 0.152850i
\(304\) −13.8875 −0.00262007
\(305\) 0 0
\(306\) −927.174 −0.173212
\(307\) 3508.64i 0.652276i 0.945322 + 0.326138i \(0.105747\pi\)
−0.945322 + 0.326138i \(0.894253\pi\)
\(308\) − 2104.80i − 0.389391i
\(309\) −5521.88 −1.01660
\(310\) 0 0
\(311\) −3133.25 −0.571287 −0.285643 0.958336i \(-0.592207\pi\)
−0.285643 + 0.958336i \(0.592207\pi\)
\(312\) 2245.47i 0.407451i
\(313\) 6389.59i 1.15387i 0.816790 + 0.576935i \(0.195751\pi\)
−0.816790 + 0.576935i \(0.804249\pi\)
\(314\) −5263.05 −0.945895
\(315\) 0 0
\(316\) −2049.92 −0.364928
\(317\) − 1634.44i − 0.289587i −0.989462 0.144794i \(-0.953748\pi\)
0.989462 0.144794i \(-0.0462518\pi\)
\(318\) − 1431.55i − 0.252445i
\(319\) 15931.3 2.79618
\(320\) 0 0
\(321\) 729.260 0.126802
\(322\) 1128.44i 0.195297i
\(323\) 154.671i 0.0266444i
\(324\) 377.205 0.0646785
\(325\) 0 0
\(326\) −1695.93 −0.288126
\(327\) − 1215.53i − 0.205562i
\(328\) 4079.36i 0.686723i
\(329\) −2416.44 −0.404932
\(330\) 0 0
\(331\) 4386.17 0.728355 0.364177 0.931330i \(-0.381350\pi\)
0.364177 + 0.931330i \(0.381350\pi\)
\(332\) − 253.389i − 0.0418872i
\(333\) 1086.09i 0.178730i
\(334\) −2001.25 −0.327854
\(335\) 0 0
\(336\) −106.236 −0.0172490
\(337\) − 1713.98i − 0.277051i −0.990359 0.138526i \(-0.955764\pi\)
0.990359 0.138526i \(-0.0442363\pi\)
\(338\) − 2104.38i − 0.338648i
\(339\) −84.5299 −0.0135429
\(340\) 0 0
\(341\) 7161.41 1.13728
\(342\) 45.1740i 0.00714249i
\(343\) − 343.000i − 0.0539949i
\(344\) −10260.4 −1.60814
\(345\) 0 0
\(346\) −3132.56 −0.486727
\(347\) 1744.83i 0.269935i 0.990850 + 0.134967i \(0.0430930\pi\)
−0.990850 + 0.134967i \(0.956907\pi\)
\(348\) 3447.03i 0.530977i
\(349\) −7046.78 −1.08082 −0.540409 0.841403i \(-0.681730\pi\)
−0.540409 + 0.841403i \(0.681730\pi\)
\(350\) 0 0
\(351\) 873.265 0.132796
\(352\) 11356.8i 1.71966i
\(353\) − 12668.5i − 1.91013i −0.296400 0.955064i \(-0.595786\pi\)
0.296400 0.955064i \(-0.404214\pi\)
\(354\) 3450.24 0.518017
\(355\) 0 0
\(356\) 4744.33 0.706317
\(357\) 1183.21i 0.175411i
\(358\) − 7433.62i − 1.09743i
\(359\) −37.7844 −0.00555483 −0.00277742 0.999996i \(-0.500884\pi\)
−0.00277742 + 0.999996i \(0.500884\pi\)
\(360\) 0 0
\(361\) −6851.46 −0.998901
\(362\) 5103.87i 0.741031i
\(363\) − 8514.29i − 1.23109i
\(364\) 1054.32 0.151817
\(365\) 0 0
\(366\) 631.938 0.0902511
\(367\) 759.829i 0.108073i 0.998539 + 0.0540364i \(0.0172087\pi\)
−0.998539 + 0.0540364i \(0.982791\pi\)
\(368\) − 446.023i − 0.0631809i
\(369\) 1586.47 0.223816
\(370\) 0 0
\(371\) −1826.86 −0.255650
\(372\) 1549.50i 0.215962i
\(373\) 719.320i 0.0998525i 0.998753 + 0.0499263i \(0.0158986\pi\)
−0.998753 + 0.0499263i \(0.984101\pi\)
\(374\) −6651.81 −0.919671
\(375\) 0 0
\(376\) 7988.81 1.09572
\(377\) 7980.19i 1.09019i
\(378\) 345.573i 0.0470221i
\(379\) −572.559 −0.0775999 −0.0388000 0.999247i \(-0.512354\pi\)
−0.0388000 + 0.999247i \(0.512354\pi\)
\(380\) 0 0
\(381\) 8222.69 1.10567
\(382\) 1159.56i 0.155310i
\(383\) 4513.18i 0.602122i 0.953605 + 0.301061i \(0.0973408\pi\)
−0.953605 + 0.301061i \(0.902659\pi\)
\(384\) −2235.25 −0.297050
\(385\) 0 0
\(386\) −466.247 −0.0614802
\(387\) 3990.26i 0.524125i
\(388\) 3372.06i 0.441213i
\(389\) 6902.13 0.899619 0.449810 0.893124i \(-0.351492\pi\)
0.449810 + 0.893124i \(0.351492\pi\)
\(390\) 0 0
\(391\) −4967.58 −0.642510
\(392\) 1133.96i 0.146107i
\(393\) 5496.11i 0.705451i
\(394\) −7628.88 −0.975475
\(395\) 0 0
\(396\) 2706.18 0.343410
\(397\) − 4124.58i − 0.521427i −0.965416 0.260714i \(-0.916042\pi\)
0.965416 0.260714i \(-0.0839579\pi\)
\(398\) − 8459.18i − 1.06538i
\(399\) 57.6485 0.00723317
\(400\) 0 0
\(401\) −1002.50 −0.124844 −0.0624219 0.998050i \(-0.519882\pi\)
−0.0624219 + 0.998050i \(0.519882\pi\)
\(402\) − 5219.14i − 0.647530i
\(403\) 3587.23i 0.443406i
\(404\) 1251.41 0.154109
\(405\) 0 0
\(406\) −3157.96 −0.386027
\(407\) 7791.89i 0.948967i
\(408\) − 3911.70i − 0.474652i
\(409\) −10335.0 −1.24947 −0.624736 0.780836i \(-0.714794\pi\)
−0.624736 + 0.780836i \(0.714794\pi\)
\(410\) 0 0
\(411\) −1148.24 −0.137807
\(412\) − 8571.53i − 1.02497i
\(413\) − 4402.99i − 0.524594i
\(414\) −1450.85 −0.172236
\(415\) 0 0
\(416\) −5688.75 −0.670466
\(417\) 9160.81i 1.07580i
\(418\) 324.091i 0.0379230i
\(419\) −3183.21 −0.371145 −0.185573 0.982631i \(-0.559414\pi\)
−0.185573 + 0.982631i \(0.559414\pi\)
\(420\) 0 0
\(421\) −6944.34 −0.803911 −0.401956 0.915659i \(-0.631669\pi\)
−0.401956 + 0.915659i \(0.631669\pi\)
\(422\) 2857.18i 0.329586i
\(423\) − 3106.85i − 0.357117i
\(424\) 6039.65 0.691772
\(425\) 0 0
\(426\) −1954.21 −0.222258
\(427\) − 806.442i − 0.0913969i
\(428\) 1132.02i 0.127846i
\(429\) 6265.05 0.705080
\(430\) 0 0
\(431\) 3868.41 0.432331 0.216166 0.976357i \(-0.430645\pi\)
0.216166 + 0.976357i \(0.430645\pi\)
\(432\) − 136.590i − 0.0152122i
\(433\) − 6132.96i − 0.680673i −0.940304 0.340336i \(-0.889459\pi\)
0.940304 0.340336i \(-0.110541\pi\)
\(434\) −1419.56 −0.157007
\(435\) 0 0
\(436\) 1886.84 0.207256
\(437\) 242.032i 0.0264942i
\(438\) 3602.48i 0.392998i
\(439\) 4090.14 0.444673 0.222337 0.974970i \(-0.428632\pi\)
0.222337 + 0.974970i \(0.428632\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 3331.97i − 0.358565i
\(443\) 12434.5i 1.33359i 0.745241 + 0.666795i \(0.232334\pi\)
−0.745241 + 0.666795i \(0.767666\pi\)
\(444\) −1685.91 −0.180203
\(445\) 0 0
\(446\) −2260.66 −0.240012
\(447\) 10681.8i 1.13027i
\(448\) − 2534.48i − 0.267283i
\(449\) −883.046 −0.0928141 −0.0464071 0.998923i \(-0.514777\pi\)
−0.0464071 + 0.998923i \(0.514777\pi\)
\(450\) 0 0
\(451\) 11381.8 1.18835
\(452\) − 131.214i − 0.0136544i
\(453\) − 9785.40i − 1.01492i
\(454\) −7646.15 −0.790422
\(455\) 0 0
\(456\) −190.587 −0.0195725
\(457\) 9068.44i 0.928235i 0.885774 + 0.464118i \(0.153629\pi\)
−0.885774 + 0.464118i \(0.846371\pi\)
\(458\) − 886.373i − 0.0904312i
\(459\) −1521.26 −0.154698
\(460\) 0 0
\(461\) 12508.9 1.26377 0.631885 0.775063i \(-0.282282\pi\)
0.631885 + 0.775063i \(0.282282\pi\)
\(462\) 2479.24i 0.249663i
\(463\) 12688.7i 1.27363i 0.771015 + 0.636817i \(0.219749\pi\)
−0.771015 + 0.636817i \(0.780251\pi\)
\(464\) 1248.20 0.124884
\(465\) 0 0
\(466\) 3804.12 0.378160
\(467\) 10136.5i 1.00442i 0.864747 + 0.502208i \(0.167479\pi\)
−0.864747 + 0.502208i \(0.832521\pi\)
\(468\) 1355.56i 0.133890i
\(469\) −6660.36 −0.655750
\(470\) 0 0
\(471\) −8635.37 −0.844791
\(472\) 14556.4i 1.41952i
\(473\) 28627.3i 2.78284i
\(474\) 2414.59 0.233979
\(475\) 0 0
\(476\) −1836.67 −0.176857
\(477\) − 2348.83i − 0.225462i
\(478\) 12459.1i 1.19219i
\(479\) 11361.1 1.08372 0.541861 0.840468i \(-0.317720\pi\)
0.541861 + 0.840468i \(0.317720\pi\)
\(480\) 0 0
\(481\) −3903.05 −0.369987
\(482\) 7170.80i 0.677637i
\(483\) 1851.50i 0.174422i
\(484\) 13216.6 1.24123
\(485\) 0 0
\(486\) −444.308 −0.0414696
\(487\) − 7929.53i − 0.737826i −0.929464 0.368913i \(-0.879730\pi\)
0.929464 0.368913i \(-0.120270\pi\)
\(488\) 2666.11i 0.247314i
\(489\) −2782.61 −0.257329
\(490\) 0 0
\(491\) 8111.51 0.745555 0.372777 0.927921i \(-0.378405\pi\)
0.372777 + 0.927921i \(0.378405\pi\)
\(492\) 2462.65i 0.225660i
\(493\) − 13901.8i − 1.26999i
\(494\) −162.341 −0.0147856
\(495\) 0 0
\(496\) 561.088 0.0507936
\(497\) 2493.85i 0.225079i
\(498\) 298.466i 0.0268566i
\(499\) −16816.6 −1.50865 −0.754324 0.656502i \(-0.772035\pi\)
−0.754324 + 0.656502i \(0.772035\pi\)
\(500\) 0 0
\(501\) −3283.55 −0.292811
\(502\) 9542.74i 0.848433i
\(503\) − 17764.6i − 1.57472i −0.616491 0.787362i \(-0.711446\pi\)
0.616491 0.787362i \(-0.288554\pi\)
\(504\) −1457.95 −0.128854
\(505\) 0 0
\(506\) −10408.8 −0.914485
\(507\) − 3452.76i − 0.302451i
\(508\) 12764.0i 1.11478i
\(509\) −13908.8 −1.21120 −0.605598 0.795771i \(-0.707066\pi\)
−0.605598 + 0.795771i \(0.707066\pi\)
\(510\) 0 0
\(511\) 4597.27 0.397987
\(512\) 1826.38i 0.157647i
\(513\) 74.1195i 0.00637905i
\(514\) −12754.6 −1.09451
\(515\) 0 0
\(516\) −6194.02 −0.528443
\(517\) − 22289.5i − 1.89611i
\(518\) − 1544.53i − 0.131009i
\(519\) −5139.76 −0.434702
\(520\) 0 0
\(521\) −8639.68 −0.726510 −0.363255 0.931690i \(-0.618335\pi\)
−0.363255 + 0.931690i \(0.618335\pi\)
\(522\) − 4060.23i − 0.340444i
\(523\) 23242.2i 1.94323i 0.236561 + 0.971617i \(0.423980\pi\)
−0.236561 + 0.971617i \(0.576020\pi\)
\(524\) −8531.53 −0.711263
\(525\) 0 0
\(526\) −6595.79 −0.546749
\(527\) − 6249.11i − 0.516538i
\(528\) − 979.932i − 0.0807691i
\(529\) 4393.66 0.361113
\(530\) 0 0
\(531\) 5660.99 0.462648
\(532\) 89.4869i 0.00729276i
\(533\) 5701.26i 0.463319i
\(534\) −5588.32 −0.452865
\(535\) 0 0
\(536\) 22019.3 1.77442
\(537\) − 12196.8i − 0.980128i
\(538\) 10.7616i 0 0.000862390i
\(539\) 3163.86 0.252833
\(540\) 0 0
\(541\) 11395.2 0.905577 0.452789 0.891618i \(-0.350429\pi\)
0.452789 + 0.891618i \(0.350429\pi\)
\(542\) − 12646.0i − 1.00220i
\(543\) 8374.19i 0.661825i
\(544\) 9910.04 0.781047
\(545\) 0 0
\(546\) −1241.88 −0.0973398
\(547\) 7870.21i 0.615184i 0.951518 + 0.307592i \(0.0995232\pi\)
−0.951518 + 0.307592i \(0.900477\pi\)
\(548\) − 1782.40i − 0.138942i
\(549\) 1036.85 0.0806045
\(550\) 0 0
\(551\) −677.329 −0.0523687
\(552\) − 6121.08i − 0.471976i
\(553\) − 3081.37i − 0.236949i
\(554\) 3875.28 0.297193
\(555\) 0 0
\(556\) −14220.2 −1.08466
\(557\) − 17769.8i − 1.35176i −0.737012 0.675880i \(-0.763764\pi\)
0.737012 0.675880i \(-0.236236\pi\)
\(558\) − 1825.15i − 0.138467i
\(559\) −14339.7 −1.08498
\(560\) 0 0
\(561\) −10914.0 −0.821370
\(562\) 438.672i 0.0329257i
\(563\) 15192.8i 1.13730i 0.822579 + 0.568651i \(0.192534\pi\)
−0.822579 + 0.568651i \(0.807466\pi\)
\(564\) 4822.72 0.360059
\(565\) 0 0
\(566\) −8304.93 −0.616753
\(567\) 567.000i 0.0419961i
\(568\) − 8244.71i − 0.609050i
\(569\) 23300.0 1.71667 0.858335 0.513090i \(-0.171499\pi\)
0.858335 + 0.513090i \(0.171499\pi\)
\(570\) 0 0
\(571\) 10638.2 0.779673 0.389837 0.920884i \(-0.372531\pi\)
0.389837 + 0.920884i \(0.372531\pi\)
\(572\) 9725.14i 0.710889i
\(573\) 1902.55i 0.138709i
\(574\) −2256.13 −0.164058
\(575\) 0 0
\(576\) 3258.61 0.235721
\(577\) 897.258i 0.0647372i 0.999476 + 0.0323686i \(0.0103050\pi\)
−0.999476 + 0.0323686i \(0.989695\pi\)
\(578\) − 3178.63i − 0.228743i
\(579\) −764.997 −0.0549088
\(580\) 0 0
\(581\) 380.885 0.0271975
\(582\) − 3971.93i − 0.282890i
\(583\) − 16851.1i − 1.19709i
\(584\) −15198.7 −1.07693
\(585\) 0 0
\(586\) −3970.79 −0.279918
\(587\) 14712.9i 1.03452i 0.855828 + 0.517261i \(0.173048\pi\)
−0.855828 + 0.517261i \(0.826952\pi\)
\(588\) 684.558i 0.0480114i
\(589\) −304.471 −0.0212997
\(590\) 0 0
\(591\) −12517.1 −0.871210
\(592\) 610.486i 0.0423831i
\(593\) − 7216.29i − 0.499726i −0.968281 0.249863i \(-0.919614\pi\)
0.968281 0.249863i \(-0.0803856\pi\)
\(594\) −3187.59 −0.220182
\(595\) 0 0
\(596\) −16581.2 −1.13958
\(597\) − 13879.4i − 0.951503i
\(598\) − 5213.91i − 0.356543i
\(599\) 20885.5 1.42464 0.712320 0.701855i \(-0.247645\pi\)
0.712320 + 0.701855i \(0.247645\pi\)
\(600\) 0 0
\(601\) −11047.7 −0.749823 −0.374911 0.927061i \(-0.622327\pi\)
−0.374911 + 0.927061i \(0.622327\pi\)
\(602\) − 5674.59i − 0.384185i
\(603\) − 8563.32i − 0.578317i
\(604\) 15189.7 1.02328
\(605\) 0 0
\(606\) −1474.03 −0.0988093
\(607\) 9434.94i 0.630894i 0.948943 + 0.315447i \(0.102154\pi\)
−0.948943 + 0.315447i \(0.897846\pi\)
\(608\) − 482.840i − 0.0322068i
\(609\) −5181.44 −0.344766
\(610\) 0 0
\(611\) 11165.0 0.739263
\(612\) − 2361.44i − 0.155973i
\(613\) − 17662.6i − 1.16376i −0.813274 0.581881i \(-0.802317\pi\)
0.813274 0.581881i \(-0.197683\pi\)
\(614\) 6415.30 0.421662
\(615\) 0 0
\(616\) −10459.8 −0.684150
\(617\) 10817.4i 0.705820i 0.935657 + 0.352910i \(0.114808\pi\)
−0.935657 + 0.352910i \(0.885192\pi\)
\(618\) 10096.3i 0.657176i
\(619\) −29073.9 −1.88785 −0.943926 0.330158i \(-0.892898\pi\)
−0.943926 + 0.330158i \(0.892898\pi\)
\(620\) 0 0
\(621\) −2380.50 −0.153826
\(622\) 5728.92i 0.369306i
\(623\) 7131.49i 0.458615i
\(624\) 490.860 0.0314906
\(625\) 0 0
\(626\) 11682.9 0.745915
\(627\) 531.754i 0.0338696i
\(628\) − 13404.5i − 0.851751i
\(629\) 6799.28 0.431009
\(630\) 0 0
\(631\) −2203.17 −0.138996 −0.0694981 0.997582i \(-0.522140\pi\)
−0.0694981 + 0.997582i \(0.522140\pi\)
\(632\) 10187.1i 0.641170i
\(633\) 4687.93i 0.294358i
\(634\) −2988.45 −0.187203
\(635\) 0 0
\(636\) 3646.05 0.227319
\(637\) 1584.81i 0.0985755i
\(638\) − 29129.3i − 1.80758i
\(639\) −3206.38 −0.198501
\(640\) 0 0
\(641\) 22466.5 1.38436 0.692180 0.721725i \(-0.256651\pi\)
0.692180 + 0.721725i \(0.256651\pi\)
\(642\) − 1333.40i − 0.0819705i
\(643\) − 12347.0i − 0.757257i −0.925549 0.378629i \(-0.876396\pi\)
0.925549 0.378629i \(-0.123604\pi\)
\(644\) −2874.05 −0.175859
\(645\) 0 0
\(646\) 282.805 0.0172242
\(647\) 24114.0i 1.46525i 0.680631 + 0.732626i \(0.261706\pi\)
−0.680631 + 0.732626i \(0.738294\pi\)
\(648\) − 1874.51i − 0.113639i
\(649\) 40613.6 2.45643
\(650\) 0 0
\(651\) −2329.15 −0.140225
\(652\) − 4319.41i − 0.259449i
\(653\) 7843.33i 0.470035i 0.971991 + 0.235018i \(0.0755148\pi\)
−0.971991 + 0.235018i \(0.924485\pi\)
\(654\) −2222.50 −0.132885
\(655\) 0 0
\(656\) 891.749 0.0530746
\(657\) 5910.78i 0.350992i
\(658\) 4418.29i 0.261767i
\(659\) −21242.8 −1.25569 −0.627846 0.778338i \(-0.716063\pi\)
−0.627846 + 0.778338i \(0.716063\pi\)
\(660\) 0 0
\(661\) −22221.7 −1.30760 −0.653801 0.756667i \(-0.726827\pi\)
−0.653801 + 0.756667i \(0.726827\pi\)
\(662\) − 8019.79i − 0.470843i
\(663\) − 5466.94i − 0.320239i
\(664\) −1259.21 −0.0735948
\(665\) 0 0
\(666\) 1985.83 0.115539
\(667\) − 21753.8i − 1.26283i
\(668\) − 5097.01i − 0.295223i
\(669\) −3709.18 −0.214358
\(670\) 0 0
\(671\) 7438.69 0.427969
\(672\) − 3693.63i − 0.212031i
\(673\) − 3787.85i − 0.216955i −0.994099 0.108478i \(-0.965402\pi\)
0.994099 0.108478i \(-0.0345976\pi\)
\(674\) −3133.88 −0.179099
\(675\) 0 0
\(676\) 5359.67 0.304943
\(677\) 11296.8i 0.641314i 0.947195 + 0.320657i \(0.103904\pi\)
−0.947195 + 0.320657i \(0.896096\pi\)
\(678\) 154.557i 0.00875474i
\(679\) −5068.75 −0.286481
\(680\) 0 0
\(681\) −12545.5 −0.705937
\(682\) − 13094.1i − 0.735190i
\(683\) 4807.14i 0.269312i 0.990892 + 0.134656i \(0.0429929\pi\)
−0.990892 + 0.134656i \(0.957007\pi\)
\(684\) −115.055 −0.00643161
\(685\) 0 0
\(686\) −627.151 −0.0349048
\(687\) − 1454.32i − 0.0807654i
\(688\) 2242.92i 0.124288i
\(689\) 8440.94 0.466726
\(690\) 0 0
\(691\) 5393.47 0.296928 0.148464 0.988918i \(-0.452567\pi\)
0.148464 + 0.988918i \(0.452567\pi\)
\(692\) − 7978.37i − 0.438283i
\(693\) 4067.82i 0.222978i
\(694\) 3190.29 0.174498
\(695\) 0 0
\(696\) 17129.9 0.932914
\(697\) − 9931.84i − 0.539735i
\(698\) 12884.5i 0.698691i
\(699\) 6241.63 0.337740
\(700\) 0 0
\(701\) 2404.77 0.129568 0.0647838 0.997899i \(-0.479364\pi\)
0.0647838 + 0.997899i \(0.479364\pi\)
\(702\) − 1596.70i − 0.0858456i
\(703\) − 331.276i − 0.0177729i
\(704\) 23378.2 1.25156
\(705\) 0 0
\(706\) −23163.4 −1.23480
\(707\) 1881.07i 0.100064i
\(708\) 8787.47i 0.466460i
\(709\) 21617.3 1.14507 0.572535 0.819881i \(-0.305960\pi\)
0.572535 + 0.819881i \(0.305960\pi\)
\(710\) 0 0
\(711\) 3961.76 0.208970
\(712\) − 23576.8i − 1.24098i
\(713\) − 9778.70i − 0.513626i
\(714\) 2163.41 0.113394
\(715\) 0 0
\(716\) 18932.8 0.988203
\(717\) 20442.3i 1.06476i
\(718\) 69.0860i 0.00359090i
\(719\) 18228.6 0.945498 0.472749 0.881197i \(-0.343262\pi\)
0.472749 + 0.881197i \(0.343262\pi\)
\(720\) 0 0
\(721\) 12884.4 0.665519
\(722\) 12527.4i 0.645736i
\(723\) 11765.5i 0.605207i
\(724\) −12999.1 −0.667278
\(725\) 0 0
\(726\) −15567.8 −0.795832
\(727\) − 20196.5i − 1.03033i −0.857092 0.515164i \(-0.827731\pi\)
0.857092 0.515164i \(-0.172269\pi\)
\(728\) − 5239.43i − 0.266739i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 24980.4 1.26393
\(732\) 1609.49i 0.0812686i
\(733\) − 15264.9i − 0.769196i −0.923084 0.384598i \(-0.874340\pi\)
0.923084 0.384598i \(-0.125660\pi\)
\(734\) 1389.29 0.0698633
\(735\) 0 0
\(736\) 15507.4 0.776643
\(737\) − 61435.7i − 3.07057i
\(738\) − 2900.74i − 0.144685i
\(739\) −13906.0 −0.692207 −0.346103 0.938196i \(-0.612495\pi\)
−0.346103 + 0.938196i \(0.612495\pi\)
\(740\) 0 0
\(741\) −266.362 −0.0132052
\(742\) 3340.29i 0.165264i
\(743\) 4592.87i 0.226778i 0.993551 + 0.113389i \(0.0361706\pi\)
−0.993551 + 0.113389i \(0.963829\pi\)
\(744\) 7700.20 0.379440
\(745\) 0 0
\(746\) 1315.22 0.0645493
\(747\) 489.709i 0.0239860i
\(748\) − 16941.6i − 0.828137i
\(749\) −1701.61 −0.0830111
\(750\) 0 0
\(751\) −8390.80 −0.407702 −0.203851 0.979002i \(-0.565346\pi\)
−0.203851 + 0.979002i \(0.565346\pi\)
\(752\) − 1746.35i − 0.0846848i
\(753\) 15657.3i 0.757747i
\(754\) 14591.2 0.704748
\(755\) 0 0
\(756\) −880.145 −0.0423420
\(757\) 1368.89i 0.0657240i 0.999460 + 0.0328620i \(0.0104622\pi\)
−0.999460 + 0.0328620i \(0.989538\pi\)
\(758\) 1046.88i 0.0501642i
\(759\) −17078.4 −0.816739
\(760\) 0 0
\(761\) −1623.77 −0.0773478 −0.0386739 0.999252i \(-0.512313\pi\)
−0.0386739 + 0.999252i \(0.512313\pi\)
\(762\) − 15034.6i − 0.714759i
\(763\) 2836.23i 0.134572i
\(764\) −2953.31 −0.139852
\(765\) 0 0
\(766\) 8252.03 0.389240
\(767\) 20343.8i 0.957722i
\(768\) 12776.6i 0.600308i
\(769\) 26842.5 1.25873 0.629366 0.777109i \(-0.283315\pi\)
0.629366 + 0.777109i \(0.283315\pi\)
\(770\) 0 0
\(771\) −20927.1 −0.977526
\(772\) − 1187.49i − 0.0553612i
\(773\) − 20961.4i − 0.975330i −0.873031 0.487665i \(-0.837849\pi\)
0.873031 0.487665i \(-0.162151\pi\)
\(774\) 7295.90 0.338819
\(775\) 0 0
\(776\) 16757.4 0.775200
\(777\) − 2534.20i − 0.117006i
\(778\) − 12620.0i − 0.581556i
\(779\) −483.902 −0.0222562
\(780\) 0 0
\(781\) −23003.5 −1.05394
\(782\) 9082.86i 0.415348i
\(783\) − 6661.85i − 0.304055i
\(784\) 247.885 0.0112921
\(785\) 0 0
\(786\) 10049.2 0.456036
\(787\) − 35333.2i − 1.60037i −0.599751 0.800187i \(-0.704734\pi\)
0.599751 0.800187i \(-0.295266\pi\)
\(788\) − 19430.1i − 0.878388i
\(789\) −10822.1 −0.488309
\(790\) 0 0
\(791\) 197.236 0.00886589
\(792\) − 13448.3i − 0.603364i
\(793\) 3726.13i 0.166858i
\(794\) −7541.49 −0.337075
\(795\) 0 0
\(796\) 21544.8 0.959342
\(797\) 8137.04i 0.361642i 0.983516 + 0.180821i \(0.0578755\pi\)
−0.983516 + 0.180821i \(0.942125\pi\)
\(798\) − 105.406i − 0.00467586i
\(799\) −19450.0 −0.861191
\(800\) 0 0
\(801\) −9169.05 −0.404460
\(802\) 1832.99i 0.0807049i
\(803\) 42405.6i 1.86359i
\(804\) 13292.7 0.583082
\(805\) 0 0
\(806\) 6558.99 0.286639
\(807\) 17.6572i 0 0.000770212i
\(808\) − 6218.87i − 0.270766i
\(809\) 36281.0 1.57673 0.788364 0.615209i \(-0.210928\pi\)
0.788364 + 0.615209i \(0.210928\pi\)
\(810\) 0 0
\(811\) −34237.2 −1.48240 −0.741202 0.671282i \(-0.765744\pi\)
−0.741202 + 0.671282i \(0.765744\pi\)
\(812\) − 8043.06i − 0.347606i
\(813\) − 20749.0i − 0.895077i
\(814\) 14246.9 0.613456
\(815\) 0 0
\(816\) −855.099 −0.0366844
\(817\) − 1217.10i − 0.0521188i
\(818\) 18896.8i 0.807717i
\(819\) −2037.62 −0.0869355
\(820\) 0 0
\(821\) −23247.8 −0.988250 −0.494125 0.869391i \(-0.664511\pi\)
−0.494125 + 0.869391i \(0.664511\pi\)
\(822\) 2099.47i 0.0890846i
\(823\) 42934.0i 1.81845i 0.416306 + 0.909225i \(0.363325\pi\)
−0.416306 + 0.909225i \(0.636675\pi\)
\(824\) −42596.0 −1.80085
\(825\) 0 0
\(826\) −8050.55 −0.339122
\(827\) 781.391i 0.0328557i 0.999865 + 0.0164278i \(0.00522938\pi\)
−0.999865 + 0.0164278i \(0.994771\pi\)
\(828\) − 3695.21i − 0.155093i
\(829\) 33493.3 1.40322 0.701611 0.712561i \(-0.252465\pi\)
0.701611 + 0.712561i \(0.252465\pi\)
\(830\) 0 0
\(831\) 6358.39 0.265427
\(832\) 11710.4i 0.487964i
\(833\) − 2760.81i − 0.114834i
\(834\) 16749.9 0.695445
\(835\) 0 0
\(836\) −825.434 −0.0341486
\(837\) − 2994.62i − 0.123667i
\(838\) 5820.27i 0.239926i
\(839\) −15155.7 −0.623639 −0.311819 0.950141i \(-0.600938\pi\)
−0.311819 + 0.950141i \(0.600938\pi\)
\(840\) 0 0
\(841\) 36489.2 1.49613
\(842\) 12697.2i 0.519686i
\(843\) 719.752i 0.0294064i
\(844\) −7277.01 −0.296783
\(845\) 0 0
\(846\) −5680.66 −0.230857
\(847\) 19866.7i 0.805935i
\(848\) − 1320.27i − 0.0534649i
\(849\) −13626.4 −0.550831
\(850\) 0 0
\(851\) 10639.6 0.428579
\(852\) − 4977.21i − 0.200137i
\(853\) − 2917.48i − 0.117107i −0.998284 0.0585537i \(-0.981351\pi\)
0.998284 0.0585537i \(-0.0186489\pi\)
\(854\) −1474.52 −0.0590832
\(855\) 0 0
\(856\) 5625.54 0.224623
\(857\) 31560.7i 1.25799i 0.777411 + 0.628993i \(0.216532\pi\)
−0.777411 + 0.628993i \(0.783468\pi\)
\(858\) − 11455.2i − 0.455797i
\(859\) 1404.81 0.0557991 0.0278995 0.999611i \(-0.491118\pi\)
0.0278995 + 0.999611i \(0.491118\pi\)
\(860\) 0 0
\(861\) −3701.76 −0.146522
\(862\) − 7073.11i − 0.279479i
\(863\) − 9808.24i − 0.386879i −0.981112 0.193439i \(-0.938036\pi\)
0.981112 0.193439i \(-0.0619643\pi\)
\(864\) 4748.96 0.186994
\(865\) 0 0
\(866\) −11213.7 −0.440018
\(867\) − 5215.35i − 0.204294i
\(868\) − 3615.50i − 0.141380i
\(869\) 28422.8 1.10952
\(870\) 0 0
\(871\) 30773.9 1.19717
\(872\) − 9376.63i − 0.364143i
\(873\) − 6516.97i − 0.252653i
\(874\) 442.537 0.0171271
\(875\) 0 0
\(876\) −9175.22 −0.353883
\(877\) 7196.05i 0.277073i 0.990357 + 0.138537i \(0.0442399\pi\)
−0.990357 + 0.138537i \(0.955760\pi\)
\(878\) − 7478.52i − 0.287458i
\(879\) −6515.10 −0.249999
\(880\) 0 0
\(881\) −3183.27 −0.121733 −0.0608667 0.998146i \(-0.519386\pi\)
−0.0608667 + 0.998146i \(0.519386\pi\)
\(882\) − 806.336i − 0.0307832i
\(883\) 25392.5i 0.967751i 0.875137 + 0.483876i \(0.160771\pi\)
−0.875137 + 0.483876i \(0.839229\pi\)
\(884\) 8486.25 0.322877
\(885\) 0 0
\(886\) 22735.6 0.862095
\(887\) 30634.2i 1.15964i 0.814746 + 0.579818i \(0.196876\pi\)
−0.814746 + 0.579818i \(0.803124\pi\)
\(888\) 8378.11i 0.316612i
\(889\) −19186.3 −0.723833
\(890\) 0 0
\(891\) −5230.05 −0.196648
\(892\) − 5757.71i − 0.216124i
\(893\) 947.648i 0.0355116i
\(894\) 19530.9 0.730660
\(895\) 0 0
\(896\) 5215.58 0.194465
\(897\) − 8554.75i − 0.318433i
\(898\) 1614.59i 0.0599994i
\(899\) 27365.8 1.01524
\(900\) 0 0
\(901\) −14704.5 −0.543704
\(902\) − 20810.7i − 0.768206i
\(903\) − 9310.61i − 0.343120i
\(904\) −652.067 −0.0239905
\(905\) 0 0
\(906\) −17891.9 −0.656091
\(907\) 28089.9i 1.02834i 0.857687 + 0.514172i \(0.171901\pi\)
−0.857687 + 0.514172i \(0.828099\pi\)
\(908\) − 19474.1i − 0.711753i
\(909\) −2418.52 −0.0882480
\(910\) 0 0
\(911\) −36102.7 −1.31299 −0.656495 0.754330i \(-0.727962\pi\)
−0.656495 + 0.754330i \(0.727962\pi\)
\(912\) 41.6624i 0.00151270i
\(913\) 3513.31i 0.127353i
\(914\) 16581.0 0.600055
\(915\) 0 0
\(916\) 2257.52 0.0814308
\(917\) − 12824.3i − 0.461826i
\(918\) 2781.52i 0.100004i
\(919\) 14533.8 0.521682 0.260841 0.965382i \(-0.416000\pi\)
0.260841 + 0.965382i \(0.416000\pi\)
\(920\) 0 0
\(921\) 10525.9 0.376592
\(922\) − 22871.6i − 0.816959i
\(923\) − 11522.7i − 0.410915i
\(924\) −6314.41 −0.224815
\(925\) 0 0
\(926\) 23200.3 0.823336
\(927\) 16565.6i 0.586933i
\(928\) 43397.6i 1.53512i
\(929\) −16539.6 −0.584118 −0.292059 0.956400i \(-0.594340\pi\)
−0.292059 + 0.956400i \(0.594340\pi\)
\(930\) 0 0
\(931\) −134.513 −0.00473522
\(932\) 9688.79i 0.340522i
\(933\) 9399.74i 0.329833i
\(934\) 18533.9 0.649301
\(935\) 0 0
\(936\) 6736.41 0.235242
\(937\) − 30212.3i − 1.05335i −0.850065 0.526677i \(-0.823438\pi\)
0.850065 0.526677i \(-0.176562\pi\)
\(938\) 12178.0i 0.423908i
\(939\) 19168.8 0.666187
\(940\) 0 0
\(941\) −26414.4 −0.915074 −0.457537 0.889191i \(-0.651268\pi\)
−0.457537 + 0.889191i \(0.651268\pi\)
\(942\) 15789.1i 0.546112i
\(943\) − 15541.5i − 0.536692i
\(944\) 3182.03 0.109710
\(945\) 0 0
\(946\) 52342.9 1.79896
\(947\) − 10187.3i − 0.349570i −0.984607 0.174785i \(-0.944077\pi\)
0.984607 0.174785i \(-0.0559231\pi\)
\(948\) 6149.77i 0.210691i
\(949\) −21241.5 −0.726583
\(950\) 0 0
\(951\) −4903.31 −0.167193
\(952\) 9127.31i 0.310733i
\(953\) 2211.39i 0.0751669i 0.999293 + 0.0375834i \(0.0119660\pi\)
−0.999293 + 0.0375834i \(0.988034\pi\)
\(954\) −4294.66 −0.145749
\(955\) 0 0
\(956\) −31732.3 −1.07353
\(957\) − 47794.0i − 1.61438i
\(958\) − 20773.0i − 0.700569i
\(959\) 2679.23 0.0902156
\(960\) 0 0
\(961\) −17489.6 −0.587077
\(962\) 7136.44i 0.239177i
\(963\) − 2187.78i − 0.0732089i
\(964\) −18263.4 −0.610193
\(965\) 0 0
\(966\) 3385.33 0.112755
\(967\) − 7955.89i − 0.264575i −0.991211 0.132287i \(-0.957768\pi\)
0.991211 0.132287i \(-0.0422322\pi\)
\(968\) − 65679.6i − 2.18081i
\(969\) 464.014 0.0153832
\(970\) 0 0
\(971\) 53071.2 1.75400 0.877001 0.480488i \(-0.159541\pi\)
0.877001 + 0.480488i \(0.159541\pi\)
\(972\) − 1131.62i − 0.0373422i
\(973\) − 21375.2i − 0.704274i
\(974\) −14498.6 −0.476965
\(975\) 0 0
\(976\) 582.813 0.0191141
\(977\) 22448.2i 0.735089i 0.930006 + 0.367545i \(0.119801\pi\)
−0.930006 + 0.367545i \(0.880199\pi\)
\(978\) 5087.80i 0.166350i
\(979\) −65781.4 −2.14748
\(980\) 0 0
\(981\) −3646.58 −0.118681
\(982\) − 14831.3i − 0.481961i
\(983\) 21712.8i 0.704509i 0.935904 + 0.352254i \(0.114585\pi\)
−0.935904 + 0.352254i \(0.885415\pi\)
\(984\) 12238.1 0.396479
\(985\) 0 0
\(986\) −25418.5 −0.820983
\(987\) 7249.33i 0.233788i
\(988\) − 413.470i − 0.0133140i
\(989\) 39089.7 1.25681
\(990\) 0 0
\(991\) −37849.2 −1.21324 −0.606620 0.794992i \(-0.707475\pi\)
−0.606620 + 0.794992i \(0.707475\pi\)
\(992\) 19508.0i 0.624373i
\(993\) − 13158.5i − 0.420516i
\(994\) 4559.82 0.145502
\(995\) 0 0
\(996\) −760.168 −0.0241836
\(997\) 39573.1i 1.25707i 0.777783 + 0.628533i \(0.216344\pi\)
−0.777783 + 0.628533i \(0.783656\pi\)
\(998\) 30748.0i 0.975261i
\(999\) 3258.26 0.103190
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 525.4.d.l.274.2 4
5.2 odd 4 105.4.a.e.1.2 2
5.3 odd 4 525.4.a.l.1.1 2
5.4 even 2 inner 525.4.d.l.274.3 4
15.2 even 4 315.4.a.k.1.1 2
15.8 even 4 1575.4.a.q.1.2 2
20.7 even 4 1680.4.a.bo.1.2 2
35.27 even 4 735.4.a.o.1.2 2
105.62 odd 4 2205.4.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.2 2 5.2 odd 4
315.4.a.k.1.1 2 15.2 even 4
525.4.a.l.1.1 2 5.3 odd 4
525.4.d.l.274.2 4 1.1 even 1 trivial
525.4.d.l.274.3 4 5.4 even 2 inner
735.4.a.o.1.2 2 35.27 even 4
1575.4.a.q.1.2 2 15.8 even 4
1680.4.a.bo.1.2 2 20.7 even 4
2205.4.a.bb.1.1 2 105.62 odd 4